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1 E M H K X Y G Q F H T d ℤ ℝ ℂ 𝔼definitionDefinitionmypropertyPropertymytheoremTheoremmylemmaLemmamyconjectureConjecturecorollaryCorollarymyproblemProblemmyobservationObservationmyremarkRemarkmyalgorithmAlgorithmmyassumptionAssumptionmyexampleExample def = = ·=x y c d b q g i s a r u m n h p z o e f v w A B C D E F G H I J K L M N S V P Q R W U X Y Z 1ψϕ0̱ 0ΦΨΓΣΔΩΛÂ A B C D F R I J N S U V W R D EB I SINR SNR MSE h I S R D SR RD =2500 NOMA based Random Access with Multichannel ALOHA Jinho ChoiThe author is with School of Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology (GIST), Gwangju, 61005, Korea (Email: [email protected]). This work was supported by the “Climate Technology Development and Application" research project (K07732) through a grant provided by GIST in 2017. today ==================================================================================================================================================================================================================================================================================================================================================== In nonorthogonal multiple access (NOMA), the power difference of multiple signals is exploited for multiple access and successive interference cancellation (SIC) is employed at a receiver to mitigate co-channel interference. Thus, NOMA is usually employed for coordinated transmissions and mostly applied to downlink transmissions wherea base station (BS) performs coordination for downlink transmissions with full channel state information (CSI). In this paper, however, we show that NOMA can also be employed for non-coordinated transmissions such asrandom access for uplink transmissions. We apply a NOMA scheme to multichannel ALOHA and show that the throughput can be improved. In particular, the resulting scheme is suitable for random accesswhen the number of subchannels is limited since NOMA can effectively increase the number of subchannels without any bandwidth expansion.random access; non-orthogonal multiple access; throughput analysis 26pt § INTRODUCTION Recently, nonorthogonal multiple access (NOMA) has been extensively studied to improve the spectral efficiency for future cellular systems, e.g.,5th generation (5G) systems, in <cit.>. In NOMA, a radio resource block is shared by multiple users and their transmission power difference plays a key role in multiple access. Successive interference cancellation (SIC) is also important in NOMA as it can mitigate co-channel interference in a systematic manner. In <cit.>, practical NOMA schemes, called multiuser superposition transmission (MUST) schemes, are considered for downlink transmissions (with two users). In <cit.>, NOMA is employed for coordinated multipoint (CoMP) downlink in order to support a cell-edge user without degrading the spectral efficiency.For multiresolution broadcast, NOMA is studied withbeamforming in <cit.>. In <cit.>, NOMA is also considered for small packet transmissions in the Internet of Things (IoT).Machine-type communications (MTC) or machine-to-machine (M2M) communications will play a crucial role in 5G or the IoT <cit.><cit.>. In <cit.><cit.>, it is shown thatuncoordinated access or random access schemes might be suitable for MTC due to low signaling overhead when devices have short packets to transmit.In general, random access for MTC is to provide access for uplink transmissions (i.e., from devices to a base station(BS) or access point (AP)).While NOMA has been actively studiedas mentioned earlier, it is mainly considered for downlink transmissions. Similarly, in the IoT to support short packet transmissions, NOMA is employed for downlink transmissions as in <cit.>. However, there are some existing works of NOMA for uplink transmissions, e.g., <cit.><cit.><cit.>.In general, NOMA requires coordinations with known channel state information (CSI) to exploit the power difference for multiple access. Since the BS can carefullyallocate powers to the signals to users with known CSI, exploiting the power difference for NOMA becomes easierfor downlink transmissions than uplink transmissions. If NOMA is employed for uplink, the BS also needs to carefully allocate powers and users over multiple channels with full CSI as in<cit.><cit.>. From this, it seems that NOMA is not suitable for any uncoordinated transmissions including random access despite its strength of providing higher spectral efficiency. In other words, NOMA may not be a suitable candidate for random access to support MTC within 5G or the IoT.In this paper, however, we consider NOMA for random access where the BS does not perform any coordination for uplink transmissions. In particular, we propose to apply a NOMA scheme to a well-known random access scheme, multichannel ALOHA<cit.><cit.>. For the NOMA scheme, we consider an approach in <cit.> thatuses a set of pre-determined power levels for multiple access. Using this NOMA scheme, the throughput of multichannel ALOHA can be improved without any bandwidth expansion. The resulting scheme might be suitable for random access when the number of subchannels is limited asNOMA can effectively increase the number of subchannels. Consequently, when MTC is considered with a limited bandwidth, the proposed scheme can be a good candidate for random access due to more available subchannels.Since the transmission power of the proposed random access scheme based on NOMA can be high, we also study a channel-dependent selection scheme for subchannel and power level, which can reduce the transmission power.The rest of the paper is organized as follows. In Section <ref>, a well-known random access scheme, multichannel ALOHA, is briefly discussed. In Section <ref>, a NOMA scheme is presented as a random access scheme. This NOMA scheme is applied to multichannel ALOHA in Section <ref> to effectively increase the number of subchannels by exploiting the power domain. Simulation results are presented in Section <ref>. The paper is concluded with some remarks in Section <ref>.§.§.§ NotationMatrices and vectors are denoted by upper- and lower-case boldface letters, respectively. The superscripts *, , anddenote the complex conjugate, transpose, Hermitian transpose, respectively. For a set , || denotes the cardinality of . [·] and Var(·) denote the statistical expectation and variance, respectively. (, ) represents the distribution of circularly symmetric complex Gaussian (CSCG) random vectors with mean vectorand covariance matrix . § MULTICHANNEL ALOHA In this section, webriefly discuss multichannel (slotted) ALOHA for uplink transmissions and its throughput. Throughout the paper, we assume a single cell with one BS and multiple users.Multichannel ALOHA is a generalization of ALOHA with multiple orthogonal subchannels <cit.><cit.>. In <cit.>, multichannel ALOHA is studiedwith orthogonal frequency division multiple access (OFDMA) where each subcarrier becomes an orthogonal subchannel.Suppose that there are B orthogonal subchannels. Denote by _i the index setof active users transmitting signalsthrough the ith subchannel. Then, the received signal at the BS over the ith subchannel can be written asy_i = ∑_k ∈_i h_i,k√(P_i,k) s_i,k + n_i,where h_i,k, P_i,k, and s_i,k represent the channel coefficient, transmit power, and signal from user k through the ith subchannel, respectively,and n_i ∼(0, N_0) is the background noise. Here, N_0 is the noise spectral density.Although it may be possible for the BS to detect some users' signals when multiple users choose the same subchannel due to the capture effect <cit.>, we ignore this possibility and employ a simple collisionmodel <cit.> for throughput analysis. In this case, if there are M active users and each active user chooses a subchannel independently and uniformly at random, the conditional throughput[The throughput is the average number of users who can successfully access a channel without collision.]can be written as η_ MA (M;B) = M (1 - 1/B)^M-1. In (<ref>), since M is a random variable, in order to find the average throughput, we need to consider a distribution of M. For convenience, we consider a uniform distribution with a large number of users in this paper. To this end, assume that there are K users and each user becomes active with access probability p_ a. In addition, let N denote the number of active users that choose a subchannel. Then, [M] =K p_a and [N] = K p_a/B. For a large K, we can use the Poisson approximation <cit.> for N. That is, N becomes a Poisson random variable as follows:N ∼ p_λ (n)= e^-λλ^n/n!,where p_λ (n) denotes the probability mass function (pmf) of a Poisson random variable with parameter λ. Here,λ = [M]/B = K p_ a/B is assumed to be constant, which is called the intensity, as K →∞. Then, the average throughput of multichannel ALOHA can be found asT_ MA (B) = [η_ MA (M;B) ] = B λ e^-λ,which is B times higher than that of single-channel ALOHA (with B = 1). The intensity that maximizes the throughput is λ = 1<cit.> and the maximum throughput is B e^-1. § RANDOM ACCESS BASED ON NOMA In this section, we only consider a single subchannel to present a random access scheme based on a NOMA scheme studied in <cit.> and derive itsconditional throughput. For simplicity, we omit the subchannel index i throughout this section.§.§ A NOMA Scheme: Power Division Multiple Access In this subsection, we consider a NOMA scheme that is suitable for random access, which is different from conventional uplink NOMA that requires central coordination including power allocation at the BS with full CSI such as the approach in <cit.>.Throughout the paper, we assume that each user knows its CSI. In time division duplexing (TDD) mode, the BS can send a beacon signal at the beginning of a time slot to synchronize uplink transmissions. This beacon signal can be used as a pilot signal to allow each user to estimate the CSI. Due to various channel impairment (e.g., fading) and the background noise, the estimation of CSI may not be perfect. However, for simplicity, we assume that the CSI estimation is perfect in this paper. The impact of CSI estimation erroron the performance needs to be studied in the future. Suppose that there are pre-determined L power levels that are denoted by v_1 >… > v_L > 0. We now assume that an active user, say user k, can randomly choose one of the power levels, say v_l, forrandom access. Then, the transmission power is decided as P_k = v_l/α_k, where α_k = |h_k|^2 is the channel power gain from user k to the BS, so that the received signal power becomes v_l. Assuming that the spectral density of the background noise is normalized, i.e., N_0 = 1, if there are no other active users, thesignal-to-noise ratio (SNR) orsignal-to-interference-plus-noise ratio (SINR) at the BS becomes v_l.Suppose that each power level in (<ref>) is decided as follows:v_l = Γ (V_l + 1), where Γ is the target SINR and V_l = ∑_m=l+1^L v_m with V_L = 0. The value of the target SINR, Γ, can be decided depending on the desired transmission rate or quality of link.It can be shown that v_l = Γ (Γ +1 )^L-l. If there exists one active user at each power level, the SINR for the active user who chooses v_1 becomes v_1/V_1 + 1, which is Γ from (<ref>). Thus, when the transmission rate, denoted by R, is given by R = log_2(1 + Γ), the signal from this user can be decoded and removed using SIC. The SINR for the active user who chooses v_2 is also Γ = v_2/V_2 + 1. Consequently, all the L signals can be decoded using SIC in ascending order if the transmission rate is given by R = log_2 (1+Γ). In other words, a total of L signals can be decoded although they are transmitted simultaneously.We can also observe that if there are M active users with M ≤ L and they choose different power levels, the M signals can be successfully decoded. This approach is referred to as power-domain multiple access (PDMA) <cit.>, which can be seen as a NOMA scheme as the power domain is exploited for multiple access and SIC is used to mitigate theco-channel interference.Note that the above approach is based on ideal SIC with capacity achieving codes. In practice, there might be decoding errors and SIC may not be perfectly carried out. Thus, a large L is not desirable due to the error propagation. §.§ Throughput Analysis With the random access scheme based on PDMA, the BS can successfully decode all signals from M active users if M ≤ L and different powers are chosen. However, if there are multiple active users who choose the same power level, the signals cannot be decoded. For convenience, the event that multiple active users choose the same power level is called power collision. Unlike conventional multichannel random access schemes, the power collision at each power level is not an independent event. That is, if power collision happens at level l, the signals at levels l+1, …, L cannot be decoded, while the signals in the signals at higher power levels can be decoded if there is no power collision. For example, suppose that L = 4 and M = 3. If one user chooses v_1 and the other two users choose v_4, the signal from the user choosing v_1 can be decoded, although the signals from the other two users cannot be decoded.For the performance of random access based on PDMA, we consider the conditional throughput that is the average number of signals that are successfully decoded for given M. A bound on the (conditional) throughput can be found as follows. The conditional throughput for given M(M ≤ L) active users, denoted by η(M;L),is bounded asη(M;L) ≥η (M;L) = M ∏_m=1^M-1(1 - m/L).If M ≤ 2, the bound is exact.The throughput, η(M;L) in (<ref>) corresponds to the case that all M signals can be decoded. Since the probability that all M signals have different power levels is ∏_m=1^M-1(1 - m/L)<cit.>, we can have (<ref>). As mentioned earlier, since it is also possible to decode some signalsin the presence of power collision, (<ref>) becomes a lower-bound.In the case of M = 2, the BS can decode two signals if two active users choose different power levels. If they choose the same power level, no signal can be decoded. Thus, the lower-bound in (<ref>) becomes exact.Note that η(M; L) ≥ 0 for any value of M. Thus, the lower-bound in (<ref>) is valid for any value of M as η(M; L) =0 for M > L.From (<ref>), we can show that the resulting random access scheme based on PDMAcan have a higher throughput as L increases. However, from (<ref>), sincethe highest power level, v_1 = Γ (Γ +1 )^L-1, grows exponentially with L, a large L becomes impractical. In Fig. <ref>, we illustrate the average transmission powerof PDMA for different values of L and target SINR.Fig. <ref> (a)shows the average transmission power of PDMA for different numbers of power levels, L. We can see that the increase of the average transmission power is significantly higher as L increases. On the other hand, the increase of the throughput with L is notsignificant as shown in Fig. <ref> (b). Thus, it may not be desirable to have a large L. Note that the conditional throughput inFig. <ref> (b) is the lower-bound in (<ref>) where the optimal value of M is chosen to maximize the bound, i.e., max_1 ≤ M ≤ Lη (M;L).§ APPLICATION OF NOMA TO MULTICHANNEL ALOHA In this section, wepropose a NOMA-multichannel ALOHA (NM-ALOHA) scheme by applying PDMA to multichannel ALOHA, and study itsthroughput. Furthermore, we study channel-dependent selection for subchannel and power level to reduce the transmission power or improve the energy efficiency. §.§ Application of PDMA to Multichannel ALOHA and Throughput Analysis As discussed at the end of Section <ref>, random access based on PDMA may not be practical in terms of its energy efficiency for a large L. However, PDMA can be used withother random access schemes to improve throughput with a small L. In this subsection, we consider NM-ALOHA using PDMA.We assume that each subchannel in multichannel ALOHA employs PDMA. Thus, there are L B subchannels. Suppose that there are M_i active usersthat choose the ith subchannel. In addition, let = [M_1… M_B]^and M = ∑_i=1^B M_i. Denote by η_ NMA (;L,B) the conditional throughput of NM-ALOHA using PDMAfor given . Then, we have η_ NMA (;L,B) =∑_i=1^B η(M_i;L).If each active user can choose a subchannel uniformly at random, the conditional throughput for given Mcan be found asT_ NMA (M;L,B)= [η_ NMA (;L,B)|M] = B [η(N, L)],where N becomes the binomial random variable with parameter M and p = 1/B, i.e., its pmf is given by(N=n) = p(n;M) = Mn(1/B)^n(1-1/B)^M-n.It can be shown thatT_ NMA (M;L,B)= ∑_η_NMA (;L,B) p() = ∑_∑_i=1^B η (M_i;L) p()where p() is the pmf of multinomial random variables that is given by p() = M/M_1! ⋯ M_B!( 1/B)^M. By marginalization, we can show that∑_∑_i=1^B η (M_i;L) p()= B ∑_n=1^M η(n;L) p(n;M).Thus, we can have (<ref>). For a large K, using the Poisson approximation, from (<ref>) and (<ref>), a lower-bound on the average throughputcan be found asT_ NMA (L,B) = [ T_ NMA (M;L,B) ]≥ B ∑_n=1^L η(n;L) p_λ(n) = B ∑_n=1^L n ( ∏_m=1^n-1(1 - m/L)) e^-λλ^n/n!.If L = 2, as mentioned in Lemma <ref>, the lower-bound is exact (because M ≤ L).Thus,we can show thatT_ NMA (2,B) = B ( e^-λλ + e^-λλ/2!) = 3/2 B λ e^-λ.In addition, if L = 1, NM-ALOHA is reduced tostandard multichannelALOHA that has the following throughput:T_ NMA (1,B) = T_ MA (B) =B λ e^-λ.From this, we can see that the average throughput of NM-ALOHA with L = 2 is 1.5 times higher than that of standard multichannel ALOHA. Furthermore, asη(n;L) increases with L, the lower-bound on the average throughput increases with L. Consequently, we can see thatNM-ALOHA can improve the throughput of multi-channel ALOHAwithout any bandwidth expansion based on the notion of NOMA. However, the increase of L results in the increase of transmission power. To mitigate the increase of transmission power, we can consider a channel-dependent subchannel/power-levelselection scheme in the following subsection.§.§ Channel-Dependent Energy Efficient Selection In above, we assume that each active user chooses a subchannel and a power level independently and uniformly at random. The selection of subchannel and power level can depend on the channel gain and it may result in the improvement in terms of energy efficiency (or the decrease of transmission power).Suppose thatusers are uniformly distributed within a cell of radius D.We assume that the large-scale fading coefficient of user k is given by <cit.>[α_i,k] = α̅_k = A_0 d_k^-κ,0 < d_k ≤ D, where α_i,k = |h_i,k|^2, κ is the path loss exponent, A_0 is constant, and d_k is the distance between the BS and user k. Thus,the large-scale fading coefficient depends on the distance. For illustration purposes, suppose that L = 2. According to the large-scale fading coefficients or distances,we can divide users into two groups as follows:_1 = {k |d_k ≤τ}_2 = {k |d_k > τ}.If an active user belongs to _1, this user selects v_1. Otherwise, the user selects v_2. That is, a user located far away from the BS tends to choose a smaller v_l to reduce the overall transmission power. We may decide the threshold value τ to satisfy the following condition:[|_1|] = [|_2|] = K/2,so that each group has the same number of users on average. In this case, we have τ = D/√(2). Consequently, the large-scale fading coefficient is used as a random number for the power level selection and the value of τ is decided to make sure that (k ∈_l) = 1/2, (i.e., for a uniform power level selection at random). The above approach can be generalized forL ≥ 2. To this end, let_l ={k | τ_l-1 < d_k ≤τ_l }.Under the assumption that users are uniformly distributed in a cell of radius D, we have τ_0 = 0 and τ_l = D √(%s/%s)lL, l = 1,…,L, to satisfy(k ∈_l) = 1/L ,l = 1,…, L,which also results in [|_l|] = K/L.To minimize the transmission power, an active user belongs to _l chooses v_l. Furthermore, when an active user in _l chooses one of B subchannels in NM-ALOHA, the user may choose the subchannel that has the maximum channel gain to further minimize the transmission power. As a result, the transmission power of user k can be decided asP_k = v_l/max_i α_i,k,k ∈_l. Note that in this case, if α_1,k, …, α_B,k are independent and identically distributed(iid), the selection of subchannelis carried out independently and uniformly at random. The selection scheme resulting in(<ref>) is referred to as the channel-dependent subchannel/power-level selection scheme.Suppose thatα_i,k = α̅_k u_i,k^2, where u_i,k is an independent Rayleigh random variable with [u_i,k^2] = 1 (i.e., small-scale fading is assumed to be Rayleigh distributed). Then, for B ≥ 2, the average transmission power is bounded as[P_k |k ∈_l]≤v_l/A_lmin{ 2 ln 2,B/B-1},where A_l = A_0 τ_l^-κ.To find an upper-bound, we consider a user of the longest distance within _l, τ_l.In this case, we have α_i,k = A_l u_i,k^2,i = 1, …, B,which are iid. According to order statistics <cit.>, we can see that[ 1/max_i α_i,k] is a nonincreasing function of B. Thus, for an upper-bound, it is sufficient to consider the case of B = 2. Since u_i,k^2 is an exponential random variable, it can be shown that[ 1/maxα_i,k] = 1/A_l∫_0^∞1/x B e^-x (1- e^-x)^B-1 dx≤2/A_l∫_0^∞1/x e^-x (1- e^-x) dx = 2 ln 2/A_l,where the last step is due to <cit.>.To find another bound for any B ≥ 2,let t = 1- e^-x. Then, it can be shown that∫_0^∞1/x B e^-x (1- e^-x)^B-1 dx= B ∫_0^1 -t^B-1/ln(1-t) dt≤ B ∫_0^1 t^B-1/t dt =B/B-1,where the inequality is due to t ≤ - ln (1-t), t ∈ (0,1). Substituting (<ref>) into (<ref>), we have[ 1/maxα_i,k] ≤1/A_lB/B-1.From (<ref>) and (<ref>),we can readily show (<ref>). From (<ref>) and (<ref>), noting that (k ∈_l) =1/L, the average transmission power is upper-bounded as[P_k] ≤min{2 ln 2, B/B-1}/L∑_l=1^L v_l/A_l= min{2 ln 2, B/B-1}/L∑_l=1^L Γ (Γ +1)^L-l/A_0(D√(%s/%s)lL)^-κ. It is noteworthy that under (<ref>), [1/α_i,k] →∞, which is the case of B = 1. Thus, the power allocation in (<ref>)may result in a prohibitively high transmission power. To avoid this problem, the truncated channel inversion power control can be used <cit.>. However, if B ≥ 2, this problem can be mitigated without any transmission power truncation, since [1/α_i,k] < ∞ from (<ref>). For comparison purposes, we now consider a random selection for subchannel and power level. If the subchannel and power level are randomly selected, the average transmission powerwould be[P_k]= 1/L∑_l=1^L [v_l/α_i,k] = 1/L∑_l=1^L v_l [1/α_i,k].In this case, even if [1/α_i,k] converges to constant, we note that [P_k]∝1/L∑_l=1^L v_l. Thus, from (<ref>) and (<ref>), we can see that the average transmission power withchannel-dependent (subchannel/power-level) selection grows slower than that withrandom selection as L increases. This shows the advantage ofchannel-dependent selectionover random selection for the selection of subchannel and power level in NM-ALOHA in terms of the transmission power for B > 1 and L > 1.It is noteworthy that the channel-dependent selection does not affect the throughput as users' locations are random, while it can greatly improve the energy efficiency.§ SIMULATION RESULTS In this section, we present simulation results to see the performance of NM-ALOHA with the fading channel coefficients, h_i,k, that are generated according to (<ref>)and (<ref>). For the path loss exponent, κ, in(<ref>), we assume that κ = 3.5. In addition, we assume that D = 1 and A_0 = 1 in (<ref>) for normalization purposes.Fig. <ref> shows the throughput of NM-ALOHA for different numbers of subchannels, B, when K = 200, p_a = 0.05, and L ∈{1, 4}. The lower-bound is obtained from (<ref>). As expected, we can observe that the throughput increases with the number of subchannels, B. More importantly, we can see thatthe throughput of NM-ALOHA (L = 4) is higher than that of (conventional) multichannel ALOHA (L = 1). In particular, when B = 4, the throughput ofNM-ALOHA becomes about 4 times higher than that of multichannel ALOHA,while the throughput gap decreases with B. This demonstrates that when the number of subchannels is limited in multichannel ALOHA, NOMA based approaches such as NM-ALOHA can help improve the throughput. For example, NM-ALOHA (L = 4) can achieve a throughput of 3.5 with B = 4, while the same throughput can be obtained by multichannel ALOHA withB = 10 (in this case, we can claim that NM-ALOHA can be 2.5 times more spectrally efficient than multichannel ALOHA).In order to see the impact of the number of power levels, L, on thethroughput of NM-ALOHA, weshow the throughput fordifferent values of L inFig. <ref>when K = 200, p_a = 0.05, and B = 6. As expected, the throughput increases with L without any bandwidth expansion (i.e., with a fixed B). For example, with L= 4, the throughput can be 3 times higher than that of (conventional) multichannel ALOHA(i.e., NM-ALOHA with L = 1). However, the improvement of throughput becomes limited when L is sufficiently large. In Fig. <ref>, we show the throughput for different values of access probability, p_a, when K = 200, L = 4, and B = 6. The performance behavior of NM-ALOHA is similar to that of multichannel ALOHA in terms of p_a. That is, the throughput increases with p_a, and then decreases, which implies that there exists an optimal access probability that maximizes the throughput. Thus, it is possible to consider the access control using the access probability as in ALOHA <cit.> or the number of subchannels <cit.>. However, this topic is beyond the scope of the paper and might be further studied in future research. From Figs. <ref>–<ref>, we can confirm that the lower-boundfrom (<ref>) is reasonably tight, while it becomes tighter as L decreasesas shown in Fig. <ref>. The main disadvantage of NM-ALOHA might bea high transmission power as mentioned earlier. To mitigate this problem, we considered the channel-dependent (subchannel/power-level) selection scheme in Subsection <ref>. To see the impact of this selectionscheme on the average transmission power, we present simulation results in Fig. <ref> where the average transmission power is shown for different values of L when K = 200, p_a = 0.05, B = 6, and Γ = 6 dB. We also show the upper-bound in (<ref>). Furthermore, for performance comparisons, we consider the random selection for subchannel and power level regardless of the channel conditions. Since the transmission power can be arbitrarily high due to the channel inversion power control in (<ref>), we assume that the transmission power is limited to be less than or equal to 10 L dB(i.e., truncated power control is assumed) in simulations hereafter. The corresponding results are shown with the legend 'Sim (Random)' in Fig. <ref>. We can observe that the average transmission power increases with L, whilethe channel-dependent selection scheme provides a much loweraverage transmission power than the (channel-independent) random selection scheme. Fig. <ref> showsthe average transmission power fordifferent numbers of subchannels, B, when K = 200, p_a = 0.05, L = 4, and Γ = 6 dB. As expected,the average transmission power decreases with B. On the other hand, the average transmission power with the random selection does not depend on B. Consequently, we can see thatalthough a large Bdoes not help improve the throughput significantly(with a fixed p_a) in NM-ALOHA as shown in Fig. <ref>, it can be effective for improving energy efficiency with channel-dependent selection. Note that the upper-bound derived in (<ref>) is tight when B is small.Fig. <ref> showsthe average transmission power fordifferent values of the target SINR, Γ, when K = 200, p_a = 0.05, L = 4, and B = 6. We can see that the average transmission power increases with Γ. Thus, the target SINR should be decided according to a given feasible average transmission power.§ CONCLUDING REMARKS In this paper, we proposed a random access schemeby applying NOMA to multichannel ALOHA. The proposed scheme has multiple subchannels and multiple power levels for random access to effectively increase the number of subchannels. It was shown that the proposed scheme can provide a higher throughput thanmultichannel ALOHA by exploiting the power difference.As a result, the proposed schemebecame suitable for random accesswhen the number of subchannels of multichannel ALOHA is limited, because NOMA can effectively increase the number of subchannels without any bandwidth expansion. A closed-form expression for a lower-bound on thethroughput was derived to see the performance.The main drawback of the proposed scheme was a high transmission power that is a typical problem of NOMA as the power domain is exploited. In order to mitigate this problem, a channel-dependentselection scheme for subchannel and power level was studied, which leads to the decrease of transmission power or improvement of energy efficiency. An upper-bound on the average transmission power was derived to see the impact of thechannel-dependentselection scheme on the average transmission power in terms of the number of power levels. ieeetr[ < g r a p h i c s > ] Jinho Choi (SM'02) was born in Seoul, Korea. He received B.E. (magna cum laude) degree in electronics engineering in 1989 from Sogang University, Seoul, and M.S.E. and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, in 1991 and 1994, respectively. He is with Gwangju Institute of Science and Technology (GIST) as a Professor. Prior to joining GIST in 2013, he was with the College of Engineering, Swansea University, United Kingdom, as a Professor/Chair in Wireless. His research interests include wireless communications and array/statistical signal processing. He authored two books published by Cambridge University Press in 2006 and 2010. Prof. Choi received the 1999 Best Paper Award for Signal Processing from EURASIP, 2009 Best Paper Award from WPMC (Conference), and is Senior Member of IEEE. Currently, he is an Editor of IEEE Trans. Communications and had served as an Associate Editor or Editor of other journals including IEEE Communications Letters, Journal of Communications and Networks (JCN), IEEE Transactions on Vehicular Technology, and ETRI journal. | http://arxiv.org/abs/1706.08799v1 | {
"authors": [
"Jinho Choi"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170627115650",
"title": "NOMA based Random Access with Multichannel ALOHA"
} |
firstpage–lastpageA solution of the dark energy and its coincidence problem based on local antigravity sources without fine-tuning or new scalesVasilios Zarikas June 19, 2017 ================================================================================================================================We present a detailed analysis of IC 4776, a planetary nebula displaying a morphology believed to be typical of central star binarity.The nebula is shown to comprise a compact hourglass-shaped central region and a pair of precessing jet-like structures.Time-resolved spectroscopy of its central star reveals periodic radial velocity variability consistent with a binary system.While the data are insufficient to accurately determine the parameters of the binary, the most likely solutions indicate that the secondary is probably a low-mass main sequence star.An empirical analysis of the chemical abundances in IC 4776 indicates that the common-envelope phase may have cut short the AGB evolution of the progenitor.Abundances calculated from recombination lines are found to be discrepant by a factor of approximately two relative to those calculated using collisionally excited lines, suggesting a possible correlation between low abundance discrepancy factors and intermediate-period post-common-envelope central stars and/or Wolf-Rayet central stars.The detection of a radial velocity variability associated with binarity in the central star of IC 4776 may be indicative of a significant population of (intermediate-period) post-common-envelope binary central stars which would be undetected by classic photometric monitoring techniques. planetary nebulae: individual (IC 4776, PN G002.0-13.4) – binaries: spectroscopic – stars: mass loss – ISM: jets and outflows§ INTRODUCTION Planetary nebulae (PNe) are the intricate, glowing shells of gas ejected by low- and intermediate- mass stars at the end of their asymptotic giant branch (AGB) evolution which are then ionized by the emerging pre-white dwarf core. With some ∼80 per cent of all PNe showing deviation from spherical symmetry <cit.>, it has proven impossible to understand their structures in terms of single star evolution <cit.>, with binarity frequently invoked to explain their diverse, often strongly axisymmetrical morphologies <cit.>. While a lower limit to the close-binary central star fraction is well constrained (at ∼20%) by photometric monitoring surveys <cit.>, it is insufficient to explain all aspherical PNe. The remaining aspherical PNe are generally understood to be the products of mergers, wider binaries and/or weaker binary interactions <cit.>. This hypothesis is supported by common-envelope (CE) population synthesis models, which predict a significant number of post-CE binaries with orbital periods of several days to a few weeks <cit.> while almost all of the known post-CE central stars have periods less than one day <cit.>. The lack of known post-CE central stars with intermediate periods <cit.> is, perhaps, not unreasonable given that the majority of the effort has been focused on photometric monitoring which becomes particularly insensitive at these longer periods <cit.>. Recent work, however, has shown that targeted radial velocity monitoring can begin to reveal these missing binary systems <cit.>. Constraining this population is of particular interest given that observations of“naked” (i.e. those with no surrounding PN) white dwarf plus main sequence binaries find a similar dearth of intermediate period systems <cit.>, strongly indicating that the lack of known systems is not purely an observational bias. Understanding to what extent this population is truly absent (rather than just difficult to detect) will greatly further our understanding of the common envelope process itself <cit.>. IC 4776 is a relatively bright, small planetary nebula comprising a central, bipolar, hourglass-like structure and an extended, jet-like structure revealed for the first time by the images presented in this paper (see Figure <ref> and Section <ref>). Both jets and hourglass structures have been strongly linked to binary central stars <cit.>.The nebula has been shown to display dual-dust chemistry <cit.>, which has also been linked to a possible binary evolution of the nebular progenitor <cit.>. The spectral type of its central star is unclear but has been classified by various authors as [WC] <cit.>.The apparent presence of narrow emission lines has also earned it a wels classification <cit.>, now generally not considered a valid classification given that in many cases the emission lines themselves do not originate from the stars but from their host nebulae <cit.> or are the product of the irradiation of a main-sequence companion <cit.>.Based on the likelihood that it hosts a binary central star, IC 4776 was selected for further study as part of a programme to search for binary central stars through time-resolved radial velocity monitoring.Here, we present the results of that radial velocity study, revealing variability consistent with an intermediate period, post-CE binary.We furthermore present spatio-kinematic and chemical analyses of the nebula itself, in order to further constrain the relationship between the nebula and the probable central binary. § NEBULAR MORPHOLOGY AND KINEMATICS §.§ FORS2 narrowband imageryNarrowband imagery of IC 4776 was obtained using the FORS2 instrument mounted on ESO's VLT-UT1 <cit.>. Exposures were acquired in the following emission lines: +[N ii] (5s exposure time on 2012 September 5, 30s on 2016 June 11),[O ii] λ3727 (20s 2012 September 3), and [O iii] λ5007 (5s 2012 September 5).In each case the seeing was better than 1. The debiased and flat-fielded images are presented in Figure <ref>.In all three bands, the nebula shows a similar morphology, namely an “X"-shape, which is most prominently visible in the light of [O ii] λ3727. At all three wavelengths the central shell appears the brightest, while only at [O ii] λ3727 it is possible to discern its detailed structure. This hourglass morphology is extremely similar to that of MyCn 18, a PN often hypothesised to have originated from a binary interaction owing to its similarity to some nebulae surrounding symbiotic binary stars <cit.>.MyCn 18, in addition to its central hourglass, displays a system of high velocity knots which may well be analogous to the jets observed in IC 4776 <cit.>, further highlighting the similarities between the two nebulae. The jets of IC 4776 are evident in all filters but most prominently in the light ofwhere their remarkable structure is clearest.The evident curvature of the jets is a strong indication of precession, which can be a natural product of a central binary system <cit.>.Such precessing jets have been observed in several close-binary PNe, e.g. Fg 1 <cit.>, ETHOS 1 <cit.>, and NGC 6337 <cit.>, providing a solid observational link between jets and binarity.§.§ FLAMES-GIRAFFE integral field spectroscopySpatially-resolved, high-resolution, integral-field spectroscopy of IC 4776 was obtained on August 8 2013 using FLAMES-GIRAFFE mounted on ESO's VLT-UT2 <cit.>.The seeing during the observations did not exceed 1. The ARGUS integral field unit, employed in its standard `1:1' scale mode (resulting in a total aperture of 11.5 x 7.3 sampled by an array of 22×14 0.52×0.52 microlenses), was used to feed the GIRAFFE spectrograph setup with the H665.0/HR15 grating (providing a resolution of R∼30 000 in the range 6470Å ≲λ≲ 6790Å). ARGUS was oriented at P.A. = 38^∘ (to coincide with the symmetry axis of the nebula) and with only two telescope pointings, offset by about 10 to each other, we were able to obtain a complete coverage of the nebula (which is ∼20 long including the faint extended emission), as shown in Figure <ref>. Two exposures of 245-s were obtained for each of the two pointings, with the resulting data reduced and combinedto produce a single, wavelength-calibrated data-cube for each pointing using the standard ESO-GIRAFFE pipeline.Finally, the data from the two pointings were corrected to heliocentric velocity and combined to form a single data-cube using specially written python routines.In the wavelength range of the ARGUS data there are several spectral lines: , [N ii] λλ6548,6584 and thedoublet.In order to probe the kinematics and morphology of the nebula, we focus on theline given that it is brighter thanand has a lower thermal width than .Channel maps for theline are shown in Figure <ref>.Each channel shows a cross-section through the nebula at a different velocity relative to the rest velocity of , starting from the top left-hand side channel being the most blue-shifted (velocity of about -130 km s^-1), ending on the bottom right-hand side channel being the most red-shifted (velocity of about 137 km s^-1). Each channel is a sum of three slices of the reduced cube where the wavelength step between slices is approximately 0.05Å (i.e., each channel represents a range of 0.15Å≡6.83 km s^-1 at ). The brightness/contrast is calculated individually for each channel using a logarithmic function in order to display the maximum amount of structural information possible. The systemic velocity of the nebula has been previously found to be v_sys=16.3±0.6<cit.>, and around this velocity the channel maps present two circular structures joined at the position of the central star.This can be interpreted as the contribution of the two lobes of an inclined bipolar shell, as inferred from the imagery presented in Section <ref>.Further from this central velocity, one lobe begins to dominate, just as would be expected from an inclined hourglass.Yet further still from the systemic velocity (e.g. v_sys±∼20), a second component begins to contribute significantly to the observed emission in each channel map, initially appearing close to the central star and then moving away at velocities furthest from the systemic velocity.This emission feature clearly originates from the jets, presenting with a similarly curved appearance at the largest velocities.The jets themselves also present clearly greater velocities than the central shell (the maximum extent of which is at roughly v_sys±35), reaching more than 100 away from the systemic velocity.Furthermore, there is an evident asymmetry in the jet velocities given that the blue-shifted jet reaches roughly v_sys-130, while the red-shifted component only reaches as far as v_sys+100.Figure <ref> reinforces this idea as in the high contrast image the Northern jet extends beyond the region encompassed by the FLAMES-GIRAFFE observations (centred on the central star), while the Southern jet is apparently well covered.Given the clear precession visible in both the imagery and IFU spectroscopy it is not possible to derive a simple correlation between line-of-sight velocity and distance from the central star which would further constrain the asymmetry.A 2 spaxel region surrounding the central star was collapsed (equivalent to a slit-width of 1.04) in order to produce a simulated longslit spectrum. The position-velocity array of this simulated longslit is shown in Figure <ref>.The position-velocity array shows a clear “X”-shape, typical of such bipolar, hourglass structures, as well as end-caps corresponding to the brightest regions of jet emission (further emission from the jets can be seen with increased contrast which blows out the central nebula).§.§ Shape modellingA spatio-kinematic model of the central hourglass of IC 4776 was constructed based on the narrowband imagery and spatially-resolved spectroscopy presented above.The shape software was used, following the standard workflow presented in <cit.>, varying both morphological and kinematic parameters over a wide-range of parameter space with the best fit being determined from a by-eye comparison of synthetic images and velocity channel maps to the observations. The jets themselves are too complex to be treated by a simple spatio-kinematic model, and their detailed modelling is reserved for a subsequent paper (Santander-García et al., in prep). Although, it is worth noting that, in principle, the data are of sufficient quality to derive the precession properties of the jets, which can then be related to the parameters of the central binary star <cit.> at the time the jet was launched (constrained by the kinematical age of the jet).The best-fitting shape model, as expected from appearance of the nebula in the narrowband imagery, comprises an hourglass-like structure whose symmetry axis is inclined at 42±4 with respect to the plane of the sky (see Figure <ref>).The velocity structure of the nebula is found to be well represented by a flow-law whereby all velocities are radial from the central star and proportional to the distance from the central star. Such flow-laws, often referred to as “Hubble flows”, are generally taken to imply an eruptive event in which the majority of the nebular shaping occurred during a brief time period.However, <cit.> showed that while deviations from a “Hubble flow” should be appreciable and easily observed in PNe formed via the classical interacting stellar winds model <cit.>, such a simple flow law will provide a good first order estimation of the overall morphological and kinematic properties of the nebula.Particularly given the relatively low spatial resolution of the FLAMES-GIRAFFE observations c.f. the size of IC 4776, we are unable to constrain the presence (or not) of such deviations from a “Hubble flow”.The systemic velocity of the nebula is found to agree well with the literature value of 16.3±0.6 km s^-1 <cit.>, and was fixed in the modelling. The kinematical age of the nebula is found to be approximately 300 yr kpc^-1 (with a significant uncertainty due to the small size of the nebula).While there is significant variation in distance determinations for IC 4776 in the literature, the kinematical age would imply a rather young nebula at all but the very largest distances.The distance of 1.01 kpc determined by <cit.> would seem to imply an impossibly young nebula,while the distances derived by <cit.> and <cit.> would give a more reasonable age of ∼1500 years.§ NEBULAR CHEMISTRY High-resolution longslit spectroscopy of the nebula IC 4776 was acquired on August 9 2016 using the Ultra-violet and Visual Echelle Spectrograph (UVES) mounted on ESO's VLT-UT2 <cit.>. Exposures were acquired using both arms of the spectrograph with the following set-ups. Using Dichroic 1 (3× 150s exposures, 1× 20s exposure), the blue-arm was set to a central wavelength of 3460Å (with the standard HER_5 blocking filter) while the red-arm was set to a central wavelength of 5800Å (with the standard SHP700 blocking filter).Using Dichroic 2 (3× 300s exposures, 1× 60s exposure, 1× 20s exposure), the blue-arm was set to a central wavelength of 4370Å (again with the standard HER_5 blocking filter) while the red-arm was set to 8600Å (with the standard OG590 blocking filter).A slit-width of 2.4 was employed, and the seeing during the observations was approximately 1.5. The spectra were reduced using the standard UVES pipeline and flux-calibrated using 300s exposures of the standard star LTT 7987 taken with the same set-ups directly following the observations of IC 4776.Collectively, the spectra provide near-continuous wavelength coverage from ∼3000–10 000Å (see Figure <ref>).To construct the final spectrum, we took the median of the flux at each wavelength from the longest exposures in each setup, and where lines were saturated, replaced the affected ranges with values from the shorter exposures in which saturation was not an issue.Emission line fluxes were measured using the alfa code <cit.>, which optimises parameter fits to the line profiles using a genetic algorithm following the subtraction of a global continuum.The effectiveness of alfa in measuring PN line fluxes has previously been demonstrated in several papers <cit.>, and is further demonstrated in Figures <ref> and <ref>.alfa assumes that all lines have a Gaussian profile.At the high resolution of the UVES spectra, velocity structure is evident, and lines of different ionisation have different profiles.For the purposes of chemical analysis, we binned the spectra by a factor of 10 in wavelength, resulting in Gaussian profiles for all lines.353 emission lines were measured, of which 329 were resolved and 24 were blends of two or more lines.Échelle spectra may contain many spurious features due to bleeding of strong lines from adjacent orders.In general, these do not affect the analysis, as ALFA only attempts to fit features close to the wavelengths of known emission lines.Spurious features not fitted by ALFA may be seen in panels (a) and (b) of Figure <ref> (though they are also present in all panels of Figure <ref>).However, in a few cases, the bleeding may be blended with nebular emission. The O ii recombination line at 4089Å is one such case, and this impacts the estimate of the temperature and density from the ratios of O ii lines, as discussed below.Table <ref> lists the observed and dereddened fluxes measured for all emission lines measured in the spectra, together with their 1σ uncertainties. Physical conditions as well as chemical abundances were calculated from the emission line fluxes using the neat code <cit.>.The code uses Monte Carlo techniques to propagate uncertainties in line flux measurements into the derived quantities <cit.>.The physical parameters determined for IC 4776 are listed in Table <ref>. The extinction towards the nebula was estimated from the flux-weighted average of the ratios of Hα, Hγ and Hδ to Hβ, and the Galactic extinction law of <cit.>.Hγ and Hδ are in the same spectrum as Hβ, while Hα is in a separate spectrum.The three lines give consistent estimates of the logarithmic extinction at Hβ, c(Hβ), which is 0.22±0.03. The nebular density is estimated using several standard collisionally excited line (CEL) diagnostics, as well as from the Balmer and Paschen decrements, and the ratios of O ii recombination lines.The CEL diagnostics give relatively high densities of 10000–30000 cm^-3; the Paschen and Balmer decrements also imply high densities, of 10^4–10^5 cm^-3, although with larger uncertainties (see Figure <ref>).The O ii recombination line ratios imply a lower density of ∼3000 cm^-3, with the caveat that the 4089 Å line flux on which the ratios rely may be overestimated due to bleeding from adjacent orders.If its flux is overestimated, the density derived is underestimated.Temperatures determined from CEL ratios are around 10kK.The Balmer and Paschen jumps are well detected and permit the derivation of temperatures from their magnitudes.They are statistically consistent with each other, but slightly lower than the temperatures from CELs, with T(BJ)=8900±800K, and T(PJ)=7200±1000K, where T([O iii])=10000±200K.The ratios of the three strong helium lines at 4471, 5876 and 6678Å are weakly temperature-sensitive, and these are well enough detected to enable their use as a diagnostic; the temperatures from the λ5876/λ4471 and the λ6678/λ4471 ratios are significantly lower than from other aforementioned methods, at 3–4kK.Finally, O ii recombination line ratios imply a temperature consistent with the helium line ratios, albeit with a large statistical uncertainty, in addition to the systematic uncertainty of the effect of bleeding, which would result in an underestimate of the temperature.Chemical abundances were derived from both collisionally excited lines and recombination lines (ORLs), with total chemical abundances derived using the ionisation correction scheme of <cit.>.Heavy element recombination lines are very well detected in the spectra, and as such it is possible to measure abundances for C^2+, C^3+, O^2+, N^2+, N^3+ and Ne^2+. The neat code employed for the analysis assumes a three-zone ionization scheme. The average of n_e([O ii]) andn_e([S ii]) was adopted as representative of the density of low-ionisation zone (Ionisation Potential, IP < 20 eV), while the average of n_e([Cl iii]) and n_e([Ar iv]) was applied for the medium ionisation zone (20 eV <IP< 45 eV). The electron temperature obtained from [N ii] was used for the temperature of the low-ionisation zone, and the average of the temperatures derived using [O iii], [Ar iii], and [S iii] lines taken as the temperature of the medium-ionisation zone.No lines of the high ionisation regime (IP>45 eV) are detected, so no high-ionisation conditions are assumed.The medium-ionisation parameters were also employed in the derivation of abundances from ORLs. The calculated abundances, from both ORLs and CELs, are listed in Tables <ref> and <ref>.O^2+ permits the best determination of the abundance discrepancy, as ORLs and CELs of the ion are both present in optical spectra.68 O ii recombination lines are detected, together with the three strong [O iii] CELs at 4363, 4959 and 5007Å.The abundances derived from the well-detected multiplets V1, V2, V10, V12, and V19 agree very well with each other; lines from multiplets V5, V20, V25, V28 and a number of 3d–4f transitions are detected, but imply much higher abundances and may be affected by noise or bleeding from strong lines in adjacent orders.As such, the total O^2+ abundance is derived based only on the 5 well detected multiplets.The abundance for each multiplet was determined from a flux-weighted average of the detected components, and the overall abundance calculated as the average of the multiplet abundances.The O^2+/H^+ abundance thus derived is (6.0±0.3)×10^-4, while that derived from the strong CELs is (3.4±0.2)×10^-4, giving an abundance discrepancy factor (adf; the ratio of ORL abundance to CEL abundance) of 1.75±0.15.In the case of oxygen, O^2+ ORLs and CELs were detected while for O^+ only CELs were detected.The He^2+/H^+ ratio implies that there should be negligible O^3+, and so the CEL total abundance is simply the sum of O^+ and O^2+.For ORLs, it was assumed that the ionisation structure is the same, and that the O^+/O ratio from CELs can be used to correct for it.Thus, the abundance discrepancy factors for O^2+ and for O are the same.Lines of Ne^2+ from both recombination and collisional excitation were also both detected in the spectra.The ORLs give abundances consistent with each other, yielding a value of (1.7±0.2)×10^-4.The CELs give (0.9±0.05)×10^-4, resulting a discrepancy factor of 1.9±0.3, consistent with that derived for O^2+.Optical spectra contain CELs only of N^+, but ORLs of both N^2+ and N^3+.Most of the N is in the form of N^2+, and so the comparison of abundances relies on the large and uncertain correction of the CEL abundance for the unseen ions. <cit.> proposed a new ICF scheme to compute N abundances; however, the differences with the classical ICF approach (N/O=N^+/O^+) can be very large, especially for relatively low-excitation PNe, as is the case of IC 4776.Using the newer ICF, we obtain N/H = 1.67× 10^ -4^+3.00× 10^ -5_ -3.60× 10^ -5, while the classical approach gives an N abundance ∼0.4 dex lower. <cit.> found that the <cit.> ICF for N showed a correlation with both He abundance and degree of ionisation and, as such, favoured the classical approach. As the ICF for N from <cit.> is strongly dependent on whether the PN is radiation or matter-bounded (providing a more realistic correction for matter-bounded PNe; Delgado-Inglada, private communication), we used the classical ICF approach to derive N abundances for IC 4776. The derived abundance discrepancy factor for N is 2.9, with statistical uncertainties of ^+0.7_-0.4, though with systematic uncertainties probably much larger (especially given the uncertainty in the choice of ICF). The net result is that, whatever ICF we chose, the N/O ratio is relatively low (-0.75 < log (N/O) < -0.35).Without UV spectra, we cannot derive an abundance discrepancy factor for carbon.However, studies have generally found that ionic and elemental ratios from ORLs are consistent with those derived from CELs <cit.>.The C/O ratio derived from recombination lines is 0.3, atypically low for a planetary nebula (which normally present values around unity[ <cit.> found an average C/O ratio of 0.85 in a sample of nebulae outside the solar circle, while <cit.> find a value of 1.15 in a sample interior to the solar circle - both measured from CELs.]). Very recently, <cit.> have demonstrated the critical role of atomic data uncertainties in the determination of chemical abundances. They showed that the transition probabilities of the commonly used density diagnostic lines of S^+, O^+, Cl^2+ and Ar^3+ as well as the collision strengths of Ar^3+ are responsible for most of the uncertainty in the derived total abundances, especially in the high-density (log (n_e) > 4.0) regime. Given that the computed density for IC 4776 is relatively high, especially for the medium ionisation zone, it is important to note that the uncertainties derived for the total nebular abundances are almost certainly underestimated. However, the consistency between the adfs derived for O^2+ and Ne^2+ provides a strong indication of the validity of the results. § CENTRAL STAR §.§ Observations and data reductionThe central star of IC 4776 was observed a total of 10 times using the FORS2 instrument mounted on ESO's VLT-UT1 <cit.>.1200s exposures were acquired in service mode, randomly distributed throughout the observing period (see Table <ref> for the exact dates).For each observation, the same instrumental setup was used, employing a longslit of width 0.5 and the GRIS_1200B grism resulting in a spectral range of 3700Å ≲λ≲ 5100Å with a spectral resolution of approximately 0.8Å.All data were bias-subtracted, wavelength-calibrated and optimally-extracted using standard starlink routines <cit.>.§.§ Radial velocity variability After the reduction, the spectra were continuum subtracted and aligned using the He i λ4471.48 nebular line, such that the systemic nebular velocity was at 0 (thus accounting for both heliocentric velocity variations between observations and small deviations in the wavelength calibration).The Balmer lines were all found to be severely contaminated by the bright surrounding nebula, essentially leaving only the absorption line of He ii λ4541 as a “clean” feature for cross-correlation.The individual spectra were all cross-correlated against a custom mask (namely a flat continuum with a deep, narrow absorption spike at the rest velocity of the He ii feature) with the resulting radial velocities (c.f. the nebular systemic velocity) listed in Table <ref>, and plotted in Figure <ref>. There is a clear variation in the radial velocity with a semi-amplitude of about 30–40strongly indicating that the central star of IC 4776 is a binary system. Unfortunately, with so few radial velocity points, it is not possible to fully constrain the orbital period of the likely binary, with several periodicities and amplitudes providing reasonable fits to the data.Our current radial velocity data rule out periods much longer than 20 days, but shorter periods are relatively well fit. In Figure <ref>, we show a tentative fit with a period of 9 days to highlight the significant radial velocity variations most likely due to orbital motion, as well as the need for further data.Interestingly, although the stellar systemic velocity (the zero of the sine curve used to fit the stellar radial velocities, often referred to as γ) was allowed to vary as part of the fitting procedure, it was found to coincide with the systemic velocity of the nebula (at zero in the plot given that the all radial velocities have been measured with respect to the nebular systemic velocity) within uncertainties (Δ v=v_neb-γ∼3 ).This indicates that the plotted fit may indeed be close to the true orbital solution.However, it must be emphasised that we find similarly good fits for a range of periods, including ∼0.3 and ∼1.2 days. It is important to note that we do not see any clear signs of a secondary component in our spectra.Previous authors have classified the central star as a wels type <cit.>, indicating that perhaps some emission lines attributable to the secondary could be present in the spectra.However, the extremely bright and compact nature of the central region of the nebula can account for the presence of these lines <cit.>.We find the same bright, narrow emission lines in our 2D spectra, but find no evidence of an appreciable contribution to their flux from the central star; rather, they originate purely from the compact nebula (although accurate nebular subtraction is a challenge).Assuming that no such stellar emission lines are present and that the secondary is a main sequence star <cit.>, short periods (⩽1 day) are unlikely given that at short periods a main-sequence star would be expected to show strong emission lines due to irradiation <cit.>.However, if the secondary is a fainter white dwarf, then short periods cannot be ruled out on the basis of a non-detection of irradiated lines in the spectrum.Assuming that the nebular inclination derived in Section <ref> is reflective of the inclination of the binary <cit.>, then it is possible to place some limits on the possible mass of the secondary.Further assuming that the amplitude of the fit shown in Figure <ref>, K_1∼40 km s^-1, is representative of the true amplitude (a seemingly reasonable assumption given the relatively even distribution of the radial velocity data points), then a 9 day period would imply a mass ratio (q=M_2/M_1) of around unity.Longer periods and/or greater radial velocity amplitudes would both imply higher mass ratios and vice versa.The mass function for the secondary can be written asf(M_2)=M_2^3 sin^3 i/(M_1+M_2)^2=P K_1^3/2πG,where M_1 and M_2 are the primary and secondary masses, i is the inclination of the orbital plane, and P is the orbital period.Taking a relatively standard white dwarf mass for the primary of M_1=0.6M_⊙ (and maintaining the assumption that K_1∼40 km s^-1), the mass function can be solved analytically for the secondary mass.Taking into account the uncertainty on the nebular inclination (based on the spatiokinematical modelling presented in Section <ref>), the possible range of secondary masses are plotted as a function of period in Figure <ref>.Under these assumptions, the tentative 9 day period would imply that the secondary is likely a late-type main sequence star (roughly of spectral type K) or a white dwarf.For the secondary to be a white dwarf, it would have to have been initially the more massive component of the binary and, as such, would be expected to leave a more massive remnant (at least more massive than the primary remnant which, in this case, was assumed to be 0.6M_⊙).This is a possibility given that the open mass range for the secondary, based on our fits, is approximately 0.6–0.7M_⊙.However, a more massive secondary would probably be required in order for it to have now cooled beyond observability in our spectra.If the true period is shorter than 9 days (Figure <ref> also shows solutions for the reasonable fits at 0.3 days and 1.2 days) andthen the open mass range would be significantly lower, ruling out an evolved white dwarf companion for the same reason, while a main sequence secondary would become less likely given the aforementioned non-detection of irradiated emission lines originating from the secondary.However, this effect may be counteracted by the implied low secondary masses (and therefore radii) at shorter periods, as the level of irradiation in such systems is principally a function of the apparent radius of the irradiated star (proportional to the ratio of the secondary radius and the orbital separation).Greater radial velocity amplitudes than the assumed ∼40 km s^-1 would open the possibility of a more massive white dwarf companion, but the currently available data provides no indication of such a large amplitude.Similarly, periods longer than 9 days open the parameter space to the possibility of a more massive, evolved white dwarf companion, however in that case the system would have to have undergone two common-envelope phases (given that the AGB radius of a more massive companion would inevitably have been larger than that of the primary) which almost certainly would result in a very short orbital period <cit.>.Therefore, on a stellar evolutionary basis, a main sequence secondary (roughly of K-type for a period of 9 day, or late M-type for periods less than 1 day) would seem the more plausible companion, although a white dwarf secondary cannot be entirely ruled out.§ DISCUSSIONPopulation synthesis models predict many post-CE systems with periods of several to a few tens of days <cit.>. However, the known population of post-CE binary central stars of PNe is very sparse in this region <cit.>, with a similar paucity observed in the general white dwarf plus main sequence binary population <cit.>.Thus far, it is unclear whether this lack of intermediate period binary central stars represents purely failure of the population synthesis models[It is important to note that population synthesis models rely on an ad hoc prescription of the common envelope process, often relying on simple parameterisations that have been proven to be less than satisfactory in reproducing observations <cit.>.Hydrodynamic models thus far fall short of being able to make the predictions required for use in population synthesis <cit.>, however tend to almost unanimously predict very short post-CE orbitals periods while the few that do predict longer periods seem to be under-resolved <cit.>.] or also an observational bias towards the discovery of short period systems.Recent observations have indicated that long-term radial velocity monitoring with modern, high-stability spectrographs may hold the key to revealing this missing intermediate-period population, should it indeed exist <cit.>.Here, we have reported on the discovery of a post-CE binary star at the centre of IC 4776 via such radial velocity monitoring.While the period of the system cannot be confirmed by the data (several periods ranging from ∼0.3–9 days provide reasonable orbital solutions), the detection further supports the hypothesis that a systematic radial velocity survey may indeed reveal the presence of many intermediate period binaries inside PNe.The central star of IC 4776 was found to be a single-lined binary with a tentative orbital period of ∼9 days (though shorter periods cannot be ruled out), where the secondary is most likely a main sequence star (of mass ∼0.1–0.7 M_⊙).The system was selected for monitoring based primarily on the presence of precessing jets, believed to result from mass transfer in a central binary system.Such mass transfer has been hypothesized to occur either prior to or immediately after the CE phase <cit.>, but may also help to prevent in-spiral during the CE as part of a grazing envelope evolution <cit.>.Such a GEE may lead to the preferential formation of wide binaries, but the range of possible periods derived for IC 4776 does not extend to long enough periods to necessitate a GEE to explain its formation (though nor does it rule out such an evolution).An empirical analysis of the chemical abundances in IC 4776 based on échelle spectra covering the entire optical range indicates a low N/O ratio as well as a low C/O ratio (both of which are ∼0.3).This may be indicative that interaction with the companion cut short the AGB evolution of the nebular progenitor <cit.>.In PNe, there is a long standing discrepancy between chemical abundances derived using ORLs and those derived using CELs, which has often been attributed to the existence of a second phase of cold hydrogen deficient material within the normal nebular gas phase <cit.>.The discrepancy is typically a factor of 2–3 but exceeds a factor of 5 in about 20% of nebulae. Recent observations have suggested that high abundance discrepancies may be strongly correlated with central stars that have undergone a CE evolution <cit.>. However, our chemical analysis reveals the adf of IC 4776 to be particularly low at ∼2, which is not just low for a PN with a binary central star, but for PNe in general.Interestingly, the only other post-CE system known to reside in a low-adf nebula is an intermediate-period system <cit.> which, if the period of IC 4776 is a long as 9 days, may imply a connection between intermediate period binaries and low nebular adfs.Furthermore, NGC 5189 is host to a [WR] type primary <cit.>, as has been claimed for the central star of IC 4776 <cit.>.In total, this means three [WR]-type central stars are known to reside in binaries, with the two previous detections (NGC 5189 and PM 1-23) both having orbital periods longer than one day[The orbital period of the [WR] central star of PM 1-23 is given as 0.6 days by <cit.>, but further observations have shown that this is, in fact, half the orbital period <cit.>.].The connection between intermediate periods, low adfs and [WR] central stars is, thus far, tenuous, given the extremely small number statistics involved, particularly given that we are unable to derive a definitive period for the central star of IC 4776.However, there may perhaps be reasonable physical grounds for such correlations. <cit.> suggest that the strong winds observed in [WR] stars may help to prevent spiral-in, via pre-CE wind interaction, leading to longer post-CE periods[A similar scenario has been suggested for the intermediate-period binary central star of NGC 2346 <cit.>, but there is no indication that its central star is of the [WR]-type.].Should high adfs be the result of a nova-like eruption which ejects a second, chemically-enriched gas phase into the nebula <cit.>, then it is not unreasonable to expect that central stars with little hydrogen on their surfaces may be less likely to experience such outbursts (i.e. without hydrogen to burn, they do not undergo an eruption).However, one formation mechanism for such [WR] stars is to undergo some form of late thermal pulse which depletes their outer layers in hydrogen, whilst also providing an explanation for the presence of hydrogen-poor ejecta in their host nebulae<cit.>.Indeed, previously connections were made between the abundance discrepancy problem and hydrogen-depleted central stars <cit.>, particularly in the case of born-again central stars which are found to show some of the highest discrepancies <cit.>.However, more recent studies have shown that in many cases PNe with [WR] central stars present with rather modest abundance discrepancies <cit.>.Assuming that high adfs are the result of late ejection of low-metallicity material, it might be expected that longer period systems do not experience any post-CE mass transfer that could lead to an outburst.There is little evidence of such post-CE mass transfer in other high adf PNe, however, in NGC 6337, jets are observed to have been launched after the ejection of the CE <cit.> while the nebular adf is observed to be highly elevated <cit.>.Furthermore, the possibility remains open for other systems based on the observation of near-Roche-lobe-filling, post-CE secondary stars <cit.>.Unfortunately, the adf of the only other PN known to host a [WR]-binary central star (PM 1-23) is not known, nor are the adfs of other PNe known to host long- and intermediate-period post-CE central stars (e.g. LTNF 1, Sp 1, 2MASS J19310888+4324577, MPA J1508-6455, NGC 2346, NGC 1360).Thus, making further assessment of the possible relationships between periodicity, abundance pattern and central star type currently impossible.In conclusion, in spite of the laborious nature of the observations <cit.>, we encourage further radial velocity studies of the central stars of planetary nebulae in search of the missing population of intermediate-period post-CE binaries.As highlighted here, the pre-selection of targets based on morphology may be the most effective means of increasing the hit-rate for detections. Whilst the possibility of a connection between [WR]-type central stars and intermediate periods is intriguing (as well as a possible link between [WR]-type and low adfs), a more rigorous survey of [WR] central stars (and their host nebulae) is required before any strong conclusions can be drawn.Furthermore, a connection between intermediate-period central stars and low adfs is on a similarly uncertain footing, and abundance studies of the known sample of PNe known to host binary central stars (of all periodicities) is strongly encouraged.Further observations of the central star of IC 4776 are essential in order to full constrain its period, and begin to place these nascent relationships on a stronger footing. § ACKNOWLEDGEMENTSWe are grateful to the referee, Orsola De Marco, for helpful comments and suggestions that further improved the paper. We would like to thank Tom Marsh for the use of his molly software package, and Aníbal García-Hernandez for the use of his UVES data. We thank Gloria Delgado-Inglada for fruitful discussions. Based on data obtained at the European Southern Observatory, Chile, under proposal numbers 097.D-0037, 091.D-0673, 089.D-0429, 087.D-0446. This research made use of Astropy, a community-developed core Python package for Astronomy <cit.>. PS acknowledges the ING Support and Research Studentship, and thanks the Polish National Center for Science (NCN) for support through grant 2015/18/A/ST9/00578. J. G-R acknowledges support from Severo Ochoa Excellence Program (SEV-2015-0548) Advanced Postdoctoral Fellowship.mnras§ UVES SPECTRUM OF IC 4776Figure <ref> shows the full UVES spectrum of IC 4776 referred to in Section <ref>, along with the alfa fit over-plotted.The observed and dereddened fluxes derived from the alfa fit are presented in Table <ref>. | http://arxiv.org/abs/1706.08766v2 | {
"authors": [
"Paulina Sowicka",
"David Jones",
"Romano L. M. Corradi",
"Roger Wesson",
"Jorge García-Rojas",
"Miguel Santander-García",
"Henri M. J. Boffin",
"Pablo Rodríguez-Gil"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170627103811",
"title": "The planetary nebula IC 4776 and its post-common-envelope binary central star"
} |
firstpage–lastpage Was the Universe Actually Radiation Dominated Prior to Nucleosynthesis? Yue Zhao^3 December 30, 2023 =======================================================================We present the serendipitous discovery of an extended cold gas structure projected close to thebrightest cluster galaxy (BCG) of the z = 0.045 cluster Abell 3716, from archival integral field spectroscopy. The gas is revealed through narrow Na D line absorption, seen against the stellar light of the BCG, which can be traced for ∼25 kpc, with a width of 2–4 kpc.The gas is offset to higher velocity than the BCG (by ∼100 km s^-1), showing that it is infalling rather than outflowing; the intrinsic linewidth is ∼80 km s^-1 (FWHM). Very weak Hα line emission isdetected from the structure,and a weak dust absorption feature is suggested from optical imaging, but no stellar counterpart has been identified. We discuss some possible interpretations for the absorber: as a projected low-surface-brightness galaxy,as a stream of gas that was stripped from an infalling cluster galaxy, or as a “retired” cool-core nebula filament. galaxies: clusters: general —galaxies: clusters: individual: Abell 3716 § INTRODUCTIONMassive galaxy clusters exhibit a wealth of astrophysical processes, driven byinteractions between gas, galaxies and the deep gravitational potential.Many gas phases are represented in emission from cluster cores, ranging from the X-ray emitting plasma down to star-forming molecular clouds seen in CO maps. Gas absorption provides a complementary window on the coldest material in thecluster environment, but because a bright backlight is necessarily required, most detections have been made in sightlines towards thecentral galaxy nucleus <cit.>. To date, there have been very few detections of extended cold gas absorbers in cluster cores. Notable cases arein the “high-velocity system”, an infalling galaxy in Perseus<cit.>, and in dusty Hα-emitting filaments inthe central galaxy in Centaurus <cit.>, both detected in the Na i D doublet. Na D absorption is a good tracer of cold neutral gas, because sodium has a low ionization potential and is not depleted onto grains in the interstellar medium. In this Letter we report the discovery of an extended cold gas structure in thez = 0.045cluster Abell 3716, revealed through Na D absorptionagainst the stellar continuum of its central galaxy ESO187-G026. X-ray observations show A3716 to be a non-cool-core cluster with two components of comparable mass, separated by ∼400 kpc in projection <cit.>. ESO187-G026 lies at the core of the northern subcluster.We adopt an angular diameter distance of 0.90 kpc arcsec^-1, valid for z = 0.045 and(h, Ω_ m, Ω_Λ) = (0.7, 0.3, 0.7), if the cluster peculiar velocity is small.§ COLD GAS ABSORPTION IN A3716 ESO187-G026 (the BCG, hereafter) was observed with the Multi-Unit Spectroscopic Explorer (MUSE) <cit.>,on the 8.2m European Southern Observatory Very Large Telescope,as part of the MUSE Most Massive Galaxies Survey (Programme 094.B-0592(A); PI: Emsellem).A stacked reduced datacube was made public by ESO as part of the “MUSE–DEEP” advanced data products release. The stack was generated from 25 individual exposures with a total integration time of 4.0 hours, and covers the standard MUSE spectral range (4750–9350 Å) over a ∼1 arcmin^2 field-of-view. The image quality in the combined frame is ∼0.8 arcsec FWHM, as measured from stars in the field. During an ongoing search for candidate lensed background sources, a visual inspection of the MUSE datacube for A3716 revealed an extended region of narrow Na D doublet absorption to the East and North of the BCG.The strongest part of the structure is already visible in a simple narrow-band image extracted in the wavelengthregion 6166–6179 Å (Figure <ref>a). To isolate the gas absorption signal from the BCG, which also has deep, but much broader, stellar Na D lines,we constructed a model datacube for the galaxy by interpolating the median counts in circular annuli,treating each wavelength channel independently.Subtracting this model before extracting the narrow-band image helps to emphasize the excess absorption East of the BCG (Figure <ref>b). Alternatively, dividing the model datacubeinto the data provides a more uniform representation of the fraction of absorbed flux (or the equivalent width),as the BCG fades with radius (Figure <ref>c).Figure <ref> shows that the Na D absorption structure extends ∼20 arcsecNorth from a point around 6 arcsec East of the BCG center (labelled as point `D'). In Figure <ref>c, there seems to be a deeper absorption at(12, 16) arcsec (point `A'), and a hint of a continuation towards the frame edge at (16, 26) arcsec, although at thisradius the BCG is too faint to detect absorption securely. The width of the structure is generally 2–4 arcsec, but slightly broadening in the southern part. While the northern limit of the absorption is necessarily poorly defined, thesouthern tip occurs in front of bright stellar continuum. There is also a hint of a faint extension further South (point `E'). The spectral characteristics of the Na D absorption are shown in Figure <ref>. The upper panel shows spectraextracted at the five positions A–E indicated in Figure <ref>, as well as from the centre of the BCG, for comparison. The line profiles show a broad absorption component from the BCG stars, superposed by anarrow component in which the doublet structure is clearly resolved. The narrow lines can beisolated by dividing out the spatially symmetric BCG model (lower panel). Fitting double gaussians to these spectra indicates a line-width of 3.1 Å FWHM, marginally larger than the 2.6 Å instrumentalresolution, suggesting an intrinsic velocity width of ∼80 km s^-1.When fitting for the ratio between the two components of the doublet,we find D_1/D_2 = 0.78±0.03 at points A–D, but D_1/D_2 = 0.60±0.08 at point E.This ratio should be ∼0.5 for absorption by optically-thin gas; in the Milky Way, larger values indicate saturation of the stronger D_2 line (the shorter wavelength component).We find no detectable absorption in the K i 7699 Å line (3σ upper limit ∼0.3 Å), which isthe next-strongest resonance line in the wavelength range covered. The profiles of Na D equivalent width and radial velocity along the absorption structure are shown in Figure <ref>.These measurements were made in 2 arcsec diameter apertures, spaced by 1 arcsec along the locus of strongest absorption,with parameters derived from double-gaussian fits with a fixed line width (3.1 Å) and fixed line ratio (D_1/D_2 = 0.8). The total (D_1 + D_2) equivalent widths are 1.5–2.0 Å for the southern part of the structure (points B–D), rising to∼2.5 Å at the northern end (point A). The gas velocity, relative to the BCG stellar absorption, is about+30 km/s at the northern end (A), rising to +150 km/s at the southern tip (D). In the Milky Way, there is an empirical correlation between Na D absorption and dust reddening towards distant objects <cit.>.Close inspection of B-band imaging from the WIde-field Nearby Galaxy-cluster Survey (WINGS) <cit.> shows a faint streak of extinction matching the position of the Na D absorption, suggesting that some dust is indeed associated with the absorbing gas(Figure <ref>, centre). The maximum apparent extinction (approximately at points A and D) is ∼10 per cent, corresponding to E(B-V) ≈ 0.03. According to the Poznanski et al. correlation, this would be associated with only 0.3 Å absorption in Na D, a factor of six smaller than observed (see Figure <ref>).Searching for emission from the filament,we find a hint of a counterpart in Hα, but this is extremely faint (Figure <ref>, right). Summing the (BCG-subtracted) spectra for MUSE pixels where the narrow Na D has equivalent width >1.0 Å,we clearly detect Hα, as well as the [N ii] 6584 Å line (Figure <ref>). The total Hα flux is 8×10^-17 erg s^-1 cm^-2, corresponding toa luminosity 4×10^38 erg s^-1. No stellar counterpart is detectable in the data from WINGS (B, V bands), or the VISTA Hemisphere Survey (J, K), or in Hubble Space Telescope WFPC2 images (F814W) from <cit.>. No counterpart is apparent, in either emission or absorption, in the Chandra X-ray Observatory image<cit.>. § DISCUSSION The key features of the absorbing gas structure described above can be summarized thus: (a) it is long (at least ∼25 kpc) and narrow (2–4 kpc); (b) there is no detectable stellar counterpart; (c) weak Hα emission is detected, with total luminosity 10^39 erg s^-1;(d) some dust is absorption is associated with the structure;(e) the absorbing gas necessarily lies in the foreground of the BCG;(f) the relative velocity of the gas is positive, i.e. it is moving towards the BCG;(g) there is a kinematic gradient, with higher velocities closer to the BCG; (h) there is a sharp termination of the absorber at its closest point to the BCG.We consider here three possible explanations to account for the features observed.The first possibility is that the absorbing object is an edge-on low-surface-brightness disk galaxy, seen in projection against the BCG.This situation would be reminiscent of the “high-velocity system” (HVS) projected onto the Perseus cluster BCG, which is detected inX-ray and dust absorption <cit.>, as well as in line emission <cit.>.Extended interstellar Na D absorption in the HVS was detected by <cit.>. In A3716, the absorbing galaxy would presumably be a cluster member, but the small difference in radial velocity with respect to the BCG(100 km s^-1, compared to a cluster velocity dispersion σ_ cl ≈ 750 km s^-1)would be coincidental.In this interpretation, the lower panel of Figure <ref> simply shows the galaxy rotation curve, with a circular velocity of V_ c ≈ 50 km s^-1. According to the low-mass Tully–Fisher relation of <cit.>, such a galaxy would haveB ≈ 20.5±1.0, which would be detectable in the WINGS data even for a large stellar half-lightradius of ∼3 kpc (typical for ultra-diffuse galaxies). For more a plausible radius, the galaxy would be easily detectableunless it is a significant outlier from the Tully–Fisher relation.(We cannot appeal to dust to obscure the stellar light, because only a few per-cent of the background BCG flux is absorbed.) Hence this scenario is disfavoured unless the absorbing galaxy is unusually faint or diffuse relative to its rotation velocity.A second interpretation is that the Na D absorption traces part of a stream of gas that was stripped from an infalling galaxy by intra-cluster medium (ICM) ram pressure <cit.>. Streams have been widely observed inHα, H i ,X-ray , and cold molecular gas , as well as through young stars formed in the tails . The stripped gas is expected to be heated and mix into the ICM through a variety of physically processes, but some cool clouds may be able to survive at large distance from the parent galaxy <cit.>.The stripped-stream scenario naturally explains the absence of a continuum counterpart, since the stellar component of the stripped galaxy is unaffected by the ICM wind, and only low-level star formation usually occurs in the trail.The weak broadband extinction may be accounted for, as dust is observed to be stripped along with the gas in at least some well-studied nearby cases <cit.>. If the stream follows a nearly-radial orbital trajectory, then the small velocity offset between the stream and the BCGwould imply that the infall is occurring mainly in the plane of the sky, along a North–South axis. There is no clearly disturbed galaxy along this track that can be identified as a potential source of the stripped material.Our third suggestion is that the structure could be a “retired” cool-core nebular filament. Giant emission regions with Hα luminosities up to ∼10^43 erg s^-1 <cit.> are a common feature of cool-core clusters but not of non-cool-core systems , showing that they are ultimately powered through accretion onto the central black hole.Detailed observations of nearby examples suggest that Hα-emitting gas filaments are lifted from the cluster centre in the wake of bouyantly rising bubbles <cit.>, before falling back towards the BCG <cit.>. In Centaurus, some filaments are associated with dust absorption <cit.>,and Na D absorption has been detected in some of these structures, co-located with Hα emission lines and with similar kinematics to the ionized gas <cit.>.As shown in Figure <ref>, the Centaurus filaments show higher reddening than we find in A3716, but display a similar offset towards stronger Na D at given E(B-V), compared to the galactic relation.Significant extended cold molecular gas has also been detected in CO emission from cool-core filaments .By contrast, A3716 is not considered to be a cool-core cluster,and has only low levels of Hα emission. The radio luminosity is also small, with a marginal detection in the SUMSS survey <cit.>,corresponding to L_0.8GHz ≈ 3×10^22 W Hz^-1,a factor of 100 lower than typical for line-emitting BCGs <cit.>.Hence the cold gas we observe is clearly in a different state to the nebular filaments observed in Perseus, Centaurus or other “active” systems.We speculate that A3716 is observed in a “post-cool-core”phase, with the corehaving been disrupted, e.g. by a possible interaction between the two subclusters .In this scenario, the Na D absorption traces a filament lifted into the ICM (or deposited by ICM cooling) during the previous active period, which is no longer excited into significant Hα emission, and is now falling back towards the BCG.The small velocity offset with respect to the BCG is naturally accounted for in this model, since velocities of 100–400 km s^-1 are typical for the H α-emitting filaments, and the length is also consistent with such objects, which canextend many tens of kpc . The Na D absorption region is much wider than the 60 pc measured for the Centaurus Hα filaments by <cit.>,perhaps partly reflecting the linear dependence of absorption EW on density, compared to the quadratic dependence of recombination lines.If the observed structure is a nebular filament, or a stream of stripped gas, it may be long enoughto pass through the BCG. This would provide an interpretation for the abrupt southern tip of the absorption (point D), representing the point of “impact”, and the reduced optical depth (D_1/D_2 ratio) of the weak absorption beyond this (point E). This interpretation can be tested with future observations of the structure in emission,e.g. in atomic hydrogen or molecular gas,which may be able to reveal emission from a continuation of the structure on the far side of the BCG. § CONCLUSIONS & OUTLOOK We have discovered a long and narrow cold gas structure in the cluster A3716, revealed throughNa D in absorption against the central galaxy light. There is no detectablestellar counterpart, but a weak broadband absorption feature seems to be coincident with the gas.Only very weak Hα emission is observed.To our knowledge this is the first detection of an object with these properties in a cluster core, and was made possible by the unprecedented capabilities of MUSE.We have described three possibilities for the nature of the absorbing gas, each of which poses some puzzles. The absence of a stellar counterpart argues against the absorption arising in a foreground galaxy, unless it is unusually faint or diffuse.The absorbing material could be part of a stripped gas stream, but its low velocity with respect tothe BCG implies a rather special orientation nearly fully in the plane of the sky, and no “parent” galaxy has been identified. Alternatively, the gas may be related to the Hα emitting filaments observed in cool-core clusters, perhaps being a “relic” from an earlier phase of cooling in A3716. In this case, the properties of this cold filament may help to distinguish amongthe various mechanisms proposed for exciting such nebulae. Because the A3716 structure is so far detected only in absorption, our evidence is limited by the extent of the background BCG light.Future observations to search forCO or H i emission from the cold gas would be a valuable step towards animproved understanding of this object. Another direction for future work is to search more systematically for Na D absorption in integral field observations of BCGs, to determine whether dark cold gas structures are a common feature of cluster cores[Sincesubmission of this Letter, we have already found a similar, though smaller, example in the cool-core group galaxy NGC 5044, possibly strengthening the association with BCG filaments.].§ ACKNOWLEDGEMENTS RJS and ACE acknowledge support from the STFC through grant ST/P000541/1. All datasets used in this Letter are available from the observatory archives. mnras | http://arxiv.org/abs/1706.08533v1 | {
"authors": [
"Russell J. Smith",
"Alastair C. Edge"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170626180003",
"title": "An extended cold gas absorber in a central cluster galaxy"
} |
Eindhoven University of Technology, The Netherlands{r.m.dijkman|p.w.p.j.grefen}@tue.nlNortheast Petroleum University, [email protected] University of Technology, [email protected] Relational Algebra for In-Database Process MiningRemco Dijkman1 Juntao Gao2 Paul Grefen1 Arthur ter Hofstede3,1 ==================================================================== The execution logs that are used for process mining in practice are often obtained by querying an operational database and storing the result in a flat file. Consequently, the data processing power of the database system cannot be used anymore for this information, leading to constrained flexibility in the definition of mining patterns and limited execution performance in mining large logs. Enabling process mining directly on a database - instead of via intermediate storage in a flat file - therefore provides additional flexibility and efficiency. To help facilitate this ideal of in-database process mining, this paper formally defines a database operator that extracts the `directly follows' relation from an operational database. This operator can both be used to do in-database process mining and to flexibly evaluate process mining related queries, such as: “which employee most frequently changes the `amount' attribute of a case from one task to the next”. We define the operator using the well-known relational algebra that forms the formal underpinning of relational databases. We formally prove equivalence properties of the operator that are useful for query optimization and present time-complexity properties of the operator. By doing so this paper formally defines the necessary relational algebraic elements of a `directly follows' operator, which are required for implementation of such an operator in a DBMS. § INTRODUCTION Enabling process mining directly on an operational database or data warehouse presents new opportunities. It provides additional flexibility, because event logs can be constructed on-demand by writing an SQL query, even if they are distributed over multiple tables, as is the case, for example, in SAP <cit.>. It even provides opportunities for fully flexible querying, allowing for the formulation of practically any process mining question. Moreover, process mining directly on a database leverages the proven technology of databases for efficiently handling large data collections in real time, which is one of the challenges identified in the process mining manifesto <cit.>. This can speed up process mining, especially when extremely large logs are going to be considered, such as call behavior of clients of a telecom provider, or driving behavior of cars on a road network.To illustrate the potential benefits of process mining on a database, Table <ref> shows a very simple event log as it could be stored in a database relation. In practice, such a log would contain thousands of events, as is the case for the well know BPI Challenge logs (e.g. <cit.>), and even millions of events in the examples of the telecom provider and the road network mentioned above. Table <ref> shows the activities that were performed in an organization, the (customer) case for which these activities were performed and the start and end time of the activities. Given such a relation, it is important in process mining to be able to retrieve the `directly follows' relation, because this relation is the basis for many process mining techniques, including the alpha algorithm <cit.> (and its variants), the heuristic miner <cit.>, and the fuzzy miner <cit.>. The directly follows relation retrieves the events that follow each other directly in some case. The SQL query that retrieves this relation from relation R in Table <ref> is:Another example query, is the query that returns the average waiting time before each activity: These queries illustrate the challenges that arise when doing process mining directly on a database: * The queries are inconvenient. Even a conceptually simple process mining request like `retrieve all directly follows relations between events' is difficult to phrase in SQL.* The queries are inefficient. The reason for this is that the `directly follows' relation that is at the heart of process mining requires a nested query (i.e. the `NOT EXISTS' part) and nested queries are known to cause performance problems in database, irrespective of the quality of query optimization <cit.>, This will be discussed in detail in Section <ref>. Consequently, measures must be taken to make process mining - and in particular extracting the `directly follows' relation - feasible on relational databases. Figure <ref> shows three possible strategies. Figure <ref>.i shows the current state of the art, in which a user constructs an SQL query to extract a log from the database. This log is written to disk (for example as a csv file) and then read again by a process mining tool for further processing. Consequently, the complete log must be read or written three times and there is some manual work involved in getting the log to and from disk. It is easy to imagine a process mining tool that does not need intermediate storage to disk. Such a tool would only need to read the log once and would not require manual intervention to get the log to and from disk. Figure <ref>.ii illustrates this strategy.This paper proposes a third strategy, in which the DBMS supports a native `directly follows' operator. This strategy has the benefit that it does not require intermediate storage on disk and that it facilitates flexible and convenient querying for process mining related information. In addition, it has the benefit that it leverages proven database technology, which can efficiently handle very large data sets in real time. To realize this strategy, the paper defines the `directly follows' operator in relational algebraic form, by defining what it does, how it behaves with respect to other operators, and what its execution costs are. By doing so this paper formally defines the necessary relational algebraic elements of a `directly follows' operator, which are required for implementation of such an operator in a DBMS.Against this background the remainder of this paper is structured as follows. Section <ref> explains relational algebra as background for this paper. Section <ref> presents a relational algebraic `directly follows' operator. Section <ref> shows the computational cost of executing this operator and the potential effects of query optimization with respect to this operator. Finally, section <ref> presents related work and section <ref> the conclusions.§ BACKGROUND Relational algebra is at the core of every relational database system. It is used to define the execution semantics of an SQL query and to define equivalence rules to determine which execution semantics (among a set of equivalent ones) is the most efficient to execute. Before we define a relational algebraic `directly follows' operator, we provide background on these two topics. §.§ Relational Algebra In this section we briefly define the basic relational algebraic operators. We refer to the many textbooks on databases (e.g. <cit.>) for more detailed definitions. An attribute is a combination of a name and a domain of possible values. A schema is a set of attributes. Any two elements s_1, s_2 of schema s with s_1 ≠ s_2 have different names. A relation is a combination of a schema and a set of tuples. Each tuple in a relation maps attribute names from the schema of the relation to values from the corresponding domain. For example, let C be the domain of case identifiers, E be the domain of activities, and T be the time domain. The relation of Table <ref> has the schema {case: C, activity: E, start_time: T, end_time: T} and the set of tuples {{case↦ 1, activity↦ A, start_time↦ 00:20, end_time↦ 00:22}, …}.In the remainder of this paper, we will also refer to R as the set of tuples of a relation R. For a relation with a schema that defines an attribute with name a, we will use t_a to refer to the value of attribute a in tuple t. Let R, S be relations with schema r, s. Furthermore, let a, b be attribute names, and ϕ a condition over attributes that is constructed using conjunction (), disjunction (), negation (¬), and binary conditions (>,≥,=,≠,≤,<) over attributes and attribute values. We define the usual relational algebra operators: * Selection: σ_ϕ R = {t | t ∈ R, ϕ(t) }, where ϕ(t) is derived from ϕ by replacing each attribute name a by its value in tuple t: t_a. The schema of σ_ϕ R is r.* Projection: π_a,b,… R = {{a↦ t_a,b↦ t_b,…} | t ∈ R }. The schema of π_a,b,… R is r restricted to the attributes with names a, b, ….* Renaming: ρ_a/b R = R. The schema of ρ_a/b R is derived from r by replacing the name of attribute a by b. In the remainder of this paper, we will also use ρ_x R to represent prefixing all attributes of R with x.In addition, we define the usual set theoretic operators R ∪ S, R ∩ S, R - S. These operators have the usual set-theoretic interpretation, but they require that R and S have the same schema. We define the Cartesian product of R with schema {r_1: R_1, r_2: R_2, …, r_n:R_n} and S with schema {s_1:S_1, s_2:S_2, …, s_n:S_n} as R × S = {{r_1↦ t_r_1,r_2↦ t_r_2,…,r_n↦ t_r_n,s_1↦ u_s_1, s_2↦ u_s_2,…,s_m↦ u_s_m}|t ∈ R, u ∈ S}. The schema of R × S is r ∪ s.Finally, a join operator is usually defined for the commonly used operator of joining tuples from two relations that have some property in common. The join operator is a shorthand for a combination of Cartesian product and selection: R ⋈_ϕ S = σ_ϕ R × S. Table <ref> shows examples of the selection, projection, and renaming operators, applied to the relation in Table <ref>. §.§ Query Optimization There are a large number of proven relational algebraic equivalences that can be used to rewrite relational algebraic equations <cit.>. In the remainder of this paper, we use the following ones. Let R, S be tables, a, b, c be attributes, x, y be attribute values, ϕ, ψ be conditions, and θ be a binary condition (>,≥,=,≠,≤,<). Then:σ_ϕψ R = σ_ϕ (σ_ψ R)σ_ϕ (σ_ψ R) = σ_ψ (σ_ϕ R) R ⋈_ϕ S = S ⋈_ϕ R (R ⋈_ϕ S) ⋈_ψ T = R ⋈_ϕ (S ⋈_ψ T)σ_ψ (R ⋈_ϕ S) = (σ_ψ R) ⋈_ϕ S , if ψ only has attributes fromRσ_ψ (R - S) = (σ_ψ R) - (σ_ψ S)σ_a θ x (ρ_b/a R) = ρ_b/a (σ_b θ x R)π_a (ρ_b/a R) = ρ_b/a (π_b R) π_a,b,… (σ_ϕ R) = σ_ϕ (π_a,b,… R) , if ϕ only has attributes froma, b, … π_a,b,… (S ⋈_ϕ R) = (π_a,… R) ⋈_ϕ (π_b,… S), ifa, b, …can be split overR, Sπ_a,b ( π_b,c R) = π_b Rπ_a,… ( π_b,… R) = π_b,… ( π_a,… R)ρ_b/a(R ⋈_bθ c S) = (ρ_b/aR) ⋈_aθ c S , ifa,bonly inRπ_Rs R = R , ifRshas only attributes fromRπ_Rs (R ⋈_ϕ S) = R , ifR ⋈_ϕ Sincludes each tuple ofR= R, andRshas exactly the attributes fromR (R-T) ⋈_a θ b S = R ⋈_a θ b S - T ⋈_a θ b S In practice these equivalences are used to generate alternative formulas that lead to the same result, but represent alternative execution strategies. For example, σ_ψ(σ_ϕ R ×σ_θ S) can be proven to be equivalent to σ_ψϕθ (R × S). However, σ_ψ(σ_ϕ R ×σ_θ S) represents the execution strategy in which we first execute the selections and then the Cartesian product, while σ_ψϕθ (R × S) represents the execution strategy where we first execute the Cartesian product and then the selection. The first execution strategy is much more efficient than the second, because it only requires the Cartesian product to be computed for a (small) subset of R and S.§ RELATIONAL ALGEBRA FOR PROCESS MINING This section defines the `directly follows' relation as a relational algebraic operator. It also presents and proves equivalences for this operator that can be used for query optimization, similar to the equivalences that are presented in Section <ref>. §.§ Directly Follows Operator The directly follows operator retrieves events that directly follow each other in some case. For a database table Log that has a column c, which denotes the case identifier, and a column t, which denotes the completion timestamp of an event, we denote this operator as >_c,t Log. For example, applying the operator to the example log from Table <ref> (i.e. >_case, end_time Log) returns Table <ref>. Similar to the way in which the join operator is defined in terms of other relational algebra operators, we define the `directly follows' operator in terms of the traditional relational algebra operators as follows. >_c,t Log =ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log- π_As ((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log)where As is the set of attributes that are in ρ_↓ Log or ρ_↑ Log. The directly follows operator can both be used in an algorithm for process mining that is based on it (or on `footprints' which are derived from it <cit.>) and for flexible querying. Some example queries include: * The two activities that precede a rejection: π_↑↑ activity = reject >_↑ case, ↑ end_time >_case, end_time Log* The activities in which the amount of a loan is changed: σ_↑ amount ≠↓ amount >_↑ case, ↑ end_time Log* The resources that ever changed the amount of a loan: π_↑ resourceσ_↑ amount ≠↓ amount >_↑ case, ↑ end_time Log§.§ Directly Follows Query Optimization To facilitate query optimization for the directly follows operator, we must define how it behaves with respect to the other operators and prove that behavior. We present this behavior as propositions along with their proofs. In each of these propositions, we use a, b, c, t as attributes (where - as convention - we use c to denote the case identifier attribute and t to denote the time attribute), θ as a binary operator from the set {>,≥,=,≠,≤,<}, and x as a value.The first proposition holds for case attributes and event attributes. We define case attributes as attributes that keep the same value for all events in a case, from the moment that they get a value. We define event attributes as attributes that have a value for at most one event in each case. Consequently, we can only use this proposition for optimizing queries that involve a selection on a case or event attribute. Selections on other types of attributes (including resource attributes) cannot be optimized with this proposition.>_c,tσ_a θ x Log = σ_↓ a θ x ↑ a θ x >_c,t Log, if a is a case or event attribute. >_c,tσ_a θ x Log =(definition <ref>) ρ_↓(σ_a θ x Log) ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑(σ_a θ x Log)- π_As(((ρ_↓ (σ_a θ x Log) ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑(σ_a θ x Log)) ⋈_↓ t < tt < ↑ t ↓ c = cσ_a θ x Log) )=(proposition <ref>) σ_↓ a θ x (ρ_↓ Log) ⋈_↓ t < ↑ t ↓ c = ↑ cσ_↑ a θ x(ρ_↑ Log)- π_As(((σ_↓ a θ x(ρ_↓ Log) ⋈_↓ t < ↑ t ↓ c = ↑ cσ_↑ a θ x(ρ_↑ Log)) ⋈_↓ t < tt < ↑ t ↓ c = cσ_a θ x Log) ) =(proposition <ref>, <ref>, <ref>) σ_↓ a θ x ↑ a θ x (ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log)- π_As( σ_↓ a θ x ↑ a θ xa θ x ((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log) ) = (assume ↓a θ x ↑a θ x ⇒ a θ x) σ_↓ a θ x ↑ a θ x (ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log)- π_As( σ_↓ a θ x ↑ a θ x ((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log) )=(proposition <ref>) σ_↓ a θ x ↑ a θ x (ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log)- σ_↓ a θ x ↑ a θ x( π_As ((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log) )=(proposition <ref>) σ_↓ a θ x ↑ a θ x( ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log- π_As ((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log) )=(definition <ref>) σ_↓ a θ x ↑ a θ x >_c,t LogNote that proposition <ref> requires that the condition only contains attributes that are also projected (in this case ↓ a, ↑ a must be in As). This condition is satisfied as per definition <ref>. Also note that the proof uses an assumption, which states that if any two events in a case have the same value for an attribute, all events for that case that are between these two (in time) must also have that value (↓a θ x ↑a θ x ⇒ a θ x). This assumption holds for case attributes and for event attributes, which are the scope of this proposition.The next proposition is a variant of the previous one, in which there is a condition on two attributes instead of an attribute and a value.>_c,tσ_a θ b Log = σ_↓ a θ↓ b ↓ a θ↑ b ↑ a θ↓ b ↑ a θ↑ b >_c,t Log, if a,b are case or event attributes.Analogous to the proof of proposition <ref> To prove that directly follows and projection commute, we first need to prove that projection and set minus commute, because the set minus operator is an important part of the definition of the directly follows operator. However, for the general case it is not true that projection and set minus commute. A counter example is easily constructed. Let R = {{a↦ 1,b↦ 2}} and S = {{a↦1,b↦3}}. For these relations it does not hold that π_a (R - S) = (π_a R) - (π_a S). However, we can prove this proposition for the special case that S is a subset of R and a uniquely identifies tuples in R. Since these conditions are satisfied for the directly follows operator, it is sufficient to prove the proposition under these conditions.π_a (R - S) = (π_a R) - (π_a S),if S ⊆ R and a uniquely identifies each tuple in R.This equivalence is proven by observing that S ⊆ R implies that a non-surjective injective function f:S→ R exists that matches each tuple s in S to a unique tuple r in R. The fact that a uniquely identifies tuples in R (and also in S, because S is a subset of R) implies that f is completely determined by the values of tuples in a, i.e., the values of attributes other than a have no consequence for f. Therefore, projecting R and S onto a does not change the tuple mapping defined by f.Now, looking at the left side of proposition <ref>, calculating the projection over the difference, means removing the attributes not in a from the selected tuples in R that are not in the range of f. Looking at the right side, calculating the difference over the projections, means removing the attributes not in a from R and S (which does not affect f) and then selecting the tuples in R that are not in the range of f. These two are equivalent. >_c,tπ_c,t,a Log = π_↓ c,↓ t,↓ a, ↑ c, ↑ t, ↑ a >_c,t Log >_c,tπ_c,t,a Log =(definition <ref>) ρ_↓(π_c,t,a Log) ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑(π_c,t,a Log)- π_As ((ρ_↓ (π_c,t,a Log) ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑(π_c,t,a Log)) ⋈_↓ t < tt < ↑ t ↓ c = cπ_c,t,a Log) =(proposition <ref>) π_↓ c,↓ t,↓ a (ρ_↓ Log) ⋈_↓ t < ↑ t ↓ c = ↑ cπ_↑ c,↑ t,↑ a (ρ_↑ Log)- π_As ((π_↓ c,↓ t,↓ a (ρ_↓ Log) ⋈_↓ t < ↑ t ↓ c = ↑ cπ_↑ c,↑ t,↑ a (ρ_↑ Log)) ⋈_↓ t < tt < ↑ t ↓ c = cπ_c,t,a Log) =(proposition <ref>) π_↓ c,↓ t,↓ a,↑ c,↑ t,↑ a (ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log)- π_As(π_↓ c,↓ t,↓ a,↑ c,↑ t,↑ a,c,t,a((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log) )=(proposition <ref>, <ref>) π_↓ c,↓ t,↓ a,↑ c,↑ t,↑ a (ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log)- π_↓ c,↓ t,↓ a,↑ c,↑ t,↑ a( π_As ((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log) ) =(proposition <ref>) π_↓ c,↓ t,↓ a,↑ c,↑ t,↑ a( (ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log)- π_As ((ρ_↓ Log ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑ Log) ⋈_↓ t < tt < ↑ t ↓ c = c Log) ) =(definition <ref>) π_↓ c,↓ t,↓ a, ↑ c, ↑ t, ↑ a >_c,t LogThe next proposition, which states that the directly follows relation and the theta join commute, makes it explicit that the directly follows relation duplicates all attributes of a log event. Table <ref> illustrates this. However, if the case, activity and start time attribute uniquely identify an event, then there is no need to duplicate the end time attribute or any other attribute. Nonetheless, the directly follows operator adds all attributes both on the ↑ and on the ↓ side of the table. This redundancy can easily be fixed later on with a project operator and in future work additional efficiency may be achieved by avoiding this redundancy altogether.>_c,t (R ⋈_a θ b S) = (>_c,t R) ⋈_↓ a θ b S ⋈_↑ a θ b S, if each tuple from R is combined with a tuple in S. >_c,t (R ⋈_a θ b S) =(definition <ref>) ρ_↓(R ⋈_a θ b S) ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑(R ⋈_a θ b S)- π_As ((ρ_↓ (R ⋈_a θ b S) ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑(R ⋈_a θ b S)) ⋈_↓ t < tt < ↑ t ↓ c = c (R ⋈_a θ b S)) =(proposition <ref>)(ρ_↓R) ⋈_↓ a θ b S ⋈_↓ t < ↑ t ↓ c = ↑ c (ρ_↑R) ⋈_↑ a θ b S- π_As (((ρ_↓R) ⋈_↓ a θ b S) ⋈_↓ t < ↑ t ↓ c = ↑ c (ρ_↑R) ⋈_↑ a θ b S) ⋈_↓ t < tt < ↑ t ↓ c = c R ⋈_a θ b S) =(proposition <ref>, <ref>, <ref>, <ref>) ρ_↓R ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑R ⋈_↓ a θ b S ⋈_↑ a θ b S- π_As (ρ_↓R ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑R ⋈_↓ t < tt < ↑ t ↓ c = c R) ⋈_↓ a θ b S ⋈_↑ a θ b S =(proposition <ref>) (ρ_↓R ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑R - π_As (ρ_↓R ⋈_↓ t < ↑ t ↓ c = ↑ cρ_↑R ⋈_↓ t < tt < ↑ t ↓ c = c R)) ⋈_↓ a θ b S ⋈_↑ a θ b S =(proposition <ref>)(>_c,t R) ⋈_↓ a θ b S ⋈_↑ a θ b S§ EXECUTION COST We determine the computational cost of executing the directly follows operation, either as part of a process mining tool or as an operation that is executed directly on the database. We also determine the effect of query optimization on the directly follows operator. §.§ Cost of computing the directly follows relation The execution cost of a database operation is typically defined in terms of the number of disk blocks read or written, because reading from or writing to disk are the most expensive database operations. In line with the strategies for process mining on a database that are presented in Figure <ref>, Table <ref> shows four execution strategies with their costs. (Note that the `with database operator' strategy from Figure <ref> is split up into two alternatives.) The cost is presented as an order of magnitude, measured in terms of the number of disk blocks B that must be read or written. The number of disk blocks is linear with the number of events in the log and depends on the number of bytes needed to store an event and the number of bytes per disk block. These measures assume that the complete log fits into memory.Process mining with intermediate storage requires that the log is read and written three times: once to query the database for the log, once to store the log to disk, and once to load the log in the process mining tool. Consequently, the complexity is 3 · B. Process mining directly on a database requires that the log is read only once. Subsequent processing can be done in memory.In many usage scenarios more flexible querying capabilities are needed, which can benefit from access to all SQL operators. For such usage scenarios, the `directly follows' relation must be extracted directly from the database. It is easy to imagine how a native `directly follows' operator would work. Such an operator would read the log, then sort it by case identifier and timestamp, and finally return each pair of subsequent rows that have the same case identifier. Such an operator would have to read the log from disk only once and consequently has linear cost. For databases that do not have a native `directly follows' operator, the operator can be emulated using the composite formula from definition <ref>. The drawback of this formula is that it requires that the intermediate results from both sides of the minus operator are stored, after which the minus operator can be applied. While this is no problem as long as the intermediate results fit into memory, the costs become prohibitive once the intermediate results must be written to disk.On a practical level, this problem manifested itself, for example, for the log of the BPI 2011 challenge <cit.> on our desktop with an i5 processor, 8GB of internal memory and an SSD drive, using MySQL and the standard MySQL buffer size. Each attempt to perform a database query that involved a composite `directly follows' relation, needed at least 10 minutes to execute, which is prohibitive for interactive exploration of an event log.On a theoretical level, the problem is illustrated in Figure <ref>. This figure shows that the problem arises when the number of events in the log is high, relative to the number of cases. The mechanism that causes this is easy to derive from definition <ref>, which shows that the intermediate results that must be stored are the pairs of events that directly or indirectly follow each other in some case (the left side and right side of the minus operator). Consequently, if there are many events per case, this number is high (cubic over the number of events per case in the right-hand side of the minus operator).The precise calculation can be performed as follows. Let V be the number of cases in the log, N be the number of events, F be the block size (i.e. the number of tuples/events that fit in a single disk block), B_Log = N/F the number of disk blocks required to store the log, and M be the total memory size in blocks. Note that the cost of a block nested join (or minus) operator on two relations R and S that take B_R and B_S disk blocks (with B_R ≤ B_S), is equal to B_R + B_S when one of the two relations fits in memory, and equal to B_R + B_S/M· B_R otherwise <cit.>. The cost is split up into five components: * The cost of the first join is denoted as B_join_1. This equals B_Log if the log fits into memory and B_Log + B_Log/M· B_Log otherwise. Note that this join appears twice, but that it only needs to be computed once.* The cost of storing the results of the first join to disk is denoted as B_result_1. This equals 0 if the result fits in memory. Otherwise, the number of tuples in the result, which we denote as |t_1|, equals the number of pairs of events that directly or indirectly follow each other in some case: V · (N/V·N/V-1)/2 on average. This fits into |t_1|· 2/F disk blocks (times 2 because each tuple in the result is a pair of tuples from the original).* The cost of the second join is denoted as B_join_2. This equals 0 if the original log fits into memory. Otherwise, the cost equals B_Log + |t_1|· 2/F/M · B_Log.* The cost of storing the result of the second join to disk is denoted as B_join_2. This equals 0 if the result fits into memory. Otherwise, the number of tuples in the result, which we denote as |t_2|, equals the number of pairs of events that indirectly follow each other. This equals the number of pairs of events |t_1| that directly of indirectly follow each other minus the number of pairs of events that directly follow each other: V· (N/V -1) on average. This fits into |t_2|· 2/F disk blocks (times 2 because each tuple in the result is a triple of tuples from the original and then reduced to a pair by projection).* The cost of the minus operator is denoted as B_minus. This equals 0 if the result of the second join fits into memory. Otherwise, it equals B_result_1 + B_result_1/M· B_result_2. To generate Figure <ref> we used a tuple size of 80 bytes, a 4 GB buffer size, and a block size of 50, such that there is a total memory size of 1 million blocks. The figure shows two `thresholds' in the computational cost. These thresholds are crossed when a particular intermediate result no longer fits into memory.The order of the cost can be determined more easily. The order of the cost is determined the cost of the set minus, because this incorporates both intermediate results, which are typically much larger than the original log. Therefore, the order of the cost of computing the intermediate results are N^2/V/F. The total order of cost is then obtained by filling these costs out in the right-hand side of the formula for computing the cost of the set minus, which yields: (N^2/V/F/M) ·N^2/V/F. If we let M be large enough to contain the log itself, but not the intermediate results (i.e. we set M=N/F), this can be simplified as: N^3/V^2/F.Summarizing, the execution cost of flexibly retrieving a directly follows relation directly from a database can be as low as retrieving it from a process mining tool, if the database supports a native `directly follows' operator and the process mining tool supports on-database process mining. However, as long as a native `directly follows' operator does not exist, the execution costs increase to third order polynomial cost if the average number of events per case is high (i.e. if intermediate results do not fit into memory anymore). §.§ The effect of query optimization An advantage of in-database process mining is that it enables query optimization. Query optimization, using the rewrite rules that are defined in section <ref> can greatly reduce the cost of executing a query. As an example, we show the cost of executing the query >_c,tσ_a θ x Log and the equivalent query σ_↓ a θ x ↑ a θ x >_c,t Log. These costs decrease at least linearly with the fraction of events that match the selection criteria. Let Q be that fraction. Table <ref> shows the different execution situations that can arise. It is possible to either first derive the directly follows relation and then do the selection, or vice versa. It is also possible that the intermediate results fit in memory, or that they must be stored on disk. If the results fit in memory (and the table is indexed with respect to the variable on which the selection is done), then the execution costs are simply the cost of reading the log, or the part that matches the selection criteria, into memory once. If the intermediate results do not fit into memory, the order of the execution cost is N^3/V^2/F as explained in the previous section. Remembering that B = N/F leads to the formulas that are shown in the table.The most dramatic increase occurs when, if the selection is done first and as a consequence the intermediate results fit into memory, while if the selection is done last, the intermediate results do not fit into memory. In practice this is likely to be the case, because the selection can greatly reduce the number of events that are considered. For example, for a log with N = 10,000 events over V = 500 cases, with a block size of F = 50 and a selection fraction Q = 0.10, the order of the cost increases from 20 to 8 · 10^4 according to the formulas from Table <ref>. The actual computed costs increase (the same order of magnitude) from 21 (plus one, because we need to read one disk block to load the index that is used to optimize the selection) to 9.5 · 10^4 using the formulas from the previous section.This shows that the way in which a query that includes the directly follows operator is executed greatly influences the execution cost. Query optimizers, which are parameterized with equivalence relations from section <ref> and the cost calculation functions from section <ref>, can automatically determine the optimal execution strategy for a particular query.§ RELATED WORK By defining an operator for efficiently extracting the `directly follows' relation between events from a database, this paper has its basis in a long history of papers that focus on optimizing database operations. In particular, it is related to papers that focus on optimizing database operations for data mining purposes <cit.>, of which SAP HANA <cit.> is a recent development. The idea of proposing domain-specific database operators has also been applied in other domains, such as spatio-temporal databases <cit.> and scientific databases <cit.>.By presenting a `directly follows' operator, the primary goal of this paper is to support computationally efficient process mining on a database. There exist some papers that deal with the computational complexity of the process mining algorithms themselves <cit.>. Also, in a research agenda for process mining the computational complexity and memory usage of process mining algorithms have been identified as important topics <cit.>. However, this paper focuses on a step that precedes the process mining itself: flexibly querying a database to investigate which information is useful for process mining.More database-related work from the area of process mining comes from shaping data warehouses specifically for process mining <cit.>. There also exists work that focuses on the extraction of logs from a database <cit.>.§ CONCLUSIONS This paper presents a first step towards in-database process mining. In particular, it completely defines a relational algebraic operator to extract the `directly follows' relation from a log that is stored in a relational database, possibly distributed over multiple tables. The paper presents and proves relational algebraic properties of this operator, in particular that the operator commutes with the selection, projection, and theta join. These equivalence relations can be used for query optimization. Finally, the paper presents and proves formulas to estimate the computational cost of the operator. These formulas can be used in combination with the equivalence relations to determine the most efficient execution strategy for a query. By presenting and proving these properties, the paper provides the complete formal basis for implementing the operator into a specialized DBMS for process mining, which can be used to efficiently and conveniently query a database for process mining information.Consequently, the obvious next step of this research is to implement the operator into a DBMS. The DBMS and the relational algebraic operators can then be further extended with additional process mining-specific operators, such as an operator to query for execution traces. In addition, more algebraic properties of those operators can be proven to assist with query optimization.There are some limitations to the equivalence relations that are presented in this paper, in particular with respect to the conditions under which they hold. These limitations restrict the possibilities for query optimization. The extent to which these theoretical limitations impact practical performance of the operator must be investigated and, if possible, mitigated.10aalst2012 van der Aalst, W., et al.: Process mining manifesto. In: Proc. of BPM Workshops. pp. 169–194 (2012)aalst2011 van der Aalst, W.: Process Mining: Discovery, Conformance and Enhancement of Business Processes. Springer (2011)aalst2004agenda van der Aalst, W., Weijters, A.: Process mining: a research agenda. Computers in Industry53(3),231–244 (2004)aalst2004 van der Aalst, W., Weijters, A., Maruster, L.: Workflow mining: Discovering process models from event logs. IEEE Transactions on Knowledge and Data Engineering16(9),1128–1142 (2004)abiteboul1995 Abiteboul, S., Hull, R., Vianu, V.: Foundations of databases: the logical level. Addison-Wesley (1995)abraham1999 Abraham, T., Roddick, J.F.: Survey of spatio-temporal databases. 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IEEE Transactions on Knowledge and Data Engineering8(6), 866–883 (1996)cudremauroux2009 Cudre-Mauroux, P., Kimura, H., Lim, K.T., Rogers, J., Simakov, R., Soroush, E., Velikhov, P., Wang, D.L., Balazinska, M., Becla, J., DeWitt, D., Heath, B., Maier, D., Madden, S., Patel, J., Stonebraker, M., Zdonik, S.: A demonstration of scidb: A science-oriented dbms. Proc. VLDB Endow.2(2), 1534–1537 (2009)dongen2011 van Dongen, B.: Real-life event logs - hospital log (2011)eder2002 Eder, J., Olivotto, G.E., Gruber, W.: A Data Warehouse for Workflow Logs, pp. 1–15 (2002)farber2012 Färber, F., Cha, S.K., Primsch, J., Bornhövd, C., Sigg, S., Lehner, W.: Sap hana database: Data management for modern business applications. SIGMOD Rec.40(4),45–51 (2012)guenther2007 Günther, C., van der Aalst, W.: Fuzzy mining: Adaptive process simplification based on multi-perspective metrics. In: Proc. of BPM 2007. pp. 328–343 (2007)ingvaldsen2007 Ingvaldsen, J.E., Gulla, J.A.: Preprocessing support for large scale process mining of sap transactions. In: Proc. of BPM. pp. 30–41 (2007)kim1982 Kim, W.: On optimizing an SQL-like nested query. ACM Trans. on Database Systems7(3),443–469 (1982)maggi2012 Maggi, F.M., Bose, R.P.J.C., van der Aalst, W.M.P.: Efficient Discovery of Understandable Declarative Process Models from Event Logs, pp. 270–285 (2012)zurmuehlen2001 zur Mühlen, M.: Process-driven manaement information systems – combining data warehouses and workflow technology. In: Proc. of ICECR. pp. 550–566 (2001)sagiv1978 Sagiv, Y., Yannakakis, M.: Equivalcen among relational expressions with the union and difference operation. In: Proc. of VLDB. pp. 535–548 (1978)weijters2006 Weijters, A., van der Aalst, W., Alves De Medeiros, A.: Process mining with the heuristics miner-algorithm. Technische Universiteit Eindhoven, Tech. Rep. WP166,1–34 (2006) | http://arxiv.org/abs/1706.08259v1 | {
"authors": [
"Remco Dijkman",
"Juntao Gao",
"Paul Grefen",
"Arthur ter Hofstede"
],
"categories": [
"cs.DB"
],
"primary_category": "cs.DB",
"published": "20170626073136",
"title": "Relational Algebra for In-Database Process Mining"
} |
Thealgorithm: non-local jumps and greedy retries improveclustering Bernd Fritzke [email protected] 61381 Germany December 30, 2023 ======================================================================================authoryear, open=(,close=)We present a new clustering algorithm calledwhich in many cases is able to significantly improve the clusterings found by , the current de-facto standard for clustering in Euclidean spaces. First we introduce thealgorithm which starts from a result ofand attempts to improve it with a sequence of non-local “jumps” alternated by runs of standard . Each jump transfers the “least useful” center towards the center with the largest local error, offset by a small random vector. This is continued as long as the error decreases and often leads to an improved solution. Occasionallyterminates despite obvious remaining optimization possibilities. By allowing a limited number of retries for the last jump it is frequently possible to reach better local minima. The resulting algorithm is calledand dominateswrt. solution quality which is demonstrated empirically using various data sets. By construction the logarithmic quality bound established forholds foras well.Clustering, Vector Quantization, Optimization, k-means,§ INTRODUCTIONIn machine learning various sub-fields can be distinguished based on the amount of feedback given to the learner. On one side there is supervised learning where the given data consists of pairs of input and corresponding output. In this case the learner is trained to approximately reproduce this mapping with the expectation that it can afterwards generalize to unseen patterns. Typical examples of supervised learning are classification (e.g. image or speech recognition) and regression (e.g. time-series prediction). Considerably less information is given in reinforcement learning where a system is expected to learn how to perform long sequences of actions (e.g. in game play) while only receiving occasional feedback (e.g. "game won" or "game lost"). Finally there is unsupervised learning where a system is expected to learn purely from unlabeled data. It has been argued recently that due to the number of connections in the human brain and the life span of humans a large part of learning must be unsupervised (G. Hinton, AMA on Reddit.com, 2015). This paper is concerned with clustering which is considered to be a fundamental unsupervised learning problem. In particular we propose a method to improve clusterings generated by thealgorithm which currently can be seen as a de facto standard for clustering numerical data.The remainder of this article is organized as follows: In <ref> the particular kind of clustering problem is described which the new algorithm (as well asand ) deal with. In <ref> the classicalalgorithm is introduced and illustrated by examples which also highlight its problematic dependency on initialization. In section <ref> we describe the more recentalgorithm which enhancesby a stepwise initialization which in general leads to much better results than e.g. the common random initialization from the data set. <Ref> investigates why is so effective and shows for a simple example problem that under certain conditionsalways finds the optimal initial placement. <Ref> gives examples of problems which are hard forin the sense thatis in general not able to find a solution close to the optimum (which is known for these specific problems by construction). It is investigated why this is the case and that for certain exampleshas difficulties if k is larger than the number of clusters in the data. In <ref> thealgorithm is introduced which considerably improves many solutions found byvia a sequence of non-local jumps alternated withphases. In <ref> an occasional problem ofis described and illustrated by an example: too early termination due to a poor (but normally still better than ) local minimum. <Ref> introduces thealgorithm which enhancesby allowing greedy retries. This allows in many cases to reach considerably better local minima. In <ref> the results of systematic simulations with various data sets are presented covering large ranges of k for each data set. In a nutshell the conclusion is:w.r.t. solution qualitydominateswhich again dominates .In <ref> the fact is pointed out that the logarithmic quality bound which has been established forholds (by construction) foras well.<Ref> provides a summary of the article. § CLUSTERING Clustering in general can be described as the problem of finding groups (clusters) in a data set such that the similarity of data items within a group is large and data items from different groups are dissimilar.The used measure of similarity can vary from a subjective perceptual criterion to precise mathematical definitions. In the context of this paper we choose the goal of clustering to be a computational one, in particular the minimization of a distance-based error function:We assume an integer k and a set of n data points 𝒳⊂ℝ^d. The goal is to select kcenters 𝒞 such that the error function ϕ(𝒞,𝒳) = ∑_x∈𝒳min_c∈𝒞 ||x-c||^2is minimized. We thus strive to position the centers in such a way that the sum of squared distances between each data point and its respective closest center is minimized. In this paper we will refer to ϕ(𝒞,𝒳) also as Summed Squared Error or SSE.The resulting set of centers 𝒞 can be used to group (cluster) the original data set but also to encode the data for transmission or storage in the sense of vector quantization. Finding the optimal solution for this problem is known to be NP-hard <cit.>, so in practice we need to use approximation algorithms. In <ref> an example of a data set is given with two clusterings which vary strongly with respect to the error criterion in <ref>.§ <cit.> is a classical method for clustering or vector quantization.Starting from an initial set of centers 𝒞 = { c_1,c_2,... ,c_k} a sequence of so-called Lloyd Iterations is performed. Each Lloyd iteration consists of two steps: * determine for each center c_i its so-called Voronoi Set C_i, which is theset of data points for which c_i is the closest center: C_i := {x ∈𝒳 | x-c_i < x-c_j ∀ j≠ i} * move all centers to the center of gravity of their Voronoi set: c_i := 1/|C_i|∑_x∈ C_ixThe completealgorithm is specified in <ref>. It consists of an initialization ("Seeding") followed by a sequence of Lloyd iterations. A common seeding method is to select the initial centers equiprobable at random from the data set 𝒳. This approach ensures that there are no unused centers ("dead units") which could occur if centers were selected from random locations not contained in X. Possible ties in defining C_i can be resolved arbitrarily. If a deterministic resolution method is used, the whole algorithm is deterministic (after the seeding) and leads to reproducible results. In summary thealgorithm performs one Lloyd iteration after another as long as the SSE decreases with each iteration[1].[1]An alternative stopping criterion is to terminate as soon as the error improvement after any Lloyd iteration falls below a threshold. in this form is very simple and known to converge in a finite number of steps. The quality of its solutions measured by the above error function ϕ can however vary strongly depending on the seeding used. Let us illustrate this by applyingto the data set A shown in <ref> (left). It consists of 36 quadratic clusters. Let us specifically consider the problem to distribute 36 centers over A, i.e. the same number as there are clusters in A. In the following we denote this particular problem as A-1 (variations of this problem where the number of centers is n times as large as the number of clusters in A will accordingly be denoted as A-n).Due to the regular structure of the data set and the fact that the number of centers is the same as the number of clusters it is obvious in this case what the optimal arrangement of centers is: One center in each cluster, positioned in the middle of the cluster (see <ref>, right side). The corresponding summed squared Error (SSE) for this optimal solution is 1.458 (numerically computed).Let us now applywith equiprobable random initialization from the given data set A to this problem. In <ref> a typical solution is shown. It has a SSE of 3.873 which is 166% percent above the known optimum (so this solution can be considered rather poor). A rather obvious way to achieve better results is to runseveral times with different random seedings and return the best result of all runs. However, even if we run it 1000 times, the best results are usually still far from the optimum (see <ref>). If we model the distribution of SSE values obtained from the simulation with a normal distribution, the optimal SSE is more than 3.4σ away from the mean indicating an occurrenceprobability of less than 0.0003. Even if it is not clear how well a normal distribution can model theSSE distribution, it seems that a brute-force method based on a large number of random seedings is a costly way to find a low-SSE configuration with . §25 years after the originalalgorithm an improvement was proposed by <cit.>, thealgorithm which today can be seen as a standard way of doing(one specific indication being that thedefault implementation ofin the popularpackage <cit.> for scientific computation is ). Thealgorithm augmentsby a stepwise seeding phase which takes into account the distance of the data points from the centers placed so far. Specifically the probability that a data point x is selected as the position of the next center is chosen to be proportional to the minimum squared distance to any of the already placed centers. In contrast to choosing the next center equiprobable from the remaining data points this method favors regions which are far from the existing centers. Placing the next center in such a region likely results is a large reduction of the overall error.The complete algorithm is depicted in <ref>. Notable details in theimplementation ofare that the algorithm is always run several times (default: 10) and only the best result is returned. Moreover, when placing new centers during seeding several candidates are tried out and the one reducing the overall error most is taken. Since no attempt is made to backtrack this can be seen as a kind of "greedy" algorithm and is already described in <cit.> without further analysis but the remark, that "it helps".In <ref> the wayworks, is illustrated. One can see there how the distance-based probability guides the selection of the respective next center. is often very effective in finding good seedings for the followingphase. According to <cit.> (but also in accordance with our own experiments) the resulting solutions are mostly significantly better than those obtained by equiprobable random seeding from the data set. There is even a proven lower bound for the quality of the solution C constructed bycompared to the optimal solution: E[ϕ] ≤ 8( k+2)ϕ_ For practical purposes this bound may often not be tight enough (for k=100, e.g., the bound guarantees that the error of the solution found differs from the optimum by not more than a factor of 52.8), but the proof of the bound as such is quite remarkable since there is no bound at all for the solutions of , i.e. they can be arbitrarily poor. To illustrate the high quality of we show - after having performed 400 simulations several times - the two most frequent results for the problem A-1 where the originalwith random seeding had problems: In 48-50% of the cases the optimal solution was found (<ref>). Also in 48-50% of the cases one cluster remained empty which is actually very good given that forthis was the best solution resulting from 1000 runs and did only occur in about 0.3% of the simulations (<ref>). Rarely (0-4%) two clusters remained empty. One should note here that In every simulation run we performedexactly once. As mentioned above the implementation ofinis configured such that per default 10 runs are performed and the best result is returned. Given the described probability of about 50% for finding the optimal solution of A-1, it is trivial to see that theimplementation would find the optimal solution with aboutP(opt)=1-(1/2)^10=0.999. § WHY DOESWORK SO WELL?Why doeswork so well? Let us analyze this for the case of a data distribution in 1-D. The calculations are readily generalized to the higher-dimensional case.Let us consider a data set A^1 consisting of n=g*h data point in 1-D space distributed in g separate regions of high density (see <ref>). Each region has a length of a and contains h=n/g points, equally distanced. The distance between neighboring high density regions is aη.Let us first consider a cluster represented by one center which is optimally placed in the center of the cluster. What is the sum of distances in this cluster? If we let n grow towards infinity we can - instead of computing a discrete sum of distances for all points in the cluster - compute the following integral expression: F_1 = 2∫_0^a/2 x^2dx = a^3/12 F_1 is proportional to the mean square distance in the cluster and to the cluster width a. For a cluster not yet covered by a center, all points have distances of at least aη to the nearest center. Let us further assume that η is so large that we can neglect the difference in distance of points within that cluster and assume all points to have a distance of exactly aη to the nearest cluster center. Therefore, the term corresponding to <ref> for the squared distances is F_2 >2*∫_0^a/2 (aη )^2dx = a^3η^2F_2 is proportional to the mean square distance of points in this uncovered cluster to the nearest center and proportional to the cluster width a.If we assume that i, 1 < i<g clusters are already covered with one center each, the probability P_f(i) that the next center is placed in one of these "false" clusters can be conservatively estimated as P_f(i) < iF_1/iF_1+(g-i) F_2 = ia^3/12/ia^3/12+(g-i)a^3η^2=1/1+12(g/i-1)η^2=c1/η^2 Therefore, for any fixed i the following holds:lim_η→∞P_f(i) = 0 In other words,if the ratio η of inter-cluster distance and intra-cluster distance keeps growing the probability of a "wrong" seeding vanishes. Accordinglywill almost always place the first k centers optimally in the assumed scenario (data set X consisting of k well-separated clusters of equal size). The above explains - for our data set A^1 with a number of well-separated clusters of similar size - the effectiveness offor the case that k is no larger that the number of clusters in A. A similar argument can be made for two two-dimensional data sets A (see <ref>) or higher-dimensional versions of it. In all cases a growing inter-cluster distance leads to dominating positioning probabilities in so far uncovered clusters.Perhaps surprisingly the situation changes a lot if one tries to position more centers than there are clusters in the given data set which actually is a typical scenario in applied data analytics where the number of true clusters is usually not known in advance.§ CLUSTERING PROBLEMS WHICH ARE HARD FOR Consider again the data set A from <ref> with 36 high-density regions. The corresponding clustering problem A-1 with k=36 has been well solved by . Let us now consider the problem A-4 where the task is to distribute k=144 centers over A (i.e. 4 times as many as there are clusters in A). Thestructure of this data set makes it obvious that for the optimal solution one should place exactly four centers in each cluster. In<ref> you find a typical result from . It can be seen that for many clusters the number of centers placed in them differs from the optimal value (4 in this case). Why is it thatcan reliably place centers for problem A-1 but not so well for problem A-4?As defined in <ref>uses the (normalized) distance of a data point x from existing centers as probability that x will be the next center. If the number g of clusters is larger than the number n of centers placed so far, this ensures that data points in uncovered clusters have a much larger probability of getting chosen due to their large squared distance to the nearest center. However, if k > g theninitially positions one center in each cluster and suddenly the situation changes. Now every data point has a center in its relative vicinity. If further centers are placed they necessarily end up in a cluster already covered by one (later possibly several) centers. Accordingly, they reduce the local error only moderately. This leads to a much higher chance thatpositions a center sub-optimally and usually leads to improvable results (compared to the optimal distribution of centers, which is unknown in most cases).In the following we analyze this behavior for the case of the one-dimensional signal distribution A^1 already used in <ref>. This data set consists of n=g*h data point in 1-D space distributed in g separate regions of high density (see <ref>). Each region has a length of a and contains h=n/g points, equally distanced. The distance between neighboring high density regions is aη. We like to investigate the problem to place 2*g centers such that the SSE is minimized. It is easy to see that the optimal solution for this problem consists of 2 centers per cluster placed in the centroid of the first and second half of each cluster. How likely is it thatdoes find this configuration?If we assume η to be large we can - according to the analysis in <ref> assume that each of the first g centers will be placed in a different cluster. The next center will necessarily beplaced in one of the existing clusters.This cluster then has two centers which is correct at that point. How likely however is it, that also the remaining g-1 centers will be placed correctly, i.e. such that no cluster contains more than 2 centers? For simplicity we will assume that in a cluster with one center this center is placedin the centroid of the cluster and that in a cluster with 2 centers these centers are already placed to minimize the SSE in this cluster, i.e. each in the centroid of one half of the cluster. As in <ref> we will consider the continuous case by interpreting each cluster as finite segment of length a and we use integrals instead of discrete sums to compare error values for different configurations.Let us first compute the integral F_1 of the squared distance for a cluster of length a covered by one single center positioned in the middle of the cluster (see <ref>, left side): F_1 = 2∫_0^a/2x^2 dx= 2[ 1/3x^3 ]_0^a/2=2*a^3/24=a^3/12=cIf a cluster is covered by two centers we assume that they are optimally positioned at 25% and 75% of its length (see <ref>, right side). The corresponding integral F_2 is the following:F_2 = 4∫_0^a/4x^2 dx= 4[ 1/3x^3 ]_0^a/4=4*a^3/192=a^3/48=c/4As defined in <ref>uses the distances as probabilities for placing further centers so F_1 is proportional to the probability that a new center is placed in a particular cluster with one center and F_2 is proportional to the probability that a new center is placed in a particular cluster with two centers. Since for the following only the relative sizes of F_1 and F_2 are needed, we can replace F_1 by c and F_2 by c/4.Let us assume that g+i centers have been placed correctly byamong the g clusters (this means that i clusters now have 2 centers). How probable is it that the next center will be placed correctly as well, i.e. in a cluster having only one center so far? To compute this we have to compare the probabilities of the (g-i) "correct" cases to those of all cases: P_corr(i,g)=c*(g-i)/c*(g-i)+c/4*n=g-i/g-3/4i To compute the probability that all of the 2*g centers are placed optimally(assuming g+1 centers were placed correctly already) we have to compute the product ofvalues P_corr(i,g) for i∈{1,..., g-1}: P_corr(g)=∏_i=1^g-1P_corr(i,g)=∏_i=1^g-1g-i/g-3/4iP_corr(g) decays exponentially quickly with g. In <ref> the values of P_corr(g) are graphically displayed and already P_corr(24) is less than 10^-4. This means that w.r.t. data set A^1- a one-dimensional data set with g clusters -is very unlikely to position the 2*g centers such that the optimal configuration with 2 centers per clusters is achieved. This is the case for all values of g which are not trivially small. This result is in sharp contrast to the result for the seemingly similar problem to distribute exactly g centers for data sets like A^1. In this case by choosing a large distance η a among the clusters it can be made arbitrarily probable thatfinds the optimal configuration (one center in each cluster).Once there is a center in each cluster, however, the inter-cluster distance is rendered meaningless. A further center in any cluster reduces the local error only moderately (e.g. by a factor of 4 if we add a second center to a cluster having one center so far, see <ref>). This again means that if we already have many clusters with two centers, it becomes very probable that a third center is placed in one of them byleading to a sub-optimal configuration.The analyzed behavior for 1-D could be extended to other data distribution and to higher data dimensions. Completely general statements however are probably difficult to establish since there is a large dependency on the particular structure of the given data set which for real data sets is usually not known. Therefore we will rather concentrate on presenting a method to improve upon the results ofand demonstrate its effectiveness by systematic comparative simulations with data sets of varying size and dimensionality. § How can the results obtained withbe further improved?employs a careful seeding step followed by standardand thus ends in a (usually local) minimum of the error function ϕ(𝒞,𝒳) from <ref>. One idea to improve such a solution is to move single centers to other positions and afterwards letfind a - hopefully better - local minimum. To make this computationally efficient and guarantee convergence, however, one needs to carefully select both the centers to be moved and their respective target positions. Also a stopping criterion needs to be defined.Which center should we move? Following the approach proposed by <cit.> we investigate how "useful" each center is for error reduction. This can be quantified by removing this center and comparing the resulting error with the current error. Thus we define the utility U(c_i) of a center c_i as U(c_i) = ϕ(𝒞∖{c_i},𝒳) - ϕ(𝒞,𝒳)The utility of a center is a measure how much we need this center or how easy its "task" of reducing error can be overtaken by neighboring centers. For example in the pathological case where two centers have the same position they both have a utility of zero since one could remove either one of them without increasing the error: the remaining center would cover the associated data points as well as the two centers did before. This case will not occur withbut all other things being equal one can expect the utility of centers with close neighbors being lower than those of more isolated centers. For an alternative but equivalent definition of the utility see <ref>. We now define the "least useful" center λ as follows: λ = _c_i∈𝒞 U(c_i) = _c_i∈𝒞 ϕ(𝒞∖{c_i},𝒳) = _c_i∈𝒞 ∑_x∈𝒳min_c_j∈𝒞∖{c_i} ||x-c_j||^2 Where should we move the center λ? Since our goal is to reduce the overall error function one rather straightforward approach is to move λ to the vicinity of that center μ currently having the maximal error sum for its Voronoi set.Recall that in <ref>, <ref>we defined for each i ∈{1, ...,k } the Voronoi set C_i to be the set of all points in 𝒳 that are closer to c_i than they are to c_j for all ji. We now define μ as the center c_i having the largest summed squared distance over its Voronoi set C_i: μ = _c_i∈𝒞∑_x∈ C_i ||x-c_i||^2It is advisable to place λ not exactly at the position of μ since that would cause all points in C_μ to have identical distances to μ and λ. We therefore place center λ at the position of μ plus some small random offset. A simple approach to define what "small" means for a given center μ is to consider the mean distance of the data points in μ's Voronoi set C_μ. We therefore define d_μ asd_μ= √(1/|C_μ|∑_x∈ C_μx-μ^2 )d_μ gives an indication of the spatial extension of μ'sVoronoi set. By choosing a small fraction of d_μ, e.g. ϵ d_μ with ϵ=0.01, as the length of our offset vector we can be confidentthat the new position of λ will be "near" μ. Having a length, we still have to choose a direction for our offset vector. A principled and informed choice would be the direction of largest variance in the Voronoi set of μ, i.e. the unit eigenvector corresponding to the largest eigenvalue of the covariance matrix of C_μ. Instead - both to have some non-determinism and to save the eigenvector computation in each step - we simply use a random vector from the d-dimensional unit hypersphere and rely on the followingphase to find good configurations. Choosing such a vector with uniform probability density is not completely trivial, however. For example normalizing a random vector from the d-dimensional hypercube would cause probability peaks in the directions of the corners of this hypercube. But it has been shown <cit.> that a uniformly distributed unit random vector from the d-dimensional unit hypersphere can be constructed as follows: * Generate d Gaussian random variables x_1, x_2,...,x_d * Return the vector ξ = 1/√(x_1^2, x_2^2,...,x_d^2) [x_1, x_2,...,x_d]^T Let u be such a random vector. Then we define our offset vector o as o=ϵd_μuThealgorithm which we now define, starts with . Thereafter repeatedly the centers μ and λ are determined and the least useful center λ is moved to the position of the center μ with maximum error (plus a small random offset o which is also applied to μ itself, but with opposite sign). After each such move standardis performed and the resulting error is measured. The algorithm terminates as soon as there is no improvement of the error measure. The complete( with utility) algorithm is specified in <ref>.The general purpose of using an offset vector o instead of simply assigning λ←μ is to break symmetry. By making the length of the offset vector depending on the local distance measure d_μ we avoid the need to a priori choose a parameter for this length which - if too large - could lead to μ and λ being moved to irrelevant areas of the input space thus contradicting the main principle of thealgorithm, namely to reduce the error where it is largest. What does locally happen during a jump in ? An example is shown in <ref>. At the previous position of λ the data points so far belonging to λ's Voronoi set are now partitioned among the neighboring centers whose Voronoi regions are correspondingly enlarged. Assuming that o is very small the centersμ and λ now both have positions very close to the previous position of μ but offset in opposite directions. The previous Voronoi set of μ is therefore partitioned into two subsets C_μ and C_λ divided by a (d-1)-dimensional hyperplane. This hyper planecontains the previous position of μ and is also a normal plane to the offset vector o. Since o is created from a random vector, the orientation of this hyperplane and the resulting partitioningis random as well. This observation will become relevant in <ref>. In <ref> a simple one-dimensional example illustrates typical results of ,and . The data set used is a variant of data set A^1 which was used in <ref> and <ref> to illustrate easy and hard problems for .<Ref> contains a 2-dimensional example wherefound a considerable improvement over the result of .In <ref> the sequence of non-local jumps leading to the result in <ref> is displayed.<Ref> demonstrates thatis also able to find improvements for more natural data sets. In this case the data is from a mixture of overlapping Gaussians. The number of data points and the number of centers is exactly as in <ref>. The improvement overis nearly 4% in this example, even though it is hard to see the difference between the two solutions. Note: In principle also another seeding method thancan be used (e.g. standardor even random seeding) but in simulationsled to the best results for the followingalgorithm.§ AN OCCASIONAL PROBLEM OF : TOO EARLY TERMINATIONSometimes we observed in simulations thatfinished very early, i.e. at a point in timewhen there were seemingly many optimization opportunities left. Initially we suspected a programming error, but indeed in all investigated cases the most recent jump had led to a particular poor - but stable - configuration causing a relative increase of the SSE and thus a termination ofaccording to its definition. In <ref> such a simulation sequence is depicted.already terminated after two jumps because the SSE had increased after performing the second jump. In <ref> a detail view of the cluster causing the error increase is shown. It does contain the optimal number (4) of centers for this problem, but their arrangement determined byis such that two centers are very close to each other and both have elongated Voronoi regions and therefore a relatively high distance to the member points of their respective Voronoi sets. Fortunately these poor configurations seem to berelative rare. Due to the associated high SSE values, however, they have the potential to deteriorate mean performance statistics. Given that inthe re-positioning of the two centers affected from a jump(μ and λ) is based on a random vector, one relatively easily comes up with the idea to re-do the positioning in such cases. This leads to an extension of our original algorithm described in the next section.§As exemplified in <ref> runs ofmay end too early due to poor local configurationsruns into after a jump. In <ref> it was discussed what happens at a jump. Let us analyze this here in more detail: Directly after the repositioning the centers μ and λ divide[2] the Voronoi set C_μ (see <ref>) of μ among them using a (d-1)-dimensional hyperplane going through the previous position of μ and having the offset vector o (<ref>) as its normal vector.[2]Strictly speaking this is true only in the limiting case when the length ofo goes to zero, since with a non-vanishing vector o there may be some data points previously associated with other centers for which now either μ or λ is the closest center. If o goes to zero however, the combination of the Voronoi regions of μ or λ approaches the old Voronoi region of μ with arbitrary precision.The orientation of o depends on a random vector drawn from a d-dimensional hypersphere and determines how the data points previously associated with μ are distributed between μ and λ.Each random choice leads with probability one (choosing two collinear vectors from a continuous distribution on the hypersphere has probability zero) to a different orientation of the hyperplane and likely to a different partitioning of the affected data points (see <ref> for an example). Thephase following every jump leads to results depending on these partitionings. Different partitionings likely lead to different results of . Based on the above observations we propose the following simple extension of thealgorithm which we call(see <ref>). Instead of immediately terminating after an error increase we allow a small finite number retry_max of retries of the most recent jump. Due to the random choice of the offset vector these retries possibly end up in a configuration with lower error and allow a continuation of , sometimes for many steps. Once a retry was successful we "reset" the retry counter so the specified number of retries is again available at a later stage which will be at a lower error level than the previous retry sequence (since we just improved our "best solution"). This leads to a strictly monotone sequence of error values of the respective best solution after every retry sequence until the algorithm terminates. Since we never try to improve a configuration with a higher error than the best solution found so far, this retry procedure can be interpreted as randomized "greedy search". In <ref> a typical simulation sequence ofis shown. Whilewould have stopped after 7 jumps,continues, in this case to the optimum. § EMPIRICAL RESULTS We performed systematic tests of ,andwith 5 different data sets. For each data set a large range of values for k was investigated. For each of these k-values 10 different simulation runs were performed. Each single simulation consisted of three phases: *(i.e.theseeding followed by ) * , starting from the result of therun and continuing until the SSE did not fall anymore (the stopping criterion of ) *starting from the result ofand allowing 2 retries for each timecame to a stop. For each data set we show an illustration of the data set itself and a performance chart (fig:figpaperdatafig:propulsionstat). If the dimension d of the data set is larger than two, we display all d^2 pairs of dimensions, each in a separate sub plot. The scaling of each sub plot is chosen such that the whole available area is used to display data points. Therefore different subplots may have different scalings, but the general nature of the data should be more visible this way.The performance chart takes the performance ofas the baseline and indicates for bothandby how many percent they did reduce the SSE obtained by . No improvement would correspond to a data point on the k-axis and any actual improvement to data points above the k-axis. Per construction the new algorithms can not deliver anything worse thanso there are no values below the k-axis. For bothandthe mean improvement (main chart) as well as the minimum and maximum improvements (error bars)are shown. While the mean indicates what to expect from one algorithm run, the maximum is an indication of what one could achieve by picking the best result of several runs.The figure captions of the performance charts contain specific remarks regarding the simulation results. In general the new algorithms were able to improve a clear majorityof theresults and often by a large margin.in particular was not only able to raise the mean improvement compared tobut also to obtain in many cases much higher maximum values (often more than 2 times as high as the maximum values of ). With the exception of data set A (<ref>fig:figpaperdata) which - as we know from earlier sections - is challenging for and data set B (<ref>)which was included as an example of an unstructured data set the other data sets have not been constructed or chosen with any result in mind but rather to provide a certain variety. Two data sets (cloud and propulsion) were taken from the UCI Machine Learning Repository as an established source of well-kept data sets. The simulations were performed in python using the optimized implementation of contained in thepackage and a (non-optimized) numpy-based implementation ofand . Since on this base the comparison of running times was difficult, we compared the number of Lloyd iterations. In fig:gaussian2DstatZZZfig:propulsionstatsZZZ we display for all performed simulations the relative overhead ofandin terms of Lloyd iterationsas well as the fraction ofsolutions whichwas able to improve. The error reduction shown earlier is repeated for reference as well. The overhead ofovermeasured as described ranged between 10% and 230%. In computing this we considered thatis executed 10 times inbefore the best result is returned. This means that for all our experiments the effort for the more complex one of our algorithms () was within a constant factor (3.3) of . Given that provides significant solution improvements for an NP-complete problem this can be seen as a very moderate effort.With few exceptions (mainly for small values of k or values of k near the number n of data points) the percentage ofsolutions improved byis at 100%. Thus according to our experiments it is highly likely that an arbitrary solution found bycan be further improved by .§ THEORETICAL BOUND FOR SOLUTION QUALITY No specific formal analysis of the new algorithm has been performed so far. However,starts off from the result ofand does only deliver a different result, if it is better than that of . Therefore (and trivially so) the same bound which has been established forholds for the new algorithm as well: E[ϕ] ≤ 8( k+2)ϕ_So it is guaranteed that the summed squared error of any solution computed bydiffers from the optimum only by a factor which is logarithmically dependent on k. The above bound holds foras well, since it also returns the initialsolution if it cannot produce a better one.§ SUMMARY In this paper we proposethealgorithm. It is based on the initially proposedalgorithm which performs non-local jumps based on a simple utility criterion to improve the clustering results obtained with the current de-facto standard method .In some cases however we observedstopping too early due to an error increase caused by a poor local minimum in which thephase ofended.This behavior was largely overcome by allowing a small and finite number of retries(e.g. 2) for the most recent jump. Due to the randomized local positioning during a jump, the resulting configuration after a retry is often different and possibly leads to a lower error which allows the algorithm to continue with the next jump. Further retries are then possible, but only on lower error levels which leads to a strictly monotone decreasing sequence of error values until the algorithm terminates. The resulting extended version ofis called thealgorithm. By construction the logarithmic quality bound established forholds foras well. Simulations with a variety of data sets (partially from the UCI Machine Learning Repository) demonstrate thatis dominated w.r.t. solution quality (SSE) bywhich from a certain value of k very often generates significantly better solutions and thatitself is dominated by ,again significantly in a large number of cases. The observed improvements overdepended on the structure of the data distribution and the number k of centers and ranged from zero to about 8% mean reduction of the summed squared error. Our method incurs only a moderate computational overhead compared to (below 230% in all of our experiments, less than 50% for the propulsion data, the largest of the data sets we used). Since the problem of minimizing SSE is NP-complete for data dimensions of two and larger, these additional costs for achieving often several percent error reduction appear to be quite reasonable. In conclusion we considera potential replacement ofin all cases where the quality of the clustering is of high importance. § EXAMPLE IMPLEMENTATIONThe complete python code used for the experiments in this paper is available fromhttps://github.com/gittar/k-means-u-star. 0.2in | http://arxiv.org/abs/1706.09059v2 | {
"authors": [
"Bernd Fritzke"
],
"categories": [
"cs.LG",
"I.5.3"
],
"primary_category": "cs.LG",
"published": "20170627215350",
"title": "The k-means-u* algorithm: non-local jumps and greedy retries improve k-means++ clustering"
} |
The AGN-Star Formation Connection: Future Prospects with JWSTAllison Kirkpatrick1, Stacey Alberts2, Alexandra Pope3, Guillermo Barro4, Matteo Bonato5, Dale D. Kocevski6, Pablo Pérez-González7, George H. Rieke2, Lucia Rodríguez-Muñoz8, Anna Sajina9, Norman Grogin10, Kameswara Bharadwaj Mantha11,Viraj Pandya12, Janine Pforr13, Paola Santini14=============================================================================================================================================================================================================================================================================================== A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by 1. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively.We give a complexity dichotomy for both problems restricted to H-free graphs, that is, graphs with no induced subgraph isomorphic to H. Moreover, we show that an edge is critical if and only if its contraction reduces the chromatic number by 1. Hence, we also obtain a complexity dichotomy for the problem of deciding if a graph has an edge whose contraction reducesthe chromatic number by 1.Keywords. edge contraction, vertex deletion, chromatic number. § INTRODUCTION For a positive integer k, a k-colouring of a graph G=(V,E) is a mapping c: V→{1,2,…,k} such that no two end-vertices of an edge are coloured alike, that is,c(u)≠ c(v) if uv∈ E. The chromatic number χ(G) of a graph G is thesmallest integer k for which G has a k-colouring.The well-known Colouring problem is to test if χ(G)≤ k for a given graph G and integer k. If k is not part of the input, then we call this problem k-Colouring instead. Lovász <cit.> proved that 3-Colouringis -complete. Due to its computational hardness, the Colouring problem has been well studied for special graph classes. We refer to the survey <cit.> for an overview of the results on Colouring restricted to graph classes characterized by one or two forbidden induced subgraphs. In particular, Král', Kratochvíl, Tuza, and Woeginger <cit.> classified Colouring for H-free graphs, that is, graphs that do not contain a single graph H as an induced subgraph. To explain their result we need the following notation. For a graph F, we write F G to denote that F is an induced subgraph of a graph G. The disjoint union of two graphs G_1 and G_2 is the graph G_1+G_2, which has vertex set V(G)∪ V(H) and edge set E(G)∪ E(H). We write rG for the disjoint union of r copies of G by rG. The graphs P_r and C_r denote the induced path and cycle on r vertices, respectively. We can now state the theorem of Král et al. Let H be a graph. If HP_4 or H P_1+ P_3, then Coloring restricted to H-free graphs is polynomial-time solvable, otherwise it is -complete.For a vertex u or edge e in a graph G, we let G-u and G-e be the graph obtained from G by deleting u or e, respectively. Note that such an operation may reduce the chromatic number of the graph by at most 1. We say that u or e is critical if χ(G-u)=χ(G)-1 or χ(G-e)=χ(G)-1, respectively. A graph is vertex-critical if every vertex is critical and edge-critical if every edge is critical. To increase our understanding of the Colouring problem and to obtain certifying algorithms that solve Colouring for special graph classes, vertex-critical and edge-critical graphs have been studied intensively in the literature, see for instance <cit.> for certifying algorithms for (subclasses of) H-free graphs and in particular P_r-free graphs.In this paper we consider the problems Critical Vertex and Critical Edge, which are to test if a graph has a critical vertex or critical edge, respectively. In addition we also consider the edge contraction variant of these two problems.We let G/e denote the graph obtained from G after contracting e=vw, that is, after removing v and w and replacing them by a new vertex made adjacent to precisely those vertices adjacent to v or w in G (without creating multiple edges). Contracting an edge may reduce the chromatic number of the graph by at most 1. An edge e is contraction-critical if χ(G/e)=χ(G)-1. This leads to the Contraction-Critical Edge problem, which is to test if a graph has a contraction-critical edge. §.§ Our Results We prove the following complexity dichotomies for Critical Vertex, Critical Edge and Contraction-Critical Edge restricted to H-free graphs.If a graph HP_4 or of H P_1+ P_3, then Critical Vertex, Critical Edge and Contraction-Critical Edge restricted to H-free graphs are polynomial-time solvable, otherwise they are -hard or co--hard. We note that the classification in Theorem <ref> coincides with the one in Theorem <ref>. The polynomial-time cases for Critical Vertex and Contraction-Critical Edge can be obtained from Theorem <ref>.The reason for this is that a class of H-free graphs is not only closed under vertex deletions, but also under edge contractions whenever H is alinear forest, that is, a disjoint union of a set of paths (see Section <ref> for further details).However, no class of H-free graphs is closed under edge deletion. We get around this issue by proving, in Section <ref>, that an edge is critical if and only if it is contraction-critical. Hence, Critical Edge and Contraction-Critical Edge are equivalent.The -hardness constructions of Theorem <ref> cannot be used for proving the hard cases forCritical Vertex, Critical Edge and Contraction-Critical Edge.Instead we construct new hardness reductions in Sections <ref> and <ref>. In Section <ref> we prove that the three problems are -hard for H-free graphs ifH contains a claw or a cycle on three or more vertices.In the remaining case H is a linear forest. In Section <ref> we prove that the three problems are co--hard even for(C_5,4P_1,2P_1+P_2, 2P_2)-free graphs. In Section <ref> we combine the known cases with our new results from Sections <ref>–<ref> in order to prove Theorem <ref>. §.§ Consequences Our results have consequences for the computational complexity of two graph blocker problems. Let S be some fixed set of graph operations, and let π be some fixed graph parameter. Then, for a given graph G and integer k≥ 0, the S-Blocker(π) problem asks if G can be modified into a graph G' by using at most k operations from S so that π(G')≤π(G)-d for some given threshold d≥ 0. Over the last few years, the S-Blocker(π) problem has been well studied, see for instance <cit.>.If S consists of a single operation that is either a vertex deletion or edge contraction, then S-Blocker(π) is called Vertex Deletion Blocker(π) or Contraction Blocker(π), respectively.By taking d=k=1 and π=χ we obtain the problems Critical Vertex and Contraction-Critical Edge, respectively.We showed in <cit.> how the results for Critical Vertex and Contraction-Critical Edge can be extended with other results to get complexity dichotomies for Vertex Deletion Blocker(χ) and Contraction Blocker(χ) for H-free graphs. §.§ Future Work A graph G is (H_1,…,H_p)-free for some family of graphs {H_1,…,H_p} and integer p≥ 2 if G is H-free for every H∈{H_1,…,H_p}. As a direction for future research we propose classifying the computational complexity of our three problems for (H_1,…,H_p)-free graphs for any p≥ 2. We note that such a classification for Coloring is still wide open even for p=2 (see <cit.>). Hence, research in this direction might lead to an increased understanding of the complexity of the Coloring problem. § EQUIVALENCE We prove the following result, which implies that the problems Critical Edge and Contraction-Critical Edge are equivalent. An edge is critical if and only if it is contraction-critical.Let e=uv be an edge in a graph G. First suppose that e is critical, so χ(G-e)=χ(G)-1. Then u and v are colored alike in any coloring of G-e that uses χ(G-e) colors. Hence, the graph G/e obtained from contracting e in G can also be colored with χ(G-e) colors. Indeed, we simply copy a (χ(G-e))-coloring of G-e such that the new vertex in G/e is colored with the same color as u and v in G-e.Hence χ(G/e)=χ(G-e)=χ(G)-1, which means that e is contraction-critical.Now suppose that e is contraction-critical, so χ(G/e)=χ(G)-1. By copying a χ(G/e)-coloring of G/e such that u and v are colored with the same color as the new vertex in G/e, we obtain a coloring of G-e. So we can color G-e with χ(G/e) colors as well. Hence χ(G-e)=χ(G/e)=χ(G)-1, which means that e is critical.§ FORBIDDING CLAWS OR CYCLES The claw is the 4-vertex star K_1,3 on vertices a,b,c,d and edges ab, ac and ad. In this section we prove that the problems Critical Vertex, Critical Edge and Contraction-Critical Edge are -hard for H-free graphs whenever the graph H containsa claw or a cycle on at least three vertices.Let G be a graph class with the following property: if G∈ G, then so are 2G and G+K_r for any r≥ 1. We call such a graph class clique-proof. If Coloringis -complete for a clique-proof graph class G, then both Critical Vertex and Contraction-Critical Edge are -hard for G.Let G be a graph class that is clique-proof. From a given graph G∈ G and integer ℓ≥ 1 we construct the graph G'=2G+K_ℓ+1. Note that G'∈ G by definition and that χ(G')=max{χ(G),ℓ+1}.We first prove that χ(G)≤ℓ if and only if G' contains a contraction-critical edge. Suppose thatχ(G)≤ℓ. Then χ(G')=χ(K_ℓ+1)=ℓ+1. In G' we contract an edge of the K_ℓ+1. This yields the graph G^*=2G+K_ℓ, which has chromatic number χ(G^*)=ℓ, as χ(K_ℓ)=ℓ and χ(G)≤ℓ. As χ(G')=ℓ+1, this means thatχ(G^*)= χ(G')-1. Hence G' contains a contraction-critical edge.Now suppose that G' contains a contraction-critical edge. Let G^* be the resulting graph after contracting this edge. Then χ(G^*)=χ(G')-1. As contracting an edge in one of the two copies of G in G' does not lower the chromatic number of G', the contracted edge must be in the K_ℓ+1, that is, G^*=2G+ K_ℓ. As this did result in a lower chromatic number, we conclude that χ(G')=χ(K_ℓ+1)=ℓ+1 and χ(G^*)=χ(2G+ K_ℓ)= max{χ(G),ℓ}=ℓ. The latter equality implies that χ(G)≤ℓ.From the above we conclude that Contraction-Critical Edge is -hard. We can prove that Critical Vertex is -hard by using the same arguments.We also need a result of Maffray and Preissmann as a lemma. The 3-Coloring problemis -complete for C_3-free graphs. We are now ready to prove the main result of this section. Let H be a graph such thatH K_1,3 or H C_r for some r≥ 3. Then the problems Critical Vertex, Critical Edge and Contraction-Critical Edge are -hard for H-free graphs.By Proposition <ref> it suffices to consider Critical Vertex and Contraction-Critical Edge. If H is not a clique, then the class of H-free graphs is clique-proof. Hence, in this case, we can use Theorems <ref> and <ref> to obtain -hardness. Suppose H is a clique.It suffices toshow -completeness for H=C_3. We reduce from 3-Coloring restricted to C_3-free graphs. This problem is -complete by Lemma <ref>.Let G be a C_3-free graph that is an instance of 3-Coloring. We obtain an instance of Critical Vertex or Contraction-Critical Edge as follows. Take the disjoint union of two copies of G and the Grötzsch graph F (see Figure <ref>), which is known to be 4-colorable but not 3-colorable (see <cit.>). Call the resulting graph G', so G'=2G+F. As G and F are C_3-free, G' is C_3-free. We claim that G is 3-colorable if and only if G' has a critical vertex if and only if G' has a contraction-critical edge. This can be proven via similar arguments as used in the proof of Theorem <ref>, with F playing the role of K_ℓ+1. Note that in Theorem <ref> we cannot prove membership in , as Coloring is -complete for the class of H-free graphs if H K_1,3 or H C_r for some r≥ 3 due to Theorem <ref>. As such, it is not clear if there exists a certificate.§ FORBIDDING LINEAR FORESTS In this section we prove our second hardness result needed to show Theorem <ref>. We first introduce some additional terminology.Let G be a graph. The graph G denotes the complement of G, that is, the graph with vertex set V(G) and an edge between two vertices u and v if and only if u and v are not adjacent in G.A subset K of vertices in G is a clique if any two vertices in K are adjacent to each other. A clique cover of a graph G is a set K of cliques in G, such that each vertex of G belongs to exactly one clique of K. The clique covering number σ(G) is the size of a smallest clique cover of G. Note that χ(G)=σ(G). The size of a largest clique in a graph G is denoted by ω(G).The hardness construction in the proof of our next result uses clique covers. Král et al. <cit.> proved that Coloring is -hard for(C_5,4P_1,P_1+2P_2, 2P_2)-free graphs. This does not give us hardness for Critical Vertex or Critical Edge, but we can use some elements of their construction. For instance, we reduce from a similar -complete problem as they do, namely the -complete problem Monotone 1-in-3-SAT, which is defined as follows. Let Φ be a formula with clause set C of size m and variable set X of size n, so that each clause in C consists of three distinct positive literals, and each variable in X occurs in exactly three clauses. The question is whether Φ has a truth assignment, such that each clause is satisfied by exactly one variable. In that case we say that Φ is 1-satisfiable. Note that m=n.Moore and Robson proved that this problem is -complete. Monotone 1-in-3-SAT is -complete. We are now ready to prove the main result of this section. The problems Critical Vertex, Critical Edge and Contraction-Critical Edge are co--hard for (C_5,4P_1,2P_1+P_2, 2P_2)-free graphs.By Proposition <ref> it suffices to consider Critical Vertex and Critical Edge. We will first consider Critical Vertex and show that the equivalent problem whether a graph has a vertex whose deletion reduces the clique covering number by 1 is co--hard for (C_4,C_5,K_4,2P_1+P_2)-free graphs.We call such a vertex critical as well.The complement of a(C_4,C_5,K_4,2P_1+P_2)-free graphs is (C_5,4P_1,2P_1+P_2, 2P_2)-free. Hence by proving this co--hardness result we will have proven the theorem for Critical Vertex.As mentioned, we reduce from Monotone 1-in-3-SAT, which is -complete due to Lemma <ref>. Given an instance Φ ofMonotone 1-in-3-SAT with clause set C and variable set X, we construct a graph G=(V,E) as follows. For every clause c∈ C, the clause gadget G_c=(V_c,E_c) is a cycle of length 7. For c=(x,y,z), we let three pairwise non-adjacent vertices c(x),c(y),c(z) of G_c correspond to the three variables x,y,z.We denote the other four vertices of G_c by a_i^c,1≤ i≤ 4, so that G_c=c(x)a_1^ca_2^cc(y)a_3^cc(z)a_4^cc(x). For each variable x∈ X we let the variable gadget Q_x consist of the triangle c(x)c'(x)c”(x)c(x), where c,c',c” are the three clauses containing x. See Figure <ref> for an illustration of the construction. We observe that |V(G)|=7n and that G is (C_4,C_5,K_4,2P_1+P_2)-free with ω(G)=3. In order to prove co--hardness we first need to deduce a number of properties of our gadget. We do this via a number of claims.Claim 1.There exists a minimum clique cover of G, in which each a_i^c is covered by a clique of size 2, and moreover, every two vertices a_1^c and a_2^c belong to the same (2-vertex) clique.We prove Claim 1 as follows. Let K be a minimum clique cover of G. Suppose two vertices a_1^c and a_2^c belong to two different cliques K, K' of K.If one of K, K' has size 1, say K={a^1_c}, then we can replace K and K' by {a_1^c,a_2^c} and K'∖{a_2^c}. This yields a new minimum size clique cover of G, in which a_1^c and a_2^c belong to the same clique. Alternatively, if K and K' each have size 2, then K={a_1^c,c(x)} and K'={a_2^c,c(y)}. Then, by construction, K containsa clique that either consists of a_3^c or of a_4^c , say a_4^c. We replace the cliques {a_4^c} and K by {a_4^c,c(x)} and {a_1^c}, respectively, and return to the previous situation. Hence we may assume without loss of generality that {a_1^c,a_2^c} is a clique in K. This means that if a_3^c or a_4^c forms a 1-vertex clique in K, then we can safely add c(y) or c(x), respectively, to it. This proves Claim 1.Now let K be a minimum clique cover. By Claim 1, we may assume without loss of generality that each a_i^c is covered by a clique of size 2, and moreover, that every two vertices a_1^c and a_2^c belong to the same (2-vertex) clique. Since the clause gadgets G_c are pairwise non-intersecting and isomorphic to C_7, it takes at least four cliques to cover the vertices of every G_c.This means that exactly 3n cliques are needed to cover the 4n vertices a_i^c. By construction, we also find that 2n vertices c(x) are covered by these cliques. Since ω(G)=3, at least n/3 other cliques are necessary to cover the n remaining vertices c(x). Hence, K has size at least 10 3n, that is, σ(G)≥10 3n. We now prove three more claims.Claim 2. Φ is 1-satisfiable if and only if σ(G)=10 3n.We prove Claim 2 as follows. First suppose Φ is 1-satisfiable. We construct a clique cover K in the following way.If x is true, then we let K contain the triangle Q_x.Since each clause c contains exactly one true variable for each G_c, exactly one vertex of G_c is covered by a variable gadget. Then K contains three cliques of size 2 covering the six other vertices of G_c.Hence K has size 10 3n. As σ(G)≥10 3n, this implies that σ(G)=10 3n.Now suppose σ(G)=10 3n.Let K be a minimum clique cover of G.By Claim 1, we may assume without loss of generality that each a_i^c is covered by a clique of size 2, and moreover, that every two vertices a_1^c and a_2^c belong to the same (2-vertex) clique. Then at least n/3 other cliques are necessary to cover the vertices c(x) that are not in a 2-vertex clique with a vertex a_i^c. Hence, asσ(G)=10 3n, these vertices are covered by exactly n/3 triangles, each one corresponding to one variable x (these are the only triangles in G). We assign the value true to a variable x∈ X if and only if its corresponding triangle Q_x is in the clique cover. Then, for each c∈ C, exactly one variable is true, namely the one that corresponds to the unique vertex of G_c covered by a triangle. So Φ is 1-satisfiable. This completes the proof of Claim 2.Claim 3. If G has a clique cover K={K_1,…,K_10 3n}, then each K_i∈ Kconsists of either two or three vertices.We prove Claim 3 as follows. As σ(G)≥10 3n and | K|=10 3n, we find that K is a minimum clique cover. With each v∈ V, we associate a weight w_v≥ 0 as follows. For K_i∈ K and v∈ K_i, we define w_v=1/| K_i|. Since ω(G)= 3 we have w_v∈{1 3,1 2,1}. So we have ∑_G_c∑_v∈ V_cw_v=∑_v∈ Vw_v= ∑_i=1^10 3n∑_v∈ K_iw_v=10 3n, where the first equality holds, because the clause gadgets G_c are vertex-disjoint. We show that for every c we have Σ_v∈ V_cw_v≥10 3. Since every a_i^c has exactly two neighbours and these neighbours are not adjacent, we have w_a_i^c∈{1 2,1}. If there exists an index i such that w_a_i^c=1, then ∑_v∈ V_cw_v≥1+3×1 2+3×1 3=7 2>10 3.Now if a_i^c has weight w_a_i^c=1 2 for each 1≤ i≤ 4, then a_i^c is covered by a clique of size 2 and the second vertex of this clique has weight 1 2 as well by definition. Thus if w_a_i^c=1 2, exactly two among c(x),c(y),c(z) have weight 1 2. It follows that ∑_v∈ V_cw_v≥ 4×1 2+2×1 2+1 3=10 3.Hence ∑_v∈ V_cw_v=10 3 if and only if each vertex of G_c is in a clique of size 2 or 3. Since ∑_G_c∑_v∈ V_cw_v=10 3n, we obtain ∑_v∈ V_cw_v=10 3 for every c∈ C. We conclude that each clique in K is of size 2 or 3.This completes the proof of Claim 3.Claim 4. If σ(G)>10 3n, then G has a minimum clique cover K that contains a clique of size 1.We prove Claim 4 as follows. Suppose σ(G)>10 3n.For contradiction, assume that every minimum clique cover of G has no clique of size 1. Let K be a minimum clique cover of G. By Claim 1, we may assume without loss of generality that each a_i^c is covered by a clique of size 2, and moreover, that every two vertices a_1^c and a_2^c belong to the same (2-vertex) clique. Hence the remaining vertex c(x) is covered bysome clique K_i∈ K,such that either K_i={c(x),c'(x)} orK_i={c(x),c'(x),c”(x)}. If K_i={c(x),c'(x)}, then c”(x) is covered by some clique K_j={c”(x),a}. However, then we can take K_i={c(x),c'(x),c”(x)} and K_j={a} to obtain a minimum clique cover with | K_j|=1, a contradiction. Hence K_i={c(x),c'(x),c”(x)}. As this holds for every G_c we find that σ(G)=10 3n, a contradiction. This completes the proof of Claim 4.We claim that Φ is a 1-satisfiable if and only if G has no critical vertex. First suppose that Φ is 1-satisfiable. By Claims 2 and 3 we find that σ(G)=10 3n and every clique in any minimum clique cover of G has size greater than 1. Hence, there is no vertex u of G with σ(G-u)≤σ(G)-1, that is, G has no critical vertex.Now suppose that Φ is not 1-satisfiable. By Claims 2 and 4we find that σ(G)>10 3n and that there exists a minimum clique cover that contains a clique {u} of size 1. Thismeans that σ(G-u)=σ(G)-1. So u is a critical vertex.We are left to consider the Critical Edge problem. We use the same construction as before except that the cycles G_c are isomorphic to C_11. To be more precise, we let G_c=c(x)a_1^ca_2^cc(y)a_3^ca_4^ca_5^cc(z)a_6^ca_7^ca_8^cc(x). Again the resulting graph G is (C_4,C_5,K_4,2P_1+P_2)-free.By using the same arguments as before we find that if Φ is 1-satisfiable, then every clique in any minimum clique cover of G has size greater than 1. Hence, as G is K_4-free, every clique in any minimum clique cover of G has size 2 or 3.Since G is 2P_1+P_2-free, we cannot merge two cliques into one by adding a new edge. So G has no critical edge. Now suppose that Φ is not 1-satisfiable. Then using the previous arguments we can prove that there exists a minimum clique cover K that contains a clique {u} of size 1. By the adjusted construction of G_c we find that u is adjacent to exactly one vertex of a 2-vertex clique {v,w} of K, say u is adjacent to v but not to w. Then by adding the edge uw,which yields the graph G+uw,we merge two cliques into one, meaning that σ(G+uw)=σ(G)-1. So uv is a critical edge of G. This completes the proof of Theorem <ref>.§ THE PROOF OF THEOREM <REF> We are now ready to prove Theorem <ref>, which we restate below.Theorem <ref>. If a graph HP_4 or of H P_1+ P_3, then Critical Vertex, Critical Edge and Contraction-Critical Edge restricted to H-free graphs are polynomial-time solvable, otherwise they are -hard or co--hard. Let H P_1+P_3 or H P_4. Let G be an H-free graph.By Theorem <ref> we can compute χ(G) in polynomial time. We note that any vertex deletionresults in a graph that is H-free as well. Hence in order to solve Critical Vertex we can compute the chromatic number of G-v for each vertex v in polynomial time and compare it with χ(G). As (P_1+P_3)-free graphs and P_4-free graphs are closed under edge contraction as well, we can follow the same approach for solving Contraction-Critical Edge.By Proposition <ref> we obtain the same result for Critical Edge.Now suppose that neither H P_1+P_3 nor H P_4. If H has a cycle or an induced claw, then we useTheorem <ref>. Assume not. Then H is a disjoint union of r paths for some r≥ 1. If r≥ 4 we use Theorem <ref>. If r=3 then either H=3P_1P_1+P_3, which is not possible, or H 2P_1+P_2 and we can apply Theorem <ref> again.Suppose r=2. If both paths contain an edge, then 2P_2 H. If at most one path has edges, then it must have at least four vertices, as otherwise H P_1+P_3. This means that 2P_1+P_2 H. In both cases we apply Theorem <ref>. If r=1, then H is a path on at least five vertices, which means 2P_2 H. We apply Theorem <ref> again.99BBPR C. Bazgan, C. Bentz, C. Picouleau and B. Ries, Blockers for the stability number and the chromatic number, Graphs and Combinatorics 31 (2015) 73–90.BTT11 C. Bazgan, S. Toubaline and Z. Tuza,The most vital nodes with respect to independent set and vertex cover,Discrete Applied Mathematics 159 (2011) 1933–1946. Bentz C. Bentz, M.-C. Costa, D. de Werra, C. Picouleau and B. 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"authors": [
"Daniël Paulusma",
"Christophe Picouleau",
"Bernard Ries"
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"categories": [
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"primary_category": "cs.CC",
"published": "20170627205826",
"title": "Critical Vertices and Edges in $H$-free Graphs"
} |
Challenges to estimating contagion effects from observational dataElizabeth L. Ogburn Johns Hopkins Bloomberg School of Public Health, 615 N. Wolfe St. Baltimore MD, [email protected]; support from ONR grant N000141512343* Elizabeth L. Ogburn Received: date / Accepted: date ===================================*A growing body of literature attempts to learn about contagion using observational (i.e. non-experimental) data collected from a single social network. While the conclusions of these studies may be correct, the methods rely on assumptions that are likely–and sometimes guaranteed to be–false, and therefore the evidence for the conclusions is often weaker than it seems. Developing methods that do not need to rely on implausible assumptions is an incredibly challenging and important open problem in statistics. Appropriate methods don't (yet!) exist, so researchers hoping to learn about contagion from observational social network data are sometimes faced with a dilemma: they can abandon their research program, or they can use inappropriate methods. This chapter will focus on the challenges and the open problems and will not weigh in on that dilemma, except to mention here that the most responsible way to use any statistical method, especially when it is well-known that the assumptions on which it rests do not hold, is with a healthy dose of skepticism, with honest acknowledgment and deep understanding of the limitations, and with copious caveats about how to interpret the results.§ BACKGROUND § MOTIVATING EXAMPLE Suppose that students attending the residential Faber College are measured and weighed at the start and close of each school year, and a complete social network census is taken, cataloguing all social ties among members of the student body. In addition, researchers have access to basic demographic covariates measured on each student. Researchers are interested in testing whether there is a contagion effect for body mass index (BMI): if one individual–the ego–gains (or looses) weight, does that make his or her social contacts–the alters–more likely to do the same? They are also interested in estimating the contagion effect if one exists: if an ego gains (or looses) weight, what is the expected increase (or decrease) in the alters' body mass indices? There are many different procedures one could use to test for or estimate a contagion effect, using different models, different assumptions, different sets of covariates, different ways of calculating intervals or uncertainty, and the list goes on. In order for a procedure to be useful, it has to satisfy two requirements. First, it has to isolate the causal effect of the ego's change in BMI on the alters' changes in BMI from potential other sources of similarity between the ego's and the alters' outcomes. This has to do with confounding, which is the subject of Section <ref>. The second requirement for a useful analysis is that it must be generalizable to populations beyond the precise student body used in the analysis. We would like to be able to extrapolate what we learn about contagion from the Faber student body to contagion of BMI in similar college populations across different colleges or even across different years at Faber College. Assume that the student body we observe at Faber College is representative of these other student populations, that is, that the true underlying contagion effect for the observed sample of Faber students is the same as the true underlying contagion effect in the other college populations to which we want to extrapolate. Then one way to determine whether we are warranted in extrapolating from Faber students to the other similar groups of students is to calculate a confidence interval for the true contagion effect, based on a model of asymptotic growth of the sample. For example, if the sample is large enough that a central limit theorem approximately holds for the contagion effect estimate, then a Gaussian confidence interval around the sample mean is approximately valid. Under the assumption of the same true underlying contagion effect, our confidence that this interval covers the true contagion effect for Faber College students is the same as our confidence that it covers the true contagion effect for students at a different college or in a different year. As in many settings for statistical inference, asymptotics are appropriate not because we care about an infinite population but because they shed light on finite samples. This requires valid statistical inference, which is the subject of Section <ref>. § DEFINING CAUSAL EFFECTS Questions about the influence one subject has on the outcome of another subject are inherently questions about causal effects: contagion is a causal effect on an ego's outcome at time t of his alter's outcome at time s for some s<t. Causal effects are defined in terms of potential or counterfactual outcomes (see e.g. ). In general, a unit-level potential outcome, Y_i(z), is defined as the outcome that we would have observed for subject i if we could have intervened to set that subject's treatment or exposure Z_i to value z. A contagion effect of interest for dyadic data might be a contrast of counterfactuals of the form Y_ego^t(y_alter^t-1), for example E[Y_ego^t(y)-Y_ego^t(y-1)] would be the expected difference in the ego's counterfactual outcome at time t had the alter's outcome at time t-1 been set to y compared to y-1. In data comprised of independent dyads this contagion effect is well-defined, but social networks represent a paradigmatic opportunity for interference, whereby one subject's exposure may affect not only his own outcome but also the outcomes of his social contacts and possibly other subjects. Under interference, the traditional unit-level potential outcomes are not well-defined. Instead, Y_i(𝐳) is the outcome that we would have observed if we could have set the vector of exposures for the entire population, 𝐙, to 𝐳=(z_1,...,z_n) where for each i, z_i is in the support of Z. The causal inference literature distinguishes between interference, which is present when one subject's treatment or exposure may affect others' outcomes, and contagion, which is present when one subject's outcome may influence or transmit to other subjects (e.g. ), but in fact they are usually intertwined. Consider three Faber students: Alex, Andy, and Ari, all friends with each other. Alex's outcome at time t depends on both Andy's and Ari's outcomes at time t-1, Andy's outcome at time t depends on Alex's and Ari's at time t-1, and Ari's outcome at time t depends on Alex's and Andy's at time t-1. This results in a situation that is hardly distinguishable from the hallmarks of interference: Y_Alex^t(y_Andy^t-1,y_Ari^t-1), Y_Andy^t(y_Alex^t-1,y_Ari^t-1), and Y_Ari^t(y_Alex^t-1,y_Andy^t-1) are potential outcomes that depend on multiple “treatments” and those treatments are overlapping across subjects. Furthermore, just as in settings with interference, a counterfactual outcome for node i that omits some of the treatments to which node i is exposed (i.e. the outcomes at time t-1 for some of i's alters) is not well-defined. This has been overlooked in most of the literature on contagion in observational social network data, which generally focuses on alter-ego pairs, thereby inherently considering ill-defined counterfactuals like Y_Alex^t(y_Andy^t-1). This points to an under-appreciated challenge for the study of contagion in a social network: simply defining the causal effect of interest. If researchers sample non-overlapping alter-ego dyads from the network then Y_ego^t(y_alter^t-1) may be well-defined, but if they wish to use all of the available data, comprised of overlapping dyads, causal effects must be defined in terms of all of the alters for a particular ego. In the latter case, we could define a contagion effect that compares the mean counterfactual outcome for an ego had the mean outcome among the alters been set to one value as opposed to a different value. For simplicity, in the remaining sections we will talk about alter-ego pairs rather than clusters of an ego with all of its alters. This is in keeping with the existing applied literature, but it is important to note that close attention should be paid in future work to the definition of causal contagion effects for non-dyadic data. Numerous papers and researchers have addressed the definition of counterfactuals and causal effects in settings with interference (e.g. ); similar attention should be paid to contagion effects. § CONFOUNDINGConfounding, is, loosely, the presence of a non-causal association that may be misinterpreted as a causal effect of one variable on another. Most commonly, confounding is due to the presence of a confounder that has a causal effect on both the hypothesized cause and the hypothesized effect. Such a confounder generates an association between the hypothesized cause and effect which, without careful analysis, could be taken as evidence of a causal effect. There are two types of confounding that are nearly ubiquitous and especially intransigent in the context of contagion effects in social networks: homophily is the tendency of people who are similar to begin with to share network ties, and environmental confounding is the tendency of people who share network ties to also share environmental exposures that could jointly affect their outcomes. We elucidate these two types of confounding below. §.§ Homophily Consider the Faber College student body. Suppose that two students, Pat and Lee, meet in September and bond over the fact that they both used to be competitive runners but recently developed injuries that prevent them from running and from participating in other active hobbies they used to enjoy. Soon Pat and Lee are close friends. Over the course of a few months, the sedentary lifestyle catches up with Pat, who gains a considerable amount of weight. It takes longer for Lee, but by the close of the school year Lee has also gained a lot of weight. If you did not have access to the back story and only observed that Pat gained weight and then Pat's close friend Lee did too, this looks like potential evidence of a causal effect of Pat's change in BMI on Lee's change in BMI. In fact, this is a case of homophily: unobserved covariates related to the propensity to gain weight (in this case, recent injury) caused Pat and Lee to become friends and also caused them to both undergo changes in BMI.Some carefully considered studies attempt to control for all sources of homophily (seefor details and references), but this is generally not possible unless researchers have a high degree of control over data collection and can collect extremely rich (and therefore expensive!) data on the covariates that affect ties. Any traits that are related to the formation, duration, or strength of ties and to the outcome of interest must be measured. For some outcomes, such as infectious diseases, it may be possible to enumerate and observe all such traits, but for other outcomes, such as BMI, endless permutations of the Pat-and-Lee story are possible (e.g. friendship based on shared body norms, shared love of sugary snacks, shared appreciation for a particular celebrity whose BMI changes could affect both Pat and Lee's, etc.), making it nearly impossible to control for all potentially confounding traits. In addition to the challenge of enumerating the potentially confounding traits, there are huge costs to collecting such rich data, and available social network data are highly unlikely to include adequate covariates.For these reasons, researchers have developed clever tricks to try to control for homophily using only data the network and the outcome of interest. One such trick is to include both the alter and the ego's outcomes ate time t-2 as covariates in a regression of the ego's outcome at time t on the alter's outcome at time t-1. The argument used to justify this method is that any traits related to tie formation and to the outcome are fully captured by the similarity in the alter and ego's outcomes at time t-2; any association between the alter's outcome at time t-1 and the ego's at time t after controlling for this baseline similarity must be due to contagion. But the story of Pat and Lee demonstrates one flaw in this argument: baseline traits can affect outcome trajectories over time and so conditioning on the outcome at a single time point does not render all future outcome measures independent of the baseline covariates. Another flaw in the argument is that homophily operates not only through the propensity to form ties, but also through the propensity to maintain ties and through the strength of the ties; neither strength nor duration can be captured by past outcomes <cit.>. Furthermore, <cit.> demonstrated that, even if a baseline trait only affects friendship formation (not strength or duration), merely conditioning on the presence of a tie, which is inherent in all analyses focused on alter-ego pairs, creates a spurious association between the alter's outcome at time t-1 and the ego's outcome at time t. This is because the presence of a tie is a collider: a common effect of two variables, conditioning on which creates a spurious association between the two causes. (For an accessible review of colliders see .) Another clever trick is to compare the strength of the association between an alter's and an ego's outcomes across different types of ties: undirected, or mutual; directed, with the ego naming the alter as a friend but not vice versa; and directed, with the alter naming the ego as a friend but not vice versa. Suppose Pat claims Lee as a friend but Lee does not claim Pat as a friend. Any similarity in baseline traits that Pat and Lee share is a symmetric relationship, the argument goes, and therefore if the regression of Pat's BMI at time t on Lee's BMI at time t-1 results in a larger coefficient than does the regression of Lee's BMI at time t on Pat's BMI at time t-1, this is evidence of contagion. Unfortunately, this argument is also flawed <cit.>. This is because, somewhat counterintuitively, similarity in baseline traits does not have to be symmetric. Suppose Pat claims Lee as a friend because Lee is the only person Pat knows who is going through a painful separation with running and other active hobbies, while Lee participates in a support group for recently injured former runners and considers only one participant, Lou, who has the exact same injury and prognosis, as a friend. By construction, even though Lee is the node with the most baseline similarity to Pat from among all of Pat's potential friends, the reverse is not true: Lou, not Pat, is the node with the most similarity to Lee from among all of Lee's potential friends. Therefore, if Lou's outcome at time t-1 has a stronger association with Lee's outcome at time t-1 than Pat's does, this could be evidence of greater similarity on baseline characteristics rather than contagion. Furthermore, it can be shown that a similar story results in reciprocated ties having the strongest association of all <cit.>. <cit.> used a slightly different data-generating process to show that purported evidence for contagion due to asymmetry in the association of an alter's outcome with an ego's outcome for different types of ties is consistent with homophily rather than contagion.§.§ Shared environment Let's turn to a different pair of Faber students, Cam and Sam, who both decided to move off campus to a neighborhood across town from the college. Over the course of the school year, both the grocery store and the gym in their neighborhood closed down and were replaced with fast food restaurants. Cam immediately starts taking every meal at the fast food joint and gains weight fairly quickly, while Sam holds out for several months, taking the bus to a distant grocery store, but when time winter weather and final exams pile on Sam, too, falls prey to the fast food marketing. By the end of the year both students have gained weight. This is confounding due to shared environment, another source of confounding that plagues attempts to learn about contagion from observational data. People who share network ties tend to live near each other, work together, pay attention to the same information, or work in the same industry, all of which can generate confounding due to shared environment (which need not be restricted to physical environment). Note that confounding due to shared environment is present whether Cam and Sam are friends because they live in the same neighborhood or they moved to the same neighborhood because they were friends. The distinction between homophily and shared environment is not always clearcut; if Cam and Sam became friends because they lived in the same neighborhood that would simultaneously be an example of homophily and of shared environment. The same strategies described above for dealing with homophily have been used in an attempt to control for confounding due to shared environment, but similar reasoning controverts their effectiveness. <cit.> proposed controlling for confounding by shared environment by including fixed effects for “community” in regressions of an ego's outcome at time t on an alter's outcome at time t-1. If all such confounding occurs due to clearly delineated and known communities, like well-defined neighborhoods in the example above, this is potentially a good solution, though in many cases the operative communities, or their membership, will likely be unknown. § DEPENDENCE Suppose confounding is not an issue, because researchers at Faber were well-funded and prescient enough to collect data on every possible confounder of the contagion effect, and further suppose that the researchers have a model–maybe a regression, maybe a propensity-score based method <cit.>, maybe some other model–that they believe gives an estimate of the causal contagion effect. We now turn to the question of how to perform valid statistical inference using a model fit to data from a social network. The issue of valid statistical inference is entirely separate from the issue of confounding or even contagion; it applies whether we want to estimate a simple mean or a complicated causal effect. The key points made in this section apply to anything that we want to estimate using social network data. Most estimators of causal effects, including The coefficient on the alter's outcome at time t-1 in a regression of the ego's outcome at time t, are closely related to sample means (to be technical, they are M-estimators), so all of the points made below apply. Going back to Faber College, administrators are now interested in the simpler problem of estimating the mean BMI for the student body at the end of the school year. There are n students, or nodes in the social network comprised of students, and each one furnishes an observed BMI measurement Y_i. Our goal is to perform valid (frequentist) statistical inference about the true mean μ of Y using a sample mean Y̅=1/n∑_i=1^nY_i of dependent observations 𝐘=(Y_1,...,Y_n), where the dependence among observations is determined or informed by network structure. But for the dependence, this is a familiar problem. In general, when we want to use a sample mean to perform inference about a true mean, we take the sample mean as our point estimate, calculate a standard error for the sample mean, and tack on a confidence interval based on that standard error. The unique challenge for the social network setting is the effect of dependence on the standard error. To keep things as simple as possible, let's assume that Y_i,...,Y_n are identically, though not independently, distributed, so the mean of Y_i is μ and the variance of Y_i is σ^2, which we assume is finite, for all i. (In fact, it is easier to deal with observations that are not identically distributed than it is to deal with observations that are dependent, so relaxing this assumption is not too difficult.)Recall that the standard error of Y̅ is the square-root of its variance, where Var(Y̅) =1/n^2Var(∑_i=1^nY_i)=1/n^2{∑_i=1^nσ^2+∑_i≠ jcov(Y_i,Y_j)}=σ^2/n+1/n^2∑_i≠ jcov(Y_i,Y_j).When Y_i,...,Y_n are independent, the covariance term cov(Y_i,Y_j) is equal to 0 for all i≠ j pairs, so the variance of Y̅ is σ^2/n, which should be familiar from any introductory statistics or data analysis class. But when Y_i,...,Y_n are dependent, in particular when they are positively correlated (which is the type of dependence that we would expect to see in just about every social network setting), the variance of Y̅ is bigger than σ^2/n because it includes the term 1/n^2∑_i≠ jcov(Y_i,Y_j). Define b_n=1/n∑_i≠ jcov(Y_i,Y_j). Then var(Y̅) =σ^2/n/(1+b_n/σ^2)and we can see that the factor by which the variance of Y̅ is bigger than what it would be if Y_i,...,Y_n were independent is (1+b_n/σ^2). We call n/(1+b_n/σ^2) the effective sample size of our sample of n dependent observations Y_1,...,Y_n. The effective sample size n/(1+b_n/σ^2) is smaller than the true sample size n; heuristically this is because each observation Y_i contains some new information about the target of inference μ and some information that is rendered redundant by dependence. Under independence each observation furnishes 1 “bit” of information about μ, whereas under dependence each observation furnishes only 1/(1+b_n/σ^2) bit of information about μ.In order to explain the impact of this dependence on statistical inference, we first review the standard inferential procedure for independent data. When Y_i,...,Y_n are independent, a typical procedure would be to calculate an approximate 95% confidence interval for μ as Y̅±1.96×σ̂/√(n), where σ̂ is the square root of an estimate of the variance of Y. The factor 1.96 is the 97.5th quantile of the standard Normal distribution; t-distribution quantiles could be used instead to account for the fact that σ is estimated rather than known. This procedure relies on several preliminaries: (1) Y̅ is unbiased for μ, (2) Y̅ is approximately Normally distributed, and (3) σ̂/√(n) is a good estimate of the variance of Y̅. These preliminaries hold, at least approximately, in most settings with independent data and moderate to large n. Dependence doesn't affect (1), but it does affect (2) and (3). When Y_i,...,Y_n are independent, the Central Limit Theorem (CLT) tells us that √(n)(Y̅-μ) converges in distribution to a Normal distribution as n→∞. The factor √(n) is called the rate of convergence and it is needed to make sure that the variance of √(n)(Y̅-μ) is not 0, in which case √(n)(Y̅-μ) would converge to a constant rather than a distribution, and is not infinite, in which case √(n)(Y̅-μ) would not converge at all. The variance of Y̅ (equivalently, the variance of Y̅-μ) is σ^2/n, so the variance of √(n)(Y̅-μ) is n×(σ^2/n)=σ^2, which is a positive, finite constant. When Y_i,...,Y_n are dependent, the rate of convergence may be different (slower) than √(n). (In fact, if the dependence is strong and widespread enough, the CLT may not hold at all; determining what types of social network dependence are consistent with the CLT is an important area for future study.) This is because the rate of convergence is determined by the effective sample size instead of by n: the variance of Y̅ is σ^2/{ n/(1+b_n/σ^2)}, so (as long as a CLT holds), √(n/(1+b_n/σ^2))(Y̅-μ) will converge to a Normal distribution as n→∞ and the rate of convergence is given by √(n/(1+b_n/σ^2)) rather than √(n). Sometimes, in particular when b_n is fixed as n→∞, this distinction will be meaningless. But sometimes, when b_n grows with n, it is a meaningfully slower rate of convergence. (Note that b_n/n must converge to 0 as n→∞ in order for a CLT to hold, so b_n must grow slower than n.) This matters because it informs when the approximate Normality of the CLT kicks in, i.e. at what sample size it is safe to assume that Y̅ is approximately Normally distributed. Many different rules of thumb exist for determining when approximate Normality holds; one popular rule of thumb is that n=30 suffices. With dependent data, this number is larger, and sometimes considerably so. The effective sample size, rather than n, should be used to assess whether the sample size is large enough to approximate the distribution of Y̅ with a Normal distribution. When researchers ignore dependence and rely on the Normal approximation in samples that have large enough n but not large enough effective sample size, there is no reason to think that their 95% confidence intervals will have good coverage properties.Ignoring dependence is most dangerous when estimating the standard error of Y̅. Any estimate of var(Y̅) that is based only on the marginal variances σ^2 of Y_i and ignore the covariances cov(Y_i,Y_j) will underestimate the standard error of Y̅, often severely. Inference that is based on an underestimated standard error is anticonservative: confidence intervals are narrower than they should be and p-values are lower than they should be, leading researchers to draw conclusions that are not in fact substantiated by the data. Even if each observation is dependent only on a fixed and finite number of other observations, so that dependence is asymptotically negligible and does not affect the rate of convergence of the CLT, in finite samples ignoring the covariance terms in var(Y̅) could still have substantial implications on inference. This is particularly a problem because no good solutions exist. Statisticians are good at dealing with dependence that arises due to space or time, or even other more complicated processes that can be expressed using Euclidean geometry. But dependence that is informed by a network is very different from these well-understood types of dependence, and, unfortunately, statisticians are only just beginning to develop methods for taking it into account. Most published research about social contagion uses regression models or generalized estimating equations (GEEs) to estimate contagion effects; though some of these models account for the dependence due to observing the same nodes over multiple time points, none of them account for dependence among nodes. §.§ Sources of network dependence In the literature on spatial and temporal dependence, dependence is often implicitly assumed to be the result of latent traits that are more similar for observations that are close in Euclidean distance than for distant observations. This type of dependence is likely to be present in many network contexts as well. In networks, edges present opportunities to transmit traits or information, and contagion or influence is an important additional source of dependence that depends on the underlying network structure.Latent trait dependence will be present in data sampled from a network whenever observations from nodes that are close to one another are more likely to share unmeasured traits than are observations from distant nodes. Homophily is a paradigmatic example of latent trait dependence. If the outcome under study in a social network has a genetic component, then we would expect latent variable dependence due the fact that family members, who share latent genetic traits, are more likely to be close in social distance than people who are unrelated. If the outcome were affected by geography or physical environment, latent variable dependence could arise because people who live close to one another are more likely to be friends than those who are geographically distant. Of course, whether these traits are latent or observed they can create dependence, but if they are observed then conditioning on them renders observations independent, so only when they are latent do they result in dependence that requires new tools for statistical inference. Just like in the spatial dependence context, there is often little reason to think that we could identify, let alone measure, all of these sources of dependence. The notions of latent sources of homophily or latent correlates of shared environment are familiar from the discussion of confounding, above, but there is an important distinction to be made between latent sources of confounding and latent sources of dependence: in order to be a source of unmeasured confounding, a latent trait must affect both the exposure (e.g. the alter's outcome at time t-1) and the outcome (ego's outcome at time t) of interest. In order to be a source of dependence, a latent trait must affect two or more outcomes of interest. Latent trait dependence is the most general form of dependence, in that it provides no structure that can be harnessed to propel inference. In order to make any progress towards valid inference in the presence of latent trait dependence, some structure must be assumed, namely that the range of influence of the latent traits is primarily local in the network and that any long-range effects are negligible.Contagion or influence arises when the outcome under study is transmitted from node to node along edges in the network. The diagram in Figure 1 depicts contagion in a network with three nodes in which node 2 is connected to nodes 1 and 3 but there is no edge between 1 and 3. Y_i^t represents the outcome for node i at time t, and the unit of time is small enough that at most one transmission event can occur between consecutive time points. Dependence due to contagion has known, though possibly unobserved, structures that can sometimes be harnessed to facilitate inference; we touch on this briefly in Section <ref>. Crucially, whenever contagion is present so is dependence, and therefore statistical analysis must take dependence into account in order to result in valid inference.§ SOLUTIONS Researchers have known for decades that learning about contagion from observational data is fraught with difficulty, perhaps most famously expressed by <cit.>. Recent years have seen incremental methodological progress, but huge hurdles remain. Most of the constructive ideas in <cit.> involve bounding contagion effects rather than attempting to point identify them; looking for bounds rather than point estimates is a general approach that could prove fruitful in the future. Indeed, <cit.> built upon the ideas in <cit.> and were able to derive bounds on the association due to homophily on traits that do not change over time (“static homophily”). Another general approach is to make use of sensitivity analyses whenever an estimation procedure relies on assumptions that may not be realistic (e.g. ). Some of the problems discussed above have solutions in some settings; below we discuss solutions that exploit features of specific settings rather than providing general approaches to the problem of estimating contagion effects. (Some of the material below was first published in .)§.§ Randomization If it is possible to randomize some members of a social network to receive an intervention, and if it is known that an alter's receiving an intervention can only affect the ego's outcome through contagion (as opposed to directly; seefor discussion), then problems of confounding and dependence can be entirely obviated. Randomization-based inference, pioneered by Fisher <cit.> and applied to network-like settings by <cit.> and <cit.>, is founded on the very intuitive notion that, under the null hypothesis of no effect of treatment on any subject (sometimes called the sharp null hypothesis to distinguish it from other null hypotheses that may be of interest), the treated and control groups are random samples from the same underlying distribution. Randomization-based inference treats outcomes as fixed and treatment assignments as random variables: quantities that depend on the vector of treatment assignments are the only random variables in this paradigm. Therefore, dependence among outcomes is a non-issue. Typically this type of inference is reserved for hypothesis testing, though researchers have extended it to estimation. We leave the details, including several subtleties and challenges that are specific to the social network context, to a later chapter (see alsofor a review).Randomizing the formation of network ties themselves obviates confounding due to the effects of homophily on tie formation. A number of studies have taken advantage of naturally occurring randomizations of this kind, such as the assignment of students to dorm rooms () or of children to classrooms (). However, this does not suffice to control for the effects of homophily on tie strength or duration, or to control for confounding due to shared environment.§.§ Parametric models If researchers are willing to commit to certain types of parametric models, it may be possible isolate contagion from confounding <cit.>. It is a reliance on strong parametric models, for example, that underpins mathematical modeling or agent based modeling approaches to contagion <cit.>.This might seem benign–after all, most statistical analyses rely on parametric models of one kind or another–but there is a fundamental difference between, for example, using a linear regression when the true underlying relationships is not linear, and relying on parametric models to identify a causal effect that is otherwise hopelessly confounded. In the first case, a misspecified model may bias the estimate we are interested in, often in ways that are well-understood, and often in proportion to the fit of the model to the data (i.e. the worse the misspecification, the greater the bias). In the latter case, at least in the absence of a model-specific proof otherwise, any hint of misspecification undermines the causal interpretation we would like to be able to justify and what looks like evidence of a causal effect could just be evidence of confounding. George Box's oft-cited aphorism, “all models are wrong but some are useful,” justifies the use of misspecified parametric models in many settings, but when the parametric form of the model is the only bulwark against confounding, the model must (in the absence of a proof to the contrary) in fact be correct in order to be useful.§.§ Instrumental variable methods <cit.> proposed an instrumental variable (IV) solution to the problem of disentangling contagion from homophily. An instrument is a random variable, V, that affects exposure but has no effect on the outcome conditional on exposure. When the exposure - outcome relation suffers from unmeasured confounding but an instrument can be found that is not confounded with the outcome, IV methods can be used to recover valid estimates of the causal effect of the exposure on the outcome. In this case there is unmeasured confounding of the relation between an alter's outcome at time t-1 and an ego's outcome at time t whenever there is homophily on unmeasured traits. <cit.>, <cit.>, and <cit.> provide accessible reviews of IV methods.<cit.> propose using a gene that is known to be associated with the outcome of interest as an instrument. In their paper they focus on perhaps the most highly publicized claim of peer effects, namely that there are significant peer effects of body mass index (BMI) and obesity <cit.>. If there is a gene that affects BMI but that does not affect other homophilous traits, then that gene is a valid instrument for the effect of an alter's BMI on his ego's BMI. The gene affects the ego's BMI only through the alter's manifest BMI (and it is independent of the ego's BMI conditional on the alter's BMI), and there is unlikely to be any confounding, measured or unmeasured, of the relation between an alter's gene and the ego's BMI.There are two important challenges to this approach. First, the power to detect peer effects is dependent in part upon the strength of the instrument - exposure relation which, for genetic instruments, is often weak. Indeed, <cit.> reported low power for their data analyses. Second, in order to assess contagion at more than a single time point (i.e. the average effect of the alter's outcomes on the ego's outcomes up to that time point), multiple instruments are required. <cit.> suggest using a single gene interacted with age to capture time-varying gene expression, but this could further attenuate the instrument - exposure relation and this method is not valid unless the effect of the gene on the outcome really does vary with time; if the gene-by-age interactions are highly collinear then they will fail to act as differentiated instruments for different time points.§.§ Data from multiple independent networks When multiple independent networks are observed, the problems of confounding due to shared environment and of dependence may be considerably easier to deal with. A large literature on interference in causal inference is dedicated to inference in the setting where independent groups of individuals interact and affect one another within, but not between, groups; this is analogous to multiple independent social networks (see, e.g., ). If environmental factors can shared within but not across networks, it may be possible to control for confounding by shared environment via a fixed effect for each network, as in <cit.>.§.§ Contagion operating alone If researchers have reason to believe that there is no unmeasured homophily or features of shared environments that contribute to confounding or to dependence, i.e. if contagion is the only mechanism giving rise to either dependence or to associations among the outcomes of interest, then there are a few recent methodological advances that can be used to estimate contagion effects (). Dependence due to contagion has known, though possibly unobserved, structures that can sometimes be harnessed to facilitate inference. Time and distance act as information barriers for dependence due to contagion, giving rise to many conditional independencies that can sometimes be used to make network dependence tractable. Two examples of the many conditional independencies that hold in Figure (1) are [Y_1^t⊥ Y_2^t| Y_1^t-2, Y_2^t-2, Y_1^t-1,Y_2^t-1] and [Y_1^t-1⊥ Y_3^t| Y_2^t-2]. The first conditional independence statement illustrates the principle that outcomes measured at a particular time point are mutually independent conditional on all past outcomes. The second conditional independence statement illustrates the fact that outcomes sampled from two nonadjacent nodes are independent if the amount of time that passed between the two measurements was not sufficiently long for information to travel along the shortest path from one node to the other, conditional any information that could have simultaneously influenced the sampled nodes (in this case Y_2^t-2). Observing outcomes in a network on a fine enough time scale to observe all transmissions requires a richness of data that will not usually be available, and if the network under a contagious process is observed at a single time point, dependence due to contagion is indistinguishable from latent variable dependence and the structure is lost.This work was funded by the Office of Naval Research grant N00014-15-1-2343.spbasic | http://arxiv.org/abs/1706.08440v2 | {
"authors": [
"Elizabeth L. Ogburn"
],
"categories": [
"stat.AP"
],
"primary_category": "stat.AP",
"published": "20170626153615",
"title": "Challenges to estimating contagion effects from observational data"
} |
1Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA; <atripathi>, <[email protected]> 2University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, GermanyWe present a sub-arcsecond resolution survey of the 340 GHz dust continuum emission from 50 nearby protoplanetary disks, based on new and archival observations with the Submillimeter Array.The observed visibility data were modeled with a simple prescription for the radial surface brightness profile.The results were used to extract intuitive, empirical estimates of the emission “size" for each disk, R_ eff, defined as the radius that encircles a fixed fraction of the total continuum luminosity, L_ mm.We find a significant correlation between the sizes and luminosities, such that R_ eff∝ L_ mm^0.5, providing a confirmation and quantitative characterization of a putative trend that was noted previously.This correlation suggests that these disks have roughly the same average surface brightness interior to their given effective radius, ∼0.2 Jy arcsec^-2 (or 8 K in brightness temperature).The same trend remains, but the 0.2 dex of dispersion perpendicular to this relation essentially disappears, when we account for the irradiation environment of each disk with a crude approximation of the dust temperatures based on the stellar host luminosities.We consider two (not mutually exclusive) explanations for the origin of this size–luminosity relationship.Simple models of the growth and migration of disk solids can account for the observed trend for a reasonable range of initial conditions, but only on timescales that are much shorter than the nominal ages present in the sample.An alternative scenario invokes optically thick emission concentrated on unresolved scales, with filling factors of a few tens of percent, that are perhaps manifestations of localized particle traps.§ INTRODUCTION Observations that help characterize the relationships among key properties of protoplanetary disks, and those of their host stars, are essential to developing a more robust and complete planet formation theory.Measurements of continuum emission at (sub-)millimeter wavelengths (hereafter “mm") are important since they trace the reservoir of planetesimal precursors.They are especially sensitive diagnostics of the growth and migration of solids during the early stages of planet formation.Despite the complexity of the planet formation process, theoretical models agree that overall efficiency depends strongly on the mass of raw material available in the disk.The most accessible diagnostic of mass is the mm continuum luminosity (L_ mm) generated by dust grains <cit.>.Because of its presumed low optical depth, L_ mm scales roughly linearly with the total mass and mean temperature of the disk solids.With reasonable assumptions for the grain properties, mm luminosity surveys have been used to construct disk mass distributions <cit.>, and study how they vary with time <cit.>, environment <cit.>, and stellar host mass <cit.>. By itself, an unresolved quantity like L_ mm provides little leverage for constraining the wide diversity of disk properties relevant to planet formation.Information on the spatial distribution of the disk material is needed for a more robust characterization of the disk population.With even crude resolution, another elementary disk property – size – becomes accessible.Disk sizes reflect some convolution of the processes involved in formation <cit.>, evolution of angular momentum <cit.>, and, perhaps most importantly, the growth and transport of the constituent solid particles <cit.>.Work in this field has been relatively restricted to detailed case studies, but nevertheless a compelling demographic trend related to sizes and masses has already been identified.Based on a small 340 GHz sample, <cit.> found that disks with lower luminosities (i.e., less massive) are preferentially smaller, but do not necessarily have lower surface brightnesses.<cit.> confirmed this tendency, with a different sample and at a slightly lower frequency (230 GHz). Crucial information about the mechanisms at play in planetesimal formation and transport in disks are potentially encoded in this putative size–luminosity relationship.Our goal is to validate and better characterize this trend.With that motivation in mind, we present a substantially larger survey of resolved protoplanetary disk continuum measurements that spans a wide range of the relevant parameter-space, in an effort to better characterize how disk sizes are related to their luminosities.We introduce this dataset in Section <ref> and describe an empirical, intuitive methodology for inferring a robust size metric for each disk in Section <ref>.The results of this homogenized analysis are presented in Section <ref> and interpreted in the context of our current understanding of the evolution of disk solids in Section <ref>.§ DATA §.§ Sample A sample of 50 nearby (d≤200 pc) disk targets was collated from the archived catalog of ∼340 GHz (880 μm) continuum measurements made with the Submillimeter Array <cit.>, since the 2004 start of science operations.We considered four primary selection criteria to include a given target in this sample: (1) a quality calibration (i.e., suitable phase stability); (2) a 340 GHz continuum flux density ≥20 mJy; (3) baseline lengths ≥200 m (to ensure sufficient resolution for estimating an emission size); and (4) no known companion within a ∼2 separation.[An exception is made for very close spectroscopic binary companions (e.g., UZ Tau E), where the dust generating the emission is confined to a circumbinary disk around both components.]The first three of these are practical restrictions to ensure reliable measurements of the metrics of interest.The fourth criterion has a physical motivation: it was designed to avoid targets where dynamical interactions in multiple star systems are known to reduce the continuum sizes and luminosities <cit.>.Of the 50 disks in our survey, 10 of these were recently observed by us expressly for the purposes of the present study.To our knowledge, the SMA observations of 18 targets have not yet been published elsewhere.The data for the remaining 32 targets have appeared previously in the literature (Sect. <ref>).Table <ref> is a brief SMA observation log, with references for where the data originally appeared. Targets in this sample are primarily located in the Taurus and Ophiuchus star-forming regions, although 9 are in other regions or are found in isolation.§.§ New and Previously Unpublished Data We observed 10 disks in the Taurus-Auriga complex (DN Tau, FY Tau, GO Tau, HP Tau, IP Tau, IQ Tau, IRAS 04385+2550, 2MASS J04333905+2227207, 2MASS J04334465+2615005, and MHO 6) using the eight 6-m SMA antennas arranged in their compact, extended, and very extended configurations (9–508 m baseline lengths).The dual-sideband 345 GHz receivers were tuned to a local oscillator (LO) frequency of 341.6 GHz (878 μm).The SMA correlator processed two intermediate frequency (IF) bands, at ±4–6 and ±6–8 GHz from the LO.Each IF band contains 24 spectral chunks of 104 MHz width.One chunk in the lower IF was split into 256 channels; all others were divided into 32 coarser channels.Each observation cycled between multiple targets and the nearest bright quasars, 3C 111 and J0510+180.Additional measurements of 3C 454.3, Uranus, and Callisto were made for bandpass and flux calibration, respectively.Another 8 disks (CI Tau, CY Tau, DO Tau, Haro 6-13, SU Aur, V410 X-ray 2, V836 Tau, and IM Lup) had previously unpublished data that are suitable for this program available in the SMA archive.These observations have slight variations on the correlator setup and calibrator sources notedabove, but in the end produce effectively similar continuum data products.The raw visibility data were calibrated using standard procedures with the facility MIR package.After calibrating the spectral response of the system and setting the amplitude scale, gain variations were corrected based on repeat observations of the nearest quasar.For observations taken over long time baselines, the visibility phases were shifted to align the data: usually this is based on the known proper motion of the stellar host, but in a few cases where these were unavailable we relied on Gaussian fits to individual measurements.All visibility data for a given target were combined, after confirming their consistency on overlapping baselines.Each visibility set was Fourier inverted, deconvolved with the clean algorithm, and then convolved with a restoring beam to synthesize an image.The imaging process was conducted with the MIRIAD software package.Table <ref> lists basic imaging parameters for the composite datasets.Figure <ref> shows a gallery of images for all targets in the sample.§.§ Supplementary Archival Data The bulk of the sample, 32 targets, have SMA continuum data that were already published in the literature.These disks are primarily in the Ophiuchus star-forming region, but also include members of the Taurus and Lupus complexes, as well as assorted isolated sources.The details of these observations and their calibration are provided in the original references (see Table <ref>), but they are generally similar to those presented in Section <ref>. § MODELING THE RESOLVED EMISSION §.§ Surface Brightness Model Definition High quality observations of disk continuum emission have previously been modeled with “broken power-law"<cit.> or “similarity solution" surface brightness profiles <cit.>.For the survey data presented here, experimentation with these brightness profiles demonstrated that some targets were much better described by one of these options, while the other left significant, persistent residuals.Instead of using different models for different targets, we elected to adopt a more flexible prescription to interpret the data for the full sample in a homogeneous context.The required flexibility in this prescription is a mechanism that treats either smooth or sharp transitions in the brightness profile.Fortunately, analogous studies of elliptical galaxy brightness profiles provide some useful guidance in this context.<cit.> introduced a prescription (the so-called “Nuker" profile) that is mathematically well-suited to this task,I_ν(ϱ) ∝( ϱ/ϱ_t)^-γ[ 1 +(ϱ/ϱ_t) ^α ] ^(γ - β)/α,where ϱ is the radial coordinate projected on the sky. The Nuker profile has five free parameters: (1) a transition radius ϱ_t, (2) an inner disk index γ, (3) an outer disk index β, (4) a transition index α, and (5) a normalization constant, which we re-cast to be the total flux density F_ν (defined so that F_ν = 2π∫ I_ν(ϱ) ϱdϱ).It is instructive to consider some asymptotic behavior.When ϱ≪ϱ_t or ϱ≫ϱ_t, the brightness profile scales like ϱ^-γ or ϱ^-β, respectively.The index α controls where these asymptotic behaviors are relevant: a higher α value pushes the profile toward a sharp broken power-law morphology.The same parameterization can reproduce the emission profiles for standard, continuous disk models as well as the ring-like emission noted for “transition" disks <cit.>.Figure <ref> shows Nuker profiles for some representative parameter values.From a given Nuker profile, we compute a set of model visibilities from the Fourier transform of Eq. (<ref>), sampled at the same discrete set of spatial frequencies as the data.Those visibilities are modified (stretched, rotated, and shifted) to account for four geometric parameters relevant to the observations: (1) a projected inclination angle i, (2) a rotation of the major axis in the plane of the sky φ (position angle), and (3)+(4) position offsets from the observed phase center {dα, dδ}.Taken together, nine parameters, θ = [ϱ_t, γ, β, α, F_ν, i, φ, dα, dδ], fully specify a set of model visibilities, 𝒱_ m(θ).It is worth highlighting two important points about the choice of this model prescription.First, the specific parameterization adopted for the brightness profile does not matter in the specific context of this study, so long as it accurately describes the data (i.e., leaves no statistically significant residuals).This will be illustrated more directly below, after a description of the full modeling procedure.Second, we have expressly avoided casting the interpretation in a physical context.Converting the results into inferences on optical depths or temperatures implicitly introduces a set of strong degeneracies and a model dependence that is not well-motivated from a physical standpoint.Our aim is instead to hew as close to the observations as possible; we will consider some connections to physical properties only in the larger context of the results from the full sample (see Sect. <ref>).§.§ Surface Brightness InferenceTo compare a given model to the visibility data, 𝒱_ d, we employ a standard Gaussian likelihood,logp(𝒱_ d | θ) = -1/2∑_i w_ d, i | 𝒱_ d, i - 𝒱_ m, i(θ) |^2,the sum of the residual moduli weighted by the standard natural visibility weights.The classic inference problem can be cast with the posterior probability distribution of the model parameters conditioned on the data aslogp(θ | 𝒱_ d) = logp(𝒱_ d | θ) + logp(θ) +constant,where p(θ) = ∏_j p(θ_j), the product of the priors for each parameter (presuming their independence).We assign uniform priors for most parameters: p(F_ν) = 𝒰(0, 10 Jy), p(ϱ_t) = 𝒰(0, 10), p(φ) = 𝒰(0, 180), and p(dα), p(dδ) = 𝒰(-3, +3).A simple geometric prior is adopted for the inclination angle, p(i) = sini.Priors for the Nuker profile's index parameters are not obvious.They were assigned based on some iterative experimentation, grounded in our findings for the targets that have data with higher sensitivity and resolution.We set p(β) = 𝒰(2, 10): the low bound forces the intensity profile to decrease at large radii (as is observed), and the high bound is practical – at these resolutions, the data are not able to differentiate between more extreme indices.To appropriately span both smooth and sharp transitions between the two power-law regimes, we set p(log_10α) = 𝒰(0, 2) (see Fig. <ref>).We sample the posterior in log_10α rather than α, since most of the diversity in Nuker profile shapes happens in the first decade of the prior-space.Finally, we set a softer-edged prior on γ that is approximately uniform over the range (-3, 2), but employs logistic tapers on the boundaries:p(γ) ∝1/1+e^-5(γ+3) - 1/1+e^-15(γ-2).The low-γ bound is again practical (like the high-β bound).The steeper bound at high γ values was designed to improve convergence for the poorly-resolved cases: when γ≳ 2, the degeneracy with ϱ_t is severe and both parameters can increasewithout bound (i.e., very steep gradients can be accommodated for very large transition radii).Since none of the well-resolved targets ended up having γ≳ 1 (regardless of the γ prior), we consider this adopted upper boundary conservative.A Markov Chain Monte Carlo (MCMC) algorithm was used to explore the posterior probability space.We employed the ensemble sampler proposed by <cit.> and implemented as the open-source code package emcee by <cit.>.With this algorithm, we used 48 “walkers" to sample in logp(θ | 𝒱_ d) and ran 50,000 steps per walker.To identify a starting point for parameter estimation, we first crudely estimate the disk geometric parameters {i, φ, dα, dδ} with an elliptical Gaussian fit to the visibility data.Using these rough values, we deproject and azimuthally average the visibilities into a one-dimensional profile as a function of baseline length.This is used as a guide to visually probe models with different Nuker profile parameters, {F_ν, ϱ_t, α, β, γ}.When a reasonable match is found (with some experience, this takes about a minute), we assign a conservative (i.e., broad) range around each of the parameters.The ensemble sampler walkers are then initialized with random draws from uniform distributions that span those ranges. While assessing formal convergence is difficult for this algorithm (since the walkers are co-dependent), we have used a variety of simple rubrics (e.g., trace and autocorrelation examinations, an inter-walkertest, and the comparison of sub-chain means and variances advocated by ) that lend confidence that we have reached a stationary target distribution.Autocorrelation lengths for all parameters are of the order 10^2 steps, implying that (after excising burn-in steps) we have ≳10^4 independent samples of the posterior for each target.We find acceptance fractions of 0.2–0.4. §.§ Disk Size MetricThe disk size is not one of the parameters that we infer directly.If we followed the traditional methodology, we would proceed by taking ϱ_t as the size metric.But that approach is problematic; ϱ_t is not really how we think about an emission size.As an illustration, consider the two disk models shown in Figure <ref> with identical ϱ_t but different γ values.Which is “larger"? Our instinct suggests the disk with a lower γ is larger, since its emission clearly has a more radially extended morphology. If we consider alternative prescriptions for the surface brightness profile, there is an analogous ambiguity.Any such model will include a “scale" parameter (like ϱ_t) that will generally have a different value than the one inferred from the data for the Nuker profile.Given the limitations of the data imposed by noise and resolution constraints, we cannot definitively determine which of these profiles, and their corresponding size metrics, is most appropriate.As a more practical concern, a degeneracy with the parameters that describe the gradients of the Nuker profile[This same degeneracy problem is present for any brightness profile <cit.>.], {α, β, γ}, makes it difficult to obtain a precise inference of ϱ_t using data with typical sensitivity and resolution (especially if α≲ 10, since the resulting smoother profile makes the transition less distinct). There is a generic definition of size that alleviates these issues.If we construct a cumulative intensity profile,f_ν(ϱ) = 2 π∫_0^ϱ I_ν(ϱ^')ϱ^'dϱ^',then we can assign an effective radius, ϱ_ eff, that encircles a fixed fraction, x, of the total flux: f_ν(ϱ_ eff) = x F_ν for some x ∈ [0,1] (note that F_ν = f_ν(∞) by definition).The inference of ϱ_ eff is considerably less affected by imprecise constraints on the gradient parameters.Moreover, ϱ_ eff more faithfully captures the intuitive intent of a size metric, as shown in Figure <ref>.Most importantly, it has the same straightforward meaning regardless of the underlying surface brightness profile; any profile that accurately reproduces the data will have the same ϱ_ eff.That said, the selection of x in the definition of ϱ_ eff is technically arbitrary.It makes sense to fix x and therefore homogenize the analysis for a sample.But, there are some practical concerns to take into consideration.If x is too low, then ϱ_ eff would be uncomfortably reliant on a sub-resolution extrapolation of the Nuker profile.And if x is too high, then ϱ_ eff would simply reflect the quality of the constraint on β (and in some cases α) based on the part of the brightness profile where we have the poorest sensitivity (large ϱ).Some of the more intuitive choices are x = 0.50 (so ϱ_ eff is the “half-light" radius) or 0.68 (so ϱ_ eff is comparable to a “standard deviation" in the admittedly poor approximation of a Gaussian brightness profile).Given the concern about extrapolation expressed above, we prefer to define ϱ_ eff based on x = 0.68.An alternative choice in the range x ∈ [0.5, 0.8] makes little difference in the analysis or results that follow. We derive posterior samples for ϱ_ eff by integrating the cumulative intensity profiles (i.e., solving Eq. <ref>) for each brightness profile sampled by the θ posteriors. We think of ϱ_ eff as a “distilled" size metric, since it reduces a complicated (and intrinsically uncertain) brightness profile into a straightforward, intuitive benchmark value.§.§ Conversion to Physical ParametersTo examine the relationship between size and luminosity, we need to work with physical (rather than observational) parameters (e.g., radii in AU rather than arcseconds, and luminosities rather than flux densities) that account for both systematics in the flux calibration and the different target distances in the sample.Our procedure for the former is to multiply each posterior sample of F_ν by a factor s, drawn from a normal distribution ∼𝒩(1.0, 0.1), that mimics the ∼10% uncertainty in the absolute flux calibration of the data.For the latter, we start with a parallax measurement for each target and, presuming a normal probability distribution, ∼𝒩(ϖ, σ_ϖ), we adopt the formalism of <cit.> to infer a distance posterior, p(d | ϖ, σ_ϖ), for a uniform distance prior.Trigonometric parallaxes from the revised Hipparcos <cit.> and Gaia DR1 <cit.> catalogs are available for AS 209, CQ Tau, UX Tau A, SU Aur, HD 163296, IM Lup, MWC 758, SAO 206462, and TW Hya.We assign the same parallax for HP Tau that was measured for its wide companion, HP Tau/G2, using VLBI radio observations <cit.>.The remaining 22 targets in Taurus are assigned parallaxes based on the values of their nearest neighbors.To do that, we compiled a list of 38 Taurus members with either Gaia DR1 or VLBI <cit.> trigonometric parallaxes (A. L. Kraus, private communication).For each target, we calculated the mean and standard deviation of the parallaxes from those members within a 5 radius.We associate WaOph 6 with its neighbor AS 209, albeit with an inflated σ_ϖ.For the remaining 14 targets in Ophiuchus, we adopt the mean ϖ measured from VLBI data for L1688 or L1689 sources by <cit.>.The parallax uncertainty for DoAr 33 was increased, since it lies north of L1688.LkHα 330, RX J1604.3-2130, and RX J1615.3-3255 are assigned parallaxes based on their affiliated cluster means <cit.>. For each posterior sample of {F_ν, ϱ_ eff}, we draw a distance from p(d | ϖ, σ_ϖ) and use it to calculate a posterior sample of the logarithms of {L_ mm, R_ eff} = {F_ν× (d/140)^2 × s, ϱ_ eff× d}, where d is in pc units.[Note that L_ mm is quantified in flux density units scaled to 140 pc, to ease comparisons with other disk samples.]The associated posteriors are summarized in Table <ref>. §.§ A Worked ExampleTo illustrate more concretely the procedure outlined above, we present a step-by-step analysis of the IQ Tau disk, which has a continuum luminosity that is roughly the median of the sampledistribution.We initialize the parameters (see Sect. <ref>) using F_ν∼𝒰(0.15, 0.19 Jy), ϱ_t ∼𝒰(0.3, 0.6), logα∼𝒰(0.3, 1.3), β∼𝒰(2, 6), γ∼𝒰(-1, 1), i ∼𝒰(44, 58), φ∼𝒰(30, 45), dα∼𝒰(0.01, 0.11), dδ∼𝒰(-0.39, -0.29).We then sample the posterior with the MCMC algorithm, and reach a stationary target distribution after 4,000 steps.The walkers have acceptance fractions of 0.26–0.30; the autocorrelation times are 80–90 steps.Figure <ref> shows the inferred covariances and marginalized posteriors for the Nuker profile parameters.We omit the geometric parameters for clarity, but they agree with independent estimates <cit.>.Confidence intervals for each parameter are derived from the marginalized posteriors and noted in Table <ref>.Figure <ref> compares the simulated observations constructed with random draws from these posteriors to the data. To construct posterior samples of ϱ_ eff, we calculate f_ν(ϱ)/F_ν (see Eq. <ref>) for each posterior sample of θ.As an illustration, Figure <ref> shows I_ν(ϱ) and f_ν(ϱ)/F_ν for 200 random draws from the posteriors.Circling back to our arguments advocating for a size metric like ϱ_ eff (see Sect. <ref>), rather than ϱ_t (or its equivalent in other model prescriptions), IQ Tau provides a clear demonstration of how the former is more precise than the latter in the face of poorly constrained gradient parameters.Because {α, β, γ} are determined with relatively poor precision in this case, the 68% confidence interval on ϱ_t is (0.46, 0.90).On the contrary, the inference on ϱ_ eff is considerably narrower, (0.53, 0.56).The ambiguity in the indices does notmatter much for ϱ_ eff: even for a diversity of I_ν(ϱ), the models that satisfactorily reproduce the data essentially have similar f_ν(ϱ).To emphasize the point that the functional form of the brightness profile does not matter, we compared these Nuker profile inferences to the more standard broken power-law and similarity solution models.For the broken power-law prescription, the radius marking the transition between the two gradients lies in the range (0.40, 0.52); for the similarity solution, the radius marking the transition between the inner power-law and the outer exponential taper is constrained to (0.22, 0.38).Despite these differences, all three models find statistically indistinguishable constraints on the effective size metric (ϱ_ eff), with a 68% confidence interval (0.53, 0.56).§ RESULTSHaving demonstrated the modeling procedure with a representative example target, we now present the results for the rest of the sample.A condensed summary of the inferred posterior distributions for the surface brightness model parameters and ϱ_ eff is provided in Table <ref>.Figure <ref> makes direct comparisons between the posteriors and the observed SMA visibilities.The surface brightness profiles, I_ν(ϱ), and cumulative intensity profiles, f_ν(ϱ)/F_ν, are shown together in Figure <ref>. Figure <ref> shows the resulting size–luminosity relation.As was hinted at in previous studies <cit.>, we find a significant correlation between these empirical variables that coarsely describe the continuum intensity profile, such that brighter disks have their emission distributed to larger radii.We employed the <cit.> linear regression mixture model to quantify the relationship and found thatlogR_ eff = (2.12±0.05) + (0.50±0.07) logL_ mm,with a Gaussian scatter perpendicular to that scaling with a standard deviation of 0.19±0.02 dex (where R_ eff is in AU and L_ mm is in Jy at an adopted distance of 140 pc; all uncertainties are quoted at the 68% confidence level).The strength of this correlation, its slope, and the amount of scatter around it are essentially the same for a considerable range of R_ eff definitions (e.g., specifically for x ∈ [0.5, 0.8], see Sect. <ref> for details[As one would expect, the intercept values depend on the adopted x.Technically the relationship inferred here is consistent with the data regardless of the choice of x, although for values of x much outside the quoted range, the uncertainties on R_ eff grow large enough that the constraints on the slope are rather poor.]). Perhaps the most surprising aspect of this relationship is the slope, which indicates that R_ eff∝√(L_ mm).That behavior has two striking (and related) implications: (1) the luminosity scales with the emitting area and (2) the surface brightness averaged over the area inside R_ eff is roughly constant for all luminosities.With respect to the latter point, we can derive ⟨ I_ν⟩|_<R_ eff = ∫_0^ϱ_ eff I_ν(ϱ) 2π ϱdϱ/∫_0^ϱ_ eff2π ϱdϱ = x F_ν/πϱ_ eff^2= x L_ mm/π (R_ eff/140)^2≈x 10^-4.3 R_ eff^2/π (R_ eff/140)^2≈0.2 Jyarcsec^-2.The first line of Eq. (<ref>) defines a surface brightness average and employs Eq. (<ref>); the second line uses the definitions of L_ mm (Jy) and R_ eff (AU; see Sect. <ref>) and folds in the approximate relationship inferred in Eq. (<ref>) (assuming the logarithmic slope is exactly 1/2); and the third line substitutes in the relevant numerical values.That average surface brightness corresponds to an average brightness temperature of ⟨ T_b ⟩|_<R_ eff≈ 8 K.The same behavior is apparent regardless of the inferred emission morphologies.Most strikingly, the “transition" disks (around LkHα 330, IRAS 04125+2902, UX Tau A, DM Tau, LkCa 15, GM Aur, MWC 758, CQ Tau, TW Hya, SAO 206462, RX J1604.3-2130, RX J1615.3-3255, SR 24 S, SR 21, WSB 60, DoAr 44, and RX J1633.9-2422) follow the same size–luminosity trend as the “normal" disks.To better quantify that finding, we compared the effective radii with a concentration parameter, defined as the ratio of effective radii for different x (e.g., the ratio of the radii that encircle the first and third quantiles of the total luminosity).But as one might expect, we find no obvious connections between R_ eff or L_ mm and such a concentration metric.[Although it should be obvious that the “transition" disks have comparatively high concentrations at a given R_ eff.] § DISCUSSIONWe have compiled a resolved survey of the 340 GHz (880 μm) continuum emission from 50 nearby protoplanetary disks using new and archival SMA observations.Our primary goal was to validate and better characterize the putative trend between continuum sizes and luminosities first identified by <cit.>, both by expanding the survey size (by a factor of 3) and extending down to intrinsically less luminous disk targets (by a factor of 2).To interpret these data in this context, we modeled the observed visibilities with a simple, flexible surface brightness prescription and used the results to establish a standardized “size" metric.These sizes, based on the radius (R_ eff) that encircles a fixed fraction of the continuum luminosity (L_ mm), are intuitive, empirical quantities that are robust to any intrinsic uncertainties in the slope(s) of the surface brightness profiles. We find a significant correlation between L_ mm and R_ eff, such that the luminosity scales with the effective emitting area.That size–luminosity relation suggests that disks have a roughly constant average 340 GHz surface brightness inside their effective radii, ∼0.2 Jy arcsec^-2 (8 K in terms of a brightness temperature). The shape of that scaling is approximately the same as was identified by <cit.> using a subset of the same data as in this sample, albeit in that case for the peripherally related quantities of disk mass and “characteristic" size.[In some sense, those radii are more similar to ϱ_t than ϱ_ eff, and thereby subject to the same issues identified in Sect. <ref>; see Fig. <ref>.]<cit.> also claimed a size–luminosity trend in their analogous 230 GHz (1.3 mm) sample <cit.>, although they did not quantify it.Using their flux densities and characteristic sizes for a regression analysis as conducted in Section <ref>, we confirm a marginal (∼3 σ) correlation that indeed also has the same slope as inferred for the SMA 340 GHz sample.It is also interesting to note that ∼0.2 dex of (presumed to be Gaussian) dispersion perpendicular to this scaling is required to reproduce the observed scatter of the individual measurements beyond their inferred uncertainties.The regression analysis used to infer that scatter does not provide information on its underlying cause.However, one compelling possibility is that it may be related to the diversity of heating the disks might experience given the wide range of stellar host (irradiation source) properties included in this sample.This stellar heating hypothesis is testable, at least in a crude sense.We can do that by re-scaling the L_ mm inferences by a factor ∝ 1/B_ν(⟨ T_d ⟩), where B_ν is the Planck function and ⟨ T_d ⟩ is a rough estimate of the average dust temperature inside the effective radius.Ideally, we could calculate ⟨ T_d ⟩ weighted by the continuum optical depth <cit.>, but since we have opted for an empirical modeling approach here, such information is not directly accessible.As an approximation, we can presume the emission is optically thin and thereby adopt a weighting function w(r) = I_ν(r)/B_ν[T_d(r)], so that⟨ T_d ⟩ = ∫_0^R_ eff w(r) T_d(r) 2πr dr/∫_0^R_ eff w(r) 2πr dr.Now we need to specify a presumed parametric form for T_d(r) that roughly captures the behavior of irradiation heating by the central star.We base that behavior on the analysis of a suite of radiative transfer models of representative disks by <cit.> and setT_d(r) ≈ T_10(L_∗/L_⊙)^1/4(r/10 AU)^-1/2where T_10 = 30±5 K is the temperature at 10 AU and L_∗ is the stellar host luminosity.We also impose a floor on T_d(r), such that it cannot go below a background level of 7 K (the adopted value makes little difference, so long as it is in a reasonable range, ≲ 12 K).Using this prescription, we collated L_∗ estimates from the literature[The adopted L_∗ values were appropriately scaled to the draws from the distance posteriors.] <cit.> and computed posterior samples of ⟨ T_d ⟩ for each posterior draw of brightness profile parameters, including a representative 20% uncertainty in L_∗ (along with the distance and T_10 uncertainties).Table <ref> includes the derived ⟨ T_d ⟩ and adopted L_∗ values.With this coarse metric of the disk heating in hand, we can then examine the relationship between R_ eff and a quantity ∝ L_ mm/B_ν(⟨ T_d ⟩) that should account for any scatter introduced by dust heating in the analysis of the size–luminosity relation.For the sake of a familiar comparison, we opt to re-cast the scaled luminosity dimension in the context of a disk mass estimate.Again assuming the emission is optically thin, we followed <cit.> to calculate logM_d = log[L_ mm/B_ν(⟨ T_d ⟩)] + 2logd^' - log(ζκ_ν),where d^' = 140 pc (the adopted reference distance for L_ mm), ζ = 0.01 (an assumed dust-to-gas mass ratio), and κ_ν = 3.5 cm^2 g^-1 (a standard 340 GHz dust opacity).Figure <ref> shows the corresponding relationship between the effective radii and these masses.It is immediately apparent that the correlation is stronger and tighter after accounting for ⟨ T_d ⟩, at least partially because much of the perpendicular spread has been redistributed along the trend.We performed the same regression analysis in the {logR_ eff, logM_d}-plane as before and foundlogR_ eff = (2.64±0.07) + (0.47±0.04)×logM_d(where again R_ eff is in AU and M_d is in M_⊙ units; the quoted uncertainties are at the 68% confidence level).The scatter perpendicular to this relation is considerably smaller than in Figure <ref>: it has a (presumed Gaussian) dispersion of ∼0.10 dex, but is consistent with the scatter arising solely from the data uncertainties (at 2.5σ).This demonstrates that the inferred slope of the size–luminosity relation is preserved (and indeed reinforced) by the temperature correction, which in this framework implies a roughly constant average surface density interior to R_ eff, ⟨Σ⟩|_< R_ eff≈ 10 g cm^-2 (following the same logic as in Eq. <ref>).That value can be re-cast into an average optical depth, ⟨τ⟩|_< R_ eff≈ 1/3.We will revisit that optical depth inference below, but it is worth pointing out that these results do not depend much on the choice for the weighting function in Eq. (<ref>).If we instead adopt an empirical w(r) = I_ν(r) or something more appropriate for optically thick emission, w(r) = B_ν(T_d), the inferred slope and scatter of the relationship are not significantly different.[Naturally the intercepts change to reflect the shifts in ⟨ T_d ⟩.] §.§ Potential Origins of a Size–Luminosity Relation While the existence and qualities of a mm continuum size–luminosity relationship are now emerging, it is not particularly obvious why we might expect such behavior.The preliminary speculations on its potential origins can be differentiated into two broad categories: (1) the scaling is a natural consequence of the initial conditions, potentially coupled with some dispersion introduced by evolutionary effects; or (2) the scaling reflects some universal structural configuration.The concepts behind these categories are not always distinct, and certainly not mutually exclusive, but in light of the new data analysis presented here we will explore them separately in the following sections.§.§.§ Initial Conditions, Evolutionary Dispersion An appeal to a specific distribution of initial conditions may seem like a fine-tuning solution, but it cannot be easily dismissed given the strong links expected between young disks and the star formation process.<cit.> noted that the size–luminosity relationship is nearly perpendicular to the direction expected for viscously evolving disks that conserve angular momentum, and because of that they speculated that it may point to the underlying spread in the specific angular momenta in molecular cloud cores <cit.>.A few other possibilities along these lines were raised by <cit.>, including tidal truncation (unlikely for this sample of primarily single stars in low-density clusters; but see ) and significantly broad age and/or viscosity distributions.But neither of these studies considered how those initial conditions in the gas disks would be manifested a few million years later in observational tracers of the solid particles. To assess this category of explanation for the size–luminosity relation, we considered the expected behavior of disks in the {L_ mm, R_ eff}-plane for simple models of the evolution of their solids when embedded in a standard viscously evolving gas disk.We adopted the methodology of <cit.> to roughly estimate how the radial surface density profiles for solids of different sizes evolve with time for a range of initial conditions, parameterized through the total disk masses (M_d), (initial) characteristic radii (R_c), and (fixed) viscosity coefficients (α_t).In these calculations, we assumed a representative 0.5 M_⊙ host star and a fragmentation velocity of 10 m s^-1.For each time and combination of parameters from a coarse grid, we constructed a radial profile of 340 GHz optical depths as the summed (over particle size) product of the computed surface densities and representative opacities (the latter using the prescription of ).We then converted that to an intensity profile with a crude approximation, I_ν (r) = B_ν [T(r)] (1 - e^-τ_ν(r)), using a (fixed) representative temperature profile appropriate for the presumed stellar host (see Eq. <ref>).The luminosities and effective radii corresponding to those intensity profiles were calculated as in Sections <ref> and <ref>. Figure <ref> shows a highly condensed summary of the models compared with the size and luminosity metrics inferred from the data.These models actually do a reasonably good job at reproducing the observed mean trend with a M_d/M_∗ distribution from ∼0.5–10%, a modest range of initial R_c of ∼60–150 AU, and relatively low turbulence levels, α_t ≲ 0.001, but preferentially at early points in the evolutionary sequence (<1 Myr).While individual young star ages are notoriously uncertain <cit.>, the nominal cluster ages from which this sample is drawn are thought to be more like 2–3 Myr.By those times, the model predictions have diverged up and to the left in the {L_ mm, R_ eff}-plane (away from the mean relation, with larger R_ eff than >90% of the sample targets for a given L_ mm) due to both viscous spreading and a relative depletion of particles that emit efficiently in the mm continuum.That timescale discrepancy is not surprising; it is another manifestation of a generic feature of such models that assume a smooth gas disk structure <cit.>.Putting aside the timescale problem, it is interesting that these models tend to predict a shallower slope than what is inferred from the data.This might suggest a correlation between the initial M_d and R_c, perhaps as a signature of the disk formation process.Alternatively, it might indicate a disk mass-dependent evolutionary timescale (perhaps a relationship between M_d and α_t).Unfortunately, it would be difficult to disentangle such effects from the stellar age and mass biases likely present in such an inhomogeneously-selected sample. §.§.§ Small-Scale, Optically Thick Substructures The timescale discrepancy highlighted above may be a telling failure of the underlying assumptions used to set up such models.The remedy often proposed for this inconsistency is the presence of fine-scale, localized maxima in the radial pressure profile of the gas disk <cit.>.Those pressure peaks attract drifting particles, can slow or stop their inward motion, and thereby produce high solid concentrations in small areas that may promote rapid growth to larger bodies.Those concentrations would likely be manifested in the mm continuum as small regions of optically thick emission.Indeed, recent studies have found that narrow rings of bright emission accompanied by darker gaps, among other features, are prevalent in the few disks that have already been observed at very high spatial resolution <cit.>.Such fine-scale optically thick features were considered by <cit.> as a potential contributor to a size–luminosity relation.Having firmly established the character of that relation here, it makes sense to revisit that possibility.In Figure <ref> we illustrate how optically thick emission could mimic the inferred distribution of disks in the {L_ mm, R_ eff}-plane.A fiducial optically thick surface brightness profile was constructed by setting I_ν(r) = B_ν[T_d(r)], where T_d(r) is specified from the approximation in Eq. (<ref>).We then consider a suite of such models where the disk extends out to some sharp cutoff radius, and for each of those profiles calculate a luminosity and effective size as described in Section <ref>.For the sample median stellar luminosity (∼1 L_⊙), this results in the red curve in Figure <ref>.In this context, it makes sense that we do not see disks much to the right of that boundary (since they would all be completely optically thick and therefore cannot produce more emission than allowed by the Planck function).If a given disk has an optically thin contribution or small-scale substructures that reduce the filling factor of optically thick emission below unity, it will lie to the left of this fiducial curve (as illustrated by the blue curve for the case of a 10% filling factor). In this scenario, the inferred distribution of disks in the size–luminosity plane could be explained if disks have optically thick filling fractions (inside a given R_ eff) of a few tens of percent.As one might expect, that is naturally in line with the rough estimate of the mean optical depth (≈1/3) inside R_ eff implied by the inferred size–luminosity relation earlier in this section.Adopting again the IQ Tau disk as an example, we find that its location in the size–luminosity plane could be explained with an optically thick filling factor of ∼10–30% inside 80 AU.Of course, the shift away from the pure optically thick curve for anydisk can be achieved in myriad ways, depending on the detailed spatial distribution of the emission.That said, it is instructive to note that the mean size–luminosity relation can, in principle, be reproduced with a toy model of rings and gaps, as shown by the green curve in Figure <ref>.There, we have intentionally tuned the model to match the inferred relation by simply removing regions of emission from the purely thick model that span ∼few AU segments at regular spacings.The intent is to crudely mimic recent high resolution images of disks.In this case, brighter disks represent systems where the local pressure maxima continue to larger distances from the host star (i.e., brighter disks are just systems that have more large rings).It is worth highlighting that one of the key hints that, for us, makes this explanation more than just plausible is that both “normal" and “transition" disks populate the same size–luminosity trend.The latter systems could be considered an especially simple form of substructure. A robust assessment of the viability of this explanation will require measurements with much higher angular resolution than are yet available.But if such data end up demonstrating that optically thick substructures are important contributors to the continuum emission, the size–luminosity relation could end up being a useful (albeit indirect) demographic metric for understanding their origins in the broader disk population (especially because it is far less observationally expensive to reconstruct the {L_ mm, R_ eff}-plane than to measure fine-scale emission distributions for large samples).If this explanation is relevant, it creates a series of other peripheral issues.For example, if optical depths are high, it means previous assessments of the disk masses might instead just reflect the underlying spatial distribution of substructures.It could imply that the inferences of particle growth from continuum spectral indices are contaminated; indeed, <cit.> already warned of essentially this same scenario.And it might indicate that the interpretation of optically thin spectral line emission from key tracer molecules near the inner disk midplane could be complicated (it could be obscured by the optically thick continuum, depending on its vertical distribution; e.g., ). §.§ Caveats and Future Work One concern that limits our conclusions stems from the inhomogeneous sample of targets available in the SMA archive.This sample is notable for its size, but is composed of disks in the bright half of the mm luminosity distribution, which belong to various clusters (and thereby potentially different ages/environments) and consider a limited range of host masses.It would be beneficial to repeat this analysis for targets with lower continuum luminosities and host masses, and that formed in the same environment at the same time (or nearly so). An investigation for an older cluster (say ∼5 Myr) would be particularly compelling.The evolutionary models described above predict that the size–luminosity relationship should flatten out: older disks have more radially concentrated solids, so smaller particles that tend to emit less overall in the mm continuum, but over more extended distributions, end up dominating the behavior in the {L_ mm, R_ eff}-plane.In the substructure scenario, we might expect little change in the shape of the relation, although effects like growth in the trapped regions could make it difficult to characterize any trend.Another important point to keep in mind is that the relationship we have quantified here is expressly only valid at ∼340 GHz.While <cit.> appear to have tentatively found a similar correlation at ∼230 GHz, we know empirically that the spatial extents of continuum emission from disks tend to increase with the observing frequency <cit.>.The frequency-dependent variation in the size–luminosity relation that this behavior might imply could prove to be crucial in better understanding its origins.In addition to improving the sample quality, complementing the analysis with analogous measurements at a lower (≲100 GHz) frequency (albeit at higher angular resolution, given the observed tendency for more compact emission at lower frequencies) should also be viewed as highly desirable. We thank John Carpenter, Antonella Natta, and an anonymous reviewer for their helpful comments on drafts of this article, as well as Ian Czekala, Mark Gurwell, and Adam Kraus for providing useful advice.This research greatly benefited from the Astropy <cit.> and Matplotlib <cit.> software suites, the emcee package <cit.> and corner module <cit.>.We are grateful for support for some of this work from the NASA Origins of Solar Systems grant NNX12AJ04G.T.B. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 714769.The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.This work has made use of data from the European Space Agency (ESA) mission Gaia (<http://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <http://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.llcc0pt 5 Observations LogTarget UT Date Config. Ref (1) (2) (3) (4)LkHα 330 2006 Nov 19 V1 2010 Nov 3C2 FN Tau2008 Jul 11 E3 IRAS 04125+2902 2011 Aug 19 E4 2011 Sep 8V4 2011 Oct 26 C4 CY Tau2013 Sep 18 E5 2013 Sep 20 E5 2013 Sep 23 E5V410 X-ray 22011 Aug 19 E5 2011 Sep 8V5 2011 Oct 26 C6 IP Tau2014 Aug 28 E5 2014 Sep 16 C5 2014 Oct 23 C5 2014 Oct 30 C5 2015 Jan 19 V5 IQ Tau2014 Aug 28 E52014 Sep 12 C5 2014 Oct 29 C5 2014 Dec 21 V5 IQ ⋮ 2014 ⋮ ⋮ ⋮ Table 1 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.The SMA configurations are S = sub-compact (6–70 m baselines), C = compact (9–75 m baselines), E = extended (17–225 m baselines), and V = very extended (24–509 m baselines).References (Col. 4): 1 = <cit.>, 2 = <cit.>, 3 = <cit.>, 4 = <cit.>, 5 = this paper, 6 = <cit.>, 7 = <cit.>, 8 = <cit.>, 9 = <cit.>, 10 = <cit.>, 11 = <cit.>, 12 = <cit.>, 13 = <cit.>, 14 = <cit.>, 15 = <cit.>, 16 = <cit.>, 17 = <cit.>, 18 = <cit.>, 19 = <cit.>, 20 = <cit.>, 21 = <cit.>.lcccccc0pt 7 Imaging ParametersTarget R.A. (J2000) Decl. (J2000) mean ν beam beam PA RMS[^ h ^ m ^ s] [] [GHz][ × ] [] [mJy beam^-1]LkHα 330 03:45:48.30 +32:24:11.99 340.6 0.37×0.32 822.5 FN Tau04:14:14.59 +28:27:58.00 339.7 0.78×0.70 741.0 IRAS 04125+2902 04:15:42.79 +29:09:59.70 340.9 0.33×0.26 511.3 CY Tau04:17:33.73 +28:20:46.90 337.1 0.75×0.58 104 1.8 V410 X-ray 204:18:34.45 +28:30:30.20 340.9 0.65×0.56 701.1 IP Tau04:24:57.08 +27:11:56.50 340.3 0.83×0.78 781.3 IQ Tau04:29:51.56 +26:06:44.90 340.2 0.50×0.42 641.1 UX Tau A04:30:03.99 +18:13:49.40 340.2 0.39×0.30 551.3 DK Tau A04:30:44.20 +26:01:24.80 341.8 0.43×0.34 691.1 Haro 6-13 04:32:15.41 +24:28:59.80 337.1 0.78×0.58 103 3.0 MHO 6 04:32:22.11 +18:27:42.64 340.2 0.86×0.80 861.4 FY Tau04:32:30.58 +24:19:57.28 340.2 0.64×0.56 791.2 UZ Tau E04:32:43.04 +25:52:31.10 335.8 0.43×0.30 491.5 J04333905+2227207 04:33:39.05 +22:27:20.79 339.7 0.41×0.31 670.7 J04334465+2615005 04:33:44.65 +26:15:00.53 340.3 0.61×0.52 761.2 DM Tau04:33:48.73 +18:10:09.90 341.7 0.69×0.49 331.2 CI Tau04:33:52.00 +22:50:30.20 337.2 0.79×0.58 105 2.3 DN Tau04:35:27.37 +24:14:58.93 340.0 0.60×0.31 931.1 HP Tau04:35:52.78 +22:54:23.11 340.2 0.48×0.40 661.2 DO Tau04:38:28.58 +26:10:49.40 337.1 0.77×0.58 104 2.6 LkCa 15 04:39:17.78 +22:21:03.50 340.2 0.41×0.32 641.0 IRAS 04385+2550 04:41:38.82 +25:56:26.75 340.2 0.59×0.48 741.1 GO Tau04:43:03.09 +25:20:18.75 340.2 0.56×0.44 771.2 GM Aur04:55:10.98 +30:21:59.40 342.3 0.32×0.27 383.8 SU Aur04:55:59.38 +30:34:01.50 341.8 1.06×0.85 861.6 V836 Tau05:03:06.60 +25:23:19.60 341.4 0.44×0.36 451.1 MWC 758 05:30:27.53 +25:19:57.10 340.2 0.86×0.58 911.8 CQ Tau05:35:58.47 +24:44:54.10 339.8 0.94×0.47 922.7 TW Hya11:01:51.87 -34:42:17.11 336.1 0.81×0.59 177 2.8 SAO 20646215:15:48.40 -37:09:16.00 339.7 0.53×0.28 133.8 IM Lup15:56:09.20 -37:56:06.20 340.6 1.24×1.00 170 2.8 RX J1604.3-2130 16:04:21.70 -21:30:28.40 339.8 0.47×0.29 291.1 RX J1615.3-3255 16:15:20.20 -32:55:05.10 341.3 0.53×0.24 173.0 SR 416:25:56.17 -24:20:48.40 339.9 0.50×0.43 152.5 Elias 2016:26:18.90 -24:28:19.80 340.2 0.44×0.29 261.7 DoAr 25 16:26:23.60 -24:43:13.20 346.9 0.49×0.37 143.0 Elias 2416:26:24.07 -24:16:13.50 340.0 0.97×0.79 233.5 GSS 3916:26:45.03 -24:23:07.74 340.0 0.71×0.66 138 3.0 WL 18 16:26:48.98 -24:38:25.20 340.0 0.77×0.66 542.5 SR 24 S 16:26:58.51 -24:45:37.00 340.0 0.38×0.28 9 3.0 SR 21 16:27:10.30 -24:19:12.00 340.0 0.41×0.29 192.5 DoAr 33 16:27:39.02 -23:58:18.70 339.9 0.52×0.43 672.3 WSB 5216:27:39.43 -24:39:15.50 339.9 0.48×0.38 312.3 WSB 6016:28:16.50 -24:37:58.00 339.9 0.63×0.41 242.5 SR 13 16:28:45.27 -24:28:19.00 340.0 0.63×0.52 373.0 DoAr 44 16:31:33.50 -24:28:37.30 339.9 0.63×0.41 242.6 RX J1633.9-2422 16:33:55.60 -24:42:05.00 334.9 0.37×0.29 8 1.3 WaOph 6 16:48:45.63 -14:16:36.00 345.2 0.54×0.40 342.8 AS 20916:49:15.30 -14:22:08.57 349.1 0.61×0.46 333.3 HD 163296 17:56:21.30 -21:57:22.20 341.0 1.71×1.27 4710lccccccc|c10 Surface Brightness Profile ParametersName F_ν [Jy] ϱ_t [] logα β γ i [] φ []ϱ_ eff []LkHα 330 0.208 ^+0.004_-0.0020.38 ^+0.45_-0.260.83 ^+0.90_-0.10 5.2 ^+2.7_-0.3-1.7 ^+0.3_-1.039 ^+3_-5 76 ^+7_-5 0.44 ^+0.03_-0.01FN Tau0.044 ^+0.002_-0.002<0.32p(logα); ↓ >3.5p(γ); ↑p(i) p(φ)<0.17IRAS 04125+2902 0.042 ^+0.001_-0.0010.27 ^+0.03_-0.02>0.74 6.8 ^+2.4_-1.0-2.5 ^+1.1_-0.326 ^+7_-13 155 ^+41_-260.30 ^+0.02_-0.01CY Tau0.207 ^+0.012_-0.0050.26 ^+0.16_-0.01p(logα); ↑4.3 ^+3.7_-0.20.3 ^+0.1_-2.6 27 ^+3_-4 136 ^+8_-80.36 ^+0.02_-0.02V410 X-ray 20.026 ^+0.002_-0.0010.14 ^+0.38_-0.04p(logα); ↓ p(β); ↓p(γ); ↑62 ^+10_-27 106 ^+19_-200.16 ^+0.05_-0.02IP Tau0.030 ^+0.002_-0.0010.25 ^+0.09_-0.04p(logα); ↑ >3.0-1.2 ^+1.4_-1.341 ^+12_-20 40 ^+93_-17 0.27 ^+0.05_-0.04IQ Tau0.163 ^+0.004_-0.0020.52 ^+0.39_-0.040.40 ^+0.57_-0.134.4 ^+4.5_-0.10.6 ^+0.1_-0.5 60 ^+1_-1 43 ^+2_-2 0.55 ^+0.01_-0.02UX Tau A0.147 ^+0.001_-0.0020.27 ^+0.01_-0.01>1.25 >8.0-2.9 ^+0.3_-0.439 ^+2_-2 167 ^+3_-20.28 ^+0.01_-0.01DK Tau A0.071 ^+0.004_-0.0060.08 ^+0.11_-0.01p(logα); ↓2.6 ^+2.0_-0.10.9 ^+0.1_-2.8 26 ^+7_-12 106 ^+24_-320.32 ^+0.15_-0.02Haro 6-13 0.441 ^+0.069_-0.0220.06 ^+0.08_-0.01p(logα); ↑ 2.3 ^+0.3_-0.1p(γ); ↑42 ^+3_-3 145 ^+6_-60.48 ^+0.43_-0.05MHO 6 0.047 ^+0.002_-0.0020.23 ^+0.17_-0.05 p(logα); ↓ p(β); ↑p(γ); ↑78 ^+7_-14 110 ^+11_-80.26 ^+0.06_-0.04FY Tau0.023 ^+0.002_-0.001<0.50p(logα); ↓ p(β); ↑p(γ); ↑>25 134 ^+20_-620.07 ^+0.04_-0.04UZ Tau E0.401 ^+0.008_-0.0070.55 ^+0.01_-0.020.83 ^+0.13_-0.08 3.3 ^+0.2_-0.10.6 ^+0.1_-0.1 55 ^+1_-1 90 ^+1_-1 0.80 ^+0.03_-0.02J04333905+2227207 0.076 ^+0.002_-0.0010.58 ^+0.14_-0.040.66 ^+0.83_-0.09 p(β); ↑0.3 ^+0.1_-0.379 ^+1_-1 115 ^+1_-20.56 ^+0.02_-0.02J04334465+2615005 0.043 ^+0.002_-0.0020.46 ^+0.20_-0.15p(logα); ↓ >4.11.2 ^+0.1_-1.272 ^+11_-17 158 ^+5_-7 0.34 ^+0.04_-0.05DM Tau0.254 ^+0.014_-0.0080.15 ^+0.07_-0.04<0.313.0 ^+0.3_-0.3-1.4 ^+0.6_-1.234 ^+2_-2 158 ^+5_-50.92 ^+0.15_-0.06CI Tau0.440 ^+0.050_-0.0141.45 ^+0.69_-0.300.38 ^+0.12_-0.20>4.10.8 ^+0.1_-0.2 44 ^+1_-311 ^+3_-2 0.89 ^+0.13_-0.04DN Tau0.179 ^+0.005_-0.0030.35 ^+0.29_-0.02p(logα); ↓p(β); ↓0.8 ^+0.1_-0.8 28 ^+2_-8 80 ^+11_-9 0.38 ^+0.02_-0.01HP Tau0.106 ^+0.001_-0.0020.44 ^+0.05_-0.04p(logα); ↑ >3.81.6 ^+0.1_-0.164 ^+2_-3 70 ^+3_-3 0.22 ^+0.02_-0.01DO Tau0.250 ^+0.012_-0.0030.12 ^+0.28_-0.01p(logα); ↓ 3.7^+4.8_-0.7 p(γ); ↑37 ^+4_-7 159 ^+7_-10 0.19 ^+0.01_-0.01LkCa 15 0.396 ^+0.005_-0.0040.62 ^+0.05_-0.020.62 ^+0.08_-0.10 5.3 ^+1.0_-0.3-1.6 ^+0.2_-0.451 ^+1_-1 62 ^+1_-1 0.77 ^+0.01_-0.01IRAS 04385+2550 0.061 ^+0.004_-0.003p(ϱ_t); ↓p(logα); ↓ p(β)>1.4p(i) 148 ^+6_-19 0.10 ^+0.08_-0.01GO Tau0.179 ^+0.019_-0.0070.89 ^+2.73_-0.050.20 ^+1.05_-0.08 p(β); ↓1.1 ^+0.1_-0.1 53 ^+2_-3 26 ^+4_-3 1.16 ^+0.23_-0.07GM Aur0.632 ^+0.011_-0.0080.92 ^+0.52_-0.27<0.26 6.7 ^+2.3_-1.1-1.1 ^+0.4_-0.755 ^+1_-1 64 ^+1_-1 0.87 ^+0.03_-0.02SU Aur0.052 ^+0.002_-0.001<0.27p(logα); ↓ >3.5p(γ); ↑p(i) 79 ^+57_-42 <0.16V836 Tau0.061 ^+0.002_-0.0020.13 ^+0.05_-0.01p(logα); ↑ >4.7p(γ)61 ^+11_-8 137 ^+9_-60.13 ^+0.02_-0.01MWC 758 0.177 ^+0.002_-0.0010.50 ^+0.01_-0.01>1.3 >7.9-3.0 ^+0.4_-0.440 ^+1_-1 168 ^+1_-10.53 ^+0.01_-0.01CQ Tau0.444 ^+0.013_-0.0090.34 ^+0.03_-0.01>0.87 >6.6-2.3 ^+1.1_-0.536 ^+3_-3 53 ^+6_-6 0.36 ^+0.01_-0.02TW Hya1.311 ^+0.003_-0.0020.98 ^+0.01_-0.020.95 ^+0.05_-0.03 >8.60.6 ^+0.1_-0.1 7 ^+1_-1 155 ^+1_-10.77 ^+0.01_-0.01SAO 2064620.596 ^+0.024_-0.0220.57 ^+0.02_-0.040.88 ^+0.59_-0.08 >5.9-1.0 ^+0.1_-0.727 ^+3_-4 33 ^+7_-9 0.56 ^+0.02_-0.01IM Lup0.587 ^+0.007_-0.0062.25 ^+0.48_-0.420.32 ^+0.12_-0.07 >5.50.9 ^+0.1_-0.1 49 ^+1_-2 142 ^+2_-31.11 ^+0.04_-0.02RX J1604.3-2130 0.194 ^+0.009_-0.0060.63 ^+0.01_-0.01>1.2 5.8 ^+0.5_-0.4-2.5 ^+0.3_-0.36 ^+1_-1 77 ^+1_-1 0.72 ^+0.01_-0.02RX J1615.3-3255 0.434 ^+0.005_-0.0031.39 ^+0.24_-0.51<0.31 >5.1-0.1 ^+0.3_-0.344 ^+2_-2 149 ^+3_-30.73 ^+0.02_-0.02SR 40.148 ^+0.003_-0.0030.19 ^+0.05_-0.01>0.45>5.2-0.9 ^+0.9_-1.543 ^+3_-6 27 ^+8_-6 0.21 ^+0.01_-0.01 Elias 200.264 ^+0.006_-0.0040.41 ^+0.05_-0.05>0.43 7.0 ^+2.1_-1.51.0 ^+0.1_-0.2 54 ^+2_-2 162 ^+2_-20.32 ^+0.02_-0.01DoAr 25 0.565 ^+0.008_-0.0081.18 ^+0.09_-0.180.58 ^+0.14_-0.07>5.50.5 ^+0.1_-0.1 63 ^+1_-1109 ^+1_-10.85 ^+0.02_-0.02Elias 240.912 ^+0.016_-0.0110.82 ^+0.09_-0.03>0.76 5.9 ^+2.2_-0.51.1 ^+0.1_-0.1 21 ^+3_-5 62 ^+13_-11 0.68 ^+0.01_-0.02GSS 390.654 ^+0.012_-0.0101.61 ^+0.05_-0.05>1.04 >7.21.1 ^+0.1_-0.1 55 ^+1_-1 120 ^+1_-11.19 ^+0.02_-0.09WL 18 0.051 ^+0.004_-0.0020.13 ^+0.08_-0.03p(logα); ↑ >3.8p(γ)60 ^+17_-29 68 ^+54_-23 0.13 ^+0.04_-0.02SR 24 S 0.509 ^+0.006_-0.0060.43 ^+0.05_-0.020.65 ^+0.87_-0.08 6.2 ^+2.9_-0.4-0.4 ^+0.1_-1.146 ^+2_-223 ^+2_-2 0.45 ^+0.08_-0.08SR 21 0.405 ^+0.003_-0.0030.44 ^+0.02_-0.02>0.95 >6.9-0.9 ^+0.2_-0.618 ^+5_-9 10 ^+45_-49 0.43 ^+0.02_-0.01DoAr 33 0.070 ^+0.008_-0.0040.19 ^+0.29_-0.04p(logα); ↓ p(β); ↑p(γ); ↑68 ^+11_-27 76 ^+21_-19 0.21 ^+0.13_-0.03WSB 520.154 ^+0.016_-0.0050.93 ^+1.26_-0.24p(logα); ↓ >3.11.6 ^+0.1_-0.3 47 ^+9_-24 165 ^+35_-260.42 ^+0.12_-0.06WSB 600.258 ^+0.015_-0.0060.26 ^+0.01_-0.010.37 ^+1.25_-0.02 3.5 ^+3.2_-0.2-0.3 ^+0.1_-2.132 ^+4_-12 121 ^+17_-170.34 ^+0.03_-0.02SR 13 0.153 ^+0.004_-0.0030.31 ^+0.03_-0.03>0.67 >5.1-2.4 ^1.3_-0.4 54 ^+4_-6 72 ^+6_-7 0.32 ^+0.03_-0.01DoAr 44 0.211 ^+0.006_-0.0050.38 ^+0.05_-0.030.69 ^+0.87_-0.06 5.6 ^+3.3_-0.4-1.8 ^+0.7_-0.816 ^+10_-6 55 ^+92_-27 0.42 ^+0.02_-0.01RX J1633.9-2422 0.225 ^+0.011_-0.0050.34 ^+0.01_-0.030.97 ^+0.80_-0.07 6.7 ^+2.3_-0.7-1.5 ^+0.2_-0.949 ^+1_-1 83 ^+2_-1 0.35 ^+0.01_-0.01WaOph 6 0.445 ^+0.024_-0.0220.47 ^+0.09_-0.03>0.33 2.9 ^+0.5_-0.21.1 ^+0.1_-0.1 41 ^+4_-3 173 ^+10_-3 0.68 ^+0.15_-0.05AS 2090.604 ^+0.007_-0.0101.65 ^+0.13_-0.50<0.16 >6.4-0.2 ^+0.3_-0.131 ^+3_-5 73 ^+6_-9 0.70 ^+0.02_-0.02HD 163296 1.822 ^+0.004_-0.0061.60 ^+0.04_-0.070.54 ^+0.04_-0.04 >8.60.8 ^+0.1_-0.1 47 ^+1_-1 133 ^+1_-10.96 ^+0.01_-0.01Quoted values are the peaks of the posterior distributions, with uncertainties that represent the 68% confidence interval.Limits represent the 95% confidence boundary.A note of “p(X)" means the posterior is identical to the prior at <68% confidence; “p(X); ↓"or “p(X); ↑" means the posterior is consistent with the prior at 95% confidence, but has a marginal preference toward the lower or upper bound of the prior on X, respectively.Note that the F_ν posterior does not include systematic flux calibration uncertainties (see Sect. <ref>) and that the ϱ_ eff posterior is not directly inferred, but rather constructed from the joint posterior on {F_ν, ϱ_t, logα, β, γ}.lcc|cccc7 Size–Luminosity Relation and Associated ParametersNamelogR_ eff/ AUlogL_ mm/ Jyϖ [mas]d [pc]L_∗ [L_⊙]⟨ T_d ⟩ [K]LkHα 330 2.13 ^+0.09_-0.06 -0.01 ^+0.17_-0.103.38±0.50 297 ^+72_-2919.6 ^+14.5_-3.7 30 ^+7_-5FN Tau<1.43 -1.40 ^+0.12_-0.09 7.62±0.86 131 ^+22_-100.65 ^+0.32_-0.13 30 ^+35_-4IRAS 04125+2902 1.60 ^+0.07_-0.05 -1.42 ^+0.12_-0.087.62±0.86 131 ^+22_-100.25 ^+0.12_-0.05 12 ^+3_-2 CY Tau1.69 ^+0.06_-0.05 -0.72 ^+0.12_-0.097.62±0.86 131 ^+22_-110.32 ^+0.15_-0.07 12 ^+3_-2 V410 X-ray 21.35 ^+0.11_-0.10 -1.62 ^+0.12_-0.09 7.62±0.86 132 ^+21_-112.09 ^+1.07_-0.40 34 ^+14_-6 IP Tau1.56 ^+0.10_-0.08 -1.56 ^+0.15_-0.10 7.63±1.01 131 ^+27_-120.37 ^+0.21_-0.08 15 ^+3_-3 IQ Tau1.90 ^+0.08_-0.05 -0.76 ^+0.16_-0.10 7.07±0.98 143 ^+30_-15 0.73 ^+0.48_-0.14 14 ^+3_-2UX Tau A1.65 ^+0.03_-0.03 -0.73 ^+0.06_-0.05 6.33±0.40 158 ^+12_-82.18 ^+0.66_-0.42 21 ^+3_-4DK Tau A1.71 ^+0.15_-0.09 -1.13 ^+0.16_-0.10 7.07±0.98 142 ^+31_-141.19 ^+0.77_-0.24 22 ^+6_-4 Haro 6-13 1.97 ^+0.21_-0.15 -0.26 ^+0.22_-0.13 6.79±1.19 147 ^+50_-150.66 ^+0.65_-0.12 17 ^+5_-4 MHO 6 1.64 ^+0.23_-0.10 -1.20 ^+0.42_-0.15 6.55±1.58 153 ^+110_-160.08 ^+0.16_-0.02 11 ^+5_-2FY Tau1.11 ^+0.14_-0.41-1.56 ^+0.23_-0.12 6.79±1.19 148 ^+49_-161.55 ^+1.50_-0.30 44 ^+36_-7 UZ Tau E2.06 ^+0.09_-0.05 -0.36 ^+0.16_-0.11 7.07±1.02 141 ^+34_-130.83 ^+0.58_-0.16 12 ^+3_-2J04333905+2227207 1.95 ^+0.15_-0.07 -0.99 ^+0.29_-0.15 6.55±1.36 153 ^+73_-170.07 ^+0.10_-0.01 8 ^+1_-1 J04334465+2615005 1.69 ^+0.10_-0.09 -1.33 ^+0.12_-0.16 7.07±1.02 141 ^+34_-120.24 ^+0.17_-0.04 14 ^+4_-3 DM Tau2.21 ^+0.22_-0.10 -0.44 ^+0.43_-0.17 6.54±1.62 153 ^+121_-160.26 ^+0.55_-0.07 9 ^+4_-1CI Tau2.18 ^+0.26_-0.08 -0.22 ^+0.35_-0.14 6.54±1.45 155 ^+81_-180.93 ^+1.66_-0.15 12 ^+4_-2 DN Tau1.81 ^+0.12_-0.07 -0.59 ^+0.23_-0.24 6.25±1.16 160 ^+60_-170.87 ^+0.99_-0.15 18 ^+5_-3 HP Tau1.56 ^+0.02_-0.03 -0.85 ^+0.01_-0.02 6.20±0.10 161 ^+3_-32.43 ^+0.49_-0.49 40 ^+7_-7 DO Tau1.46 ^+0.11_-0.07 -0.52 ^+0.21_-0.12 6.79±1.16 147 ^+47_-151.33 ^+1.25_-0.25 28 ^+10_-5 LkCa 15 2.12 ^+0.23_-0.08 -0.23 ^+0.46_-0.15 6.28±1.58 161 ^+130_-180.95 ^+2.33_-0.20 12 ^+4_-2IRAS 04385+2550 1.31 ^+0.22_-0.12 -1.06 ^+0.25_-0.13 6.25±1.16 161 ^+59_-180.42 ^+0.47_-0.08 42 ^+15_-9 GO Tau2.30 ^+0.15_-0.07 -0.59 ^+0.25_-0.13 6.25±1.16 160 ^+60_-170.32 ^+0.36_-0.06 8 ^+3_-1 GM Aur2.07 ^+0.08_-0.05 -0.25 ^+0.15_-0.09 7.64±1.02 131 ^+27_-120.96 ^+0.58_-0.20 12 ^+2_-3 SU Aur<1.43 -1.26 ^+0.10_-0.07 7.02±0.67 142 ^+19_-109.92 ^+4.18_-1.93 62 ^+67_-9 V836 Tau1.24 ^+0.06_-0.05 -1.30 ^+0.10_-0.07 7.93±0.72 126 ^+15_-90.41 ^+0.17_-0.08 21 ^+6_-3MWC 758 1.90 ^+0.04_-0.03 -0.69 ^+0.08_-0.05 6.63±0.47150 ^+14_-88.52 ^+2.49_-1.84 21 ^+4_-4CQ Tau1.76 ^+0.04_-0.03 -0.24 ^+0.07_-0.05 6.25±0.40159 ^+13_-816.7 ^+4.8_-3.4 31 ^+6_-4TW Hya1.66 ^+0.01_-0.01 -0.63 ^+0.02_-0.02 16.80±0.40 59 ^+2_-20.33 ^+0.08_-0.06 9 ^+1_-1 SAO 2064621.95 ^+0.04_-0.04 -0.12 ^+0.08_-0.07 6.41±0.54156 ^+17_-109.62 ^+3.24_-2.09 22 ^+4_-4 IM Lup2.26 ^+0.02_-0.03 -0.11 ^+0.08_-0.06 6.20±0.47161 ^+16_-100.90 ^+0.28_-0.19 9 ^+4_-1 RX J1604.3-2130 2.03 ^+0.08_-0.05 -0.67 ^+0.17_-0.10 6.90±0.97144 ^+34_-130.55 ^+0.36_-0.11 9 ^+2_-1 RX J1615.3-3255 2.03 ^+0.11_-0.06 -0.33 ^+0.23_-0.12 7.14±1.26140 ^+48_-140.68 ^+0.70_-0.12 12 ^+3_-2 SR 41.45 ^+0.03_-0.03 -0.85 ^+0.01_-0.01 7.28±0.06 137 ^+1_-11.22 ^+0.33_-0.25 22 ^+4_-4 Elias 201.65 ^+0.02_-0.02 -0.59 ^+0.01_-0.01 7.28±0.06 137 ^+1_-12.31 ^+0.44_-0.48 25 ^+4_-5 DoAr 25 2.07 ^+0.01_-0.01 -0.26 ^+0.01_-0.01 7.28±0.06 137 ^+1_-10.96 ^+0.20_-0.18 11 ^+3_-2 Elias 241.97 ^+0.01_-0.01 -0.06 ^+0.01_-0.01 7.28±0.06 137 ^+1_-16.16 ^+1.24_-1.23 22 ^+5_-3GSS 392.21 ^+0.01_-0.01 -0.20 ^+0.01_-0.01 7.28±0.06 137 ^+1_-11.21 ^+0.34_-0.24 11 ^+2_-2 WL 18 1.29 ^+0.07_-0.12 -1.30 ^+0.03_-0.03 7.28±0.06 137 ^+1_-10.37 ^+0.07_-0.08 20 ^+8_-3 SR 24 S 1.79 ^+0.01_-0.01 -0.31 ^+0.01_-0.01 7.28±0.06 137 ^+1_-15.26 ^+1.12_-1.01 22 ^+4_-4 SR 21 1.78 ^+0.02_-0.01 -0.41 ^+0.01_-0.01 7.28±0.06 137 ^+1_-113.3 ^+2.6_-2.7 27 ^+5_-5 DoAr 33 1.48 ^+0.19_-0.07 -1.17 ^+0.05_-0.03 7.28±0.12 137 ^+1_-11.45 ^+0.30_-0.29 23 ^+7_-4 WSB 521.77 ^+0.10_-0.08 -0.82 ^+0.04_-0.02 7.28±0.06 137 ^+1_-10.72 ^+0.15_-0.14 20 ^+4_-4 WSB 601.67 ^+0.04_-0.03 -0.60 ^+0.02_-0.02 7.28±0.06 137 ^+1_-10.24 ^+0.05_-0.05 12 ^+2_-2 SR 13 1.67 ^+0.02_-0.04 -0.83 ^+0.01_-0.01 7.28±0.06 137 ^+1_-10.48 ^+0.10_-0.09 13 ^+3_-2 DoAr 44 1.79 ^+0.03_-0.02 -0.63 ^+0.03_-0.02 6.79±0.16 147 ^+4_-31.81 ^+0.37_-0.37 17 ^+3_-3 RX J1633.9-2422 1.72 ^+0.01_-0.02 -0.60 ^+0.03_-0.02 6.79±0.16 147 ^+4_-31.04 ^+0.24_-0.20 16 ^+3_-3 WaOph 6 1.99 ^+0.13_-0.08 -0.41 ^+0.22_-0.12 7.87±1.37 128 ^+41_-142.79 ^+2.68_-0.54 20 ^+5_-4 AS 2091.96 ^+0.07_-0.04 -0.30 ^+0.13_-0.08 7.87±0.91 127 ^+22_-101.49 ^+0.75_-0.30 15 ^+3_-3 HD 163296 2.06 ^+0.04_-0.04 0.13 ^+0.09_-0.08 8.43±0.78118 ^+15_-833.1 ^+13.0_-6.7 28 ^+6_-5 Quoted values are the peaks of the posterior distributions, with uncertainties that represent the 68% confidence interval.Inferences of R_ eff, L_ mm, ϖ, and d are described in Sect. <ref>; L_∗ and ⟨ T_d ⟩ are defined in Sect. <ref> (the latter in Eq. <ref>). | http://arxiv.org/abs/1706.08977v1 | {
"authors": [
"Anjali Tripathi",
"Sean M. Andrews",
"Tilman Birnstiel",
"David J. Wilner"
],
"categories": [
"astro-ph.EP",
"astro-ph.SR"
],
"primary_category": "astro-ph.EP",
"published": "20170627180006",
"title": "A Millimeter Continuum Size-Luminosity Relationship for Protoplanetary Disks"
} |
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA Finite-range interacting spin models are the simplest models to study the effect of beyond nearest-neighbour interactions and access new effects caused by the range of the interactions. Recent experiments have reached the regime of dominant interactions in Ising quantum magnets via optical coupling of trapped neutral atoms to Rydberg states. This approach allows for the tunability of all relevant terms in an Ising Hamiltonian with 1/r^6 interactions in a transverse and longitudinal field. This review summarizes the recent progress of these implementations in Rydberg lattices with site-resolved detection. The strong correlations in this quantum Ising model have been observed in several experiments up to the point of crystallization. In systems with a diameter small compared to the Rydberg blockade radius, the number of excitations is maximally one in the so-called superatom regime. Finite-range interacting Ising quantum magnets with Rydberg atoms in optical lattices - From Rydberg superatoms to crystallization Peter Schauss December 30, 2023 ================================================================================================================================== § INTRODUCTIONSpin Hamiltonians are often introduced as tractable simplifications of realistic condensed matter Hamiltonians <cit.>. Quantum simulation of these spin Hamiltonians can be performed by ultracold atoms in optical lattices <cit.>. Heisenberg-type spin Hamiltonians have been implemented in optical lattices by second order tunnelling, the so-called superexchange process <cit.>. In fermionic systems superexchangeleads to antiferromagnetic correlations <cit.>. Progress in site-resolved imaging of ultracold bosonic atoms in optical lattices has enabled the study of spin Hamiltonians with single spin resolution in a very controlled setting <cit.> and has recently been extended to fermions allowing to study antiferromagnetic correlations in the repulsive Hubbard model <cit.>. The main advantage of these single-site resolution techniques is their single-particle sensitivity in density-snapshots of the full many-body wavefunction and therefore the direct access to spatial correlation functions <cit.>. Typically the interactions in these systems are limited to nearest neighbour interactions. Spin systems with power-law interactions are predicted to show new physics not observable in systems with at most nearest neighbour interactions <cit.>. Examples include spin glasses <cit.>, quantum crystals <cit.> and modified light cone dynamics <cit.>. In order to investigate longer range interacting spin models new experimental techniques are required. One direction is to employ dipolar molecules in optical lattices <cit.> or magnetic atoms <cit.>. Long range interactions have also been implemented with ions in one-dimensional systems <cit.> and extended to two dimensions <cit.>. Another route is to exploit the power-law interactions between Rydberg atoms <cit.>. While molecules and magnetic atoms are expected to have a very long lifetime in a suitably designed trapping configuration, Rydberg atoms are relatively short-lived but have much stronger interactions. Currently the lifetime to interaction timescale ratio ends up in a comparable range and none of these systems has shown to be superior yet <cit.>. However, long-range interacting systems based on trapped ions currently show the longest coherence times where analog simulation is possible <cit.>. This review focuses on the approach to use Rydberg atoms to implement quantum Ising systems with beyond nearest neighbour interactions in lattice configurations. Rydberg atoms are atoms with at least one electron in a highly excited state and exhibit surprisingly long lifetimes of typically few ten microseconds to few milliseconds due to the low overlap of the excited electronic state with the ground state <cit.>. Their large electronic wavefunction leads to stronginduced dipole-dipole interactions. These van der Waals interactions typically show a 1/r^6 dependence with distance r in the absence of special resonances. Rydberg states exist for all atoms and therefore constitute a general concept to generate long-distance interactions that is applicable for all atoms. This review is organized as follows. It starts with a short introduction to the mapping of the naturally arising Rydberg Hamiltonian to a spin Hamiltonian and then discusses the experimental implementation.Thereafter, we discuss systems in the regime of strong interactions, where the interaction range exceeds the system size. As a consequence, the system can only host a single excitation and is well described by a so-called “superatom". The following part focuses on larger systems where correlations and crystallization have been observed. In the last part future directions are discussed.It is not intended to give a general introduction to Rydberg physics here and the reader is referred to other reviews on this topic <cit.>. § FROM THE RYDBERG HAMILTONIAN TO THE ISING MODEL Here we consider a setup where many ground state atoms are coupled with a single laser field to a Rydberg state (Fig. <ref>). When coupling atoms with a laser near-resonantly with Rabi frequency Ω, the experimental timescales are maximally on the order of the Rydberg atom lifetime τ. This leads for ultracold atoms to the observation that the motion of the atoms is in most cases negligible for the excitation dynamics. The corresponding theoretical model describing this system is the so-called “frozen Rydberg gas", in which only the internal electronic degrees of freedom are considered and all atomic motion is neglected <cit.>. A very important concept for the understanding of these systems is the so-called dipole blockade <cit.>. The interactions between Rydberg atoms lead to a relevant resonance shift of the Rydberg excitation line in ground state atoms close to a Rydberg atom. This interaction-induced detuning of the transition frequency suppresses the excitation of Rydberg atoms at small distances and equating the interaction potential, V(r), with the excitation bandwidth gives an estimate for the length scale of this effect. For resonant excitation the bandwidth is typically given by the Rabi frequency and the blockade radius R_b is therefore defined by V(R_b) = ħΩ. The blockade constraint in the relative distance between Rydberg excitations by itself leads to strongly correlated many-body states <cit.>. In this review, we focus on systems where single atoms, arranged on a square lattice with positions i⃗, in a ground state |g_i⃗⟩ are coupled to a Rydberg state |e_i⃗⟩. We assume homogeneous Rabi frequency Ω(t) and detuning Δ(t)=ω_l(t)-ω_0 of the coupling for all atoms, both considered time-dependent. The lattice settingcomes with the inherent advantage that molecular loss between atoms can be suppressed by choosing a Rydberg state that has no molecular lines in the range of Δ(t) of interest and beyond the minimum spacing between two atoms, the lattice constant<cit.>.For the basic model no detailed knowledge about Rydberg atoms is required, as the main property of interest here is the interaction between two Rydberg atoms in the same state which can be approximated by a van-der-Waals potential of type V(r) = -C_6/r^6 for distances r ≥ and large compared to the electronic wavefunction. The interaction potential can become more complicated in the presence of Förster resonances which are not discussed here <cit.>. More details on Rydberg atoms can be found in the reviews <cit.>. We note that the interactions between Rydberg and ground state atoms are not relevant here under the assumption that the Rydberg state is chosen as not to overlap with ground state atoms on neighbouring sites. Ground state atoms would interact only if two of them were on the same site.The Hamiltonian describing the system is given in rotating wave approximation by Ĥ = ħΩ(t)/2∑_i⃗( σ̂^(i⃗)_eg + σ̂^(i⃗)_ge) -ħΔ(t) ∑_i⃗σ̂^(i⃗)_ee + ∑_i⃗≠j⃗V_i⃗j⃗/2σ̂^(i⃗)_eeσ̂^(j⃗)_ee.Here, the vectors i⃗ = (i_x, i_y) label the lattice sites in the plane. The first term in this Hamiltonian describes the coherent coupling of the ground and excited states withRabi frequency Ω(t), with σ̂^(i⃗)_ge = |e_i⃗⟩⟨g_i⃗| and σ̂^(i⃗)_eg = |g_i⃗⟩⟨e_i⃗|. The second term takes into account the effect of the laser detuning Δ(t). The projection operator σ̂^(i⃗)_ee = |e_i⃗⟩⟨e_i⃗| measures the population of the Rydberg state at site i⃗. The third term is the interaction potential between two atoms in the Rydberg state V_i⃗j⃗ = -C_6/r_i⃗j⃗^6, with van der Waals coefficient C_6 and r_i⃗j⃗ =|i⃗ - j⃗| the distance between the two atoms at sites i⃗ and j⃗. By identifying |g_i⃗⟩ = |↓_i⃗⟩ and |e_i⃗⟩ = |↑_i⃗⟩ the Hamiltonian can be rewritten as a spin Hamiltonian for which we introduce spin-1/2 operators on each site as follows (Fig. <ref>): We define Ŝ^(i⃗)_x = (|↑_i⃗⟩⟨↓_i⃗| + |↓_i⃗⟩⟨↑_i⃗|)/2 and Ŝ^(i⃗)_z = (|↑_i⃗⟩⟨↑_i⃗| - |↓_i⃗⟩⟨↓_i⃗|)/2 and note that σ̂^(i⃗)_ee = 1/2+Ŝ^(i⃗)_z with 1 being the identity. The operators |↑⟩⟨↓| and |↓⟩⟨↑| describe a spin flip from the ground state |↓⟩ to the Rydberg state |↑⟩ and vice versa, while the operators |↑⟩⟨↑| = n̂_↑ and |↓⟩⟨↓| = n̂_↓ represent the local Rydberg and ground state population, respectively. Using these expressions and neglecting a constant offset, the Hamiltonian becomes: Ĥ = ħΩ(t) ∑_i⃗Ŝ^(i⃗)_x + ∑_i⃗ [ℐ_i⃗ - ħΔ(t)] Ŝ^(i⃗)_z + ∑_i⃗≠j⃗V_i⃗j⃗/2Ŝ^(i⃗)_zŜ^(j⃗)_z.The first two terms of this spin Hamiltonian describe a transverse and longitudinal magnetic field. The former is controlled by the coherent coupling between the ground and the Rydberg state with the time-dependent Rabi frequency Ω(t). The detuning Δ(t) determines the longitudinal field and can be used to counteract the energy offset ℐ_i = ∑_j⃗,(i⃗≠j⃗)V_i⃗j⃗/2. In finite systems, ℐ_i has a spatial dependence with negligible consequences besides favouring the pinning of Rydberg excitations at the edge of the system. In infinite systems, ℐ_i just leads to a constant offset of the detuning. This spin Hamiltonian has a dynamic aspect in the sense that it requires the presence of driving to a Rydberg state which inherently limits its study to timescales shorter than the Rydberg lifetime. But the interactions can be very strong, and therefore the time evolution fast enough that the properties of the system can be explored before the decay processes become relevant. Experimentally the strength of the interactions can easily exceed the linewidth of the Rydberg excitation line and the maximum achievable Rabi frequency. Therefore the relevant figure of merit for the observation of coherent dynamics is the product of system lifetime τ and Ω. Values of Ωτ > 100 are experimentally feasible <cit.>. While the lifetime is fundamentally limited by natural decay rates, an increase in Rabi frequency is mainly a technical challenge. § EXPERIMENTAL TECHNIQUES Combining an optical lattice experiment with the excitation of Rydberg atoms introduces a new length scale in the system, but it also leads to challenges on the experimental side. It is desirable to prepare atom distributions with one atom per lattices site with spatial structures on the order of the blockade radius of the Rydberg atoms. Another difficulty are the extremely different energy scales of Rydberg excitation and lattice physics. Forces between the Rydberg atoms can be larger than all energy scales of the lattice and can lead to movement of the atoms before imaging. In the following, we give a short overview of experimental techniques from preparation of ground state atoms to imaging of Rydberg atoms in an optical lattice. §.§ Preparation of atoms in optical lattice Densely filled systems with one atom per site can be prepared in optical lattices by driving the superfluid-Mott-insulator transition with ultracold bosonic atoms with repulsive interactions <cit.>. In this way more than 95% filling can be reached in the central region of the lattice. Employing single-site addressing techniques, the atom distribution can be tailored to the needs <cit.>. The single-site addressing implemented in ref. <cit.> starts from atoms that are prepared spin-polarized in the optical lattice with one atom per site. Then a spin-selective optical light shift with the desired pattern is imaged onto the atoms. This will shift a microwave-transition between two hyperfine ground states for targeted atoms compared to the ones in the dark and allows for a selective microwave transfer via a narrow transition to another hyperfine spin state.In the end, the possible shapes are only limited by the size of the initial unit-filling region in the lattice and the resolution of the imprinted light pattern. Recently also other techniques were demonstrated to achieve a near-perfect array arrangement of atoms. Progress in the deterministic loading of microtraps allows for loading many traps with exactly one atom <cit.>. Compared to the lattice approach discussed above, the larger spacing between these traps can be an advantage for Rydberg experiments. Progress on Rydberg excitation in microtraps is reviewed in refs. <cit.>.Another promising approach is to use atoms in an array of magnetic traps <cit.> though the deterministic preparation of single atoms in these traps has not been demonstrated yet. §.§ Rydberg excitation and detection The most common Rydberg excitation schemes are either a two-photon excitation to the Rydberg state or a direct excitation with a single laser from the ground state (Fig. <ref>). Both schemes have their advantages and drawbacks shortly discussed here. The two-photon scheme typically requires laser wavelengths that are well accessible with diode lasers or frequency-doubled diode lasers. In contrast, the direct excitation wavelengths are typically in the ultraviolet wavelength range and require more challenging laser systems. Another big advantage of the two-photon excitation scheme is the accessibility of both Rydberg S- and D-states for alkali atoms leading to more flexibility compared to the direct excitation using a single laser, which only gives access to P-states. Two-photon excitation is limited by the smallest of the two transition matrix elements of both transitions. For Rydberg excitation the limiting matrix element is the final one to the Rydberg state and for the two-photon excitation this matrix element is larger than the direct S-P matrix elements to Rydberg states of the same principal quantum numbers. But there are some drawbacks of the two-photon excitation, which are light shifts and scattering from the intermediate state. Scattering can be reduced by more intensity for the second excitation step or a narrow linewidth of the intermediate state (if available in the atom) and light shifts are in principle no problem as long as they are stable. This stability requirement can become challenging as it increases requirements for laser intensity stability tremendously. For the direct excitation light shifts are minimized. However, for experiments that use resonant excitation of Rydberg atoms and have enough intensity for the second excitation step available by either strong focussing or a lot of laser power, none of the schemes has big advantages over the other. There are many ways Rydberg atoms were detected historically. The standard way is to apply electric fields and ionize the Rydberg atoms and guide the resulting ions and/or electrons to detectors <cit.>. Gaining spatial resolution from this approach is possible but challenging <cit.>. The required in-vacuum components are also hard to combine with a high-resolution imaging and an optical lattice setup. Another approach is the detection of Rydberg atoms as loss, a technique that is often used in optical tweezer experiments <cit.> (Fig. <ref>b). In an optical lattice this technique is on the one hand problematic as the Rydberg atoms are typically not lost with high fidelity and on the other hand it requires high fidelity imaging of the atom distribution before and after Rydberg excitation. This double imaging is challenging in a quantum gas microscope because the heating of the atoms needs to be kept low enough during the first imaging phase that they do not tunnel in the lattice during the experiment. Optimized Raman sideband cooling might allow for such a scheme in the future. The requirement of double imaging can be avoided by detecting the Rydberg atoms themselves after de-exciting them in a controlled way to the ground state and blowing the other ground state atoms <cit.> (Fig. <ref>(a)). In this way spatial correlations between the Rydberg excitations can be extracted with high fidelity because features in spatial correlation functions are insensitive to uncorrelated loss of Rydberg atoms in the process. Alternative ways to optically image Rydberg atoms are direct imaging on another transition available in alkaline-earth-like atoms <cit.> or EIT-imaging <cit.>. For both of these techniques there are no major obstacles expected when extending them to imaging of single Rydberg atoms with single-site resolution in a lattice configuration. It is also possible to detect the photons from the decay of Rydberg atoms <cit.> but it seems challenging to achieve site-resolved detection of single Rydberg atoms in this way.§ LIMIT OF STRONG INTERACTIONS - ISOLATED SUPERATOMS The time-evolution in the Rydberg Hamiltonian leads in general to complicated entangled many-body states and is barely tractable numerically in mesoscopic systems. It is illustrative to first focus on a simple limit. One interesting case is the limit of dominating Rydberg-Rydberg interactions. In this setting, the blockade radius is larger than the diameter of the system and there can be only maximally one Rydberg excitation (Fig. <ref>(a)). As a consequence, this system behaves as a collective two-level system with basis states of zero and one Rydberg excitation <cit.>. Experiments with two atoms in optical tweezers have first demonstrated this concept <cit.>. Here we focus on a system with many ground state atoms. Such a two-level system offers promising perspectives for quantum computing as it allows for the storage of a qubit while being insensitive to atom loss <cit.>. Moreover, gaining excellent control over such collective Rabi oscillations even constitutes one promising approach to implement quantum gates via collective encoding <cit.>. §.§ Superatoms as effective two-level systemThe isolated superatom arises in a finite system in the limit of dominant interactions. Because there is only maximally one Rydberg atom in the whole system the state space is dramatically reduced and the full system can be described as a collective two-level system in the symmetric subspace of zero (n_↑=0) and one (n_↑=1) excitations (Fig. <ref>b).Here the ground state is the state with all N atoms in the ground state |0⟩=|↓_1,…,↓_N⟩ and the excited state is the entangled first Dicke state |W⟩=1/√(N)∑_i=1^N|↓_1,…,↑_i,…,↓_N⟩, where ↓_i and ↑_i label the i-th atom in the ground or Rydberg state <cit.>.In the state |W⟩ the single Rydberg excitation is symmetrically shared among all N atoms under the assumption that both coupling and interaction are uniform. In this ideal situation, the W-state is the only state coupled by light from the ground state.With these definitions and assuming resonant coupling with Δ=0, the Hamiltonian can be rewritten in the simple form H=ħ√(N)Ω/2 ( |0⟩⟨W|+|W⟩⟨0|). In this collective two-level system, the coupling is not given by the bare coupling Ω but by the symmetry-induced collectively enhanced coupling Ω_coll = √(N)Ω.§.§ Implementations The first implementations of isolated superatoms were created to demonstrate a two-qubit gate with Rydberg atoms <cit.>. These experiments implement the minimal superatom with exactly two atoms and observed the √(2) enhanced collective Rabi oscillation<cit.>. Other implementations were driven by the idea to study single photon, single electron or single ion sources <cit.> and exploit the fact that the superatom can have maximally one Rydberg atom. For these purposes the coherence of the superatom is not required and in some cases even disadvantageous. In contrast, for quantum computing applications the coherence of the superatoms is essential. Collective Rabi oscillations were observed in larger ensembles <cit.> and for few atoms with preselected atom number <cit.> (Fig. <ref>). The spatial ordering of the atoms is not important for superatoms as long as all atoms are within the blockade radius and ground state-Rydberg atom interactions can be neglected.Recently systems with many atoms allowed to confirm the √(N) scaling of the collective Rabi frequency over about two orders of magnitude <cit.>. Here the initial atom number was known with sub-Poissonian fluctuations by either employing the Mott insulator transition in an optical lattice, or measuring the number of atoms before Rydberg excitation.The entangled nature of the excited state in the superatom has been also inferred <cit.>. The underlying idea is to determine the population in the coherent superatom sector by determining the amplitude of the Rabi oscillation. Based on this quantity an entanglement witness can be constructed to show k-particle entanglement for a significant part of the ground state atom number. But the predicted W-state has not been directly confirmed via quantum state tomography yet. §.§ Challenges Collective Rabi oscillations have been observed with fully-blockaded Rydberg ensembles in various experiments. The coherence time of the collective Rabi oscillation in all of them is on the order of a few oscillations and shorter than expected and the different influences of some contributing effects are not fully understood. Next to effects caused by technical noise that can scale with the collective rather than the bare Rabi frequency, there are other more complicated dephasing processes. One is for example the coupling of the excited state to the set of dark states with one excitation that grows linearly with the number of spins. Another is the off-resonant coupling to the doubly excited states, discussed in the next section. There are also contributions from black body radiation and imperfect blockade due to coupling to molecular states. A more systematic study of these effects is necessary to understand the dephasing effects in more detail and improve the coherence times. Another technical difficulty is the preparation of a deterministic atom number in the fully-blockaded ensemble. Variations in atom number are hard to distinguish from dephasing in the measurements. The recent work on reduction of temperature and noise in quantum gas microscopes will also lead to improvements here. But also new techniques for deterministic preparation of arbitrary atom configurations have been developed for tweezer arrays <cit.>.§ BEYOND ISOLATED SUPERATOMS In experiments the Rabi coupling in the superatom is typically not fully negligible compared to the interaction shift caused by two Rydberg excitations in the system. As the coherent collective Rabi oscillation will off-resonantly couple to a large two-excitation space growing quadratically with the number of spins, even small couplings lead to relevant dephasing. Understanding of these effects can be gained by breaking the full blockade condition on purpose by spreading the atoms out over an area with a diameter close to or larger than the blockade radius.This regime has been explored experimentally with systems on the order of the blockade radius <cit.>. Even continuum systems much larger than the blockade radius still show a collectively enhanced Rabi frequency <cit.>. To illustrate the effects of breaking the full blockade condition slightly in a superatom we consider a simple model. We imagine a square of atoms in a lattice where the diagonal is comparable to the blockade radius. In such a setup the first pairs of Rydberg atoms are expected along the diagonal of the system since these doubly excited states exhibit the smallest detuning. To describe this system we can extend the simple two-level model discussed above and add a state with two excitations aligned along the diagonals |╱⟩ and |⟩. Only the symmetric superposition |2⟩ = 1/√(2)(|╱⟩+|⟩) of these states is coupled with a Rabi frequency ⟨W|Ω̂|2⟩ = 4/√(2 N)Ω to the W-state (⟨↓|Ω̂|↑⟩ = Ω). More generally one finds a coupling of C/√(N M)Ω with C the number of coupled pairs of states with Ω between a N-fold degenerate singly excited state and a M-fold degenerate doubly excited state <cit.>. We end up with a Hamiltonian with the states {|0⟩, |W⟩,|2⟩}: H = ħΩ( [ 0√(N)/2 0;√(N)/2 0 C/2√(N M); 0 C/2√(N M)-C_6/d^6 ]) ,where d is the distance between the two atoms in the doubly excited state.Surprisingly, this model fails to predict the coupling to doubly excited states quantitatively. The reason is that many two-excitation states with different energy have a considerable influence. Taking all of them into account in the same way as above leads already to a reasonable approximation. For that purpose we introduce one additional state for each appearing distance between two atoms. As simplification we couple the W-state equally to each of these states which consist of a symmetric superposition of all doubly excited states with same energy. The result of this approximation is shown together with experimental data in Fig. <ref>. But there is another important effect which is the spatially inhomogeneous coupling of the singly excited states to the doubly excited ones. This leads to a deviation from the W-state in the singly excited subspace (Fig. <ref>). It manifests itself as a spatially dependent variation of the amplitude factors around 1/√(N). This spatial dependence can be taken into account in the model by treating all states with n_↑=1 separately and not only as one W-state. Thereby one can describe the dynamics on a nearly blockaded 15×15 square with a 164×164 Hamiltonian as well as the reduced basis set calculation that takes into account many more states (Fig. <ref>). These approximations are, of course, only reasonable for short evolution times in the special case of weak breaking of the full blockade condition where the interaction energy of the lowest two-excitations states is on the order of the Rabi-frequency. For long evolution times weak couplings to more states have to be considered. The reduced basis set calculation considers only states that have a minimal distance between excitations larger than a critical distance R_c <cit.>. This idea relies on the observation that the states with closer excitations have extremely high energy and are typically far off-resonant and not coupled. Care has to be taken for anti-blockade configurations <cit.>. In any case the validity of this approximation can be checked by looking at the convergence of the solution for R_c → 0. § SUDDEN QUENCH IN THE RYDBERG HAMILTONIAN For systems larger than the blockade radius the approximations discussed in the previous sections fail and in general the full Hamiltonian needs to be considered. The most commonly used technique to investigate the Rydberg Hamiltonian is a sudden switch-on of the coupling to the Rydberg state for a certain time. This corresponds to a quench of the transverse field in the spin Hamiltonian. The sudden quench of the coupling from the ground state with n_↑=0 leads to dynamics because it is not an eigenstate of the Hamiltonian. The initial state is projected to the basis states leading to a superposition with wide spread of energies that show different time evolution. However, experiments show that for atom numbers larger than two, the system ends up in a quasi steady-state on a timescale on the order of 2π/Ω_coll. In the following, we look in more detail at the dynamics and then at the properties of the quasi-steady state.§.§ Excitation dynamicsHere we consider the following excitation sequence. Initially, all atoms are spin-polarized in the ground state and at time t = 0 the coupling Ω to the Rydberg state is switched on resonantly (detuning Δ = 0). After an evolution time, the system is detected at a time t = t_0. This evolution is equivalent to a quench of the transverse magnetic field in a long-range interacting Ising Hamiltonian. The excitation dynamics in the Rydberg Hamiltonian is quite complicated, especially in the general case where the system is neither fully-blockaded nor in the low-excitation limit. The van-der-Waals interaction leads to a huge variety of energy scales that cause interaction-induced dephasing of the system. The coherence is preserved for longer than Rabi oscillations can be observed in the system average. Butcoherence is hard to measure as it becomes only accessible in experiments that are sensitive to the phases <cit.>. All density-related observables seem to show a steady-state behaviour that has been investigated <cit.> and observed in many experiments <cit.>. For resonant driving of the Rydberg transition, this state is reached in a time on the order of 2π/√(N_b)Ω <cit.>, where N_b is the number of atoms per blockade sphere. Theory calculations show that the observation of quasi steady states can be explained by fully coherent dynamics and does not rely on external decoherence sources. The time evolution can be seen intuitively as a superposition of many Rabi oscillations from the ground state to all singly excited states which are then further coupled to doubly excited states and so on. For small system sizes, the number of excited states is limited and the relevant dynamics of the system can be understood by the approximations discussed in the previous section.The superatom idea can be extended to systems with larger number of excitations leading to a picture of a dense packing of Rydberg blockade spheres for large excitation numbers <cit.>. In this regime it becomes hard to approximate the full dynamics but it provides access to the excitation statistics of the steady-state. The typical timescales of the excitation dynamics are still given by a collectively enhanced Rabi frequency where the effective number of atoms participating is roughly N_b <cit.>. §.§ Spatial correlationsSpatially resolved detection of the Rydberg atoms allows to directly measure correlation functions and thereby also the blockade radius. The discussion of the excitation dynamics in the last section shows that the spatial correlation function evolves to a steady-state.In this regime the Rydberg atoms arrange randomly but are subject to the blockade constraint and thereby show liquid-like distance correlations. There has been a variety of theoretical work on the steady state patterns expected in the Rydberg gas <cit.>. It has been argued that high density initial states entropically favour ordered Rydberg excitations <cit.>. Another view on the ordering is that the spin-↑-distribution can be modelled in driven-dissipative steady-state by particles interacting with logarithmic potentials <cit.>. One signature for the appearance of correlations is the excitation statistics of Rydberg atoms in the system. It has been observed experimentally that the Rydberg numbers are sub-Poissonian <cit.>.A more direct way to measure spatial correlations is to extract them from high-resolution images.These correlations have been studied by ionizing the Rydberg atoms <cit.> and in-situ with optical lattices <cit.> (Fig. <ref>) and optical micro-traps <cit.> (Fig. <ref>). The spatial correlations have been characterized by the pair correlation function(r) = ∑_i⃗≠j⃗δ_r, r_i⃗j⃗ ⟨σ̂^(i⃗)_eeσ̂^(j⃗)_ee⟩/∑_i⃗≠j⃗δ_r, r_i⃗j⃗ ⟨σ̂^(i⃗)_ee⟩⟨σ̂^(j⃗)_ee⟩.It measures the joint probability of two excitations at a distance r. Here δ_r, r_i⃗j⃗ is the Kronecker symbol that restricts the sum to sites (i⃗,j⃗) for which r_i⃗j⃗ = r.The basis for the calculation of this correlation function is a set of data of positions of all spin-↑ atoms in many realizations of the experiment (Fig. <ref>(a)). A first idea of the correlations can be gained by aligning center of mass and angle of events with same spin-↑ number (Fig. <ref>b). For the calculation of the correlation function all realizations independent of the number of spin-↑ atoms are included (Fig. <ref>c). This correlation function shows the expected features from theory <cit.>, which are the blockade effect for small distances and a peak slightly beyond the blockade radius. Imaging imperfections lead to non-vanishing correlations below the blockade radius and a slight smearing out of the edge. The sharp rise at very short distance is caused by tunnelling during imaging that leads to an atom to be falsely detected as two atoms with probability of about 1 <cit.>. A signal below the blockade radius could in principle also arise from pair excitations <cit.> but it is unlikely that pairs excited in that way can be detected in the applied imaging technique considering the forces acting between the two atoms in such a pair. Imperfect blockade is typically rather caused by imaging imperfections that lead to detection of Rydberg atoms in places where they not have been during excitation. This can either happen by detecting a ground state atom wrongly as Rydberg atom or due to movement of the Rydberg atoms during the imaging sequence. The corresponding pair correlation function in a 1D system has been measured with an array of equidistant atoms in microtraps on a larger spatial scale but the correlation function still shows qualitatively the same features (Fig. <ref>). In the same study also the influence of anisotropic interactions on the 2D correlation function has been observed <cit.>.§ ADIABATIC PREPARATION OF RYDBERG CRYSTALS The sudden switching of the coupling to the Rydberg state and the resulting time evolution discussed in the previous section poses the question if it is possible to perform adiabatic sweeps into collective Rydberg states and thereby deterministically excite a certain number of Rydberg excitations in the system. These states show interesting properties, such as crystalline ordering of the Rydberg excitations at low energy <cit.>.Adiabatic preparation has been proposed to deterministically prepare these states <cit.>. In the following we first discuss the general adiabatic preparation scheme in a many-body system and then the details of an implementation to prepare ordered Rydberg many-body states. §.§ Adiabatic preparation in a many-body system The adiabatic preparation in the many-body system follows the basic scheme of the Landau-Zener sweep in a two-level system <cit.> (Fig. <ref>(a)). In both cases the underlying idea is to use a Rabi coupling to open a spectral gap, which allows to connect initial and final state by an adiabatic path. This path can be followed by changing the detuning and the coupling with time. Compared to the two-level system, the interacting many-body system has more energy scales which leads to a breakdown of the Landau-Zener picture. In the end the main problem is the appearance of the new interaction gaps in the many-body Hamiltonian (Fig. <ref>(b)). The smallest gap in the process is not only determined by the Rabi frequency any more but by interactions <cit.>. So the simple recipe of the Landau-Zener sweep that an increase of Rabi frequency always improves the transfer is not true any more. In the many-body case one either has to adapt the duration of the adiabatic sweep to be slow compared to the gaps, enlarge the gaps or reduce the number state crossings. Larger gaps can be mainly achieved by decreasing the number of atoms, decreasing the distance between Rydberg atoms in the crystal or byoptimizing the path in (Ω,Δ)-space to avoid regions with small gaps <cit.>. The number state crossings depends on the spatial configuration of the ground state atom distribution and is, for example, much smaller in 1D than in 2D. §.§ CrystallizationAdiabatic preparation techniques have been proposed to deterministically prepare Rydberg crystals <cit.>. For an experimental implementation of these proposals it is important to first look at the energy scales in the system that determine the timescale for the preparation. As the time for Rydberg experiments is typically limited to about 10 due to mechanical motion and lifetime of the Rydberg atoms, this imposes strict limits on the set of systems with large enough gaps to make the adiabatic preparation realistic. In contrast to adiabatic passage in two-level systems the transfer here is not limited by Rabi frequency but by the interaction gaps. Experimentally the preparation was demonstrated in one-dimensional systems with up to about 20 ground state atoms and up to three Rydberg excitations <cit.> (Fig. <ref>(a-b)), which is an implementation in an optical lattice with spacing = 532nm using the rubidium Rydberg state 43S. The spin-↑ number in the system changes by one when increasing the length by about 10 sites. Therefore a 1D system with length fluctuations much smaller than that is required. Average fluctuation of less than one site can be achieved using single-site addressing in a Mott insulator <cit.>. Experimentally, the number of Rydberg excitations is limited here to about three due to breakdown of adiabaticity for more than about 25 ground state atoms. Supporting theory calculations for the experimental parameters show that there is essentially no parameter space where one can prepare four excitations but not three for these experimental parameters (Fig. <ref>(c)). Crystals with more excitations are realizable in an optimized setup with larger interaction gaps, for example by reducing the blockade radius, or by coupling to Rydberg states with longer lifetime.For the experimental implementation an optimized time-dependence of Ω(t) and Δ(t) needs to be determined which is done via numerical optimization.Typically, the sweep starts with a rise of the Rabi frequency followed by a sweep of the detuning over the resonance. This part can be relatively fast if sufficiently high Rabi frequency can be reached in the experiment. In the end the Rabi frequency needs to be reduced to reach the final state at Ω=0. The timescale of this part is limited by the interaction gaps in the system. In Fig. <ref>(a) the spectrum during the sweep is shown for a simplified small system. The gap in the many-body spectrum during the adiabatic sweep is comparably large during the first half of the sweep. In the end during the reduction of the Rabi frequency the gap becomes very small. This shows the importance of a soft switch-off of the Rabi frequency in the experiment. The approximate path of the adiabatic sweep through the (Ω/Δ)-space together with the average number of spin-↑ atoms along the path is illustrated in Fig. <ref>(b). The sweep starts at Δ<0 in the classical limit (Ω=0) where the system contains no spin-↑ atoms. The target state is one of the classical crystalline states at Δ>0. There the many-body ground state corresponds to crystalline states with vanishing fluctuations in the total magnetization M=2 N_↑ - N, which, for fixed total atom number N, is determined by the spin-↑ component N_↑=∑_i⃗⟨n̂^(i⃗)_↑⟩. In a one-dimensional chain of ℓ≫ N_↑ lattice sites, the number of spin-↑ atoms increases by one at the critical detunings ℓ^6ħΔ_ c≈ 7 C_6 N_↑^6 / ^6 separating successive crystal states with a lattice spacing ℓ/(N_↑-1) <cit.>. The laser coupling introduces quantum fluctuations that can destroy the crystalline order <cit.>. One clear signature for the successful implementation of the adiabatic preparation is a staircase of the number of spin-↑ atoms versus length of the system (Fig. <ref>). On the plateaus the number of spin-↑ atoms is insensitive to changes of the experimental parameters. Due to finite detection efficiency of the Rydberg atoms the plateaus do not show integer values. To confirm the number of excitations on a plateau one can look at the average spin-↑ density which shows three spots for the stair of three spin-↑ atoms (Fig. <ref> inset). We note that although a macroscopic population of the many-body ground state is reached in these experiments the system is not prepared with high fidelity in the absolute ground state <cit.>. Reaching the absolute ground state of a mesoscopic quantum many-body system is challenging, and simulations show that obtaining near unit fidelity would be only possible if the sweep is longer than about 100 which is beyond the lifetime of the Rydberg state and therefore impossible in this experimental setting. Slight deviations from adiabaticity leak population to states very close in energy and the states populated in this way are experimentally effectively indistinguishable from the ground state. A typical example is the shift of one spin-↑ by one site in a one-dimensional crystal. From these results in 1D directly the question arises if the deterministic preparation of ordered states can be also implemented in two-dimensional systems. Experimental parameters are much more challenging in 2D and it becomes hard to observe the staircase of excitations in the same lattice with 43S rubidium Rydberg states. This becomes obvious when looking at the interaction energy for the different configurations of different spin-↑ numbers (Fig. <ref>c). In the case of a circular initial system the radius needs to be adjusted to less than a lattice site to observe a plateau in the staircase. This is effectively impossible in an experimental setting with a preparation uncertainty of about 80 % filling in the initial state. The imperfect filling leads already to effective radius variations of the order of one site even if slight alignment errors of the addressing pattern with the lattice are neglected. But it is experimentally possible to prepare low energy configurations of the spin-↑ numbers although the spin-↑ number is not fully deterministic. One hint that lower energy states are reached is provided by comparing the correlation function (Fig. <ref>) from data with adiabatic sweep with the previously measured one without adiabatic sweep (Fig. <ref>). Another way is to directly look at the average magnetization density without any configuration alignment. Low energy states will show a ring-like structure in the spin-↑ density which has not been observed for the simple pulsed excitation (Fig. <ref>). When a state with mainly n_↑=3 is excited the blockade radius requirement leads automatically to a hole in the spin-↑ density in the center. This demonstrates that the adiabatic preparation is partly working in 2D but the precision and stability of the sweep and initial atom configurations are not sufficient to reach deterministic preparation of states with certain spin-↑ number. The quality of the sweep in these experiments was mainly limited by fluctuations in the lightshift caused by laser intensity noise and decoherence due to a combined two-photon laser linewidth of approximately 50kHz. In addition the large spacing of the Rydberg crystals in these experiments are unfavorable for adiabatic preparation but experimental constraints did not allow to reduce the blockade radius significantly in the described setting without deteriorating detection efficiency.§.§ Challenges and Limits The vastly different energy scales of Rydberg atoms and ground state atoms in the optical lattice can lead to interesting physics but also pose experimental challenges. This starts with high-fidelity imaging of Rydberg atoms and also experiments as simple as determining the resonance frequency of the Rydberg transition can become complicated in the presence of strong interactions. In a dense ensemble the Rydberg line exhibits significant shifts and broadening, requiring to work with very dilute atomic ensembles or Rydberg excitation numbers on the order of one to determine the Rydberg line with high precision. But the position of the Rydberg line has to be determined very precisely for the adiabatic sweeps as the relative frequency offset of the sweep with respect to the resonance needs to be adjusted to few ten Kilohertz to excite certain excitation configurations.Another experimental issue is that Rydberg atoms experience a different potential landscape than ground state atoms in optical lattices. The conflicting requirements of targeted lattice spacing and necessary lattice wavelength for trapping of ground and Rydberg state make it very hard to design optical lattices with reasonable parameters that trap Rydberg and ground states of alkaline atoms in the same places. But for optical tweezers equal trapping for ground and Rydberg state (magic trapping) has been achieved <cit.>. For alkaline atoms in optical lattices the options are mainly to avoid the effects of light shifts on the Rydberg atom by minimizing the time spent in the Rydberg state, switching the potential off during Rydberg excitation, working with Rydberg states that experience less forces <cit.> or phase-shifting the lattice <cit.>. For alkaline-earth-like atoms magic trapping in optical lattices is promising <cit.>. Many challenges for the preparation of crystalline configurations are caused by the very fast fall-off of the Rydberg van der Waals interaction with 1/r^6, leading to stringent requirements for the precision of the size of the initial atom distribution and exact tuning of the frequency, as the interaction energy drops very fast to small values that still need to be resolved for the adiabatic preparation. The very fast rise of the interaction energy for distances below the blockade radius leads to problems with energy scales as the energy pumped into the system by an adiabatic sweep can lead to strong forces between the Rydberg atoms that effectively heat up the system due to interactions between Rydberg atoms. In particular in optical lattices a slight heating or shift of the atoms leads already to occupation of higher bands in the lattice and therefore increased tunnelling during imaging of the Rydberg atom positions. The adiabatic preparation as discussed here is fundamentally limited to small excitation numbers as the gaps in the adiabatic preparation drop exponentially with increasing atom number <cit.> leading to immense requirements in terms of lifetime and coherence time in the experiments that cannot be fulfilled with the lifetime of typically used S/P/D Rydberg states. The extension to anisotropic interactions for the creation of small crystals is possible <cit.>. Current experimental techniques also allow for a detailed investigation of the other phases in the phase diagram which are theoretically more interesting than the mainly classical crystalline state, for example floating crystal phases <cit.>.By optimizing time dependence of detuning and Rabi frequency by optimal control techniques it might become possible to not only create crystalline states with higher fidelity but also to prepare entangled states deterministically <cit.>. § CONCLUSIONRecently there has been tremendous progress in controlling Rydberg excitation coherently and employing them to prepare entangled states of neutral atoms. The combination of Rydberg experiments in large ensembles with high-resolution imaging in optical lattices opens a wide range of new experiments in long-range interacting spin models. In this review we focussed on a finite-range Ising spin Hamiltonian that emerges naturally when exciting an ensemble of atoms on a lattice to a Rydberg state.§.§ Possible improvementsOne of the experimentally most important open questions about working with Rydberg atoms is a precise characterization of the sources for decoherence and dephasing. They are plentiful and it is hard to track down certain experimental uncertainties and eliminate them. This is partly caused by the sensitivity of Rydberg atoms to electromagnetic fields, which is inherently connected to their strong interactions. Static electric fields, microwave fields in the Gigahertz range and black body radiation have all a strong influence on Rydberg atoms and are rarely controlled in typical ultracold atom experiments due to their negligible influence on neutral atoms. Specialized Rydberg experiments control the electric fields but reduction of black body radiation is currently limited to only very few experiments. The effects that might be caused by black body radiation are still barely understood. Black body radiation effectively leads to diffusion of Rydberg state population to nearby Rydberg states. As many experiments cannot resolve population of neighbouring Rydberg states in detection, these repopulation effects show up as loss of Rydberg atoms in many experiments and effectively reduce detection efficiency. Minimizing effects of black body radiation is challenging, the obvious solution is to cool the whole vacuum chamber to low temperatures. There might be ways to use Rydberg states with lower coupling to black body radiation. Selecting a special Rydberg state with particular neighbourhood of other states might lead to an improvement but the influence of the level splitting on interactions limits this approach. In some experiments, the use of additional lasers to depump unwanted Rydberg states that were populated by black-body radiation might be a solution. There is also the general question to what extent the lifetime of systems with finite-range interactions can be improved. Typical interactions between Rydberg atoms lead to interaction forces on short scales larger than the trapping of the atoms which turn the system intrinsically unstable if Rydberg atoms can come close to each other. Additionally optical driving to these interacting states leads to extreme broadening of the excitation lines which becomes even worse when unwanted states are populated in the system <cit.>. This is in particular a problem in van-der-Waals-interacting systems which rely on second-order interactions where impurities in other Rydberg states can introduce first-order dipole-dipole interactions that are much stronger. Developing techniques to keep these systems under control still needs more detailed understanding. This will also increase understanding in other systems as some of the effects are quite generic for dipolar interacting systems.§ FUTURE PROSPECTS The investigation of spin models created by the coupling of atoms in lattices to Rydberg states just started and many fields have not been explored yet.But even many predicted effects in the Ising spin model discussed here have not been seen in experiments. In the following we discuss an exemplary set of open questions. Entanglement One big open field is still the investigation of entanglement properties of mesoscopic Rydberg-excited states. This requires very good detection and preparation, and even then, typically a specifically designed entanglement witness is required to show entanglement in reasonably large systems. Besides the demonstration of entanglement of two atoms <cit.> and the special case of the Rydberg superatoms <cit.> entanglement experiments with Rydberg atoms are scarce. It has been proposed to generate spin squeezing and non-Gaussian states with Rydberg atoms which can be directly applied in optical lattice clocks <cit.>. Lately also interest came up in the study of the entanglement growth in quench experiments <cit.>.Phase diagram of the Ising model with finite-range interactions The crystalline phase is only a small region of the phase diagram of the Ising model with finite-range interactions. A more detailed measurement of correlation functions and excitation fractions should allow to pin down phase transitions experimentally. This includes the dynamics of the build up of order and the investigation of the path to long-range order <cit.>.Dipole interactions Next to van-der-Waals interactions Rydberg atoms also allow for the investigations of spin models based on the direct dipole-dipole interaction between different Rydberg states. First experiments on the exchange interactions <cit.> look promising for the investigation of larger spin systems.Exotic spin systems Spin systems with exotic interactions can be designed with Rydberg atoms <cit.>. These ideas widen the scope of Rydberg spin systems to a much larger class of spin models and employ the Rydberg dressing technique discussed in the following. §.§.§ Beyond spin models - Rydberg dressingOne of the big targets is to implement interacting many-body systems combining atomic motion with tunable long-range interaction via Rydberg atoms <cit.>. This could be achieved by engineering a tunnelling term for the atoms in the lattice of the Ising model dicussed here. The main experimental challenge is to bridge the mismatch in energy and timescales between the Rydberg excitation and the dynamics of ground state atoms. A possible solution is the so-called Rydberg dressing where ground state atoms are coupled off-resonantly to Rydberg states leading to effectively weaker interaction with lower decay rates. Rydberg dressing is the off-resonant admixture of a Rydberg state to the ground state in the limit Δ≫Ω. But there is a continuous transition from off-resonant Rydberg excitation such that the description is similar also closer to resonance. The motivation for Rydberg dressing is to tune the lifetime of Rydberg atoms to another independent timescale like the tunnelling in the optical lattice. In this case the lifetime of the Rydberg-dressed ground state atom has to be long compared to the tunnelling in the lattice.The main difficulty in this approach is that decay and loss processes of Rydberg atoms have to be controlled on these timescales that are much longer than for near-resonant experiments such that also more exotic loss processes become relevant. Rydberg dressing has been proposed to implement interactions in quantum gases <cit.> together with techniques to detect weak dressing interactions <cit.>. One of the exotic states that might be possible to realize via Rydberg dressing is a supersolid droplet crystal <cit.>. Rydberg dressing also allows to implement local constraints that are at the heart of the implementation of models related to gauge theories like the quantum spin ice <cit.>.But in addition, it is also possible to design interaction terms in spin Hamiltonians that are quite unusual, for example terms that conserve the parity of the spin but not the magnitude <cit.>.Another prediction are cluster Luttinger liquids in 1D <cit.> and the implementation of glassy phases <cit.>. It might be even possible to implement a universal quantum simulator <cit.> and quantum annealer based on Rydberg dressing <cit.>.Experimental implementation of Rydberg dressing turned out to be challenging and schemes based on two-photon excitation in large systems very close to resonance suffered from strong loss <cit.>. Rydberg dressing has recently been implemented first with two atoms <cit.> and in a many-body setting <cit.>. Both of these experiments used a direct excitation to Rydberg states with an ultra-violet laser. This direct coupling scheme is superior in experimental parameters to two-photon excitation schemes in alkali atoms. An experimental implementation of many-body systems with finite-range interactions and hopping in the lattice would open another new field for Rydberg physics.§ ACKNOWLEDGMENTS I thank I Bloch, S Kuhr and C Gross for support of the Rydberg project and supervision of my PhD thesis, J Zeiher, M Cheneau and A Omran for the joint work on the Rydberg experiments and the theory support by T Pohl and T Macrì. I also thank C Weitenberg, M Endres, S Hild, T Fukuhara, J-y Choi, F Seeßelberg, D Bellem for their contributions to the experiments in Munich. I am grateful for comments on the manuscript by A Browaeys, C Gross, T Lahaye, A Omran, T Macrì, and J Zeiher. P.S. was supported by a Dicke fellowship from Princeton University. | http://arxiv.org/abs/1706.09014v1 | {
"authors": [
"Peter Schauss"
],
"categories": [
"physics.atom-ph",
"cond-mat.quant-gas",
"quant-ph"
],
"primary_category": "physics.atom-ph",
"published": "20170627190150",
"title": "Finite-range interacting Ising quantum magnets with Rydberg atoms in optical lattices - From Rydberg superatoms to crystallization"
} |
Facultad de Física, Universidad de la Habana,San Lázaro y L, Vedado, La Habana 10400, CubaInstituto de Cibernética, Matemática y Física (ICIMAF),Calle E esq a 15 Vedado 10400 La Habana Cuba Instituto de Cibernética, Matemática y Física (ICIMAF),Calle E esq a 15 Vedado 10400 La Habana Cuba The thermodynamical properties of a neutral vector boson gas in a constant magnetic field are studied starting from the spectrum given by Proca formalism. Bose Einstein Condensation (BEC) andmagnetization are obtained, for the three and one dimensional cases, in the limit of low temperatures.In three dimensions the gas undergoes a phase transition to an usual BEC in which the critical temperature depends on the magnetic field. In one dimension a diffuse condensateappears as for the charged vector boson gas. In both cases, the condensation is reached not only by decreasing the temperature but also by increasing the magnetic field. In three and one dimensions self-magnetization is possible. The anisotropy in the pressures due to axial symmetry imposed to the system by the magnetic field is also discussed. The astrophysical implications are commented.98.35.Eg, 03.75Nt, 13.40Gp, 03.6Thermodynamical properties of a neutral vector boson gas in a constant magnetic field H. Pérez Rojas[[email protected]] December 30, 2023 ===================================================================================== § INTRODUCTION There is a diversity of structures associated with a wide range of magnetic fields (10^-9-10^15 G) cohabiting in our Universe. Salient examples are galaxies (radius 1.5 × 10^18km) and compact objects (radius 20km). The internal composition of neutron stars (still poorly understood) is described by all sorts of exotic dense matter in form of hyperons, a Bose Einstein Condensate (BEC) of mesons or deconfined quark matter in presence of strong magnetic fields <cit.>. Size and shape of a compact object depends on its composition but also on the magnetic field <cit.>. There are also some phenomena at astrophysical scale without explanation, as jets of pulsars, where magnetic fields might be relevant <cit.>.Even though some theories have been proposed to explain the origin of such magnetic fields, this issue is far from being exhausted and it is still under great debate. In this regard, spin one bosons seems to be good candidates as magnetic field sources since, at low temperature, they are known to show an spontaneous magnetization. As a consequence, under certain conditions, a gas of bosons can generate and sustain its own magnetic field <cit.>.The study of BEC and magnetization for a charged scalar or vector boson gas in presence of a constant magnetic field was tackled in <cit.>. For low temperatures, the charged vector boson gas is paramagnetic, can be self magnetized and undergoes a phase transition to a diffuse BEC <cit.>.For a diffuse phase transition there is not a critical temperature, but an interval of temperatures along which it occurs gradually <cit.>. In particular, a diffuse BEC phase is characterized by the presence of a finite fraction of the total particle density in the ground state and in states on its neighborhood at some temperature T>0. In this sense, the criterion for defining a diffuse BEC is weaker than the used for the usual one, which requiresthe existence of a critical temperature below which there is a macroscopic amount of particles in the ground state. Whether a BEC is usual or diffuse is strongly related to the dimension of the system <cit.>.Although both, charged and neutral vector bosons, could be relevant participants of astronomical phenomena, the thermodynamics of the neutral vector boson gas has been less studied. An effect analogous to self-magnetization, named BE-ferromagnetism, was founded in <cit.> for a gas of non relativistic ideal neutral boson with spin one. In <cit.> and <cit.> magnetic field induced superconductivity and superfluidity are obtained for a gas of charged an neutral vector mesons, but it ignores the weak coupling between the neutral mesons and the magnetic field. More recently, BEC for an gas of interacting vector bosons at zero magnetic field was studied in <cit.>.Hence, the aim of this paper is to study the thermodynamical properties of a neutral vector bosongas (NVBG) in a constant magnetic field. We will concern with its phenomenology in the framework of Proca theory and independently of the realistic conditions in which it may appear. Neutral vector bosons can be mesons, atoms and other paired fermions with total integer spin. For numerical calculations we use a positronium gas parameters, characterized by a mass approximately 2 m_e, (m_e is the electron mass) and twice the electron magnetic moment κ = 2μ_B being μ_B the Bohr magneton. Since we are focussed on possible astrophysical applications we deal with systems of densities in the range of 10^30-10^34cm^-3.The thermodynamical properties of the NVBG are studied in three and one dimensions. The BEC phase transition turns out to be diffuse in one dimension and usual in three. In both cases, for low enough temperatures self-magnetization arises. An analysis of this phenomenon lead us to the conditions for the appearance of a self sustained field. The axial symmetry of the magnetic field is reflected in the particle spectra and in the energy-momentum tensor of the system which becomes anisotropic. For that reason, we also study the splitting of the parallel and perpendicular pressures with respect to the direction of the magnetic field.Our paper is organized as follows. In Section <ref>, we present the equation of motion and spectrum of neutral vector boson with MM. Section <ref> contains a derivation of the thermodynamical potential, particle density,BEC, internal energy, entropy and specific heat for the three and one dimensionalNVBG. Magnetization, self-magnetization and anisotropic pressures are also discussed.Section <ref> is devoted to conclusions. Appendix A and B contains some details of the calculations. § EQUATION OF MOTION OF A NEUTRAL VECTOR BOSON BEARING A MAGNETIC MOMENT Neutral spin-one bosons with magnetic moment that moves in a magnetic field can be described by an extensionof the original ProcaLagrangian for spin one particles that includes particle-field interactions <cit.> L = -1/4F_μνF^μν-1/2ρ^μνρ_μν+ m^2 ρ^μρ_μ +i m κ(ρ^μρ_ν-ρ^νρ_μ) F_μν. In Eq. (<ref>) the indices μ and ν run from 1 to 4, F^μν is the electromagnetic tensor, andρ_μν, ρ_μ are independent field variables that follow <cit.> ∂_μρ_μν-m^2 ρ_ν+ 2i κ m ρ_μ F_μν=0,ρ_μν = ∂_μρ_ν - ∂_νρ_μ. A variation of the Lagrangianwith respect to the field ρ_μ give us the equations of motion that in the momentum space read ((p_μ^2+ m^2)δ_μν -p_μ p_ν- 2i κ m F_μν)ρ_μ = 0. Thus, the boson propagator is D_μν^-1=((p_μ^2+ m^2)δ_μν-p_μ p_ν- 2i κ m F_μν). Considering the magnetic fielduniform, constant and in p_3 direction B=Be_3 one can start from Eq. (<ref>) and obtain the generalized Sakata-Taketani hamiltonian for the six component wave equation of the system <cit.> following the same procedure of Ref. <cit.>. The hamiltonian reads H = σ_3 m + (σ_3 + i σ_2) p^2/2 m - i σ_2 (p·S)^2/m -(σ_3 - i σ_2) κS·B, with p=(p_⊥,p_3) and p_⊥=p_1^2 + p_2^2.σ_i are the 2×2 Pauli matrices, S_i are the 3×3 spin-1 matrices in a representation in which S_3 is diagonal and S = {S_1,S_2,S_3}[ [S_1=1/√(2)( [ 0 1 0; 1 0 1; 0 1 0 ]), S_2=i/√(2)( [0 -10;10 -1;010 ]), S_3= ( [100;000;00 -1 ]) ]]. The eigenvalues of (<ref>) areε(p_3,p_⊥, B,s)=√(m^2+p_3^2+p_⊥^2-2κ s B√(p_⊥^2+m^2)), where s=0, ± 1 are the spin eigenvalues. Although we are dealing with neutral bosons, as happens for the charged ones, the magnetic field intensity B enters in the energy coupled with the perpendicular momentum (see the last term in the previous equation). This coupling reflects the axial symmetry imposed to the system by the magnetic field. The ground state energy of the neutral spin one boson (s=1 and p_3=p_⊥=0) isε(0, B)=√(m^2-2κ B m)=m√(1-b).being b=B/B_c and B_c=m/2κ.From Eq. (<ref>) follows that the rest energy of the system decreases with the magnetic field and becomes zero for B=B_c. For the values or m and κ we are considering, m=2m_e and κ=2μ_B, B_c=m_e^2/e=4.41 × 10^13 G which is the Schwinger critical field. Let us note that the ground state of the charged vector boson has a similar instability (see <cit.>). Eq.(<ref>) allows us to define and effective magnetic moment asd=-∂ε(0,B)/∂ B=κ m/√(m^2-2mκ B)=κ/√(1-b).The system has a paramagnetic behavior because d>0. It will be also important for the discussion below the fact that d grows with the increasing of the magnetic field and diverges for b → 1 (B → B_c). § THERMODYNAMICAL PROPERTIES The general expression for the thermodynamical potential of the NVBG has the formΩ(B,μ,T)= ∑_s=-1,0,11/β[∑_p_4∫_-∞^∞p_⊥dp_⊥dp_3/(2π)^2ln D^-1(p^*)].Here D^-1(p^*) is the neutral bosonpropagator given by(<ref>),β = 1/Tdenotes the inverse temperature, μ the boson chemical potential and p^*=(ip_4-μ,0,p_⊥,p_3). After doing the Matsubara sum, Eq.(<ref>) becomes Ω(B,μ,T)= Ω_st+Ω_vac, where Ω_st is the statistical contribution of bosons/antibosons that depends on B, T and μ Ω_st(B,μ,T)=∑_s=-1,0,11/β(∫_0^∞p_⊥dp_⊥dp_3/(2π)^2ln((1-e^-(ε(p_3,p_⊥, B,s)- μ)β)(1-e^-(ε(p_3,p_⊥, B,s)+ μ)β)) ),and Ω_vacis the vacuum term which is only B-dependent [The vacuum term is important for instance, in the positronium case. It would represent a correction to the usual Euler-Heisenberg term in which the electron positron pairs bosonize by coupling, for instance, through Coulomb force.] Ω_vac=∑_s=-1,0,1∫_0^∞p_⊥dp_⊥dp_3/(2π)^2ε(p_3,p_⊥, B,s). We can rewrite Ω_st as Ω_st(B,μ,T)=∑_s=-1,0,1Ω_st(s), with Ω_st(s)=1/β(∫_0^∞p_⊥dp_⊥dp_3/(2π)^2ln((1-e^-(ε(p_3,p_⊥, B,s)- μ)β)(1-e^-(ε(p_3,p_⊥, B,s)+ μ)β)) ) being the contribution of each spin state to the statistical part of the potential.Using the Taylor expansion of the logarithm, Eq. (<ref>) can be written as Ω_st(s)= - 1/4 π^2 β∑_n=1^∞e^n μβ+e^- n μβ/n∫_0^∞ p_⊥dp_⊥∫_-∞^∞ dp_3 e^-n βε(p_3,p_⊥, B,s).The term e^n μβ stands for the particles and the term e^- n μβ for the antiparticles.In Eq. (<ref>) the integration in p_3 can be completely carried out, while the integration in p_⊥ can be only partially done and we obtain for the thermodynamical potential the expression Ω_st(s)= - z_0^2/2 π^2 β^2∑_n=1^∞e^n μβ+e^- n μβ/n^2 K_2 (y z_0) - α/2 π^2 β∑_n=1^∞e^n μβ+e^- n μβ/n∫_z_0^∞ dz z^2/√(z^2+α^2) K_1 (y z), where K_n(x) is the McDonald function of order n, y = n β, z_0= m √(1-s b) and α=s m b/2. To compute the integral in the second term of Eq. (<ref>) I = ∫_z_0^∞ dz z^2/√(z^2+α^2) K_1 (y z), we follow the procedure described in Appendix A. Finally Ω_st (s) reads Ω_st(s)= - z_0^2/2 π^2 β^2 (1+α/√(z_0^2 + α^2)) ∑_n=1^∞e^n μβ+e^- n μβ/n^2 K_2 (y z_0) - α z_0^2/π^2 β^2 √(z_0^2 + α^2) ×∑_n=1^∞e^n μβ+e^- n μβ/n^2∑_w=1^∞(-1)^w(2 w -1)!!/(z_0^2 + α^2)^w(z_0/y)^w K_-(w+2)(y z_0). In the low temperature limit T ≪ m (which for m=2 m_e≅1 MeV means T ≪ 10^10 K),Ω_st(-1) and Ω_st(0) vanish. Therefore, in this limitΩ_st≃Ω_st(1). This means that all the particles are in the spin ground state s=1. The leading term of Ω_st(1)is the first one in Eq.(<ref>). Since it admits a further simplification, for the assumed low temperatures the statistical part of the thermodynamical potential is Ω_st(B,μ,T) = -ε(0,B)^3/2/2^1/2π^5/2β^5/2 (2-b) Li_5/2(e^βμ^'), whereLi_n(x) is the polylogarithmic function of order n and μ^' = μ - ε(0,B). The quantity μ^' is a function of the temperature and the magnetic field, and it leads to the critical condition because the existence of a non-zero temperature T_cond for whichμ^' = 0 is the requirement for the usual BEC.The vacuum contribution to the thermodynamical potential after being regularized is (see appendix B)Ω_vac = -m^4/288 π( b^2(66-5 b^2)-3(6-2b-b^2)(1-b)^2 log(1-b). . -3(6+2b-b^2)(1+b)^2log(1+b) ). Adding Eqs. (<ref>) and (<ref>) we get the total thermodynamical potential for the NVBG in the limit of low temperatures Eq.(<ref>) Ω(B,μ,T) =-ε(0,B)^3/2/2^1/2π^5/2β^5/2 (2-b) Li_5/2(e^βμ^')-m^4/288 π( b^2(66-5 b^2) . . -3(6-2b-b^2)(1-b)^2 log(1-b)-3(6+2b-b^2)(1+b)^2log(1+b) ). §.§ Particle density and Bose Einstein Condensation To obtain the particle density we compute the derivative of Eq.(<ref>) with respect to the chemical potential μ N = -∂Ω(B,μ,T)/∂μ= ε(0,B)^3/2/2^1/2π^5/2β^3/2 (2-b) Li_3/2(e^μ^'β). This expression allows the substitution μ^' = 0 (because Li_3/2(1)=ζ(3/2), where ζ(x) is the Riemann zeta function). Consequently, a critical temperature can be defined (Eq. <ref>) and the neutral vectorboson gas shows usual Bose Einstein Condensation T_cond = 1/ε(0,B) ( 2^1/2π^5/2 (2-b) N/ζ(3/2))^2/3. Although this behaviour resembles the one obtained for bosons at zero magnetic field -the functional relation between T_cond and N is the same-, when the field is present the critical temperature depends on it (through ε(0,B) and b), and, what is more interesting, T_cond diverges when b → 1 (B → B_c). The dependence of the critical temperature on the field put in evidence that the condensation can be reached not only by decreasing the temperature or augmenting the density, but also by increasing the magnetic field. This can be easily seeing if we compute the density of particles out of the condensate N_oc (for T<T_cond) N_oc = ε(0,B)^3/2T^3/2/2^1/2π^5/2 (2-b) Li_3/2(e^μ^'β) = N (T/T_cond)^3/2,because from Eq. (<ref>) follows that N_oc→ 0 when T → 0 but also when b → 1 (ε(0,B)→ 0).Given that a critical temperature is well defined for each value of the field (as well as there is a critical field for each temperature), it is possible to draw a T vs bphase diagram. We did so in Fig. 1 for two fixed values of the densities: N=10^30 cm^-3 and N=10^32 cm^-3,respectively. These high boson densitiesmay be assumed for compact objects. The values of the critical temperatures (the dotted lines that separate the region where the BEC appears from that where there is noBEC) are in the range of T=10^7-10^9 K which are also typical of compact objects. We can see from the graphics how T_cond grows with the augment of the density and diverges when b→1 (B→ B_c). We can also examine the transition to the condensate through the behavior of specific heat. In particular we consider the specific heat at constant volume, defined by C_v = ∂ E/∂ T, where E = T S+Ω+μ N is the internal energy of the system. The entropy of the vector bosons gas is S=-∂Ω/∂ T= -β(μ^' N +5/2Ω_st + β∂μ^'/∂β N ),with μ^'≅ -ζ(3/2)T/4 π ( 1-(T_cond/T)^3/2 )Θ(T-T_cond) in the low temperature limit. Here Θ(x) is the Heaviside theta function.With the use of Eq. (<ref>) the internal energy can be written as E = ε(0,B) N + Ω_vac- 3/2Ω_st -β∂μ^'/∂β N. Eq. (<ref>) allows us to obtain for the specific heat the following expression C_v = -β(15/4Ω_st + 3/2μ^' N +1/2β∂μ^'/∂β N- β^2 ∂^2 μ^'/∂β^2N ).The specific heat has been plotted in Fig. 2 as a function of the temperature for a fixed value of the density N=10^32 cm^-3 and three values of the magnetic field. As it is apparent, it has a maximum that is a fingerprint of the BEC phase transition. The maximum decreases and smoothes with the increment of the magnetic field, and is expected to disappear for B → B_c (b → 1), because when B=B_c the gas is condensed at any temperature and density.At this point, it is worthwhile to comment on the reasons for the difference in the nature of the BEC showed by the charged and the neutral vector boson gas in a constant magnetic field. As it was already known, the BEC in the charged gas is diffuse <cit.>, while we just found it is usual for the neutral gas. The difference arises from the reduction in the dimension suffered by the charged gas for lows temepratures or strong fields. This reduction is a consequence of the quantized spectrum, since the perpendicularmomentum component is replaced by Landau levels due to the coupling between the charge and the magnetic field. In the low temperature/high field limit the charged bosons concentrate in the lowest Landau Level (LLL) i.e, n=0 (which implies p_⊥=0), and therefore the three dimensional charged boson gas behaves as a one dimensional system <cit.>. Once the dimension is reduced, a further decrease in the temperature, or a increase in the magnetic field increases the population of the states around p_3 ≈ 0 until it becomes a macroscopic quantity. But, due to the one dimensionality of the system, a critical temperature can not be defined, and the condensation is diffuse<cit.>.For the neutral boson gas, the coupling between the field and the spin magnetic moment does not imply quantization for any momentum component. All of them are preserved as quantum observable and the three dimensionality of the neutral gas is kept for any temperature and magnetic field. For a Bose gas in three dimensions a critical temperature is well defined, and that is why it undergoes a phase transition to a usual BEC. However, it is still possible to find a formal analogy between the charged and the neutral spin-one gas in what respect to the condensation if we restrict the later to move in one dimension. §.§.§ Condensation in the one dimensional system Let us consider agas of neutral vector bosons with p_⊥=0. The statistical contribution to the thermodynamical potential has the form Ω^1D_st(B,μ,T)= -∑_s=-1,0,11/2 πβ∑_n=1^∞e^n μβ+e^- n μβ/n∫_0^∞ dp_3 e^-n βε(s), where, as for the three dimensional gas, the terms e^n μβ and e^- n μβ stand for the particles and the antiparticles respectively. The integration over p_3 can be carried out, and we obtain for Ω^1D_st(B,μ,T) the expression Ω^1D_st(B,μ,T)= -∑_s=-1,0,1m √(1- s b)/πβ∑_n=1^∞e^n μβ+e^- n μβ/n K_1(n β m √(1-s b)). K_1(x) is the first order McDonald function.Performing the sum over the spin we have Ω^1D_st(B,μ,T)= - m √(1+ b)/πβ∑_n=1^∞e^n μβ+e^- n μβ/n K_1(n β m √(1+b))-m/πβ∑_n=0^∞e^n μβ+e^- n μβ/n K_1(n β m)-m √(1-b)/πβ∑_n=0^∞e^n μβ+e^- n μβ/n K_1(n β m √(1-b)). As for the three dimensional gas, in the low temperature limit the leading term of Eq. (<ref>) comes from the particles with s=1. This leading term can be re-written as Ω^1D(B,μ,T)= - √(ε(0,B))/√(2 π)β^3/2 Li_3/2(e^μ^' 1Dβ), with μ^' 1D =μ-ε(0,B). (Besides they have the same definition, to avoid a confusion, the distinction between μ^' for three dimensions and μ^' 1D is needed.)The vacuum contribution to the 1D-thermodynamical potential after regularization is (see Appendix B) Ω^1D_vac(B)= -m^2/2 π ((1-b) log (1-b)+ (1+b)log (1+b)), and finally for the 1D-thermodynamical potential in the limit of low temperature we have Ω^1D(B,μ,T) =- √(ε(0,B))/√(2 π)β^3/2 Li_3/2(e^μ^' 1Dβ) -m^2/2 π ( (1-b) log (1-b)+ (1+b)log (1+b)). The particle density obtained from Eq. (<ref>) is N^1D = -∂Ω^1D(B,μ,T)/∂μ= √(ε(0,B))/√(2 πβ^1/2) Li_1/2(e^μ^' 1Dβ). This expression is similar to the one obtained for a three dimensional charged vector boson gas in a constant magnetic field (see Eq. (23) of<cit.>). It does not admit the substitution μ^' 1D = 0 (because Li_1/2(1)→∞), therefore, a critical temperature can not be defined and the gas does not have BEC in the usual sense. Nevertheless, in the very low temperature limit, in which μ∼ε(0,B)≫ T but T ≫μ^' 1D expression (<ref>) can be approximated as N^1D≃1/2 β√(2 ε(0,B)/-μ^' 1D). Again, an expression equivalent toEq. (<ref>) for a charged vector bosons gas has been already obtained in Ref. <cit.>. Eq. (<ref>) has a divergence when μ'→ 0 but the particle density must remain finite, thus its right interpretation is as a manner to compute μ^' 1D μ^' 1D = -ε(0,B)T^2/ N^2= -m √(1-b)T^2/ N^2. From Eq. (<ref>) follows that μ^' is a decreasing function of T, as occurs in the usual Bose-Einstein condensation, but also that μ^' 1D is a decreasing function of the field that goes to zero when B → B_c. Since μ^' 1D = 0 is the condition for the BEC to occur, μ^' 1D∼ 0 means that p_3 ∼ 0. In this regard, by the use of Eq. (<ref>) we can approximate the Bose-Einsten distribution n^+(p_3) in a vicinity of p_3 =0 as n^+(p_3) = 1/e^β(ε(p_3,B)-μ)-1≃2 ε(0,B) T/p^2 +ε(0,B)^2 - μ^2≃2 ε(0,B) T/p^2 -2 ε(0,B) μ^' 1D,n^+(p_3) ≃ 2 N γ/p_3^2 + γ^2, where γ = √(- 2 ε(0,B) μ^' 1D). Eq.(<ref>) means that for p_3 ≈ 0, n^+(p_3) tends to a Cauchy distribution centered in p_3=0 (as happens for the charged vector bosons distribution <cit.>). Now the equivalent to the limitsT → 0 orB→ B_cis the limit γ→ 0and lim_γ→ 0 n^+(p_3) = 2 π N δ(p_3).Using Eq.(<ref>) we have the following expression for the particle density N in the vicinity of p_3=0 N ≃1/2 πlim_γ→ 0∫_-∞^∞ 2 π N γ/p_3^2 + γ^2 dp_3 = N ∫_-∞^∞δ(p_3)dp_3.The delta behaviour of the Bose-Einstein distribution n^+(p_3) for low temperatures or high fields is depicted in Fig. 3. Left panel of Fig. 3 shows the boson distribution as a function of p_3 for a fixed value of temperature T=10^7 K and several values of the magnetic field. The maximum of the curves increases and move to the left when the magnetic field grows. Right panel of Fig. 3 shows the boson distribution as a function of p_3 for a fixed value of the magnetic field B=0.1 B_c anddifferent values of the temperatures.The maximum of the curves increases and shift to the left with the decrease of the temperature. Fig. 3 illustrates that although there is not a critical temperature (or field), as the limits T → 0 or B → B_c are approached, the bosons concentrate in the ground state and its neighboring states. Therefore, the system has a diffuse BEC. The region of temperatures around which the diffuse phase transition occurs may be estimated from the specific heat. To compute the specific heat, we need the energy and the entropy of the one dimensional gas. They read as S^1D=-∂Ω^1D/∂ T= -1/T(μ^' N +3/2Ω^1D_st+ 2 ε(0,B)T^2/N),E^1D = T S^1D+Ω^1D+μ N = ε(0,B)( N - 2 T^2/N ) + Ω^1D_vac- 1/2Ω^1D_st. From Eq.(<ref>) the specific heat is C^1D= ∂ E^1D/∂ T = -1/2 T ( μ^' N +3/2Ω_st + 10 ε(0,B) T^2/N ).Eq.(<ref>) has been plotted as a function of temperature in the right panel of Fig. 4. From this figure it can be seen that the specific heat has a maximum. Is position on the abscise axis can be computed as a function of b T_max = (ζ (3/2) N)^2/144 ε(0,B). As T_cond, T_max increases with the density and diverges with the magnetic field. Left panel of Fig. 4 shows T_max as a function of b for N = 10^34cm^-3. Although the condensed and not condensed region were shaded, it is worth not to forget that T_max does not define a critical temperature. Therefore, left panel of Fig. 4 is only an approximate phase diagram whose provides a range of temperatures around which the population of the ground state starts to grow. The values of these temperatures are in the order of those of several astronomical objects. §.§ Magnetization We can obtain the magnetization of the three and one dimensional systems from Eq. (<ref>) and Eq. (<ref>) if we derive in both expressions with respect to the magnetic field M = d N_0 -∂Ω_st/∂ B-∂Ω_vac/∂ B,M^1D=-∂Ω^1D_st/∂ B-∂Ω^1D_vac/∂ B. In Eq. (<ref>) N_0 is the number of particles in the condensate and d is the effective magnetic moment Eq.(<ref>). This term has to be added because in the low temperature limit all the bosons are aligned to the field and contributes to the magnetization, but for temperatures under T_cond, Ω_st only accounts for the particles that are out of the condensate.For the statistical contributions to the magnetization we have the expressions M_st=d N_0 -∂Ω_st/∂ B= κ m/ε(0,B) N - 2 κ m T^5/2/(4 π)^5/2 (2-b)^2 ε(0,B)^1/2 Li_5/2(e^βμ^'),M^1D_st=-∂Ω^1D_st/∂ B= κ m/ε(0,B) N - κ m T^3/2/2^3/2π^1/2ε(0,B)^3/2 Li_3/2(e^βμ^'),while the vacuum contributions are M_vac= -κ m^3/72 π( 7 b(b^2-6) + 3(2 b^2+2 b-7)(1-b)log(1-b)-3(2b^2-2b-7)(1+b)log(1+b) ),M^1D_vac= κ m/πlog(1+b/1-b ). It is possible to show that for T≪ m the second terms in Eq.(<ref>) and Eq.(<ref>) are negligible, as well as one can prove that the vacuum magnetization (Eqs. (<ref>) and (<ref>)) is only relevant for low particle densities at very high fields, so it can be also neglected. Finally, for both, one and three dimensions, the total magnetization of the NVBG is M = κ m/ε(0,B) N =d N. The previous expression is expected because at T≪ m all the particles are in the s=1 state.It is nothing else but the product of the effective magnetic moment by the particle density. However, an increase in the field still augment the magnetization because the effective magnetic moment d grows with B and diverges when B→ B_c (b → 1). Since d is strictly positive for all values of B, the magnetization is always positive and different from zero even if B=0 (M(B=0)=κ N).This is an evidence of ferromagnetic response of the NVBG at low temperature. This behavior described for M is shown in left panel of Fig. 5. In the seek of one of our main motivations, the search for astrophysical magnetic field sources, we are interested in exploring if the system reaches the self-magnetization condition, i.e. whether or not the solid line in left panel of Fig. 5 intersects the curves of the magnetization. To do that we consider H = B-4 π M with no external magnetic field H=0, and solve the self-consistent equation B=4π M. This is a cubic equation due to the non linear dependency of the magnetization on the field. In right panel of Fig. 5 its three solutions have been plotted but only one of them is physically meaningful. For one of the roots, the magnetic field is negative (see dotted line), while for another, it decreases with the increasing density, reaching B_c when N goes to zero (dot dashed line). These solutions implies that the magnetization also decreases with N, hence, they are contrary to Eq. (<ref>) and must be discarded. Therefore, the only admissible solution of the self-magnetization equation is the one given by the solid line. The points of this line are the values of the self-maintained magnetic field. Nevertheless, this solution becomes complex for densities higher than N_c = 7.14 × 10^34cm^-3. N_c bounds the values of particle densities for which self-magnetization is possible. The maximum field that could be self sustained by the gas corresponds to the critical density and has a magnitude of 2/3× B_c. The values of B and N for which a self-magnetization may occur are in the order of those typical of compact objects. The maximum field that can be self maintined by the NVBG is the same obtained for a gas of charged vector bosons with the same mass and magnetic moment, but in this case the critical particle density is of the order of 10^32cm^-3 <cit.>.§.§ Anisotropic Pressures We will consider the energy momentum tensor and the anisotropic pressures of the system. The total energy momentum tensor of matter plus vacuum will be obtained as a diagonal tensor whose spatial part contains the pressures and the time component is the internal energy density E. One gets from the thermodynamical potentialT^i_j=∂Ω/∂ a_i,λa_j,λ-Ωδ_j^i, T_4^4=-E, wherea_i denotes the boson or fermion fields <cit.>.For a thermodynamical potential that depends on an external field, Eq.(<ref>) leads to pressure terms of form T^i_j=-Ω-F_k^i( ∂Ω/∂ F_k^j),i=j. Computing the pressures along each direction makes the anisotropy explicit P_3=T_3=-Ω = -Ω_st -Ω_vac,P_⊥=T_1^1=T_2^2=T_⊥=-Ω-BM = P_3-BM.In left panel of Fig. 6 the perpendicular and parallel pressures are depicted as function of the field (Eqs. (<ref>)) for T=10^9 K and N=10^33 cm^-3. We also shows the statistical and the vacuum parts of the parallel pressure in dashed and dot-dashed lines respectively. The values of the parallel pressure and its statistical part (-Ω_st) coincides for B=0, but their behavior is different when the field grows. Both are always positive but the total parallel pressure increases with the field and tends to the vacuum contribution -Ω_vac, while its statistical part decreases and goes to zero for B=B_c -when all the particles are condensed the gas exerts no pressure. We would like to remark that the parallel pressure remains different form zero due to the vacuum contribution.On the contrary, the perpendicular pressure decreases (dashed line in left panel of Fig. 6) whit the magnetic field and eventually reaches negative values. This is because the main contribution to P_⊥ comes from the term -M B which is always negative and diverges in the critical field. A similar result is obtainedfor fermion gases in a magnetic field <cit.>-<cit.>. In this frame, a negative pressure can be interpreted as the system becoming unstable. Because the effect of the negative perpendicular pressure is to push the particles inward to the magnetic field axis, we could be in presence of a transversal magnetic collapse <cit.>.Whether the transversal pressure is negative or not depends on the field but also on the temperature and the particle density. This can be seeing if we examine this pressure in more detail for the self-magnetized NVBG. To do thatwe substitute in P_⊥the solution of the self-magnetization condition B = 4 π M, and plot the perpendicular pressure as a function of the particle density for several values of T (rihgt panel of Fig. 6). If we start from the lower values of N adding particles to the system increments the parallel pressure. But it also increases the self produced magnetic field and T_cond. Once T_cond becomes higher than the gas temperature, the BEC phase appears and the pressure diminishes because a fraction of the particles fall in the condensate. Besides, as the self-generated field becomes higher, thecontribution to P_⊥ of the negative term -M B becomes more and more relevant until eventually adding more particles makes the system unstable. A decrease in the temperature lowers the value of particle density where the instability starts.When the gas is not self-magnetized, but subject to an external magnetic field, an increment in the density continuously also leads the system to the instability. In this case, the increase of N does not augment the field, but it still increments the magnetization and the condensation temperature. Therefore, the NVBG will be unstable or not depending on the values of the temperature, the density and the magnetic field, regardless this field is self produced or not.The arising of an instability in the magnetized NVBG might be relevant in the description of some phenomena, as jets, that are related to the exertion of mass and radiation out of astronomical objets <cit.>. § CONCLUSIONSStarting from the Proca formalism<cit.> we computed the spectrum of a gas of neutral vector bosons in a constant magnetic field. The effective rest energy Eq. (<ref>) turns out to be a decreasing function of the magnetic field that becomes zero when it reaches certain critical value B_c=m/2 k. When the temperature is low enough, the NVBG undergoes a phase transition to a Bose-Einstein Condensation. In dependence on whether the gas is three or one dimensional, this transition is usual or diffuse. However, in one dimension as well as in three, the phase transition to the BEC is driven not only by the temperature or the density, but also by the magnetic field.The magnetization of the gas is a positive quantity that increases with the field and diverges when B=B_c for both, the three and the one dimensional cases. For particle densities under a critical value N_c ≅ 7.14 × 10^34cm^-3 the self-magnetization condition is fulfilled and the gas can maintain a self-generated magnetic field. The maximum field that can be reached by self-magnetization turns out to be 2/3 × B_c ∼ 10^13 G.The change of spherical to axial symmetry induced by the magnetic field is explicitly manifested in the spectrum of the NVBG (through the asymmetry in the momentum components) and also in the splitting of the pressures in the parallel and perpendicular directions to the field. For low values of the field, the pressure exerted by the particles has the main role in both components. However, when the magnetic field grows, the increasing parallel pressure is dominated by the positive vacuum pressure term, while the decreasing perpendicular pressure is determined by the negative magnetic pressure term -M B. For magnetic fields and particle densities high enough, or low enough temperatures, the perpendicular pressure becomes negative and an instability emerges in the system that turns out to be susceptible to suffer a transversal magnetic collapse. All these phenomena undergone by NVBG (BEC, self-sustained magnetic field and the collapsingof the transverse pressure) appear for typical values of densities and magnetic fields in compact objects. Therefore they could be relevant in modeling jets as well as the mechanism thatsustain the strong magnetic field in compact objects. These models deserve a separated treatment which is in progress.§ ACKNOWLEDGEMENTS The authors thank the comments of Maxim Chernodub to the first version of this work. G.Q.A, A.P.M and H.P.R have been supported by the grant CB0407 and acknowledge the receipt of the grant from the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. § CALCULATION OF I In order to compute the integral of the second term of Eq.(<ref>) I=∫_z_0^∞ dz z^2/√(z^2+α^2) K_1 (y z), let's introduce the following form for K_1 (yz) K_1 (yz) = 1/y z∫_0^∞ dt e^-t-y^2 z^2/4 t. If we substitute (<ref>) in (<ref>), the integration over z can be carried out I= √(π)/y^2∫_0^∞ dt √(t) e^-t+y^2 α^2/4 t erfc (y √(z^2+α^2)/2 √(t)).To integrate over t in (<ref>) we replace the complementary error function erfc(x) by its series expansion erfc(x) ⋍e^-x^2/√(π) x(1 - ∑_w=1^∞(-1)^w(2 w -1)!!/(2 x^2)^w). After the replacement and integration, I_3 can be written as I = z_0^2/y √(z_0^2 + α^2) K_2 (y z_0) - z_0^2/y √(z_0^2 + α^2)∑_w=1^∞(-1)^w(2 w -1)!!/(z_0^2+α^2)^w(z_0/y)^w K_-(w+2) (y z_0).§ VACUUM THERMODYNAMICAL POTENTIAL To obtain Eq.(<ref>) for the vacuum contribution to the thermodynamical potential we start from its definition Ω_vac=∑_s=-1,0,1∫_0^∞p_⊥dp_⊥dp_3/(2π)^2εwhere ε(p_⊥,p_3, B,s)=√(p_3^2+p_⊥^2+m^2-2κ s B√(p_⊥^2+m^2)).To integrate over p_3 and p_⊥ we use the equivalence√(a)= -1/2 √(π)∫_0^∞ dy y^-3/2 (e^- y a-1) and introduce the small quantity δ as lower limit of the integral to regularize the divergence of the a dependent term and eliminate the term that does not depends on a√(a(δ))= -1/2 √(π)∫_δ^∞ dy y^-3/2 e^-y a.Now, let's make a(δ) = ε^2 = p_3^2+p_⊥^2+m^2-2κ s B√(p_⊥^2+m^2). Consequently ε = -1/2 √(π)∫_δ^∞ dy y^-3/2 e^- y(p_3^2+p_⊥^2+m^2-2κ s B√(p_⊥^2+m^2)). By substituting Eq.(<ref>) in Eq.(<ref>) we obtain for the vacuum thermodynamical potential Ω_vac=-1/8 π^5/2∑_s=-1,0,1∫_δ^∞ dy y^-3/2∫_0^∞dp_⊥ p_⊥∫_-∞^∞dp_3 e^- y(p_3^2+p_⊥^2+m^2-2κ s B√(p_⊥^2+m^2)) After doing the gaussian integral over p_3 Eq.(<ref>) reads Ω_vac=-1/8 π^2∑_s=-1,0,1∫_δ^∞ dy y^-2∫_0^∞dp_⊥ p_⊥ e^- y(p_⊥^2+m^2-2κ s B√(p_⊥^2+m^2)). If we introduce the new variable z = √(m^2+p_⊥^2) - s κ B, Eq.(<ref>) becomes Ω_vac=-1/8 π^2∑_s=-1,0,1{∫_δ^∞ dy y^-3 e^- y(m^2-2 m s κ B) + s κ B ∫_δ^∞ dy y^-2∫_z_1^∞dz e^- y(z^2 - s^2 κ^2 B^2)}, where z_1 = m - s κ B.Eq.(<ref>) admits a further simplification if we perform a second change of variables w=z-z_1 in its last term, sum over the spin and remember that b = B/B_c with B_c = m / 2 κ Ω_vac=-1/8 π^2{∫_δ^∞ dy y^-3 e^- ym^2 (1+2 cosh[m^2 b y]) + m b ∫_δ^∞ dy y^-2∫_0^∞dw e^- y(m - w)^2sinh[m b (m - w) y] }. To take the limit δ→ 0 we subtract from 1+2 cosh[m^2 b y] and sinh[m b (m - w) y] the first terms in their series expansion and obtain for the vacuum thermodynamical potential the expression Ω_vac=-1/8 π^2∫_0^∞ dy y^-3 e^- ym^2{2 cosh[m^2 b y] - 2 - m^4 b^2 y^2 }--m b/8 π^2∫_0^∞ dy y^-2∫_0^∞dw e^- y(m - w)^2{sinh[m b (m - w) y]- m b (m - w) y - [m b (m - w) y]^3/6} that leads to Eq.(<ref>) after integration.A similar procedure was used to obtain the vacuum contribution to the thermodynamical potential for the one dimensional gas (Eq. (<ref>)). 1 Lattimer:2004pg J. M. Lattimer and M. Prakash,Science 304 (2004) 536 doi:10.1126/science.1090720 [astro-ph/0405262]. 0954-3899-36-7-075202 R. G. Felipe and A. P. Martínez, Journal of Physics G: Nuclear and Particle Physics 36, p. 075202(2009).MNL2:MNL2848 J. Charbonneau, K. Hoffman and J. Heyl, Monthly Notices of the Royal Astronomical Society: Letters 404, L119(2010).Charbonneau:2009ax J. Charbonneau and A. Zhitnitsky, JCAP 1008, p. 010(2010). Yamada Keiji Yamada, Prog Theor Phys (1982) 67 (2): 443-453. DOI:https://doi.org/10.1143/PTP.67.443 Elizabeth H. Perez Rojas, E. Rodriguez Querts, A. Perez MartinezConference Series (Quantum Relativistic electron gas expanding in one dimension), (2017). ROJAS1996148 H. Rojas, Physics Letters B 379, 148 (1996).PEREZROJAS2000 H. Perez Rojas and L. Villegas-Lelovski, Brazilian Journal of Physics 30, 410(06 2000). Khalilov:1999xd V. R. Khalilov and C. L. Ho,Phys. Rev. D 60 (1999) 033003 doi:10.1103/PhysRevD.60.033003 [hep-th/0001120]. Khalilov1997 V. R. Khalilov, C. L. Ho and C. Yang Modern Physics Letters A 12, 1973 (1997). BarguenoR. L.Delgado, R. L, P. Bargueño, F.Sols. Physical Review E, vol. 86, Issue 3, 10.1103/PhysRevE.86.031102 (2012). Jian X. Jian, J. Q.Gu J. Phys. Condens. Matter, 23, 026003 (2011).ROJAS1997 H. Rojas, Physics Letters A 234, 13 (1997).Chernodub:2010qx M. N. Chernodub,Phys. Rev. D 82 (2010) 085011 doi:10.1103/PhysRevD.82.085011 [arXiv:1008.1055 [hep-ph]]. Chernodub:2012fi M. N. Chernodub, J. Van Doorsselaere and H. Verschelde,Phys. Rev. D 88 (2013) 065006 doi:10.1103/PhysRevD.88.065006 [arXiv:1203.5963 [hep-ph]]. Satarov:2017jtu L. M. Satarov, M. I. Gorenstein, A. Motornenko, V. Vovchenko, I. N. Mishustin and H. Stoecker,arXiv:1704.08039 [nucl-th]. PhysRev.131.2326 J. A. Young and S. A. Bludman, Phys. Rev. 131, 2326 (Sep 1963).PhysRevD.89.121701 A. J. Silenko, Phys. Rev. D 89, p. 121701 (Jun 2014). PerezRojas:2006dq H. Perez Rojas and E. Rodriguez Querts,Int. J. Mod. Phys. A 21 (2006) 3761 doi:10.1142/S0217751X06031715 [hep-ph/0603254].Chaichian M. Chaichian, S. S. Masood, C. Montonen, A. Perez Martinez and H. Perez Rojas, Phys. Rev. Lett. 84 (2000) 5261;A. Perez-Martinez, H. Perez-Rojas, and H. J. Mosquera-Cuesta, Eur. Phys. J. C29, 111 (2003); S. Chakrabarty, Phys. Rev. D 54 (1996) 1306; R. Gonzalez-Felipe, A. Perez Martinez, H. Perez Rojas, and M. Orsaria, Phys. Rev. C 77, 015807 (2008); Ferrer:2015wca E. J. Ferrer, V. de la Incera, D. Manreza Paret, A. Pérez Martínez and A. Sanchez,Phys. Rev. D 91 (2015) no.8,085041 doi:10.1103/PhysRevD.91.085041 [arXiv:1501.06616 [hep-ph]]. | http://arxiv.org/abs/1706.08994v2 | {
"authors": [
"G. Quintero Angulo",
"A. Pérez Martínez",
"H. Pérez Rojas"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170627181818",
"title": "Thermodynamical properties of a neutral vector boson gas in a constant magnetic field"
} |
The optimal particle-mesh interpolation basis Xingyu Gao December 30, 2023 =============================================We consider the exit event from a metastable state for the overdamped Langevin dynamics dX_t = -∇ f(X_t) dt + √(h) dB_t. Using tools from semiclassical analysis, we prove that, starting from the quasi stationary distribution within the state, the exit event can be modeled using a jump Markov process parametrized with the Eyring-Kramers formula, in the small temperature regime h → 0. We provide in particular sharp asymptotic estimates on the exit distribution which demonstrate the importance of the prefactors in the Eyring-Kramers formula. Numerical experiments indicate that the geometric assumptions we need to perform our analysis are likely to be necessary. These results also hold starting from deterministic initial conditions within the well which are sufficiently low in energy. From a modelling viewpoint, this gives a rigorous justification of the transition state theory and the Eyring-Kramers formula, which are used to relate the overdamped Langevin dynamics (a continuous state space Markov dynamics) to kinetic Monte Carlo or Markov state models (discrete state space Markov dynamics). From a theoretical viewpoint, our analysis paves a new route to study the exit event from a metastable state for a stochastic process. § MOTIVATION AND PRESENTATION OF THE RESULTS In materials science, biology and chemistry, atomistic models are now used on a daily basis in order to predict the macroscopic propertiesof matterfrom a microscopic description. The basic ingredient is a potential energy function f: ^d → which associates to a set of coordinates of particles the energy of the system. In practice, d is very large, since the system contains many particles (from tens of thousands to millions).Using this function f, two types of models are built: continuous state space Markov models (stochastic differential equations), such as the Langevin or overdamped Langevin dynamics, and discrete state space Markov models (jump Markov processes). The objective of the analysis presented in this work is to make a rigorous link between these two types of approaches, and in particular to provide a justification of the use of Eyring-Kramers laws to parameterize jump Markov models, by studying the exit event from a metastable state for the overdamped Langevin dynamics. Jump Markov models are used by practitioners for many reasons. From a modelling viewpoint, new insights can be gained by building such coarse-grained models, that are easier to handle than a large-dimensional stochastic differential equation. From a numerical viewpoint, it is possible to simulate a jump Markov model over timescales which are much larger than the original Langevin dynamics. Moreover, there are many algorithms which use the underlying jump Markov model in order to accelerate the sampling of the original dynamics <cit.>.This section is organized as follows.First, the two models under consideration are introduced, namely the overdamped Langevin dynamics inSection <ref>, and the underlying jump Markov processin Section <ref>. Next, Section <ref> is devoted to a review of the mathematical literature dealing with metastable processes and the exit event from a metastable state. In Section <ref>, the notion of quasi stationary distribution is reviewed. This is a crucial tool in our analysis, in order to connect the overdamped Langevin dynamics to a jump Markov process. Then, in Section <ref>, our main result (Theorem <ref>)is stated.In Section <ref>, we generalize Theorem <ref> in various directions and discuss the geometric assumptions used to state Theorem <ref>. Finally, in Section <ref>, we give an outline of the proof of Theorem <ref>, together with the general organization of the paper. §.§ Overdamped Langevin dynamics and metastability The continuous state space Markov model we consider in this work is the so-called overdamped Langevin dynamics in ℝ^dd X_t = -∇ f(X_t) d t + √(h)d B_t,driven by the potential function f: ℝ^d →ℝ. We assume in the following that f is a C^∞ Morse function (all the critical points are non degenerate). The parameter h=2k_B T>0 is proportional to the temperature T and (B_t)_t≥ 0 is a standard d-dimensional Brownian motion. One henceforth assumes that∃ h_0, ∀ h < h_0, ∫_ℝ^d e^-2/hf(x) dx<∞.The invariant probability measure of (<ref>) ise^-2/hf(x)dx/∫_ℝ^d e^-2/hf(y) dy.The basic observation which motivates the use of a jump Markov model to obtain a reduced description of the dynamics (<ref>) is the following. In many practical cases of interest in biology, physics or chemistry, the dynamics (<ref>) is metastable, meaning that the process (X_t)_t ≥ 0 remains trapped for very long times in some regions (called metastable states).If a state is metastable, the way the process leaves this state should not depend too much on the way it enters the state.It is thus tempting to introduce an underlying jump process among these metastable states.Let us consider a region Ω⊂^d and the associated exit event from Ω. More precisely, let us introduceτ_Ω=inf{ t≥ 0 | X_t ∉Ω} the first exit time from Ω. The exit event from Ω is fully characterized by the couple of random variables (τ_Ω, X_τ_Ω).The focus of this work is the justification of a jump Markov process to model the exit event from the region Ω, in the small temperature regime h → 0.§.§ From the potential function to a jump Markov processThe potential function f can also be used to build a jump Markov process to describe the evolution of the system. Jump Markov models are continuous-time Markov processes with values in a discrete state space. In molecular dynamics such processes are known as kinetic Monte Carlo models <cit.> or Markov state models <cit.>.Kinetic Monte Carlo models. The basic requirement to build a kinetic Monte Carlo model is a discrete collection of states D⊂ℕ, with associated rates k_i,j≥ 0 for transitions from state i to state j,where (i,j)∈ D× D and i ≠ j. The neighboring states of state i are those states j such that k_i,j>0.The dynamics is then given by a jump Markov process (Z_t)_t ≥ 0 with infinitesimal generator L ∈^D × D, where L_i,j=k_i,j for i ≠ j.To be more precise, let us describe how to build the jump process (Z_t)_t ≥ 0 by defining the residence times (T_n)_n ≥ 0 and the subordinated Markov chain (Y_n)_n ≥ 0. Starting at time 0 from a state Y_0 ∈ D, the model consists in iterating the following two steps over n ≥ 0: given Y_n,* first sample the residence time T_n in Y_n as an exponential random variable with parameter ∑_j ≠ Y_n k_Y_n,j: ∀ i ∈ D, ∀ t>0,(T_n>t | Y_n=i) = exp(-∑_j ≠ i k_i,jt ), * and then sample independently from T_n the next visited state Y_n+1 using the following law:∀ j ≠ i,(Y_n+1=j | Y_n=i)=k_i,j/∑_j ≠ i k_i,j. The continuous time Markov process (Z_t)_t ≥ 0 is then defined as:∀ n ≥ 0,∀ t ∈[∑_m=0^n-1 T_m,∑_m=0^n T_m ), Z_t=Y_nwith the convention ∑_m=0^-1 T_m = 0.Transition rates and Eyring Kramers law. Starting from the potential function f:^d →, one approach to build a kinetic Monte Carlo model is to consider a collection of disjoint subsets (Ω_k)_k ∈ D which form a partition of the space ^d and to set the transition rates k_i,j by considering transitions between these subsets, see for example <cit.>.The concept of jump rate between two states is one of the fundamental notions in the modelling of materials. Many papers have been devoted to the rigorous evaluation of jump rates from a full-atom description. The most famous formula is probably the rate derived in the harmonic transition state theory <cit.>, which gives an explicit expression for the rate in terms of the underlying potential energy function: this is the Eyring-Kramers formula, that we now introduce.Let us consider a subset Ω of ^d, which should be thought as one of the subsets (Ω_k)_k ∈ D introduced above, say the state k=0. If Ω is metastable (in a sense which will be made precise below), it seems sensible to model the exit event from Ω using a jump Markov model, as introduced in the previous paragraph. As explained above, this requires to define jump rates (k_0,j) from the state 0 to the neighboring states j. The aim of this paper is to prove thattheexit from Ω forthe dynamics (<ref>) can be approximated using a kinetic Monte-Carlo model with transition rates computed withthe Eyring-Kramers formula:k_0,j=A_0,je^-2/h( f(z_j) - f(x_0) )where A_0,j >0 is a prefactor, x_0=min_x ∈Ω f(x) is the global minimum of f on Ω which is assumed to be unique and z_j=min_z ∈∂Ω_j f(z) where ∂Ω_j denotes the part of the boundary ∂Ω which connects the region Ω (numbered 0) with the neighboring region numbered j (see Figure <ref>for a schematic representation when Ω has 4 neighboring states). The prefactors A_0,j depend on the dynamics under consideration and on the potential function f around x_0 and z_j. If Ω is the basin of attraction of x_0 for the gradient dynamics ẋ=-∇ f(x) so that the points z_j are order one saddle points, the prefactor writes for the overdamped Langevin dynamics (<ref>)A_0,j=1/2π | λ^-(z_j)| √( det (Hess f)(x_0)/ |det (Hess f)(z_j) | ) , where λ^-(z_j) is the negative eigenvalue of the Hessian off at z_j. Theformulas (<ref>)–(<ref>) have been obtained byKramers <cit.>, but also by many authors previously, see theexhaustive review of the literature reported in <cit.>. Wealso refer to <cit.> for generalizations to the Langevin dynamics. §.§ Review of the mathematical literature on the Eyring-Kramers formulaLet us give the main mathematical approaches to rigorously derive the Eyring-Kramers formula or tostudy of the exit event from a domain for a stochastic process. See also <cit.> for a nice review.Global approaches.Some authors adopt a global approach: they look at the spectrum of the infinitesimal generator of the continuous space dynamics in the small temperature regime h → 0.It can be shown that there are exactly m small eigenvalues, m being the number of local minima {x_1,…,x_m} of f, and that each of these eigenvalues writes asymptotically when h→ 0, A_ie^-2/h (f(z_i)-f(x_i)), for i∈{1,…,m}. One thus recoversthe form of the Eyring-Kramers formula (<ref>).Here, the saddle point z_i attached to the local minimum x_i is defined by (it is here implicitly assumed that the inf sup value is attained at a single saddle point z_i)f(z_i)=inf_γ∈𝒫(x_i,B_i)sup_t ∈ [0,1] f(γ(t))where 𝒫(x_i,B_i) denotes the set of continuous paths from [0,1] to ^d such that γ(0)=x_i and γ(1) ∈ B_i with B_i the union of small balls around local minima lower in energy than x_i. For the dynamics (<ref>), we refer for example to the work <cit.> based on semi-classical analysis results for Witten Laplacian and the articles <cit.>where a potential theoretic approach is adopted. In the latter results, a connection is made between the small eigenvalues and mean transition times between metastable states. Let us also mention the earlier results <cit.>. These spectral approaches give the cascade of relevant time scales to reach from a local minimum an other local minimum which is lower in energy. They do not give any information about the typical time scale to go from one local minimum to any other local minimum (say from the global minimum to the second lower minimum for example).These global approaches can be used to build jump Markov models using a Galerkin projection of the infinitesimal generator of (X_t)_t ≥ 0 onto the first m eigenmodes, which gives an excellent approximation of the infinitesimal generator. This has been extensively investigated by Schütte and his collaborators <cit.>, starting with the seminal work <cit.>. Local approaches. In this work, we are interested in a local approach, namely the study of the exit event (exit time and exit point) from a fixed given metastable state Ω. The most famous approach to study the exit event in the small temperature limit is the large deviation theory <cit.>. It relies essentially on the study of slices ofthe processdefined with a suitable sequence of increasing stopping times. In the theory of large deviations, the notion of rate functional is fundamental and gives the cost of deviating from a deterministic trajectory. In the small temperature regime, large deviation results provide the exponential rates (<ref>), but without the prefactors and without precise error bounds. It can also be proven that the exit time is exponentially distributed in this regime, see <cit.>. For the dynamics (<ref>), a typical result on the exit point distribution is the following (see <cit.>): for all Ω' compactly embedded in Ω, for any γ >0, for any δ > 0, there exists δ_0 ∈ (0,δ] and h_0 > 0 such that for all h < h_0, for all x ∈Ω' such that f(x) < min_∂Ωf, and for all y ∈∂Ω,e^-2/h( f(y)-min_∂Ωf + γ)≤_x (X_τ_Ω∈𝒱_δ_0(y))≤ e^-2/h (f(y)-min_∂Ωf- γ)where 𝒱_δ_0(y) is a δ_0-neighborhood of y in ∂Ω. Here and in the following, the subscript x indicates that the stochastic process starts from x ∈^d: X_0=x. The strength of large deviation theory is that it is very general: it applies to any dynamics (reversible or non reversible) and in a very general geometric setting, even though it may be difficult in such general cases to make explicit the rate functional, and thus to determine the exit rates. See for example <cit.> for recent contributions to the non reversible case.Many authors have developed partial differential approachto the same problem. We refer to <cit.> for a comprehensive review. In particular, formal approaches to study the exit time and the exit point distribution have been developed by Matkowsky, Schuss and collaborators in <cit.> and by Maier and Stein in <cit.>, using formal expansions for singularly perturbed elliptic equations. Some of the results cited above actually consider more general dynamics than (<ref>). Rigorous version of these derivations have been obtained in <cit.>. Rescaling in time and convergence to a jump process.Finally, some authors prove the convergence to a jump Markov process using a rescaling in time. See for example <cit.> for a one-dimensional diffusion in a double well, and <cit.> for a similar problem in larger dimension. In <cit.>, a rescaled in time diffusion process converges to a jump Markov process living on the global minima of the potential f, assuming they are separated by saddle points having the same heights. We also refer to the recent review paper <cit.> for related results. In this work, we are interested in precise asymptotics of the distribution of X_τ_Ω. Our approach is local, justifies the Eyring-Kramers formula (<ref>) with the prefactors and provides sharp error estimates (see (<ref>)). It uses techniques developed in particular in the previous works <cit.>. Our analysis requires to combine various tools from semiclassical analysis to address new questions: sharp estimates on quasimodes far from the critical points for Witten Laplacians on manifolds with boundary, a precise analysis of the normal derivative on the boundary of the first eigenfunction of Witten Laplacians, and fine properties of the Agmon distance on manifolds with boundary.Let us finally mention that a summary of the results of this work appeared in <cit.>. §.§ Quasi stationary distributionThe quasi stationary distributionis the cornerstone of our analysis. The quasi stationary distribution can be seen as a local equilibrium for a metastable stochastic process when it is trapped in a metastable region. It is actually useful in order to make precise quantitatively what a metastable domain is. In all what follows, we focus on the overdamped Langevin dynamics (<ref>) and a domain Ω⊂^d. For generalizations to other processes, we refer to <cit.> and in particular to <cit.> for the spectral analysis of theLangevin dynamics on a bounded domain. §.§.§ Definition and a first property of quasi stationary distributionsLet us first define the quasi stationary distribution. Let Ω⊂ℝ ^d and consider the dynamics (<ref>).A quasi stationary distribution is a probability measure ν_h supported in Ω such that∀ t ≥ 0,ν_h(A)=∫_Ω_x[X_t ∈ A, t<τ_Ω]ν_h(dx) /∫_Ω_x[t<τ_Ω] ν_h(dx). In words, if X_0 is distributed according toν_h, then ∀ t>0,X_t is still distributed according toν_h conditionally onX_s ∈Ω for all s ∈ (0,t). It is important to notice that ν_h is not the invariant measure (<ref>) of the original process restricted to Ω, i.e.ν_h≠ 1_Ω(x)e^-2/hf(x)dx/∫_ℝ^d e^-2/hf(y) dy.In all the following, we will consider that Ω is a bounded domain in ^d. In this context, we have the following results from <cit.>.Let Ω⊂ℝ ^d be a bounded domain and consider the dynamics (<ref>). Then, there exists a probability measure ν_h with support in Ω such that, whatever the law of the initial condition X_0 with support in Ω, lim_t →∞ Law(X_t| t < τ_Ω) - ν_h _TV = 0. Here and in the following, Law(X_t| t < τ_Ω) denotes the law of X_t conditional to the event {t<τ_Ω}. A corollary of this proposition is that the quasi stationary distribution ν_h exists and is unique.This proposition also explains why it is relevant to consider the quasi stationary distribution for a metastable domain. The domain Ω is metastable if the convergence in (<ref>) is much quicker than the exit from Ω.In the following of this paper, we will assume that Ω is a metastable domain, and we will thus consider the exit event from Ω, assuming that X_0 is distributed according to the quasi stationary distribution ν_h. Let us also mention that we will obtain results starting form deterministic initial conditions X_0=x ∈Ω which are sufficiently low in energy (see Section <ref>).§.§.§ An eigenvalue problem related to the quasi stationary distribution In this section, a connection is made between the quasi stationary distribution and an eigenvalue problem for the infinitesimal generator of the dynamics (<ref>) L^(0)_f,h=-∇ f ·∇+ h/2 Δ with Dirichlet boundary conditions on ∂Ω. In the notation L^(0)_f,h, the superscript (0) indicates that we consider an operator on functions, namely 0-forms.Here and in the following, we assume that the domain Ω is a connected open bounded C^∞ domain in ℝ^d.Let us introduce the weighted L^2 spaceL^2_w(Ω)={u:Ω→,∫_Ω u^2(x)e^-2/h f(x)dx < ∞}(the weighted Sobolev spaces H^k_w(Ω) are defined similarly). The subscript w in the notation L^2_w(Ω) and H^k_w(Ω) refers to the fact that the weightfunction x∈Ω↦ e^-2/h f(x) appears in the inner product.The basic observation to define our functional framework is that the operator L^(0)_f,h is self-adjoint on L^2_w(Ω).Indeed, for any smooth test functions u and vwith compact supports in Ω, one has∫_Ω (L^(0)_f,hu) v e^-2/h f = ∫_Ω (L^(0)_f,hv) u e^-2/h f = - h/2∫_Ω∇ u·∇ ve^-2/h f.This gives a proper framework to introduce the Dirichlet realization L^D,(0)_f,h(Ω) on Ω of the operator L^(0)_f,h: The Friedrichs extension associated with the quadratic form ϕ∈ C^∞_c(Ω)↦h/2∫_Ω|∇ϕ|^2 e^-2/hf(x) dx, on L^2_w(Ω),is denoted -L^D,(0)_f,h(Ω). It is anon negative unbounded self adjoint operator on L^2_w(Ω) with domainD(L^D,(0)_f,h(Ω))=H^1_w,0(Ω)∩ H^2_w(Ω)where H^1_w,0(Ω)={u ∈ H^1_w(Ω), u=0on ∂Ω}.The compact injectionH^1_w(Ω)⊂ L^2_w(Ω) (which follows from <cit.>together with the fact that if a sequence (u_n)_n∈ℕ is bounded inH^1_w(Ω) then, (e^-1/hfu_n)_n∈ℕ is bounded inH^1(Ω)), implies that the operator L^D,(0)_f,h(Ω) hascompact resolvent.Consequently, its spectrum is purely discrete. Let us introduce λ_h >0 the smallest eigenvalueof -L^D,(0)_f,h(Ω). One has the following proposition, whichfollows from standard results for the first eigenfunction of anelliptic operator, see for example <cit.> and <cit.>.The smallest eigenvalue λ_hof -L^D,(0)_f,h(Ω)is non degenerate and its associated eigenfunction u_hhas a sign on Ω.Moreover u_h ∈ C^∞ (Ω).Withoutloss ofgenerality, one can assume that:u_h > 0on Ωand ∫_Ω u_h^2(x) e^-2/hf(x) dx=1. The eigenvalue-eigenfunction couple (λ_h,u_h) satisfies:{-L^(0)_f,hu_h=λ_h u_honΩ, u_h = 0 on∂Ω. .The link between the quasi stationary distribution ν_h (see Definition <ref>) and u_h is given by the following proposition (see for example <cit.>): The unique quasi stationary distribution ν_h associated with the dynamics (<ref>) and the domain Ω is given by:ν_h(dx)=u_h(x) e^-2/hf(x)/∫_Ω u_h(y) e^-2/hf(y)dydx, where u_h is the eigenfunction associated with the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>). §.§.§ Back to the jump Markov process As explained in Section <ref>, if the process remains for a sufficiently long time in the domain Ω, it is natural to consider the exit event starting from the quasi stationary distribution attached to Ω. The next proposition characterizes the law of this exit event (see for example <cit.>).Let us consider the dynamics (<ref>) and the quasi stationary distribution ν_h associated with the domain Ω. If X_0 is distributed according to ν_h, the random variables τ_Ω and X_τ_Ω are independent. Furthermore, τ_Ω is exponentially distributed with parameter λ_h and the law of X_τ_Ω has a density with respect to the Lebesgue measure on ∂Ω given byz∈∂Ω↦ - h/2λ_h∂_n u_h(z) e^-2/h f(z)/∫_Ω u_h(y) e^-2/h f(y)dy,where u_h is the eigenfunction associated with the smallest eigenvalue λ_hof -L^D,(0)_f,h(Ω) (see Proposition <ref>). Here and in the following, ∂_n=n·∇ stands for the normal derivative and n is the unit outward normal on ∂Ω.This proposition shows that, starting from the quasi-stationary distribution in the domain Ω, the exit event can be modeled by a jump Markov process without any approximation. Indeed, using the notation of Section <ref>, let us consider that Ω⊂^d is associated with the state 0. Let us assume that Ω is surrounded be n neighbourding states, associated with domains (Ω_i)_i=1,…, n (see Figure <ref> for a schematic representation when n=4). Let us define the transition rates: ∀ i ∈{1, … n},k_0,i=_ν_h(X_τ_Ω∈∂Ω∩Ω_i)/_ν_h(τ_Ω). Then, by Proposition <ref>, the exit event is such that: * The residence time τ_Ω is exponentially distributed with parameters ∑_i=1^n k_0,i.* The next visited state is independent of the residence time and is i with probability k_0,i/∑_j=1^n k_0,j.This is exactly the two properties (<ref>) and (<ref>) which are required to define a transition using a jump Markov process. The quasi stationary distribution can thus be used to parameterize the underlying jump Markov process if the domains are metastable. The question we would like to address in this work is now the following: what is the error introduced when one approximates the exact rates (<ref>) using the Eyring-Kramers formula (<ref>)–(<ref>). From Proposition <ref>, since _ν_h(τ_Ω)=1/λ_h, one has the following formula for the exact rates:k_0,i=- h/2∫_∂Ω∩∂Ω_i(∂_n u_h )(z)e^-2/h f(z) σ (dz) /∫_Ω u_h(y) e^-2/h f(y)dywhere σ denotes the Lebesgue measure on ∂Ω. We will be able to prove that in the small temperature regime h → 0, the exact rates (<ref>) can indeed beaccurately approximated by the Eyring-Kramers formula (<ref>) with explicit error bounds. The asymptotic analysis is done directly on the rates, and not only on the logarithm of the rates (which is the typical result obtained with the large deviation theory for example, see Section <ref>). §.§ Statement of the main resultWe state in this section the main result of this work (Theorem <ref>) on the asymptotic behavior of the normal derivative ∂_n u_h in the regime h → 0, as well as its corollary on the exit point density and the accuracy of the approximation of the exit rates by the Eyring-Kramers formula. This section is organized as follows. We introduce in Section <ref> a crucial tool in our analysis, the Agmon distance. Then, in Section <ref>,we give the set of hypotheses which will be needed throughout this work. Finally, Section <ref> is dedicated to the statement of our main result.§.§.§ Agmon distance Our results hold under some geometric assumptions which require to introduce the so-called Agmon distance. The objective of this section is to introduce this distance, which is particularly useful to quantify the decay of eigenfunctions away from critical points <cit.>. We introduce the Agmon distance in a general setting, namely for Ω a Riemannian manifold, but one could thinkof Ωas a C^∞ connected open bounded subset of ℝ^d.Let Ω be a C^∞ oriented connectedcompactRiemannianmanifold of dimension d with boundary ∂Ω and f: Ω→ℝ be C^∞. Define g : Ω→ℝ by∀ x ∈Ω,g(x)= |∇ f(x)| and∀ x ∈∂Ω,g(x)= |∇_T f(x)|,where for any x ∈∂Ω, ∇_Tf(x) denotes the tangential gradient of the function f on ∂Ω, i.e.∇_Tf(x)=∇ f(x)-(∇ f(x)· n) n, where n is the unit outward normalto Ω at x.One defines the length L of a Lipschitz curve γ: I→Ω, where I⊂ℝ is an interval, byL(γ,I):= ∫_I g(γ(t)) |γ'(t) |dt ∈ [0+∞]. Let us recall that theRademacher theorem (see for example <cit.>) states that every Lipschitz function admits almost everywhere a derivative (which is then bounded by the Lipschitz constant). Therefore, if I is bounded, then L(γ,I)<∞. Let us now define the Agmon distance. Let g be the function introduced in Definition <ref>. The Agmon distance between x∈Ω and y ∈Ω is defined byd_a(x,y)=inf _γ∈ Lip(x,y) L(γ,(0,1)),where Lip(x,y) is the set ofcurve γ :[0,1] →Ω which are Lipschitz with γ(0)=x, γ(1)=y.The Agmon distance is obviously symmetric, non negative and satisfies the triangular inequality. It is a distance if the critical points of f and f_ | ∂Ω are isolated (see Proposition <ref> below). Let us mention that in the case when Ω is a manifold without boundary, the Agmon distance introduced in Definition <ref> coincides with the Agmon distance defined in <cit.>. We will give in Section <ref> more details about the Agmon distance we consider. In particular, it will be shown that the Agmon distance to the critical points of f|_∂Ω coincides with the solution to the eikonal equation |∇Φ| ^2=|∇ f |^2 in neighborhoods of the critical points. This requires to use the tangential gradient of f on ∂Ω in the definition of the Agmon distance (see (<ref>)).§.§.§ Notations and hypotheses As already stated above, we assume that Ω is a connected open bounded C^∞ domain of ℝ^d and f :Ω→ℝ is a C^∞ function.[Actually, as explained in Section <ref>, we will perform the analysis in a more general setting, namely when Ω is a C^∞ oriented connected compact Riemannian manifold. In this introductory section, we stick to a simpler presentation, with Ω a subset of ^d.] We will need the following set of assumptions:[H1] The function f : Ω→ℝ is a Morse function on Ω and the restriction of f to the boundary of Ω denoted by f | _∂Ω, is a Morse function.The function f does not have any critical point on ∂Ω.[H2] The function f has a unique global minimum x_0∈Ω in Ω:min_∂Ωf>min_Ωf= min_Ωf=f(x_0). The point x_0 is the unique critical point of f in Ω.The function f|_∂Ω has exactly n ≥ 1 local minima denoted by (z_i)_i=1,…,n such that f(z_1)≤ f(z_2)≤…≤ f(z_n).[H3] ∂_n f>0 on ∂Ω.In the following, n_0 ∈{1, …, n} denotes the number of points in min f|_∂Ω:f(z_1)=… = f(z_n_0) < f(z_n_0+1) ≤…≤ f(z_n). We will need to define the basins of attraction of the local minima z_i for the dynamics ẋ=-∇_T f(x) in ∂Ω, where, we recall, for any x ∈∂Ω, ∇_T f(x) denotes the tangential gradient of f on ∂Ω.Assume that [H1]holds. For each local minimum z ∈∂Ω, one denotes by B_z⊂∂Ω the basin of attraction of z for the dynamics in ∂Ω ẋ=-∇_T f(x): denoting by φ_t(y) the solution to d/dtφ_t(y)=-∇_T f(φ_t(y)) with initial condition φ_0(y)=y∈Ω,one has B_z:= {y∈Ω,lim_t →∞φ_t(y)=z }. Notice that B_z is anopen subset of Ω. Additionally, one defines B_z^c:=∂Ω∖ B_z.From this definition, one obviously has that foreach local minimum z ∈∂Ω, for anyx ∈ B_z, f(x) ≥ f(z). On Figure <ref>, one gives a schematic representation in dimension 2 of a function f satisfying the assumptions [H1], [H2], and [H3], and of its restriction to Ω, in the casen=4 and n_0=2. As a consequence of the assumption [H1], the determinants of the Hessians of f (resp. of f|_∂Ω) at the critical points of f (resp. off|_∂Ω) are non zero. These quantities appear in the prefactors of the Eyring-Kramers law (see Equation (<ref>) below). Let us recall how the Hessians are defined.Let ϕ: N → be a C^∞ function defined on a Riemannian C^∞ manifold N of dimension d. By standard results of Riemannian geometry, the Hessian ϕ(x) of ϕ at a point x ∈ N is defined as a bilinear symmetric form acting on vectors in the tangent space T_x N as:∀ X,Y ∈Γ(T N),ϕ(X,Y) = ∇_X dϕ(Y)where ∇ is the covariant derivative (Levi-Civita connection) and dϕ is the differential of ϕ. Then, ϕ(x) is defined as the determinant of the bilinear form ϕ(x) in any orthonormal basis of T_x N.In practice, ϕ(x) can be computed at a critical point of ϕusing a local chart as follows. Let us assume that x_0 is a critical point ofϕ: d_x_0ϕ = 0. Let us introduce ψ: y∈ U ↦ψ(y)∈ V a local chart around x_0, where U ⊂^d is a neighborhood of 0, V ⊂ N is a neighborhood of x_0 and ψ(0)=x_0. Let us assume in addition that the vectors (e_i)_i=1,…,d:=(∂ψ/∂ y_i(0))_i=1,…,d are orthonormal (thus defining an orthonormal basis of T_x_0 N). Let us introduce the symmetric matrix H associated with the second order differential of ϕ∘ψ at point 0: ∀ (u,v) ∈^d ×^dD^2_0 (ϕ∘ψ) (∑_i=1^d u_i e_i, ∑_i=1^d v_i e_i )=u^T H v.Then ϕ(x_0)= H. This formula is only valid at a critical point and is a direct consequence of the definition (<ref>) of the Hessian and the explicit expression of the Levi Civita connection in the local chart ψ:∇_X dϕ(Y)|_x= ∑_i,j=1^d ( ∂^2 (ϕ∘ψ)/∂y_i ∂ y_j(y) -∑_k=1^d Γ^k_i,j(ψ(y))∂ (ϕ∘ψ)/∂y_k(y))Y_iX_jwhere x=ψ(y)∈ V, Γ^k_i,j(x) are the Christoffel symbols of the connection ∇ associated with the basis ((∂_y_jψ)(ψ^-1(x))_j=1,…,n of T_xN and (X_j)_j=1,…,n (respectively (Y_j)_j=1,…,n) are the coordinates of X (respectively Y) in this basis. §.§.§ Main result In view of equations (<ref>) and (<ref>), we need to give an estimate of three quantities in order to analyze the exit point density and the asymptotic of the transition rates in the regime h → 0: ∫_Σ (∂_n u_h)e^-2/h f for a subset Σ of ∂Ω, ∫_Ω u_h e^-2/h f and λ_h, where, we recall (λ_h,u_h) is defined by (<ref>). We will consider a subset Σ such that Σ⊂ B_z_i for a local minimum z_i (see Definition <ref> for the definition of B_z_i).Assume that [H1], [H2] and [H3] hold. Moreover assume that * ∀ i∈{1,…,n},inf_z∈ B_z_i^c d_a(z,z_i) >max[f(z_n)-f(z_i),f(z_i)-f(z_1)], * andf(z_1)-f(x_0)>f(z_n)-f(z_1). Then, for all i∈{1,…,n} and all open set Σ_i ⊂∂Ω containing z_i and such that Σ_i ⊂ B_z_i, in the limit h→ 0∫_Σ_i (∂_n u_h ) e^- 2/hf dσ=A_i(h)e^-2f(z_i)-f(x_0)/h( 1+ O(h) ), where u_h is the eigenfunction associated with the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>) which satisfies (<ref>) and A_i(h)=- ( detHessf (x_0))^1/4∂_nf(z_i) 2π^d-2/4/√( detHessf|_∂Ω (z_i) ) h^d-6/4.Let us mention that the importance of thegeometric assumptions(<ref>) and (<ref>) will bediscussed in Section <ref>. As will become clear in the proof of Theorem <ref>, it can actually be proven that for all i∈{1,…,n}, the residual r_i(h)=O(h) appearing in (<ref>) admits a full asymptotic expansion in h: there exists a sequence (b_k,i)_k ≥ 0∈ℝ^ℕ such that for all N∈ℕ, in the limit h → 0, r_i(h) = h ∑_k=0^N b_k,ih^k+ O(h^N+2). We do not state our main result with this expansion since, for general domains Ω, the explicit computations of the sequence (b_k,i)_k ≥ 0 is not possible in practice. This remark also holds for all the residuals O(h)in the next results. Assume that [H1], [H2] and [H3] hold. Then when h→ 0∫_Ω u_h(x)e^- 2/h f(x) dx= π ^d/4/( detHessf (x_0))^1/4h^d/4 e^-1/hf(x_0) (1+O(h) ),where u_h is the eigenfunction associated with the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>) which satisfies (<ref>). Assume that [H1], [H2], and [H3] hold. Then, in the limit h → 0,λ_h= √( detHessf (x_0) )/√(π h)∑_i=1^n_0∂_nf(z_i)/√( detHessf|_∂Ω (z_i) ) e^-2/h(f(z_1)-f(x_0))( 1+ O(h) ),where λ_h is the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>).Theorem <ref> is the main contribution of this work. Actually Theorem <ref> will be proven in a more general framework: namely when Ω is a C^∞ connected compact oriented Riemannian d-dimensional manifold withboundary ∂Ω. Theorem <ref>, Proposition <ref> and Proposition <ref>are respectively proved in Sections <ref>,<ref> and <ref>. For the sake of completeness, we provide a proof of Proposition <ref> in our specific setting, but this result actually holds under weaker geometric assumptions, see <cit.> or <cit.>.These results have the following consequence on the first exit point distribution and the estimate of the exact rates (k_0,i)_i=1,…,n using the Eyring-Kramers formula (see Section <ref>). We recall that (X_t)_t≥ 0 denotes the solution to (<ref>), τ_Ω is the exit time from the domain Ω and ν_h is the quasi stationary distribution associated with (X_t)_t≥ 0 and Ω.Under the hypotheses of Theorem <ref>, for i∈{1,…,n} andfor allopen sets Σ_i ⊂∂Ω containing z_i andsuch that Σ_i ⊂ B_z_i, in the limit h→ 0:ℙ_ν_h[ X_τ_Ω∈Σ_i] =∂_nf(z_i)/√( detHessf_|∂Ω (z_i) ) ( ∑_k=1^n_0∂_nf(z_k) /√( detHessf_|∂Ω (z_k) ))^-1× e^-2/h (f(z_i)-f(z_1))( 1+ O(h)).As a simple consequence of Corollary <ref>, we obtain the expectedresult that (X_t)_t ≥ 0 leaves Ω around the global minima of f on ∂Ω:for any collection of open sets (Σ_j)_1 ≤ j ≤ n_0 such that for all j∈{1,…,n_0}, Σ_j ⊂ B_z_j and z_j ∈Σ_j, inthe limit h→ 0, ℙ_ν_h [ X_τ_Ω∈⋃_j=1^n_0Σ_j] =1+ O (h).Actually, this latterresult can be proven with an exponentially small residual (O (h) is replaced by O (e^-c/h) for some positive c) in a more general setting (see for instance <cit.>). Let us also refer to <cit.> where we discuss this result in a more general setting (for example f can have several critical points in Ω and the assumptions (<ref>) and (<ref>) are not needed). Let us assume that the hypothesesofTheorem <ref> are satisfied.Leti∈{1,…,n} and Σ_i ⊂∂Ω be an open setcontaining z_isuch that Σ_i ⊂ B_z_i.Using the notation of Section <ref>, assume that Σ_i is the common boundary between Ω and another domain Ω_i⊂ℝ^d.Under the hypotheses of Theorem <ref>, the transition rate given by (<ref>), to go from Ω to Ω_i satisfies, in the limit h → 0,k_0,i=1/√(π h)∂_nf(z_i)√( detHessf(x_0) )/√( detHessf_|∂Ω (z_i) ) e^-2/h(f(z_i)-f(x_0)) (1+ O (h)).This corollary thus gives a justification of the Eyring-Kramers formula and the Transition State Theory to build Markov models. As stated in the assumptions, the exit rates are obtained assuming ∂_n f > 0 on ∂Ω: the local minima z_1, … ,z_n of f on ∂Ω are therefore not saddle points of f but so-called generalized saddle points (see <cit.>). Thisappellation ”generalized saddle points” is justified by the fact that, under[H1], [H2], [H3] and when f is extended by -∞ outside Ω (whichis consistent with the Dirichlet boundary conditions used to define L^D,(0)_f,h), the points (z_i)_i=1,…,n are geometrically saddle points of f: z_i is a local minimum of f|_Ω and alocal maximum of f|_D_i, where D_i isthe straight line passing through z_i and orthogonal toΩ at z_i.In a future work, we intend to extend these results to the case where the points (z_i)_1 ≤ i ≤ n are saddle points of f, in which case we expect to prove the same result (<ref>) for the exit rates, with a modified prefactor:A_0,i=1/π|λ^-(z_i)| √( f(x_0))/√(|f(z_i)|)(this formula can be obtained using formal expansions on the exit time and Laplace's method). Notice that the latter formula differs from (<ref>)–(<ref>) by a multiplicative factor 1/2 since λ_h is the exit rate from Ω and not the transition rate to one of the neighboring state. Concerning this multiplicative factor 1/2,we refer for exampleto the remark on page 408 in <cit.>, <cit.>, and the results on asymptotic exit times in <cit.>. This factor is due to the fact that once on the saddle point, the process has a probability one half to go back to Ω, and a probability one half to effectively leave Ω, in the limit h→0. This multiplicative factor does not have any influence on the law of the next visited state which only involves ratio of the rates k_0,i, see Section <ref> and Equation (<ref>). §.§ Discussion and generalizations As explained above, the interest of Theorem <ref> is that it justifies the use of the Eyring-Kramers formula to model the exit event using a jump Markov model including the prefactors. It gives in particular the relative probability to leave Ω through each of the local minima z_i of f on the boundary ∂Ω. Moreover, one obtains an estimate of the relative erroron the exit probabilities (and not only on the logarithm of the exit probabilities as in (<ref>)): it is of order h, see Equation (<ref>).In Section <ref>, we explain how this result can be generalized to a situation where the process (X_t)_t ≥ 0 is assumed to start under another initial condition than the quasi stationary distribution.The importance of the geometric assumption (<ref>)-(<ref>) (resp. assumption (<ref>)) to obtain theasymptotic result of Corollary <ref> (resp. its generalization to deterministic initial conditions, see Corollary <ref>) is discussed in Section <ref>. Finally, in Section <ref>, we discuss extensions to less stringent conditions than (<ref>)-(<ref>). Moreover the exit through subsets of ∂Ω which do not necessarily contain one of the local minima z_i of f|_∂Ω is considered: this shows in particular the interest of estimating the prefactors in the asymptotic approximations of the exit rates.§.§.§ Extension of the result to other initial conditions The question we would like to address in this section is how to generalize Corollary <ref>,to a deterministic initial condition: X_0=x for x ∈Ω. Let us assume that all the hypotheses of Corollary <ref> are satisfied, and that in addition there exists i_0∈{2,…,n} such that 2( f(z_i_0)-f(z_1))<f(z_1)-f(x_0). Let j∈{1,…,i_0} and α∈ be such that f(x_0) < α< 2 f(z_1)-f(z_j).Then, for i∈{1,…,j} and for allopen sets Σ_i ⊂∂Ω containing z_i andsuch that Σ_i ⊂ B_z_i, we have uniformly in x∈ f^-1( (-∞, α ]) ∩Ω, in the limit h→ 0:_x [ X_τ_Ω∈Σ_i]=∂_nf(z_i)/√( detHessf_|∂Ω (z_i) ) ( ∑_k=1^n_0∂_nf(z_k) /√( detHessf_|∂Ω (z_k) ))^-1e^-2/h (f(z_i)-f(z_1)) ( 1 + O(h) ). Let us give a simple example to illustrate this result. In a situation where n=2, this corollary shows that the estimates we have obtained on the probability to exit in a neighborhood of z_2 under the assumption X_0 ∼ν_h are still valid if X_0=x for x ∈ f^-1((-∞,2f(z_1)-f(z_2))) ∩Ω under the assumption f(z_1) - f(x_0) > 2 (f(z_2)-f(z_1)), which is a stronger assumption than (<ref>).§.§.§ On the geometric assumptions (<ref>), (<ref>) and (<ref>) On the geometric assumption (<ref>). The question we would like to address is the following: isthe assumption (<ref>) necessary for the result on the exit point density (<ref>) to hold?In order to test this assumption numerically, we consider the following simple two-dimensional setting. The potential function isf(x,y)=x^2+y^2-ax,with a∈(0,1/9), and the domain Ω is defined by (see Figure <ref>):Ω=(-1,1)^2∪{(x,y)| x^2+(y-1)^2<1}∪{(x,y)| x^2+(y+1)^2<1}.The two local minima of f on ∂Ω are z_1=(1,0) and z_2=(-1,0). Notice that f(z_2)-f(z_1)=2a>0. The potential f has a unique critical point in Ω, namely the global minimum x_0=(a/2,0). Let us check that the assumptions of Theorem <ref> are satisfied in this setting (i.e. for a∈ (0,1/9)). Indeed, the inequality f(z_1)-f(x_0)>f(z_2)-f(z_1) is satisfied if and only if 1-3a+a^2/4>0 i.e. if and only if a∉(2(3-√(8)), 2(3+√(8))). Moreover, using Proposition <ref>, the inequality d_a(z_1,B_z_1^c)>f(z_2)-f(z_1) is satisfied. Finally, to check that the inequalityd_a(z_2,B_z_2^c)>f(z_2)-f(z_1) is satisfied we use Proposition <ref> with W={(x,y)∈ℝ^2,| (x,y)-z_2|≤1/3}∩Ω and W'={(x,y)∈ℝ^2,| (x,y)-z_2|≤2/3}∩Ω. In that case, one has α=1/3 (where α is defined by (<ref>)) and thusthe inequality αinf_x∈W'∖W g(x)=1/3min ( 2/3, | 2 (-1+2/3)-a|)=1/3min( 2/3, 2/3+a ) >f(z_2)-f(z_1) =2ais satisfied if and only if a<1/9.Let us consider the segment Σ_2 joining the two points (-1,-1) and (-1,1). This subset of ∂Ω contains the highest saddle point z_2 and is included in B_z_2. From Theorem <ref>, we expect that, in the limit h → 0,_ν_h[X_τ_Ω∈Σ_2] = exp(G(2/h)) (1+O(h))whereG(x)=ln[∂_n f(z_2) √( detHessf|_∂Ω(z_1) )/∂_n f(z_1) √( detHessf|_∂Ω (z_2) )]-x(f(z_2)-f(z_1)).The function G is compared for various value sod hto the numerically estimated function F defined byF(2/h)=ln( _ν_h[X_τ_Ω∈Σ_2]).In practice, the quasi stationary distribution ν_h is sampled using a Fleming-Viot particle system (the convergence diagnostics is based ona Gelman-Rubin statistics, see <cit.>) composed of 10^5 particles. The probability ℙ_ν_h(X_τ_Ω∈Σ_2) is estimated using a Monte Carlo procedure using 6× 10^5 particles distributed according to the quasi stationary distribution ν_h. The dynamics (<ref>) is discretized in time using an Euler-Maruyama scheme with a timestep Δ t which is made precise in the captions of the figures. On Figures <ref> and <ref>, we observe an excellent agreement between the theory and the numerical results.Now, the potential function f is modified such that the assumption (<ref>) is not satisfied anymore. More precisely, the potential function is f(x,y)=(y^2-2a(x) )^3, witha(x)=a_1x^2+b_1x+ 0.5 where a_1 and b_1 are chosen such thata(-1+δ)=0, a(1)=1/4for δ=0.05. We have f(z_1)=-1/8 and f(z_2)=-8 (a(-1))^3> 0 > f(z_1). Moreover, two 'corniches' (which are in the level set f^-1({0}) of f, and on which |∇ f|=0) on the 'slopes of the hills' of the potential f join the point (-1+δ,0) to B_z_2^c (at the points (1,-1/√(2))∈ B_z_2^c and (1,1/√(2))∈ B_z_2^c) so that inf_z∈ B_z_2^c d_a(z,z_2) < f(z_2)-f(z_1). Indeed, in that case assumption (<ref>) is not satisfied since inf_z∈ B_z_2^c d_a(z,z_2) ≤ d_a(z_2, (1,1/√(2)))≤ d_a(z_2, (0,-1+δ))+ d_a((0,-1+δ), (1,1/√(2)))=f(z_2)-f(0,-1+δ)+0=f(z_2) < f(z_2)-f(z_1).Notice that the Hessians ( f|_∂Ω)(z_1) and (f|_∂Ω)(z_2) are nonsingular. The functions f_|Ω and f|_∂Ω are not Morse functions, but an arbitrarilysmall perturbation (which we neglect here)turns them into Morse functions. When comparing the numerically estimated probability _ν_h(X_τ_Ω∈Σ_2), with the theoretical asymptotic result in the limit h→ 0, we observe a discrepancy on the prefactors, see Figure <ref>. Therefore, it seems that assumption (<ref>) is indeed required to get an accurate description of the dynamics by the jump Markov process using the Eyring-Kramers law to estimate the rates between the neighboring states. On the geometric assumptions (<ref>) and (<ref>). To discuss the necessity of the assumptions (<ref>) in Corollary <ref> and (<ref>) in Corollary <ref>, we consider a one-dimensional case, where the law ofX_τ_Ω when X_0=x has an explicit expression. Let f:ℝ→ℝ be C^∞ and let z_1,z_2∈ℝ such that z_1<z_2. Let us assume that f'(z_1)<0, f'(z_2)>0, f(z_1)<f(z_2) and f has only one critical point in (z_1,z_2) denoted by x_0. This implies in particular that f(x_0)=min_[z_1,z_2]f<f(z_1). Moreover let us assume that f”(x_0)>0. Therefore, the hypotheses [H1]-[H2]-[H3] hold. For x∈ [z_1,z_2], let us denote by w_h(x)=_x[X_τ_(z_1,z_2)=z_2]. It is standard that using a Feynman-Kac formula, w_h solves the elliptic boundary value poblemh/2 w_h” - w_h'f'=0,w_h(z_1)=0, w_h(z_2)=1. Therefore, one has for x∈ [z_1,z_2]: w_h(x)=∫_z_1^x e^2/h f/∫_z_1^z_2 e^2/h f.Let x∈ [z_1,z_2]. Using Laplace's method, if f(x)<f(z_1), one obtains in the limit h→ 0:_x[X_τ_(z_1,z_2)=z_2]=-f'(z_2)/f'(z_1)e^-2/h (f(z_2)-f(z_1))(1+O(h)),if f(x)=f(z_1), x≠ z_1,it holds in the limit h→ 0:_x[X_τ_(z_1,z_2)=z_2]=f'(z_2) (1/f'(x)-1/f'(z_1) )e^-2/h (f(z_2)-f(z_1))(1+O(h)),and if f(x)>f(z_1), it holds in the limit h→ 0:_x[X_τ_(z_1,z_2)=z_2]=f'(z_2)/f'(x)e^-2/h (f(z_2)-f(x))(1+O(h)).Therefore, in dimension one, the estimate (<ref>) holds if and only if x∈ f^-1( (-∞, f(z_1))). In accordance with Corollary <ref>, the asymptotic (<ref>) only holds for initial conditions which are sufficiently low in energy. However, we observe that in this simple one-dimensional setting, the assumption (<ref>) is not needed. We do not know if the result of Corollary <ref> would hold in general without the assumption (<ref>). Let us now discuss the assumption (<ref>) in the framework of Theorem <ref> and Corollary <ref>.From (<ref>), one has:_ν_h[X_τ_(z_1,z_2)=z_2]=∫_z_1^z_2 u_h w_h e^-2/h f/∫_z_1^z_2 u_he^-2/h f.Using Lemma <ref>, Lemma <ref> and (<ref>), one hasfor some c>0, for any δ>0 and for h small enough: u_h(x)= χ(x)/√(∫_z_1^z_2χ^2 e^-2/h f)(1+α_h)+r(x),for x∈Ω with α_h∈ℝ, α_h=O(e^-c/h),∫_z_1^z_2 r^2 e^-2/h f=O(e^-2/h (f(z_1)-f(x_0)-δ)) and where χ∈ C^∞_c(z_1,z_2) is given by Lemma <ref>.Therefore, one has: _ν_h[X_τ_(z_1,z_2)=z_2] =1/∫_z_1^z_2 u_he^-2/h f [ ∫_z_1^z_2χ (x) ∫_z_1^x e^2/h (f(y)-f(x))dydx /∫_z_1^z_2 e^2/h f√(∫_z_1^z_2χ^2 e^-2/h f) (1+α_h) + ∫_z_1^z_2 rw_he^-2/h f].Using Proposition <ref> and Laplace's method, one gets for any δ>0, in the limit h→ 0:_ν_h[X_τ_(z_1,z_2)=z_2]=-f'(z_2)/f'(z_1)e^-2/h (f(z_2)-f(z_1))(1+O(h))+O(e^-1/h (f(z_2)-f(x_0)+f(z_1)-f(x_0)-δ)). Therefore, the result of Corollary <ref> holds if 2(f(z_1)-f(x_0))>f(z_2)-f(z_1).This explicit computation in dimension one shows that the result of Corollary 1 indeed requires an assumption of the type: the height of the energy barrier to leave the well f(z_1)-f(x_0) is sufficiently large compared to the largest difference in energy of the saddle points f(z_2)-f(z_1). Notice that (<ref>) differs from (21) by a multiplicative factor 1/2. We do not know if the result of Corollary <ref> would hold in general under the weaker assumption (<ref>). Finally, let us mention that when d=1, (<ref>) is always satisfied. §.§.§ Extension of the results to a subset of generalized saddle points and to more general subsets of ∂Ω It is actually possible to generalize the result of Theorem <ref> and Corollary <ref> to less stringent conditions than (<ref>)-(<ref>) and to more general subsets Σ⊂∂Ω.Assume that [H1], [H2] and [H3] hold. Assume that there exist k_0∈{1,…,n} and f^*∈ℝ such that f(z_k_0) ≤ f^*≤ f(z_k_0+1)(with the convention f(z_k_0+1)=+∞ if k_0=n), {[c]∀ i∈{1,…,k_0}, inf_z∈ B_z_i^c d_a(z,z_i) >max[f^*-f(z_i),f(z_i)-f(z_1)],∀ i∈{k_0+1,…,n},inf_z∈ B_z_i^c d_a(z,z_i) >f^*-f(z_1),.and, f(z_1)-f(x_0)>f^*-f(z_1).Let u_h be the eigenfunction associated with the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>) which satisfies (<ref>). * For all i∈{1,…,k_0} and for all smooth open set Σ_i ⊂∂Ω containing z_i and such that Σ_i ⊂ B_z_i, the limit (<ref>) holds for ∫_Σ_i( ∂_n u_h)e^- 2/hf dσ and the limit (<ref>) holds for _ν_h[ X_τ_Ω∈Σ_i]. Moreover, if f(z_k_0+1)>f(z_k_0), for all i∈{k_0+1,…,n} and for all smooth open set Σ_i ⊂∂Ω containing z_i and such that Σ_i ⊂ B_z_i, there exist >0 and h_0>0 such that for all h∈ (0,h_0) ∫_Σ_i (∂_nu_h)e^-2/hf dσ = (∫_Σ_k_0 (∂_nu_h)e^-2/hf dσ) O ( e^-/h),and_ν_h[ X_τ_Ω∈Σ_i]=_ν_h[ X_τ_Ω∈Σ_k_0] O ( e^-/h) * Let j_0 ∈{1, …, k_0} and Σ⊂∂Ω be a smooth open set such that Σ⊂ B_z_j_0 and inf_Σ f=f^*. Let (B^*,p^*)∈ℝ_+^*×ℝ be such that∫_Σ (∂_nf) e^-2/h fdσ = B^*h^p^* e^-2/h f^*( 1+ O(h) ). Then, one obtains in the limit h→ 0 ∫_Σ(∂_n u_h)e^- 2/h f dσ=- 2B^* ( detHess f(x_0))^1/4/π^d/4h^p^*-d/4-1e^-1/h(2f^*-f(x_0))( 1+ O(h) ) and _ν_h [ X_τ_Ω∈Σ]= B^*/π^d-1/2 ( ∑_k=1^n_0∂_nf(z_k) /√( detHessf_|∂Ω (z_k) ))^-1h^p^*-d-1/2 e^-2/h( f^*-f(z_1))( 1+ O(h) ).In practice, the expansion (<ref>) is given by Laplace's method. Theorem <ref> is a generalization of Theorem <ref>. Indeed, (<ref>)-(<ref>) is weaker than (<ref>)-(<ref>) ((<ref>)-(<ref>) implies (<ref>)-(<ref>) for k_0=n and f^*=f(z_n)) and item 2 gives an asymptotic result on the exit probability through Σ⊂ B_z_j_0 even if z_j_0∉Σ.As an illustration, let us state a corollary of this theorem, which demonstrates the importance of obtaining a precise asymptotic result including the prefactors.Let us consider a simple situation with only two local minima z_1 and z_2 on the boundary, with f(z_1) < f(z_2). Let us now compare the two exit probabilities (see Figure <ref> for a schematic representation of the geometric setting): * The probability to leave through Σ_2 such that Σ_2⊂ B_z_2 andz_2 ∈Σ_2;* The probability to leave through Σ such that Σ⊂ B_z_1 and inf_Σ f=f(z_2).By classic results from the large deviation theory (see for example (<ref>)) the probability to exit through Σ and Σ_2 both scale like a prefactor times e^-2/h(f(z_2)-f(z_1)):the difference can only be read from the prefactors. Actually, using item 2 in Theorem <ref>, one obtains the existence of C>0 such thatin the limit h → 0 (see Corollary <ref> below),_ν_h(X_τ_Ω∈Σ)/_ν_h(X_τ_Ω∈Σ_2)∼ C √(h) .The probability to leave through Σ_2 (namely through the generalized saddle point z_2) is thuslarger than through Σ, even though the two regions are at the same height. This result explains why the local minima of f on the boundary (namely the generalized saddle points) play such an important role when studying the exit event. Let us now state the precise result. Assume [H1], [H2],[H3]. Assume that f|_∂Ω has only two local minima z_1 and z_2 such that f(z_1)<f(z_2) and, for j∈{1,2}, inf_z∈ B_z_j^c d_a(z,z_j) >f(z_2)-f(z_1),and f(z_1)-f(x_0)>f(z_2)-f(z_1).Let Σ⊂∂Ω be a smooth open set such that Σ⊂ B_z_1. Assume moreover that inf_Σ f=f(z_2) and that the infimum is attained at a single point z^*: inf_Σf=f(z^*) (necessarily z^*∈∂Σ). Finally, let us assume that z^* is a non degenerate minimumof f_|∂Σ and ∂_n(∂Σ) f_|∂Σ(z^*)<0 where n(∂Σ) is the unit outward normal to ∂Σ⊂∂Ω. Then, one has the following asymptotic expansion of _ν_h[ X_τ_Ω∈Σ]in the limit h → 0:_ν_h[ X_τ_Ω∈Σ] =-√(h)/2√(π)∂_nf(z^*)/∂_n(∂Σ)f(z^*) √( detHessf_|∂Σ (z^*) ) ( ∑_k=1^n_0∂_nf(z_k) /√( detHessf_|∂Ω (z_k) ))^-1× e^-2/h(f(z_2)-f(z_1))(1 +O(h) ), with by convention, detHessf_|∂Σ (z^*) =1 if d=2.Corollaries <ref>, <ref> and <ref> imply the result (<ref>) announced above.By using Laplace's method, one can check that the asymptotic results obtained in Corollaries <ref>, <ref> and <ref> on the law of X_τ_Ωimply that the density of X_τ_Ω with respect to the Lebesgue measure on ∂Ω is, in the limit h → 0,z ∈∂Ω↦∂_nf(z) e^-2/h f(z)/∫_∂Ω∂_nf e^-2/h fdσ (1+O(h)).This indeed yieldsthe same asymptotic limits on the exit distribution. This is reminiscent of previous results obtained in<cit.>, where the authors proved that, starting from a deterministic initial condition in Ω, X_τ_Ω has a density with respect to the Lebesgue measure on ∂Ω which satisfies, in the limit h → 0, z ∈∂Ω↦∂_nf(z) e^-2/h f(z)/∫_∂Ω∂_nf e^-2/h fdσ + o(1), which is however a less precise estimate than (<ref>).§.§ Strategy for the proof of Theorem <ref> and outline of the paperThe aim of this section is to give an overview of the strategy for the proof of Theorem <ref>. In view of (<ref>), we would like to identify the asymptotic behavior of the normal derivative ∂_n u_h on ∂Ω in the limit h → 0. We recall that (λ_h,u_h) satisfiesthe eigenvalue problem (<ref>). By differentiating (<ref>), we observe that ∇ u_h satisfies{ L^(1)_f,h∇ u_h= - λ_h ∇ u_hon Ω, ∇_T u_h = 0on ∂Ω, (h/2 div - ∇ f ·) ∇ u_h = 0on ∂Ω, .whereL^(1)_f,h= h/2Δ - ∇ f ·∇ -Hessfis an operator acting on 1-forms (namely on vector fields). Therefore ∇ u_h is an eigen-1-formofthe operator -L^(1)_f,h with tangential Dirichlet boundary conditions (see (<ref>)), associated with the small eigenvalue λ_h. It is known (see for example <cit.>) that in our geometric setting, -L^D,(0)_f,h(Ω) admits exactly one eigenvalue smaller than √(h)/2, namely λ_h with associated eigenfunction u_h (this is because f has only one local minimum in Ω) and that -L^D,(1)_f,h(Ω) admits exactly n eigenvalues smaller than √(h)/2(where, we recall, n is the number of local minima of f on ∂Ω). Actually, all these small eigenvalues are exponentially small in the regime h → 0, the other eigenvalues being bounded from below by a constant in this regime. The idea is then to construct an appropriate basis (with so called quasi-modes, which are localized on the generalized saddle points (z_i)_i =1, … ,n) of the eigenspace associated with small eigenvalues for L^D,(1)_f,h(Ω), and then to decompose ∇ u_h along this basis.The article is organized as follows. In Section <ref>, we introduce the general setting for the proof of our results, and the Gram-Schmidt procedure which allows, starting from a set of quasi-modes, to compute the projection of (an approximation of) ∇ u_h along the quasi-modes. In order toquantify the distance between the space spanned by these quasi-modes and the eigenspace of L^D,(1)_f,h(Ω) associated with small eigenvalues, we need to use so-called Agmon estimates. Section <ref> is devoted to a presentation of the main properties of the Agmon distance which intervenes in these estimates. The most technical part is the effective construction of the quasi-modes using auxiliary simpler eigenvalue problems associated with each of the local minima (z_i)_i =1, … ,n. This is explained in Section <ref> which concludes the proof of Theorem <ref>. Finally, Section <ref> concludes the paper by providing the proofs of all the other results stated above, in particular Theorem <ref>. For the ease of the reader, a list of the main notation used in this work is provided at the end of this work.§ GENERAL SETTING AND STRATEGY FOR THE PROOF OF THEOREM <REF> The general setting for proving the results presented in Section <ref> will be the following: Ω is a C^∞ oriented connectedcompactRiemannianmanifold of dimension d with boundary ∂Ω and the function f is a C^∞ real valued function defined on Ω. One defines Ω:=Ω∖∂Ω. In particular, Theorem <ref> will actually be proven in this framework. Notice that the assumptions [H1], [H2] and [H3]are still meaningful in this more general setting. In order to use previous results from the literature on semi-classical analysis, we will transform the original problem (<ref>) on (λ_h,u_h) associated with weighted Hilbert space H^q_w(Ω) to an eigenvalue problem on the standard (non-weighted) Hilbert spaces H^q(Ω), by using a unitary transformation which relates the operator L^(p)_f,h to the Witten Laplacians Δ^(p)_f,h. This is explained in Section <ref>, together with some first well-known results on the spectrum of Witten Laplacians. Then, in Section <ref>, we explain what are the requirements on the quasi-modes we will build in order to obtain the estimate (<ref>), see Proposition <ref>. Section <ref> is finally devoted to the proof of Proposition <ref>. §.§ Witten Laplacians §.§.§ Notations for Sobolev spaces For p∈{0,…,d}, one denotes by Λ^pC^∞(Ω) the space of C^∞ p-forms on Ω. Moreover, Λ^pC^∞_T(Ω) is the set of C^∞ p-forms v such that tv=0 on ∂Ω, where 𝐭 denotes the tangential trace on forms (see for instance <cit.>). Likewise, the set Λ^pC^∞_N(Ω) is the set of C^∞ p-forms v such that n v=0 on ∂Ω, where 𝐧 denotes the normal trace on forms defined by: for all w∈Λ^pC^∞(Ω), 𝐧 w:=w|_∂Ω-𝐭 w. For p∈{0, …,d} and q∈ℕ, one denotes by Λ^pH^q_w(Ω) the weighted Sobolev spaces of p forms with regularity index q, for the weight e^-2/h f(x)dx on Ω: v∈Λ^pH^q_w(Ω) if and only if for all multi-index α with |α|≤ q, the α derivative of v is in Λ^pL^2_w(Ω) where Λ^pL^2_w(Ω) is the completion of the space Λ^pC^∞(Ω) for the norm w∈Λ^pC^∞(Ω)↦√(∫_Ω| w|^2e^-2/h f). See for example <cit.> for an introduction to Sobolev spaces on manifolds with boundaries. For p∈{0,…,d} and q> 1/2, the set Λ^pH^q_w,T(Ω) is defined by Λ^pH^q_w,T(Ω):= {v∈Λ^pH^q_w(Ω)| 𝐭v=0 on ∂Ω}. Notice that Λ^pL^2_w(Ω) is the space Λ^pH^0_w(Ω), and that Λ^0H^1_w,T(Ω) is the space H^1_w,0(Ω) that we introduced in Proposition <ref> to define the domain of L^D,(0)_f,h(Ω). Likewise for p∈{0,…,d} and q> 1/2, the set Λ^pH^q_w,N(Ω) is defined by Λ^pH^q_w,N(Ω):= {v∈Λ^pH^q_w(Ω)| 𝐧v=0 on ∂Ω}.We will denote by ‖ . ‖_H^q_wthe norm on the weighted space Λ^pH^q_w (Ω). Moreover ⟨· , ·⟩_L^2_w denotes the scalar product in Λ^pL^2_w (Ω).Finally, we will also use the same notation without the index w to denote the standard Sobolev spaces defined with respect to the Lebesgue measure on Ω .§.§.§ Witten Laplacians on a manifold with boundary Let us first recall some basic properties of Witten Laplacians, as well as the link between those and the operators L^(p)_f,h introduced above (see (<ref>) and (<ref>)). As already explained above, we will actually need in this article to work only with 0 and 1-forms (p∈{0,1}). For an introduction to the Hodge theory and the Hodge Laplacians on manifolds with boundary, one can refer to <cit.>.Denote by d the exterior derivative on Ω,d^(p): Λ^pC^∞(Ω )→Λ^p+1C^∞(Ω),and (d^(p))^* its adjoint. The exterior derivative is 2 nilpotent, d^(p+1)∘ d^(p) =0,and therefore (d^(p))^*∘ (d^(p+1))^* =0. In all what follows, the superscript (p) may be omitted when there is no ambiguity.Let us now introduce the so called distorted exterior derivative d^(p)_f,h:=e^-f/hhd^(p)e^f/h=hd^(p)+df∧and its formal adjoint(d^(p)_f,h)^*:=e^f/h h (d^(p))^*e^-f/h=h(d^(p))^*+𝐢_∇ f.The distorted exterior derivative was firstly introduced by Witten in <cit.>.The Witten Laplacian is the non negative differential operatorΔ^(p)_f,h:=(d^(p)_f,h+(d^(p)_f,h)^*)^2. Anequivalentexpression of the Witten Laplacians is Δ^(p)_f,h=h^2 Δ_H^(p) + |∇ f|^2 + h(ℒ_∇ f + ℒ^*_∇ f),where ℒ stands for the Lie derivative, ∇ is the covariant derivative associated to the metric on Ω and Δ_H^(p) is the Hodge Laplacian acting on p-forms, defined by:Δ_H^(p):= (d^(p)+(d^(p))^*)^2.We recall that Δ_H^(p) is a positive operator (in ^d, Δ_H^(0)=-∑_i=1^d ∂^2_x_i,x_i). The operator ℒ_∇ f + ℒ^*_∇ f is an operator of order 0 (namely a multiplicative operator). On 0-forms, namely on functions, the Witten Laplacian has the following expression Δ^(0)_f,h=h^2 Δ_H^(0) + |∇ f|^2 + h Δ_H^(0)f.Let usnow make precise the natural Dirichlet and Neumann boundary conditionsfor Witten Laplacians on a manifold with boundary(see <cit.>). The Dirichlet realization of Δ^(p)_f,h on Ω is the operator Δ^D,(p)_f,h(Ω) with domainD(Δ^D,(p)_f,h(Ω))={v ∈Λ^pH^2(Ω)|𝐭 v=0,𝐭 d^*_f,h v=0 }.The Neumann realization of Δ^(p)_f,h on Ω is the operator Δ^N,(p)_f,h(Ω) with domainD(Δ^N,(p)_f,h(Ω))={v ∈Λ^pH^2(Ω)|𝐧 v=0,𝐧 d_f,hv=0 }. The operators Δ^D,(p)_f,h(Ω) and Δ^N,(p)_f,h(Ω) are both self adjoint operators with compact resolvent. Werecall that t denotes the tangential trace on forms and that nω=ω -tω is the normal trace. The proof of Proposition <ref> can be found in <cit.> and in <cit.>. It is a generalization of what is stated in <cit.> for the Hodge Laplacians. One can check that the operator Δ^D,(p)_f,h (Ω ) is actually the Friedrichs extension associated to the quadratic formv ∈Λ^pH^1_T (Ω ) ↦‖ d^(p)_f,hv‖_ L^2^2 +‖ (d^(p)_f,h)^*v‖_ L^2^2. The following properties are easily checked for v∈ D(Δ^D,(p)_f,h(Ω)) such that d_f,hv∈ D(Δ^D,(p+1)_f,h(Ω)) and d_f,h^*v∈ D(Δ^D,(p-1)_f,h(Ω)):d_f,hΔ^D,(p)_f,h(Ω)v=Δ^D,(p+1)_f,h(Ω)d_f,hvand d_f,h^*Δ^D,(p)_f,h(Ω)v=Δ^D,(p-1)_f,h(Ω)d_f,h^*v.Similar relations hold for Δ^N_f,h(Ω).One can relate the infinitesimal generator L^(0)_f,h of the diffusion (<ref>) to the Witten Laplacian Δ^(0)_f,h through the unitary transformation:ϕ∈ L^2_w(Ω) ↦ e^-f/hϕ∈ L^2(Ω).Indeed, one can check thatΔ^D,(0)_f,h(Ω) = -2h e^-f/h L^D,(0)_f,h(Ω)e^f/h.Let us now generalize this to p-forms, using extensions of L^(0)_f,h to p-forms. The Friedrichs extension associated with the quadratic form v ∈Λ^pC^∞_T(Ω)↦h/2[‖ d^(p)v‖_L^2_w(Ω)^2 +‖e^2f/h(d^(p))^* e^-2f/h v‖_ L^2_w(Ω)^2] on Λ^pL^2_w(Ω), is denoted (-L^D,(p)_f,h(Ω), D(-L^D,(p)_f,h(Ω))). The operator -L^D,(p)_f,h(Ω) is apositive unbounded self adjoint operator on Λ^pL^2_w(Ω). Besides, one has D(-L^D,(p)_f,h(Ω) )={v ∈Λ^pH^2_w(Ω) |𝐭v=0,𝐭 d^*(e^-2f/hv)=0 }.For p=0, the differential operator L_f,h^(0) =- h/2Δ^(0)_H-∇ f·∇is the infinitesimal generator (<ref>) of the overdamped Langevin dynamics (<ref>).For p=1 one gets the operator already introduced in (<ref>):L_f,h^(1)= - h/2Δ^(1)_H-∇ f·∇ -f,where we recall f is the Hessian of f, see Remark <ref>. The generalisation of (<ref>) to p-forms is:Δ^D,(p)_f,h(Ω) = -2h e^-f/h( L^D,(p)_f,h(Ω) )e^f/h.The intertwining relation (<ref>) writes on L_f,h^D,(p)(Ω):L_f,h^D,(p+1)(Ω) d = d L_f,h^D,(p)(Ω)andL_f,h^D,(p-1)(Ω) d_2f,h^* = d^*_2f,h L_f,h^D,(p)(Ω).Thanks to the relation (<ref>), the operators L_f,h^D,(p)(Ω) and Δ^D,(p)_f,h(Ω) have the same spectral properties. In particular the operators L_f,h^D,(p)(Ω) and Δ^D,(p)_f,h(Ω) both have compact resolvents, and thus a discrete spectrum. The generalization of Proposition <ref> is the following:The smallest eigenvalue of -L_f,h^D,(0)(Ω), denoted byλ_h, is positive and non degenerate. The associated eigenfunction u_h has sign on Ω. Moreover u_h ∈ C^∞(Ω,ℝ).Withoutloss of generality, one can assume that u_h satisfies(<ref>). Thanks to the relation (<ref>), the couple (μ_h,v_h):=(2hλ_h, u_h e^-f/h) is the first eigenvalue and eigenfunction of Δ^D,(0)_f,h(Ω). The couple (μ_h,v_h) satisfies{Δ^(0)_f,hv_h=μ_hv_honΩ, v_h = 0 on∂Ω. . Moreover, v_h>0 on Ω and ∫_Ω v_h^2(x) dx=1.The following lemma, which is a direct consequence of the spectral theorem (see for instance <cit.>), will be instrumental throughout this work. Let (A,D(A)) be a non negative self adjoint operator on a Hilbert Space ( H, ‖.‖) with associated quadratic form q_A(u)=(u,Au) with domain D(q_A). Thenfor any u∈ D(q_A) and b>0‖π_[b,+∞) (A) u‖^2 ≤q_A(u)/bwhere, for E ⊂ a Borel set, π_E(A) is the spectral projection of the operator A on E.This lemma will be in particular applied to the non negative self adjoint operators Δ^D,(p)_f,h(Ω) and - L^D,(p)_f,h(Ω) and their associated quadratic forms. §.§.§ Small eigenvalues of Δ^D,(0)_f,h(Ω) and L^D,(0)_f,h(Ω)According to <cit.>, the following relations hold for all v∈Λ^pH^1_T(Ω):π_[0,h^3/2) (Δ^D,(p+1)_f,h(Ω))d_f,hv=d_f,h π_[0,h^3/2) (Δ^D,(p)_f,h(Ω)) v,andπ_[0,h^3/2) (Δ^D,(p-1)_f,h(Ω))d_f,h^*v=d_f,h^* π_[0,h^3/2) (Δ^D,(p)_f,h(Ω)) v. For p∈{0,…,n}, let us define F^(p)_h:= Ran( π_[0,h^3/2) (Δ^D,(p)_f,h(Ω)) ).Then, according to the previous intertwining relations, one can define afinite dimensionalDirichlet complex structure (see <cit.>, <cit.> and <cit.>):{0}⟶ F^(0)_hF^(1)_h⋯F^(d)_h {0} {0}F^(0)_hF^(1)_h⋯ F^(d)_h⟵ {0}.For p∈{0,…,n}, the dimension of the vector space F^(p)_h in the regime h → 0 have been studiedin <cit.> when ∇ f≠ 0 on Ω and when f:Ω→ℝ and f_|Ω are Morse functions. In particular, it is proved there thatthe dimension of F^(0)_h (respectively F^(1)_h) is equal to the number of local minima of f (respectively to the number of generalized critical points of index 1). A generalized critical point of index 1 for Δ^D,(1)_f,h(Ω) iseither a local minimum of f|_∂Ω such that ∂_n f(z_i)>0 or a saddle point of index 1 of f inside Ω. In our setting, thanks to assumptions [H1], [H2] and [H3], there are n generalized critical points of index 1, which are the local minima (z_i)_i=1,…,n of f|_∂Ω.Under [H1], [H2], and [H3], there exists h_0>0 such that for all h ∈ (0,h_0),F^(0)_h =1 and F^(1)_h =n. We refer to <cit.> for the proof of this proposition. From <cit.>, each eigenvalueμ of Δ^D,(1)_f,h(Ω) which is smaller than h^3/2 is exponentially small when h→ 0, i.e.lim sup_h→ 0 hlnμ <0. Thanks to (<ref>), similar results hold for L^D,(p)_f,h(Ω): there exists h_0>0 such that for all h ∈ (0,h_0)π_[0,√(h)/2)(-L^D,(0)_f,h(Ω)) =1and π_[0,√(h)/2)(-L^D,(1)_f,h(Ω))=n.The spectral projection π_[0,√(h)/2)(-L^D,(0)_f,h(Ω)) is the orthogonal projection in L^2_w(Ω) onspan( u_h) and thanks to the intertwining property (<ref>), we have the following crucial property:∇ u_h ∈π_[0,√(h)/2)(-L^D,(1)_f,h(Ω)).For the ease of notation, for p∈{0,1}, we use in the following the notation:π^(p)_h:=π_[0,√(h)/2)(- L^D,(p)_f,h(Ω)). §.§ Statement of the assumptions required for the quasi-modes§.§.§ Assumptions on quasi-modes for L_f,h^D,(p), p∈{0,1} The next proposition gives the assumption we need on the quasi-modes (ψ̃_i)_i=1,…,n whose span approximates π_h^(1), and ũ whose span approximates π_h^(0), in order to prove Theorem <ref>. Assume [H1], [H2] and [H3]. As in the statement of Theorem <ref>, for all i∈{1,…,n}, Σ_i denotes an open set included in ∂Ω containing z_i and such that Σ_i ⊂ B_z_i.Let us assume in addition that there exist n quasi-modes (ψ̃_i)_i=1,…,n and a family of quasi-modes (ũ=ũ_δ)_δ>0 satisfying the following conditions: * For all i∈{1,…,n}, ψ̃_i∈Λ^1 H^1_w,T(Ω) and ũ∈Λ^0 H^1_w,T(Ω). The function ũ is non negative in Ω.Moreover, one assumes the following normalization: for all i∈{1,…,n}, ψ̃_i _L^2_w= ũ _L^2_w = 1. * (a) There exists _1>0such that for all i∈{1,…,n}, in the limit h→ 0:(1-π_h^(1)) ψ̃_i _H^1_w^2=O (e^-2/h( max[f(z_n)-f(z_i), f(z_i)-f(z_1)] +_1)).(b) For any δ>0, in the limit h→ 0: ∇ũ_L^2_w^2 =O (e^-2/h(f(z_1)-f(x_0) - δ)).* There exists _0>0 such that∀ (i,j) ∈{1,…,n}^2 with i<j, in the limit h→ 0: ψ̃_i, ψ̃_j_L^2_w=O( e^-1/h(f(z_j)- f(z_i)+ε_0)). *(a) There exist constants (C_i)_i=1,…,n andp which do not depend on h such that for all i∈{1,…,n}, in the limit h→ 0:∇ũ,ψ̃_i_L^2_w=C_ih^pe^-1/h(f(z_i)- f(x_0))(1+ O(h ) ) . (b) There existconstants (B_i)_i=1,…,n and m which do not depend on h such that for all (i,j) ∈{1,…,n}^2, in the limit h→ 0:∫_Σ_i ψ̃_j · n e^- 2/h fdσ = B_ih^me^-1/h f(z_i)(1+ O(h )) ifi=j0 ifi≠ j. Then,for all i∈{1,…,n}, in the limit h→ 0:∫_Σ_i(∂_nu_h)e^-2/hf dσ =C_i B_i h^p+m e^-1/h(2f(z_i)- f(x_0)) (1+O(h) ), where u_h is the eigenfunction associated with the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>) which satisfies (<ref>).Let us comment on the assumptions made on the quasi-modes. Assumption 1 gives the proper functional setting and the normalization. Assumption 2 will be used to show that span(ψ̃_i, i=1, …, n) (respectively span(ũ)) is included in (π^(1)_h) (respectively in (π^(0)_h)= span(u_h)) up to exponentially small terms.Assumption 3 states the quasi-orthonormality of the quasi-modes (ψ̃_i)_i=1, …, n. Finally, Assumption 4(a) gives the components of the decomposition of ∇ũ over span(ψ̃_i, i=1, …, n), and Assumption 4(b) is then used to find the asymptotic behavior of ∫_Σ_i(∂_nu_h)e^-2/hf dσ, knowing those of ∫_Σ_iψ̃_j · ne^- 2/hf dσ.Theorem <ref> is a consequence of the existence of quasi-modes satisfying this proposition. The construction of such quasi-modes ũ and (ψ̃_i)_i=1,…,n satisfying the requirements of Proposition <ref> will be the focus of Section <ref>, where explicit values for the constants B_i, C_i, p and m will be given in (<ref>) and (<ref>). Let us mention that, from (<ref>) and (<ref>), it holdsfor all i∈{1,…,n}, C_iB_i<0, which isconsistent with the fact that_nu_h< 0 on Ω (due to the first statement in (<ref>) and the HopfLemma, see <cit.>). §.§.§ Assumptions on quasi-modes for Δ_f,h^D,(p), p∈{0,1}The quasi-modes (ψ̃_i)_i ∈{1,…,n} will be built using eigenforms of some Witten Laplacians. It will thus be more convenient to work in non weighted Sobolev spaces, and to actually consider the 1-forms (see (<ref>)): for i ∈{1, … ,n},ϕ̃_i :=e^-1/h fψ̃_i ∈Λ^1H^1_T(Ω).For further references, let us rewrite the hypotheses on the 1-forms (ψ̃_j)_j=1,…,n stated in Proposition <ref> in terms of the 1-forms (ϕ̃_i)_i=1,…,n defined by (<ref>): * For all i∈{1,…,n}, ϕ̃_i∈Λ^1 H^1_T(Ω) and ϕ̃_i _L^2 = 1. * There exist _1>0such that for all i∈{1,…,n}, in the limit h→ 0:( 1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω) ) ) ϕ̃_i_H^1^2 =O (e^-2/h(max[f(z_n)-f(z_i), f(z_i)-f(z_1)] +_1)). * There exists _0>0 such that ∀ (i,j) ∈{1,…,n}^2, i<j, in the limit h→ 0:∫_Ωϕ̃_i(x)·ϕ̃_j(x)d x= O ( e^-1/h[f(z_j)- f(z_i)+ε_0]). * (a) There exist constants (C_i)_i = 1,…,n andp which do not depend on h such thatfor all i ∈{1,…,n}, in the limit h→ 0:∫_Ω∇ũ·ϕ̃_ie^-1/h f=C_ih^pe^-1/h(f(z_i)- f(x_0))(1+ O(h ) ) .(b) There existconstants (B_i)_i = 1,…,n and m which do not depend on h such that for all (i,j) ∈{1,…,n}^2, in the limit h→ 0:∫_Σ_i ϕ̃_j · n e^- 1/h f dσ = B_ih^me^-1/h f(z_i)(1+ O(h )) ifi=j,0 ifi≠ j. As mentioned above, the construction of quasi-modes ũ and (ϕ̃_i)_i=1, …,n satisfying those estimates will be the purpose of Section <ref>.Let us comment on the equivalence between the first assumption here (namely (<ref>)) and assumption 2(a) in Proposition <ref> (namely (<ref>)). This equivalence is a consequence of the following relation between the projectors which comes from (<ref>):π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω) )= e^-1/hfπ_h^(1) e^1/hf.Indeed, using this relation, one has: e^-1/hf(1-π^(1)_h) ψ̃_i_H^1=( 1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω) ) ) ϕ̃_i _H^1.Moreover, one easily checks that there exists C>0 such that, for all h∈ (0,1) and for all u ∈Λ^p H^1(Ω),u_H^1_w≤C/hu e^-1/hf_H^1.Therefore(1-π^(1)_h) ψ̃_i_H^1_w≤C/h ( 1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω) ) ) ϕ̃_i _H^1which shows that (<ref>) (with _1) implies (<ref>) (with _1/2). A similar reasoning shows that (<ref>) also implies (<ref>), but this will not be used in the following. §.§ Proof of Proposition <ref>In all this section, we assume that the hypotheses [H1], [H2] and [H3] hold and we assume the existence of n+1 quasi-modes (ũ, ( ψ̃_i)_i=1,…,n) satisfying the conditions of Proposition <ref>. In the following, ε denotes a positive constant independent of h, smaller than min (ε_1, ε_0), and whose precise value may vary (a finite number of times) from one occurrence to the other. Let us start the proof with two preliminary lemmas relating ũ with u_h on the one hand, and span(ψ̃_j,j=1,…,n) with π_h^(1) on theother hand.Let us assume that the assumptions of Proposition <ref> hold. Then there exist c>0 and h_0>0 such that for h ∈ (0,h_0),π_h^(0)ũ_L^2_w=1+O ( e^-c/h).Since ũ∈Λ^0H^1_w,T, ‖ (1-π_h^(0) ) ũ‖ _L^2_w is bounded from above by h^1/4‖∇ũ‖_L^2_w thanks to Lemma <ref>. In addition since∇ũ_L^2_w^2 = O(e^-2/h[f(z_1)-f(x_0) - δ]) (see assumption 2(b) in Proposition <ref>), by taking δ∈ (0,f(z_1)-f(x_0)), one gets thatπ_h^(0)ũ_L^2_w=1+O ( e^-c/h).As a direct consequence of Lemma <ref>, one has that for h small enough π_h^(0)ũ≠ 0 and therefore (since moreover ũ is non negative in Ω: u_h, π_hũ_L^2_w=u_h, ũ_L^2_w≥ 0),u_h=π_h^(0)ũ/π_h^(0)ũ_L^2_w.Additionally, one has the following lemma concerning the 1-forms. Let us assume that the assumptions of Proposition <ref> hold. Then there exists h_0 such that for all h ∈ (0,h_0),span( π_h^(1)ψ̃_i, i=1, …,n)=π_h^(1).The determinant of the Gram matrix of the 1-forms(π_h^(1)ψ̃_i)_i=1,…,n is 1+O(e^-c/h) thanks to the following identity π_h^(1)ψ̃_i , π_h^(1)ψ̃_j_L^2_w =-(π_h^(1)-1) ψ̃_i ,(π_h^(1) -1) ψ̃_j _L^2_w + ψ̃_i,ψ̃_j _L^2_wand the fact that, from assumptions 1, 2(a) and 3 in Proposition <ref>, there exist h_0>0, c>0 such that for h∈ (0,h_0),ψ̃_i, ψ̃_j_L^2_w=(1-δ_i,j)O(e^-c/h) + δ_i,j and (1-π_h^(1)) ψ̃_i_H^1_w^2=O (e^-c/h). Moreover, from Proposition <ref>, π_h^(1)=n. This proves Lemma <ref>.Thanks to Lemma <ref>, one can build on orthonormal basis (ψ_i)_i=1,…,n of Ran(π_h^(1)) using a Gram-Schmidt orthonormalization procedure on (π_h^(1)(ψ̃_i)) _i=1,…,n. This will be done in Section <ref> below. Then, since∇ u_h ∈ Ran(π_h^(1))=span(ψ_j,j=1,…,n)(see (<ref>))and‖ψ_j‖_L^2_w=1,one has∫_Σ_k (∂_nu_h)e^-2/h fdσ= ∑_j=1^n ⟨∇ u_h , ψ_j_L^2_w∫_Σ_kψ_j · ne^- 2/hfdσ . The proof of Proposition <ref> then consists in replacing, in the right-hand side of (<ref>), the function u_h by its expression (<ref>) in terms of ũ, and the (ψ_i)_i=1,…,n by the (ψ̃_i)_i=1,…,n, and to use the assumptions of Proposition <ref> to get an asymptotic equivalent of ∫_Σ_k (∂_nu_h)e^-2/h fdσ when h→ 0. In Section <ref>, oneprovides asymptotic equivalents on ⟨∇ u_h , ψ_j_L^2_w for j∈{1,…,n}. In Section <ref>, one givesasymptotic equivalentson ∫_Σ_kψ_j · ne^- 2/hf for (k,j)∈{1,…,n}^2. All these results are then gathered to conclude the proof of Proposition <ref> in Section <ref>. §.§.§ Gram-Schmidt orthonormalization Using a Gram-Schmidt procedure, one obtains the following result.Let us assume that the assumptions of Proposition <ref> hold. Then for all j∈{1,…,n}, there exist (κ_ji)_i=1,…,j-1∈ℝ^j-1 such that the 1-formsv_j:=π_h^(1)[ ψ̃_j + ∑_i=1^j-1κ_ji ψ̃_i],(with the convention ∑_i=1^0=0) satisfy * for all k∈{1,…,n}, span(v_i,i=1,…,k )= span( π_h^(1)ψ̃_i,i=1,…,k),* for all i≠ j, v_i , v_j _L^2_w=0. In the following, we denote by Z_j:= ‖ v_j ‖_L^2_w and ψ_j:=v_j/Z_jthe normalized 1-forms. We are first interested in estimating κ_ji and Z_j.Let us assume that the assumptions of Proposition <ref> hold. There exist >0 and h_0>0 such that for all (i,j) ∈{1,…,n}^2 with i<j and all h∈ (0,h_0)π_h^(1)ψ̃_i,π_h^(1)ψ̃_j _L^2_w=O(e^-1/h(f(z_j) - f(z_i)+)).Using assumption 2(a) in Proposition <ref>, one gets: for i<j,(1- π_h^(1))ψ̃_j ,(1-π_h^(1)) ψ̃_i_L^2_w ≤(1-π_h^(1)) ψ̃_k _H^1_w(1-π_h^(1)) ψ̃_k _H^1_w= O (e^-1/h(f(z_n)-f(z_i)+f(z_j)-f(z_1)+))= O (e^-1/h(f(z_j)-f(z_i)+)).The result is then a consequence of assumption 3 in Proposition <ref> and the identity (<ref>). Notice that since π_h^(1) is an L^2_w-projection, π_h^(1)ψ̃_i,π_h^(1)ψ̃_j _L^2_w=π_h^(1)ψ̃_i,ψ̃_j _L^2_w. This will be used extensively in the following.Let us assume that the assumptions of Proposition <ref> hold.Then there exist h_0>0, >0 and c>0 such that for allj∈{1,…,n}, i∈{1,…,j-1} and h∈ (0,h_0)κ_ji= O (e^-1/h(f(z_j)-f(z_i)+))andZ_j=1+O ( e^-c/h). Let us introduce the notation: for all i ∈{1, … ,n},r_i:=(1-π_h^(1)) ψ̃_i _H^1_w^2.Let us prove this lemma by induction. Concerning ψ_1, one has from Lemma <ref>ψ_1 = v_1/Z_1 withv_1=π_h^(1)ψ̃_1.Since ‖ψ̃_1 ‖_L^2_w=1, one has Z_1=‖π_h^(1)ψ̃_1 ‖_L^2_w≤ 1. In addition, by PythagoreanTheorem and by assumption 2(a) in Proposition <ref> on r_1, there exists c>0 such that for h small enoughZ_1^2 ≥ 1-‖(1- π_h^(1))ψ̃_1 ‖^2_L^2_w≥ 1-r_1 ≥ 1-e^-c/h.ThusZ_1= 1+O ( e^-c/h). Now, concerning ψ_2, one hasψ_2=v_2/Z_2 withv_2=π_h^(1)ψ̃_2 -π_h^(1)ψ̃_2 , ψ_1_L^2_w ψ_1,and thus κ_21=-1/Z_1^2π_h^(1)ψ̃_1, ψ̃_2_L^2_w=O( e^-1/h(f(z_2)-f(z_1)+ε)) (by Lemma <ref>). Then one obtains that Z_2=1+O ( e^-c/h) by a similar reasoning as the one we used above for Z_1. In order to prove Lemma <ref> by induction, let us now assume that for k∈{1,…,n} and for allj∈{1,…,k}, i∈{1,…,j-1},κ_ji= O (e^-1/h[f(z_j)-f(z_i)+]) andZ_j=1+O ( e^-c/h).One gets by the Gram-Schmidt procedure which defines the (ψ_i)_i=1,…,n,ψ_k+1=v_k+1/Z_k+1where, using the notation κ_ii=1,v_k+1 = π_h^(1)ψ̃_k+1 -∑_j=1^kπ_h^(1)ψ̃_k+1, ψ_j_L^2_wψ_j = π_h^(1)ψ̃_k+1 -∑_j=1^k ∑_l,q=1^j 1/Z_j^2π_h^(1)ψ̃_k+1, π_h^(1)ψ̃_l_L^2_w κ_jlκ_jq π_h^(1)ψ̃_q = π_h^(1)ψ̃_k+1 -∑_q=1^kπ_h^(1)ψ̃_q∑_j=q^k ∑_l=1^j 1/Z_j^2π_h^(1)ψ̃_k+1, π_h^(1)ψ̃_l_L^2_w κ_jlκ_jq.Then for q∈{1, … ,k},κ_(k+1) q =- ∑_j=q^k ∑_l=1^j1/Z_j^2π_h^(1)ψ̃_k+1, π_h^(1)ψ̃_l_L^2_w κ_jlκ_jq.Since Z_j=1+O ( e^-c/h), one gets Z_j^-1=1+O ( e^-c/h).Using Lemma <ref>, one obtains π_h^(1)ψ̃_k+1, π_h^(1)ψ̃_l _L^2_w = O(e^-1/h[f(z_k+1) - f(z_l)+]), since l<k+1. Moreover,since l≤ j and q≤ j, it holds:κ_jl=O (e^-1/h[f(z_j)-f(z_l)])and κ_jq=O (e^-1/h[f(z_j)-f(z_q)]).Consequently, one obtains that 1/Z_j^2π_h^(1)ψ̃_k+1, π_h^(1)ψ̃_l_L^2_w κ_jlκ_jq=O (e^-1/h[f(z_k+1) - f(z_q) + 2(f(z_j)-f(z_l))+]).Thus, since f(z_j)≥ f(z_l), one obtains from (<ref>), that for q∈{1, …,k}, there exists >0 such that for h small enough: κ_(k+1) q =O(e^-1/h[f(z_k+1) - f(z_q)+]).The estimate Z_k+1=1+O ( e^-c/h) is a consequence of the fact that (κ_(k+1) q)_q ∈{1,… ,k} are exponentially small together withthe estimate ‖π_h^(1)ψ̃_k+1‖_L^2_w = 1+O ( e^-c/h).This concludes the proof of Lemma <ref> by induction.§.§.§ Estimates on the interaction terms (∇ u_h , ψ_j _L^2_w)_j ∈{1,…,n}Let us assume that the assumptions of Proposition <ref> hold. Then for j∈{1,…,n}, one has∇u_h , ψ_j _L^2_w =C_j h^p e^-1/h(f(z_j)- f(x_0))( 1+ O(h)).From (<ref>), for any ϕ∈ H^1_w,T(Ω) and v∈ L^2_w(Ω), it holds,∇π^(0)_hϕ , π^(1)_h v_L^2_w=∇ϕ , π^(1)_h v _L^2_w.Using (<ref>)–(<ref>)–(<ref>)–(<ref>), for all j∈{1,…,n}, one has∇ u_h , ψ_j _L^2_w =Z^-1_j/π_h^(0)ũ_L^2_w[ ∇ũ, π_h^(1)ψ̃_j_L^2_w +∑_i=1^j-1κ_ji ∇ũ,π_h^(1)ψ̃_i_L^2_w ]=Z^-1_j/π_h^(0)ũ_L^2_w[ ∇ũ,ψ̃_j_L^2_w +∇ũ, ( π_h^(1)-1 ) ψ̃_j_L^2_w] +Z^-1_j/π_h^(0)ũ_L^2_w[∑_i=1^j-1κ_ji (∇ũ,ψ̃_i_L^2_w + ∇ũ, ( π_h^(1)-1 ) ψ̃_i_L^2_w) ]. From Lemmata <ref> and <ref>, one hasZ_j^-1/π_h^(0)ũ_L^2_w=1+O ( e^-c/h). Using assumptions 2 and 4(a) in Proposition <ref> and Lemma <ref>, there exists δ_0>0 such that for all δ∈ (0,δ_0),∇ u_h , ψ_j _L^2_w =C_j h^pe^-1/h[f(z_j)- f(x_0)] ( 1 + O(h ) ) + O(e^-1/h[f(z_1)- f(x_0) -δ+ f(z_j)-f(z_1) + ])+∑_i=1^j-1O(e^-1/h[f(z_j)-f(z_i)+ +f(z_i)- f(x_0) -δ])+∑_i=1^j-1O(e^-1/h[f(z_j)-f(z_i)+ +f(z_1)- f(x_0) -δ+f(z_i)-f(z_1)+]).Therefore choosing δ<ε, there exists '>0 such that∇ u_h , ψ_j _L^2_w =C_j h^pe^-1/h[f(z_j)- f(x_0)] ( 1+ O(h ) ) + O (e^-1/h[f(z_j)- f(x_0) + ']). This concludes the proof of Lemma <ref>.§.§.§ Estimates on the boundary terms (∫_Σ_kψ_j · ne^- 2/h fdσ)_(j,k)∈{1,…,n}^2One denotes in this section,for k∈{1,…,n}, K_k:=max(f(z_n)-f(z_k), f(z_k)-f(z_1))≥0. Let us assume that the assumptions of Proposition <ref> hold.Then for all (j,k)∈{1,…,n}^2, there exists >0 such that it holds ∫_Σ_kψ_j · n e^- 2/h fdσ = O( e^-1/h[f(z_j)+])ifk<j, B_j h^m e^-1/h f(z_j)( 1 + O(h) )ifk=j, O(e^-1/h[K_j+f(z_k)+]) +∑_i=1^j-1 O(e^-1/h[f(z_j)-f(z_i)+K_i+f(z_k)+]) ifk>j.Using (<ref>)–(<ref>) and writing π_h^(1)ψ̃_i=ψ̃_i +( π_h^(1)-1 )ψ̃_i, one obtains that∫_Σ_kψ_j · n e^- 2/h f dσ=a_jk+b_jk+∑_i=1^j-1(c_jki+d_jki)with for (j,k) ∈{1, … ,n}^2 and i ∈{1, …, j-1},a_jk=Z_j^-1∫_Σ_kψ̃_j · n e^- 2/hf dσ ,b_jk=Z_j^-1∫_Σ_k( π_h^(1)-1 ) ψ̃_j · n e^- 2/hf dσ c_jki=Z_j^-1κ_ji∫_Σ_kψ̃_i · n e^- 2/h fdσ andd_jki=Z_j^-1κ_ji∫_Σ_k( π_h^(1)-1 )ψ̃_i · n e^- 2/h fd σ.Using the trace theorem and assumption 2(a) in Proposition <ref>, one has, for some universal constant C,∫_Σ_k( π_h^(1)-1 ) ψ̃_j · n e^- 2/hfdσ ≤C/h(π_h^(1)-1) ψ̃_j _H^1_w√(∫_Σ_k e^-2/hf)=O ( e^-1/h[K_j+f(z_k)+]).If k=j and i ∈{1, …, j-1}, one gets, using (<ref>), Lemma <ref> and assumption 4(b) in Proposition <ref>:a_jk=B_j h^m e^-1/h f(z_j) ( 1+ O(h)),b_jk=O ( e^-1/h[K_j+f(z_j)+]), c_jki= 0andd_jki=O(e^-1/h[f(z_j)-f(z_i)+K_i+f(z_j)+]) .If k≠ j and i ∈{1, …, j-1}, one gets using again (<ref>), Lemma <ref> and assumption 4(b) in Proposition <ref>:a_jk=0,b_jk=O(e^-1/h[K_j+f(z_k)+]), c_jki=O ( e^-1/h[f(z_j)+])ifk=i ,0 ifk≠ i, andd_jki=O( e^-1/h[f(z_j)-f(z_i)+K_i+f(z_k)+]).Notice that c_jki=0 ifj< k and that if j>k there exists only one i such that c_jki≠0, namely i=k. This concludes the proof of the Lemma <ref>.§.§.§ Estimates on (∫_Σ_k( ∂_nu_h)e^- 2/h f dσ)_k∈{1,…,n } We are now in position to conclude the proof of Proposition <ref>, by proving that for k∈{1,…,n}, one has ∫_Σ_k (∂_nu_h)e^-2/hfdσ =C_kB_kh^p+m e^- 1/h(2f(z_k)-f(x_0))(1+ O(h ) ). Let us assume that the assumptions of Proposition <ref> hold. Let us recall the decomposition (<ref>):∫_Σ_k (∂_nu_h)e^-2/h fdσ= ∑_j=1^n ⟨∇ u_h , ψ_j_L^2_w ∫_Σ_kψ_j · ne^- 2/hfdσ.Using Lemmas <ref> and <ref>, we can now estimate the terms in the sum in the right-hand side. If j>k, there exist >0 and h_0>0 such that for all h∈ (0,h_0)⟨∇ u_h , ψ_j_L^2_w ∫_Σ_kψ_j · n e^- 2/h fdσ =C_jh^p O( e^-1/h[f(z_j)-f(x_0)]e^-1/h [f(z_j)+])=C_jh^p O(e^-1/h[2f(z_j)-f(x_0)+])=C_jh^p O (e^-1/h[2f(z_k)-f(x_0)+]).If j<k, there exist >0 and h_0>0 such that for all h∈ (0,h_0)⟨∇ u_h , ψ_j_L^2_w ∫_Σ_kψ_j ·ne^- 2/hfdσ=O(e^-1/h[f(z_j)-f(x_0)+K_j+f(z_k)+]) +∑_i=1^j-1 O( e^-1/h[f(z_j)-f(x_0)+ f(z_j)-f(z_i)+K_i+f(z_k)+] )=O(e^-1/h[f(z_j)-f(x_0)+f(z_n)-f(z_j)+f(z_k)+]) +∑_i=1^j-1 O( e^-1/h[f(z_j)-f(x_0)+ f(z_j)-f(z_i)+f(z_n)-f(z_i)+f(z_k)+])=O(e^-1/h[f(z_n)+f(z_k)-f(x_0)+]) +∑_i=1^j-1 O( e^-1/h[f(z_n)+f(z_k)-f(x_0)+2(f(z_j)-f(z_i)) +] )=O(e^-1/h[2f(z_k)-f(x_0)+]).Finally if j=k, ∃>0 and ∃ h_0>0 such that for all h∈ (0,h_0)⟨∇ u_h , ψ_k_L^2_w ∫_Σ_kψ_k·ne^- 2/hfdσ=C_kB_k h^p+m e^- 1/h(2f(z_k)-f(x_0)) (1+ O(h ) ).From these estimates, for a fixed k∈{1,…,n}, thedominant term in the sum in the right-hand sideof (<ref>) is the one with index j=k, namely (<ref>).This concludes the proof ofProposition <ref>. § ON THE AGMON DISTANCEIn this section, we present the main properties of the Agmon distance introduced in Definition <ref>. The Agmon distance is useful to quantify the decay of eigenforms of Witten Laplacians away from critical points of f and f|_∂Ω. The Agmon distance on a domain without boundary has been extensively analyzed in many previous works (see in particular <cit.>). The aim of this section is to generalize well-known results in the case without boundary to our context, namely for bounded domains. Indeed, to the best of our knowledge, this has not been done in the literature before in a comprehensive way.For simplicity, all the proofs in this section are made for a bounded connected open connected C^∞ domain Ω⊂ℝ^d (equipped with the geodesic Euclidean distance (<ref>)) and for a C^∞functionf:Ω→ℝ. The generalization to a C^∞ compact connected Riemannian manifold of dimension d with boundary is straightforward. The notation | x-y |denotes the Euclidean distance between x and y in ℝ^d. If one deals with a compact connected Riemannian manifold of dimension d with boundary, this distance has to be replaced by thegeodesic distance on Ω for the initial metric and the scalar product between two vectors of ℝ^d has to be replaced by the one induced by the initial metric on the tangent space of Ω.This section is organized as follows. Section <ref> is devoted to an equivalent definition of the Agmon distance, which will be crucial in the following. In Section <ref>, we then give a few useful properties of the Agmon distance. As already mentioned in Section <ref>, there is a link between the Agmon distance and the eikonal equation. This is explained in Sections <ref> and <ref>. This link is useful in order to build explicit curves realizing the Agmon distance, as explained in Section <ref>. §.§ The set A(x,y) and an equivalent definition of the Agmon distanceIn order to study the Agmon distance, it will be more convenient for technical reasonsto restrict the class of curves appearing in the definition of the Agmon distance (see Definition <ref>).For x,y∈Ω,we denote by A(x,y) the set ofcurves γ :[0,1] →Ω which are Lipschitz with γ(0)=x, γ(1)=y and such that the set ∂{t∈ [0,1] | γ(t)∈∂Ω} is finite.Here,∂{t∈ [0,1],γ(t)∈∂Ω} denotes the boundary of the set {t∈ [0,1],γ(t)∈∂Ω}. The main result of this section is that, under assumption [H3], the Agmon distance d_a satisfies (compare with (<ref>)):∀(x,y)∈Ω×Ω , d_a(x,y)=inf_γ∈ A(x,y) L(γ,(0,1)).See Corollary <ref> below. The following lemma will be needed several times throughout this section. It motivates the use of the set A(x,y) appearing in Definition <ref>. Let x,y ∈Ω and γ∈ A(x,y). Then for any h: Ω→ℝ which is C^1, one getsh(y)-h(x)=∫_{ t ∈ [0,1],γ(t)∈Ω}(∇ h)(γ)·γ ' +∫_ int{ t ∈ [0,1],γ (t)∈∂Ω}(∇_T h)(γ )·γ'.Here, the notation int stands for the interior of a domain. Since γ is Lipschitz, h∘γ is Lipchitz and thus absolutely continuous. Therefore, one has: h(y)-h(x)=∫_0^1 d/dt (h∘γ )= ∫_{ t ∈ [0,1],γ(t)∈Ω}d/dt (h∘γ )+∫_ int{ t ∈ [0,1], γ(t)∈∂Ω}d/dt (h∘γ ) + ∫_∂{ t ∈ [0,1],γ(t)∈∂Ω}d/dt (h∘γ ).By definition of the set A(x,y) (see Definition <ref>) the set ∂{ t ∈ [0,1],γ(t)∈∂Ω} has Lebesgue measure zero, and thus ∫_∂{ t ∈ [0,1],γ(t)∈∂Ω}d/dt (h∘γ )=0. The curve γ is continuous andthus the set { t ∈ [0,1],γ(t)∈Ω} is open in [0,1]. As a consequence, using in addition that since γ is Lipschitz, it is almost everywhere differentiable (by the Rademacher Theorem), one has for almost every t∈ [0,1]:d/dth(γ)(t)={ (∇ h)(γ(t))·d/dtγ(t)a.e.on{ t ∈ [0,1], γ(t)∈Ω}(∇_T h)(γ(t))·d/dtγ(t)a.e. on int{ t ∈ [0,1],γ(t)∈∂Ω}. .This proves (<ref>). Notice that there exist Lipschitz curves γ such that∂{ t ∈ [0,1],γ(t)∈∂Ω} has a positive Lebesgue measure. Let us give an example. Consider Ω=(0,1)× (0,2) and the curve γ: t∈ [0,1]↦(t,inf_y∈ K| t-y|)∈ [0,1]^2,where K is the Smith-Volterra-Cantor set in [0,1], such that K is closed, has a positive Lebesgue measure and has an empty interior (see <cit.>). Notice that the distance inf_y∈ K| t-y| to K is a Lipschitz function of t ∈ (0,1), so that γ is a Lipschitz function. The set K is closed and thus{ t ∈ [0,1],γ(t)∈∂Ω}={ t ∈ [0,1],γ(t)=0}=K. Therefore ∂{ t ∈ [0,1], γ(t)∈∂Ω}=K∖( int K)=K. The following results will be useful to prove the equality (<ref>) and to prove the existence of curves realizing the Agmon distance inSection <ref>. Assume that [H3] holds. Let γ : [0,1]→Ω be a Lipschitz curve and assume that there exists a time t^*∈ [0,1] such that γ(t^*)∈∂Ω. Then there exists (a,b)∈ [0,1]^2, with a≤ t^*≤ b and a<b such that for all (t_1,t_2)∈ [0,1]^2, with a≤ t_1<t_2≤ b, there exists a Lipschitz curve γ_12: [t_1,t_2]→Ω satisfying γ_12(t_1)=γ(t_1)and γ_12(t_2)=γ(t_2),L(γ, (t_1,t_2))≥L(γ_12, (t_1,t_2)) and, either { t ∈ [t_1,t_2],γ_12(t)∈∂Ω} is empty, or its boundary ∂{ t ∈ [t_1,t_2], γ_12(t)∈∂Ω} consists of isolated points in { t ∈ [t_1,t_2],γ_12(t)∈∂Ω}. Moreover, if the following is satisfied:∃ (s_1,s_2,s_3)∈ [t_1,t_2]^3, s_1<s_2<s_3, γ(s_1)∈∂Ω, γ(s_2)∈Ω and γ(s_3)∈∂Ω,then the inequality (<ref>) is strict. Notice that if t^*∈∂{ t ∈ [0,1], γ(t)∈∂Ω} is not isolated in ∂{ t ∈ [0,1],γ(t)∈∂Ω}, then there exists a neighborhood [t_1,t_2] of t^* in [0,1] such that (<ref>) is satisfied and thus the inequality (<ref>) is strict. Therefore if a Lipschitz curve γ realizes the infimum of L on Lip(x,y), then ∂{ t ∈ [0,1], γ(t)∈∂Ω} is finite. This motivates the introduction of the set A(x,y). Let t^*∈ [0,1] be such that γ(t^*)∈∂Ω. The proof is divided into three steps. Step 1: Introduction of a normal system of coordinates and definition of a and b.Let us consider a neighborhood V_∂Ω of γ(t^*) in ∂Ω, and asmooth local system of coordinates in V_∂Ω⊂∂Ω, denoted by x_T:V_∂Ω→^d-1. Let us now extend it to a tangential and normal system of coordinates around γ(t^*) in Ω, denoted by ϕ(x)=(x_T,x_N). The function ϕ is defined from a neighborhood ofγ(t^*) in Ω to ^d. Moreover, one has x_N(x)≥ 0 and for all x, x_N(x)=0 if and only if x∈∂Ω. One can assume without loss of generality that ϕ is defined on a neighborhood V_α ofγ(t^*) in Ω such that ϕ(V_α)=U × [0,α] for α >0, and U ⊂^d-1. For this normal system of coordinates, the metric tensor G is such that:∀ (x_T,x_N) ∈ U×[0,α], G(x_T,x_N)=[G̃ (x_T,x_N) 0; 0 G_NN(x_T,x_N) ],whereG̃ (x_T,x_N) ∈^(d-1)×(d-1) and G_NN(x_T,x_N) ∈ are smooth functions of (x_T,x_N).The existence of such a system of coordinates is a consequence of the collar theorem, see <cit.>. Under assumption [H3] (namely ∂_nf>0 on ∂Ω), there exist constants A>1 and ε_1>0 such that for all x_N ∈ (0,ε_1] andfor all x_T ∈ U, (see (<ref>) for the definition of g)g(ϕ^-1(x_T,x_N))≥ A g(ϕ^-1(x_T,0)).Since the local change of coordinates is smooth, for all ε∈ (0,1), there exists η>0 such that for allx_N ∈ [0,η] and for all x_T ∈ U, one hasG̃(x_T,x_N)≥ (1-ε)^2G̃(x_T,0).Let us choose ε>0 such that (1-ε) A>1. One can reduce V_α such that 0≤ x_N(x) ≤min (η,ε_1) for all x∈ V_α. By continuity of γ, there exist (a,b)∈ [0,1]^2, with a≤ t^*≤ b and a<b such that for all t∈ [a,b], γ(t)∈ V_α. Let us introduce the components of γ in the normal system of coordinates: (γ_T(t),γ_N(t))=ϕ(γ(t)). Let us now define: for t∈ [a,b],γ̃(t):= ϕ^-1(γ_T(t),0)∈∂Ω.For almost every t ∈ (a,b), if γ(t)∈∂Ω, γ(t)=ϕ^-1 (γ_T(t),0)=γ̃(t), g(γ(t))=g(γ̃(t)) and |γ'(t)|^2=[(γ_T,γ_N)']^Tr G(γ_T(t),0) (γ_T,γ_N)'=[γ_T'(t)]^Tr G̃(γ_T(t),0) γ_T'(t) + G_NN(γ_T(t),0) γ_N'(t)^2≥ [γ_T'(t)]^Tr G̃(γ_T(t),0)γ_T'(t) =|γ̃'(t)|^2,where the supersript ^Tr stands for the transposition operator.For almost everyt ∈ (a,b), if γ(t)∈Ω,|γ'(t)|^2=[(γ_T,γ_N)']^TrG((γ_T,γ_N))(γ_T,γ_N)'=[γ_T'(t) ]^Tr G̃((γ_T,γ_N))γ_T'(t) + G_NN((γ_T,γ_N)) γ_N'(t)^2≥(1-ε)^2[γ_T'(t) ]^Tr G̃(γ_T(t),0) γ_T'(t) =(1-ε)^2 |γ̃'(t)|^2.Step 2: Definition of γ_12. Let (t_1,t_2)∈ [0,1]^2, with a≤ t_1<t_2≤ b.Let us distinguish between two cases. * If the set {t ∈ [t_1,t_2],γ(t)∈∂Ω} isnon empty, let us consider t_1^+:=inf{t ∈ [t_1,t_2],γ(t)∈∂Ω} and t_2^-:=sup{t ∈ [t_1,t_2],γ(t)∈∂Ω}. The curve γ_12: [t_1,t_2]→Ω is then defined by γ_12(t)= {[γ(t) t∈ (t_1,t_1^+),; γ̃(t) t∈ (t_1^+,t_2^-),;γ(t) t∈ (t_2^-,t_2). ].Observe that for all t∈ (t_1,t_1^+)∪ (t_2^-,t_2), γ(t)= γ_12(t), which implies g(γ(t))|γ'(t)|=g(γ_12(t))|γ̃_12'(t)| almost everywhere in (t_1,t_1^+)∪ (t_2^-,t_2). Moreover, using the fact that A(1-ε)>1, for almost everyt∈ (t_1^+,t_2^-),g(γ(t))|γ'(t)|≥ A(1-ε) g(ϕ^-1(γ_T(t),0)) |γ̃'(t)|>g(γ̃)|γ̃'(t)|=g(γ_12)|γ_12'(t)|.Therefore (<ref>) is satisfied.* If the set {t ∈ [t_1,t_2],γ(t)∈∂Ω} isempty, which means that ∀ t∈ [t_1,t_2], γ(t)∈Ω, then one simply defines the curve γ_12: [t_1,t_2]→Ω by γ_12=γ. In both cases, the curve γ_12 is Lipschitz, γ_12(t_j)=γ(t_j) for j∈{1,2} and (<ref>) is satisfied. Moreover by construction of γ_12, the set ∂{ t ∈ [t_1,t_2],γ_12(t)∈∂Ω} consists of isolated points in { t ∈ [t_1,t_2],γ_12(t)∈∂Ω}, or is empty.Step 3: On the strict inequality in (<ref>).Assume that (<ref>) holds and let us show that the inequality (<ref>) is strict. Indeed, in that case t_1^+≤ s_1 < s_3 ≤ t_2^- and by continuity of γ, there exists (u_1,u_2)∈ (s_1,s_3)^2 such that u_1<s_2<u_2 and γ([u_1,u_2]) ⊂Ω. Thus, the inequality (<ref>) holds almost everywhere on the open nonempty interval (u_1,u_2) which implies that L(γ, (u_1,u_2))> L(γ_12, (u_1,u_2)). This concludes the proof of Proposition <ref>. A consequence of the previous proposition is the following result.Let x,y ∈Ω and assume that [H3] holds. For any Lipschitz curve γ:[0,1]→Ω with γ(0)=x and γ(0)=y, there exists γ_1∈ A(x,y) such that L(γ,(0,1))≥ L(γ_1,(0,1)).The set ∂{ t ∈ [0,1],γ(t)∈∂Ω} is closed, so its limit points are its non isolated points. Let us define Ad(γ) as the set of limit points of ∂{ t ∈ [0,1],γ(t)∈∂Ω}. If Ad(γ) is empty, then ∂{ t ∈ [0,1],γ(t)∈∂Ω} is empty or consists of isolated points in ∂{ t ∈ [0,1],γ(t)∈∂Ω} and since ∂{ t ∈ [0,1],γ(t)∈∂Ω} is compact, this implies thatγ∈ A(x,y) and Proposition <ref> is thus proved by simply taking γ_1=γ.If Ad(γ) is non empty, we will construct a curve γ_1∈ A(x,y) such that L(γ,(0,1))≥ L(γ_1,(0,1)).Without loss of generality, one can assume that 0 and 1 are not in Ad(γ). Otherwise one could modify γ in neighborhoods of 0 and 1without increasing L(γ,(0,1)) and without changing the end points using Proposition <ref>. To prove the result, we will show by induction on N ≥ 1 the following property 𝒫_N: for all {t_1,…,t_N}⊂ Ad(γ), denote by (a_j,b_j)_j=1,…,N the open intervals givenby Proposition <ref> for each t_i; then, it is possible to change γto construct a Lipschitz curve γ_1: [0,1]→Ω with γ_1(0)=x and γ_1(0)=y, such that γ_1=γ on [0,1]∖ (⋃_j=1^N(a_j,b_j) )^c, Ad(γ_1) ∩⋃_j=1^N(a_j,b_j)=∅ and L(γ,(0,1))≥ L(γ_1,(0,1)).The first step to prove 𝒫_1 is just a straightforward application of Proposition <ref> (choosing t_1=a_1 and t_2=b_2). Now, let us prove𝒫_N+1 assuming 𝒫_N. Let us consider {t_1,…,t_N,t_N+1}⊂ Ad(γ) and denote by (a_j,b_j)_j=1,…,N,N+1 the open intervals given by Proposition <ref> for each t_i.Applying 𝒫_N, it is possible to change γ to construct a Lipschitz curve γ_1: [0,1]→Ωwith γ_1(0)=x and γ_1(0)=y, such that Ad(γ_1) ∩⋃_j=1^N(a_j,b_j) =∅ and L(γ,(0,1))≥ L(γ_1,(0,1)). If (a_N+1,b_N+1)⊂⋃_j=1^N(a_j,b_j), then 𝒫_N+1 holds taking γ_1. Otherwise, there exist K∈ℕ^* and (q_1,…,q_K,d_1,…,d_K)∈ [0,1]^2K such that (a_N+1,b_N+1)∩[⋃_j=1^N(a_j,b_j)]^c =⋃_i=1^K[(q_i,d_i)],with 0<q_1<d_1<q_2<d_2<…<q_K<d_K<1; the notation [( and )] mean that the extremities can or not belong to the interval. In addition, for i ∈{1, …,K}, q_i∈{a_N+1}∪{ b_1,…,b_N} and d_i∈{b_N+1}∪{ a_1,…,a_N}. Then applyingProposition <ref> to γ_1 on each interval (q_i,d_i)⊂ (a_N+1,b_N+1), one gets that it is possible to construct a Lipschitz curve γ_2 with γ_2(0)=x and γ_2(0)=y, such that L(γ_1,(0,1))≥ L(γ_2,(0,1)), Ad(γ_2) ∩⋃_i=1^K(q_i,d_i)=∅.If for some k∈{1,…, K} and j∈{1,…,N} q_k=b_j, then q_k is isolated from the left in {t∈ [0,1],γ_1(t)∈∂Ω} from the construction of γ_1 (see Proposition <ref>) and,by construction of γ_2, q_k is also isolated from the right in {t∈ [0,1],γ_2(t)∈∂Ω}. Thus, since there exists s∈ [0,q_k) such thatγ_2=γ_1 on [s,q_k], one has q_k∉ Ad(γ_2). A similar reasoning holds for the points d_k. ThusAd(γ_2) ∩(a_N+1,b_N+1)∩[ ⋃_j=1^N(a_j,b_j)]^c =∅. This proves that Ad(γ_2) ∩⋃_j=1^N+1(a_j,b_j)=∅ and thus 𝒫_N+1. By induction, we thus have proven 𝒫_N for all N≥ 1. Now, notice that by a compactness argument, there exist N≥ 0 and {t_1,…,t_N}⊂ Ad(γ) such that if one denotes by(a_j,b_j)_j=1,…,N the open intervals given by Proposition <ref> for each t_i, thenAd(γ) ⊂⋃_i=1^N(a_i,b_i).Applying 𝒫_N yields the desired result.A direct consequence of Proposition <ref> is the following. Assume that [H3] holds. Then the Agmon distance d_a introduced in Definition <ref> satisfies (<ref>): for all (x,y)∈Ω^2 d_a(x,y)=inf _γ∈ A(x,y)L(γ,(0,1)). §.§ First properties of theAgmon distance In this section, we aim at giving the basic properties of the Agmon distance. §.§.§ Upper bounds on d_a and topology of (Ω,d_a)The Agmon distance (x,y)∈Ω×Ω↦ d_a(x,y) is symmetric and satisfies the triangular inequality. Moroever,there exists a constant C such that for all x,y ∈Ω,d_a(x,y) ≤ C | x-y | . For any fixed y ∈Ω,x ∈Ω↦ d_a(x,y) is Lipschitz. Its gradient is well defined almost everywhere and satisfies for y ∈Ω and for almost every x ∈Ω,|∇_x d_a(x,y) |≤|∇ f (x) |. Moreover, if[H3]holds, for all x,y ∈Ω, we have| f(x)-f(y)|≤ d_a(x,y). Theinequality (<ref>) is proved below in Lemma <ref>.The proof of (<ref>) is standard, see for instance <cit.>. For the ease of the reader, let us recall the proof. Let y ∈Ω and x ∈Ω. Since Ω is open, there exists an open ball B ⊂Ω with center x. Let z ∈ B. For t∈ [0,1], the path γ(t)=tz+(1-t)x is included in B. Then, one obtains| d_a(x,y)-d_a(z,y) |≤ d_a(x,z)≤| x-z|∫_0^1 g(tz+(1-t)x) dt=| x-z|∫_0^1 |∇ f | (x+t(z-x)) dt.Since f is smooth, up to considering a smaller ball B centered at x, there exists a constant c>0 such that for all z∈ B, for all t ∈ [0,1], |∇ f | (x+t(z-x)) ≤|∇ f | (x)+c| x-z|.Thus, for all z ∈ B, it holds | d_a(x,y)-d_a(z,y)|≤| x-z|( |∇ f |(x)+c| x-z|).As a consequence, for any fixed y ∈Ω and for almost x ∈Ω one gets (<ref>), by considering the limit z → x in the previous inequality.Let us now prove the inequality (<ref>). For any γ∈ A(x,y), using Lemma <ref>, one has:| f(x)-f(y) | =|∫_0^1 d/dt (f∘γ )dt |=| ∫_{ t ∈ [0,1],γ (t)∈Ω}(∇ f)(γ )·γ ' dt+∫_ int{ t ∈ [0,1],γ (t)∈∂Ω}(∇_T f)(γ )·γ 'dt |≤∫_0^1g(γ(t) )|γ '(t)|dt.Therefore (<ref>) is proved by taking the infimum over γ∈ A(x,y) in the right-hand side (see Corollary <ref>).Let us now give the proof of (<ref>).The function (x,y) ∈Ω×Ω↦d_a(x,y) is bounded and satisfiessup_(x,y) ∈Ω×Ω, x≠ y d_a(x,y)/ | x-y|<∞.Let us first prove by contradiction that (x,y) ∈Ω×Ω↦d_a(x,y) is bounded. Let us assume that there exists asequence (x_k,y_k)_k≥1∈Ω×Ω such that for all k≥ 1, d_a(x_k,y_k)≥ k.Up to theextraction of a subsequence, it can be assumed that lim_k →∞ (x_k,y_k)= (x,y) ∈Ω×Ω (the convergence being for the Euclidean metric).Notice that d_a(x_k,y_k)≤ d_a(x_k,x)+d_a(x,y)+d_a(y_k,y).Let us consider d_a(x_k,x). If x∈Ω then there exist an openball B⊂Ω centered on x and an integer N such that for all k≥ N, x_k ∈ B and therefore γ(t)=tx_k+(1-t)x ∈ B for all t∈ (0,1). Then, by definition of the Agmon distance, for all k≥ 1,d_a(x_k,x) ≤‖ g‖_L^∞(B) | x-x_k |.If x∈∂Ω then there existr>0 and a C^∞ bijective map ϕ :B(x,r) → B(0,1) such that ϕ(x)=0, ϕ ( B(x,r) ∩∂Ω)=Q_0 and ϕ( B(x,r) ∩Ω)=Q_-, where Q_0:={y=(y_1,…,y_d),| y |≤ 1,y_d=0} and Q_-:={y=(y_1,…,y_d), | y |≤ 1, y_d≤0}. Moreover, there exists N such that for all k≥ N, x_k ∈ B(x,r) ∩Ω. Now, for any k ≥ N, let us considerγ(t)=ϕ^-1((1-t)ϕ(x_k)+tϕ(x)).Notice that γ∈ A(x_k,x) and satisfies γ([0,1]) ⊂ B(x,r). Moreover c(t)=ϕ(γ(t))=(1-t)ϕ(x_k)+ t ϕ(x) ∈Q_- for t∈ [0,1]. Then, one has:d_a(x_k,x) ≤∫_0^1g(γ) |γ' | ≤g_L^∞(B(x,r))∫_0^1 |γ' | = g_L^∞(B(x,r))∫_0^1 | Jac(ϕ^-1)(c) c' |≤g_L^∞(B(x,r)) Jac(ϕ^-1)_L^∞(B(0,1)) |ϕ(x_k)-ϕ(x)|,and therefore, since ϕ is Lipschitz,d_a(x_k,x) ≤ C | x_k-x|,where C is a constant independent of k ≥ N.This shows that d_a(x_k,x) is bounded. The same reasoning shows that d_a(y,y_k) is bounded. This yields a contradiction, considering the inequality (<ref>) and (<ref>).To show (<ref>), one proceeds in the same way. Assume that thereexists a sequence (x_k,y_k)∈Ω×Ω such that d_a(x_k,y_k)≥ k |x_k-y_k|.Up to the extraction of a subsequence, it canbe assumed thatlim_k →∞ (x_k,y_k)= (x,y) ∈Ω×Ω. If x≠ y, then, for k sufficiently larged_a(x_k,y_k)/| x_k-y_k|≤ 2 sup_(x,y)∈Ωd_a(x,y)/| x-y|<∞which contradicts (<ref>). If x=y∈Ω, then, for all k sufficiently large, the curve γ(t)=tx_k+(1-t)y_k is with values in Ω and therefore for all k sufficiently large, d_a(x_k,y_k) ≤‖ g ‖_L^∞(Ω)| y_k-x_k |.This again leads to a contradiction when k →∞. Finally, if x=y∈∂Ω, using the same reasoning as above to prove (<ref>), one has the existence of a constant C such that for all k sufficiently large, d_a(x_k,y_k) ≤ C | x_k-y_k|,which again contradicts (<ref>). This concludes the proof of Lemma <ref>.A consequence of the previous lemma is the following proposition. Assume that [H1] holds.The space (Ω, d_a) is a compact separated metric space. Moreover the topology of the metric space (Ω, d_a) is equivalent to the topology induced by the Euclidean metric on Ω. Let us show that for any (x,y) ∈Ω×Ω, if x≠ y then d_a(x,y)>0.Let us denote by d_e the geodesic distance on Ω fortheEuclidean metric: for all x,y ∈Ω,d_e(x,y):=inf _γ∫_0^1|γ'(t) |dt,where the infimum is taken over all the paths γ :[0,1] →Ω which are Lipschitz with γ(0)=x and γ(1)=y. Since [H1] holds, the functions f and f|_∂Ω have a finite number of critical points, and thus, there exist 0<r_1<r_2<d_e(x,y) such that the infimum of g on the set C(r_1,r_2):= { z∈Ω, r_1<d_e(x,z)<r_2 } is positive i.e. c(r_1,r_2):=inf_C(r_1,r_2) g>0. For any path γ∈ A(x,y), one has∫_0^1 |γ'(t)| g(γ(t)) dt ≥ c(r_1,r_2) r(C(r_1,r_2)),where r(C(r_1,r_2)):=inf_z∈ C(r_1,r_2)sup_y∈ C(r_1,r_2)d_e(z,y)>0. Then d_a(x,y)>0. If x=y, it is clear that d_a(x,y)=0 since L(γ,(0,1))=0 where γ(t)=x for all t∈ [0,1]. This shows that (Ω, d_a) is separated. The fact that (Ω, d_a) is compact comes from the inequality (<ref>) proved in Lemma <ref>. Indeed, since(Ω, d_a) is a metric space, it is sufficient to prove the sequential compactness. Let (x_n)_n ≥0 be a sequence in Ω. Since Ω is compact for the Euclidean metric, one can extract a converging subsequence (x'_n)_n ≥ 0 for the Euclideanmetric. From (<ref>), this subsequence is also converging for d_a, which ends the proof. Let us finally prove the equivalence of the topologies on Ω. From Lemma <ref>, it is obvious that if a sequence (x_n)_n ≥ 0 converges to x in Ω for the Euclidean metric, then d_a(x_n,x) converges to 0. Conversely, let us assume that (x_n)_n ≥0 is a sequence ofΩ such that d_a(x_n,x) converges to 0, for a point x ∈Ω. Since Ω is compact for the Euclidean metric, it is enough to show that x is the only limit point of the sequence to show that (x_n)_n ≥0 converges to x for the Euclidean metric. From Lemma <ref>, any limit point y for the Euclidean metric is also an limit point for the Agmon distance, and thus, since (Ω, d_a) is a separated space, y=x. This concludes the proof.Notice that from the proof, it is obvious that the topology is separated as soon as f and f|_∂Ω have a finite number of critical points, which is a weaker assumption than [H1]. Finally, the following lemma will be useful in the following.Assume that[H3]holds. Let I⊂ℝ be an interval and γ: I →Ω a Lipschitz curvesuch that ∂{t∈ I,γ(t)∈∂Ω} is finite and such that x:=lim_t→(inf I)^+γ(t) and y:=lim_t→(sup I)^-γ(t) exist. Then one hasd_a(x,y)≤ L(γ, I).Let (a,b)∈ I^2 with a<b and define for u∈ [0,1], γ_ab(u)=γ(a+u(b-a)). Then γ_ab∈ A(γ(a),γ(b)).By definition of the Agmon distance (see Definition <ref>), d_a(a,b)≤ L(γ_ab,(0,1))=L(γ,(a,b)). Taking the limits a→(inf I)^+,b→(sup I)^- and using the continuity of the Agmon distance, one obtains that d_a(x,y)≤ L(γ,I). Lemma <ref> is proved. As a simple consequence of this lemma, we have the following simple remark. Let x^* be a local minimum of f (from [H3], x^*∈Ω). Then, for all x in the basin of attraction of x^* for the dynamics γ'={ - ∇ f(γ)inΩ- ∇_T f(γ) on∂Ω.it holds x ∈Ω andd_a(x^*,x)=f(x)-f(x^*).Indeed, from (<ref>), we already have d_a(x^*,x)≥ f(x)-f(x^*). To prove the reverse inequality, from Lemma <ref>, it is enough to exhibit a Lipschitz curveγ̃: I →Ω such that ∂{t∈ I,γ̃(t)∈∂Ω} is finite, L(γ̃,I)=f(x)-f(x^*) and lim_t→(inf I)^+γ̃(t)=x^* and lim_t→(sup I)^-γ̃(t)=x. Such a curve is given on the interval I=(-∞,0] by γ̃: t∈ I↦γ(-t) where γ is the solution to (<ref>) with initial condition γ(0)=x. Notice that if ∃ t_0 such that γ̃(t_0) ∈∂Ω, then ∀ t ≤ t_0, γ̃(t) ∈∂Ω. Thus, since lim_t→ -∞γ̃(t)=x^*∈Ω, one has: for all t≤ 0,γ̃(t)∈Ω. Therefore,f(x)-f(x^*) =∫_-∞^0d/dt f ∘γ̃(t) dt=∫_-∞^0 ∇ f (γ̃(t)) ·γ̃ '(t) d t=∫_-∞^0|∇ f (γ̃(t)) |^2 dt=∫_-∞^0 |∇ f (γ̃(t))||γ̃'(t)|d t=∫_-∞^0 g(γ̃(t)) |γ̃'(t) |dt= lim_t→ -∞ L(γ̃, (t,0)).This concludes the proof of (<ref>). Notice that for >0 small enough, the set f^-1 ([f(x^*),f(x^*)+) )∩ B(x^*,)⊂Ω is a neighborhood of x^* which is includedin the basin of attraction of x^* for the dynamics (<ref>). Therefore (<ref>) holds in a neighborhood of x^*. §.§.§ A lower bound on the Agmon distance In this section, easily computable lower bounds on the Agmon distance are provided. This is in particular useful to check if the hypothesis (<ref>) appearing in Theorem <ref> is satisfied, see for example Section <ref>. Let z∈Ω and denote by W and W' twoclosed neighborhoods of z in Ω with W⊂ W'. Defineα:=inf{d_e(x,y),x∈Ω∖ W' , y ∈ W },where d_e denotes the geodesic distance for the Euclidean metric, see (<ref>). Assume that α>0 and that there exists K>0 such thatinf_x∈W'∖W g(x) >K /α,where g has been introduced in Definition <ref>.Then, for all set B ⊂Ω such that B∩W'=∅, inf_y∈ B d_a(z,y) >K,where d_a is the Agmon distance (see Definition <ref>). By assumption, there exists ε>0 such thatinf_x∈W'∖W g(x) ≥ K /α+ ε. Let y ∈ B and γ∈ Lip(z,y). Let us define t_2=inf{ t∈ [0,1],γ(t)∉ W'},t_1=sup{ t∈ [0,1],t<t_2, γ(t) ∈ W}. Since γ is continuous and α>0, it holds 0<t_1<t_2<1 and one has γ(t)∈ W'∖W for all t∈ [t_1,t_2], γ(t_1)∈W=W andγ(t_2)∈Ω∖ W'. Then, one has: ∫_0^1g(γ(t)) |γ'(t) |dt≥∫_t_1^t_2g(γ(t)) |γ'(t) |dt≥( K /α+ε) ∫_t_1^t_2|γ'(t) |dt≥( K /α+ε) α =K+εα.Since εα is independent of y∈B and since εα is also independent of the curve γ, one can take the infimum over γ and y∈ B. Thus inf_y∈ B d_a(z,y) >K.We now give a simple sufficient condition for the hypotheses (<ref>) to hold,in the case where f|_∂Ω has only two local minima. This result is based on Proposition <ref> that will be proven in Section <ref> below and which shows thatd_a(z_1,z_2)>f(z_2)-f(z_1).In particular, the condition stated in the following proposition has been used in Section <ref> in order to checkhypothesis (<ref>). Assume that [H1] and [H3] hold and assume in addition that f|_∂Ω has only two local minima z_1 and z_2 (with f(z_1)≤ f(z_2)) on ∂Ω. Then, if z_2 is the only global minimum of f|_∂Ω on B_z_1^c, one hasinf_z∈ B_z_1^c d_a(z_1,z) > f(z_2)-f(z_1). Proposition <ref>and the continuity of the Agmon distance ensure that there exist an open ball B_2⊂ B_z_1^c centered at z_2, and >0 such that for all z∈ B_2d_a(z_1,z) ≥f(z_2)-f(z_1) +.Since z_2 is the only global minimum of f|_∂Ω on B_z_1^c, there exists '>0, such that for all z∈ B_z_1^c ∖ B_2, f(z)≥ f(z_2) +'. In addition, from theinequality (<ref>), for all z∈ B_z_1^c ∖ B_2, it holdsd_a(z_1,z) ≥f(z)-f(z_1) ≥ f(z_2)-f(z_1) +'.Consequently inf_z∈ B_z_1^c d_a(z_1,z) > f(z_2)-f(z_1). §.§ Agmon distance near critical points of f or f|_∂Ω and eikonal equation We will show that the Agmon distance d_a locally solves the eikonal equation in a neighborhood of any critical point of f|_∂Ω or f (or equivalently, any point x such that g(x)=0, see (<ref>)).§.§.§ The Agmon distance near critical points of fLet us assume that [H1] holds. Let x^*∈Ω be such that ∇ f(x^*)=0. Let us denote by (μ_1,…,μ_d)∈ (ℝ^*)^d the eigenvalues of the Hessian of f at x^*. Then there exist a neighborhood V^* of x^* in Ω and a C^∞ function Φ:V^* → such that{|∇Φ| ^2 =|∇ f | ^2, Φ(x_1,…,x_d) =1/2∑_i=1^d |μ_i | (x_i-x_i^*)^2 +O(| x-x^*|^3). . Moreover, one has the following uniqueness result: if Φ is a C^∞ real valued function defined on a neighborhood V^* of x^* satisfying (<ref>), then Φ=Φ on V^* ∩V^*. Let us notice that Φ(x^*)=0. In addition, up to choosing a smaller neighborhood V^* of x^*, one can assume that Φ ispositive on V^*∖{x^*}. The point x^* is then anon degenerate minimum of Φ. The proof is made in <cit.> in the more general setting where |∇ f|^2 is replaced in (<ref>) by a smooth positive function W around a non degenerate minimum y^* of W such that W(y^*)=0. Here W= |∇ f |^2 and y^*=x^*. This leads to ∇ W= 2 Hess f (∇ f) and thus HessW(x^*)=2 ( Hessf)^2 (x^*) is a non degenerate matrix. Therefore x^* is indeed a non degenerate minimum of W= |∇ f |^2. Let us assume that [H1] and [H3]hold. Let x^*∈Ω be such that ∇ f(x^*)=0. Then there exists a neighborhood U^* of x^* in Ω such that for all x∈ U^*d_a (x^*,x)=Φ(x),where Φ is the smooth solution to (<ref>) and d_a is the Agmon distance.For the ease of the reader, let us give the proof of Proposition <ref> which is similar to the proof of <cit.>. Notice that hypothesis [H3] allows us to use Corollary <ref>. Let Φ be a smooth solution to (<ref>) on a neighborhoodV^* of x^*, as defined in Proposition <ref> and such thatΦ is positive on V^*∖{x^*}. There exists ε>0 such thatU^*:=Φ^-1([0,ε))⊂ V^* is a neighborhood ofx^* (consider for example ε=inf{Φ(x), x∈ V^* ∖ B(x^*,r)}>0 where r>0is such that B(x^*,2r) ⊂V^*). Let us first prove that for x∈ U^*, Φ(x)≤ d_a(x,x^*). For x∈ U^*, one has Φ(x)< and thus Φ^-1([0,Φ(x)))⊂ U^*. Let γ belong to A(x^*,x). Let us define the time t_0:=inf{t∈ [0,1],γ(t)∉Φ^-1([0,Φ(x)))}. By continuity of the curve γ, one has t_0>0, Φ(γ(t_0))=Φ(x) and for all t∈ [0,t_0), γ(t)∈Φ^-1([0,Φ(x)))⊂ U^*. Thus, since the curve γ is Lipschitz and since for all t∈ [0,t_0), γ(t)∈Ω, one hasΦ(x) = ∫_0^t_0 d /dt Φ (γ)(t) dt=∫_0^t_0∇Φ (γ(t)) ·γ'(t) dt ≤∫_0^t_0|∇Φ (γ(t)) ||γ'(t) |dt ≤∫_0^t_0|∇ f (γ(t)) ||γ'(t) |dt ≤∫_0^1 g(γ(t)) |γ'(t) |dt= L(γ,(0,1)).Taking the infimum on the right-hand side over γ∈ A(x^*,x), one gets Φ(x)≤ d_a(x^*,x), for all x∈ U^*. Let us now prove the reverse inequality: for x∈ U^*, Φ(x)≥ d_a(x,x^*). For x∈ U^*, let us define a curve γ:_+ → U^* by∀ t ≥ 0,γ'(t)=- ∇Φ(γ(t))and γ(0)=x.Since the function t↦Φ(γ(t)) is decreasing, the curve γ always belongs to U^* and is defined on ℝ_+.Moreover γ is C^∞ and satisfies lim_t→ +∞γ(t)=x^*.Since γ is with values in U^* ⊂Ω, one has-Φ(x) =∫_0^+∞d/dtΦ∘γ(t) dt=∫_0^+∞∇Φ (γ(t)) ·γ'(t) d t= -∫_0^+∞|∇Φ (γ(t)) |^2 dt=-∫_0^+∞|∇Φ (γ(t))||γ'(t) |d t=- ∫_0^+∞ g(γ(t)) |γ'(t) |dt=- lim_t→ +∞ L(γ, (0,t)).Then, thanks to Lemma <ref>, it holds: d_a(x,x^*)≤ L(γ, (0,+∞))=Φ(x).Therefore Φ(x)=d_a(x^*,x) for all x∈ U^*. Let usmention a simple consequence of the previous proof that will be useful in the following. If x^*∈Ω is such that ∇ f(x^*)=0, there exists a neighborhood U^* of x^* in Ω such that for all x∈ U^*, there exists a C^∞ curve γ: ℝ_+→Ω such that d_a (x^*,x)=∫_0^+∞|∇ f(γ(t)) ||γ'(t) |dt,with γ(0)=x and lim_t→ +∞γ(t)=x^*. The curve γ is defined by γ'(t)=-∇Φ(γ(t)), γ(0)=x,where Φ solves (<ref>).In addition { t∈ [0,∞),γ(t)∈∂Ω} is empty. §.§.§ The Agmon distance near critical points of f|_∂Ω Let us first define the Agmon distance in the boundary ∂Ω. The Agmon distance between x∈∂Ω and y ∈∂Ω in the boundary ∂Ω is defined byd_a^∂Ω(x,y)=inf _γ∫_0^1|∇_Tf (γ(t))||γ'(t) |dt, where the infimum is taken over all the paths γ :[0,1] →∂Ω which are Lipschitz with γ(0)=x and γ(1)=y. Similarly to Remark <ref>, one has:If x^* is a local minimum of f|_∂Ω, one has d_a^∂Ω(x^*,x)=f(x)-f(x^*) for all x ∈∂Ω which is in the basin of attraction of x^* in ∂Ω for the gradient dynamics γ'=-∇_T f(γ).The next proposition is the equivalent of Proposition <ref> for that Agmon distance in ∂Ω. Since ∂Ω is a smooth manifold without boundary, the next result is a direct consequence of well known results from<cit.>, <cit.> and <cit.>.Let us assume that [H1] holds. Let x^*∈∂Ω be such that ∇_Tf(x^*)=0. Then there exists a neighborhood U^* of x^* in ∂Ω such that y↦ d_a^∂Ω (x^*,y) is smooth on U^* and∀ x∈ U^*,|∇_T d_a^∂Ω (x^*,x)|^2=|∇_T f (x)|^2. The boundary ∂Ω is a C^∞ compact manifold andx^* is a non degenerate minimum of |∇_Tf|^2. Theproof is made in <cit.> in the more general setting where|∇_Tf|^2 is replaced in (<ref>) and in(<ref>) by a smooth non negative function W around anon degenerate minimum y^* of W such that W(y^*)=0. Here W= |∇_Tf |^2 and y^*=x^*. This leads to ∇ W= 2 Hess(f|_∂Ω) (∇_T f) andtherefore x^* is a critical point of W= |∇_T f |^2(which turns out to be a minimum). In addition, since ∇_Tf(x^*)=0, one gets that Hess W(x^*)=2 (Hess(f|_∂Ω))^2(x^*) which is a nondegenerate matrix. Let us assume that[H1]and [H3]hold. Let x^*∈∂Ω be such that ∇_Tf(x^*)=0. Then,there exist a neighborhood V^* of x^* in Ω and a C^∞ function Φ:V^* → such that {|∇Φ|^2=|∇ f |^2in Ω∩ V^*, Φ = d_a^∂Ω(x^*,.)on ∂Ω∩ V^*,∂_n Φ < 0on ∂Ω∩ V^*. .Moreover, one has the following uniqueness results: if Φ is a C^∞ real valued function defined on a neighborhood V^* of x^* satisfying (<ref>), then Φ=Φ on V^*∩ V^*. Finally, up to choosing a smaller neighborhood V^* of x^*, one can assume that:* The function Φ is positive on V^*∖{x^*}, so that x^* is a non degenerate minimum of Φ on V^*.* According to (<ref>) and (<ref>), it holds on V^*∩Ω,|∇_T Φ| =|∇_T f |. From Proposition <ref>, the function x∈∂Ω↦ d_a^∂Ω(x^*,x) is smooth near x^*.Then, the result statedcan be proven using the method of characteristics, see <cit.> or <cit.>. Let us mention that the proof crucially relies on the assumption ∂_nf(x^*)>0. The fact that one can reduce V^* such that Φ is positive on V^*∖{x^*} is a consequence of ∂_n Φ<0 on ∂Ω∩ V^* together with the fact that x^* is the only minimum of d_a^∂Ω(x^*,.) (which is positive on ∂Ω∖{x^*}). Let us state a simple corollary of Proposition <ref> and Remark <ref>. Let us assume that [H1] and [H3]hold. Let x^*∈∂Ω be a local minimum of f|_∂Ω. Then there exist a neighborhood V^* of x^* in Ω and a C^∞ function Φ:V^* → such that {|∇Φ|^2=|∇ f |^2in Ω∩ V^*, Φ = f-f(x^*)on ∂Ω∩ V^*,∂_n Φ < 0on ∂Ω∩ V^*. .Moreover, one has the following uniqueness results: if Φ is a C^∞ real valued function defined on a neighborhood V^* of x^* satisfying (<ref>), then Φ=Φ on V^*∩ V^*. Finally, up to choosing a smaller neighborhood V^* of x^*, one can assume that Φ is positive on V^*∖{x^*}, and thatΦ-f > -f(x^*) in V^* ∩ (∂Ω)^c. As a consequence,{x ∈ V^*, Φ(x)=f(x)-f(x^*)}⊂∂Ω.All the statements but (<ref>) are direct consequences of Proposition <ref> and the fact that d_a^∂Ω(x^*,x)=f(x)-f(x^*), thanks to Remark <ref>. Now, notice that on ∂Ω∩ V^*, Φ-f=-f(x^*) and ∂_n(Φ -f)<0 so that, up to choosing a smaller neighborhood V^* of x^*, one can assume that Φ-f > -f(x^*) in V^* ∩ (∂Ω)^c. This concludes the proof of (<ref>).We are now in position to state the main result of this section. Let us assume that [H1] and [H3]hold. Let x^*∈∂Ω be such that ∇_Tf(x^*)=0. Then,there exists a neighborhood U^* of x^* in Ω such that for all x∈ U^*d_a(x,x^*)=Φ(x),where Φ solves (<ref>) and d_a is the Agmon distance. Notice that hypothesis [H3] allows us to useCorollary <ref>. The proof follows the same lines ofthe proof ofProposition <ref>. Let x^*∈∂Ω be such that ∇_Tf(x^*)=0. Let Φ be the smooth solution to (<ref>) on a neighborhood V^* of x^* and such that Φ is positive on V^*∖{x^*} and it holds |∇_T Φ| =|∇_T f | on V^*∩Ω, as defined in Proposition <ref>. One chooses ε>0 sufficiently small such that U^*:=Φ^-1([0,ε))⊂ V^*. Notice that U^* is a neighborhood of x^* in Ω.Step 1. Let us first prove that for all x ∈ U^*, Φ(x) ≤ d_a(x^*,x). For x∈ U^*, one has Φ(x)< and thus Φ^-1([0,Φ(x)))⊂ U^*. Let γ belong to A(x^*,x). Let us define the time t_0:=inf{t∈ [0,1], γ(t)∉Φ^-1([0,Φ(x)))}. By continuity of the curve γ, one has t_0>0, Φ(γ(t_0))=Φ(x) and for all t∈ [0,t_0), γ(t)∈Φ^-1([0,Φ(x)) )⊂ U^*. Thus, usingLemma <ref>, one obtainsΦ(x) = ∫_0^t_0 d /dtΦ∘γ(t) dt = ∫_0^t_0∇Φ (γ(t)) ·γ'(t) dt =∫_ int{t∈ (0,t_0),γ(t) ∈∂Ω}∇_T Φ (γ(t)) ·γ'(t) dt+∫_{t∈ (0,t_0),γ(t) ∈Ω}∇Φ (γ(t)) ·γ'(t) dt.On the one hand, ∫_ int{t∈ (0,t_0),γ(t) ∈∂Ω}∇_T Φ (γ(t)) ·γ'(t) dt ≤∫_ int{t∈ (0,t_0),γ(t) ∈∂Ω}|∇_TΦ( γ(t) ) ||γ'(t) |dt ≤∫_ int{t∈ (0,t_0),γ(t) ∈∂Ω}|∇_T f (γ(t))||γ'(t) |dt,where one used the last statement inProposition <ref>. On the other hand, using (<ref>), one obtains∫_{t∈ (0,t_0),γ(t) ∈Ω}∇Φ (γ(t)) ·γ'(t) dt≤∫_ int{t∈ (0,t_0),γ(t) ∈Ω}|∇Φ (γ(t)) ||γ'(t) |dt ≤∫_ int{t∈ (0,t_0),γ(t) ∈Ω}|∇ f (γ(t))||γ'(t) |dt.Thus one getsΦ(x) ≤∫_ int{t∈ (0,t_0),γ(t) ∈∂Ω}|∇_T f (γ(t))||γ'(t) |dt+∫_{t∈ (0,t_0),γ(t) ∈Ω}|∇f (γ(t))||γ'(t) |dt= ∫_0^t_0 g(γ(t)) |γ'(t) |dt ≤∫_0^1 g(γ(t)) |γ'(t) |dt =L(γ,(0,1)).Taking the infimum on the right-hand side over γ∈ A(x^*,x), one gets Φ(x)≤ d_a(x^*,x), for all x∈ U^*. Step 2. Let us now prove the reverse inequality: ∀ x∈ U^*, d_a(x,x^*)≤Φ(x). Let us define the following vector field on U^*,X:={ - ∇ΦinΩ∩ U^*, - ∇_T Φon∂Ω∩ U^*. .For x∈ U^*, let us define the curve γ by∀ t ≥ 0,γ'(t)=X(γ(t))and γ(0)=x.This curve is well defined for all positive time using the Cauchy-Lipschitz theorem and the fact that (γ(t))_t ≥ 0 remains in U^* for all positive time.Indeed, if x ∈∂Ω∩ U^*, then γ solvesγ'(t)=- ∇_T Φ (γ(t)) (with γ(0)=x). Since the function t↦Φ(γ(t)) is decreasing, the curve γ remains in ∂Ω∩ U^*=∂Ω∩Φ^-1([0,ε)) and is defined on ℝ_+.Moreover, lim_t→ +∞γ(t)=x^*. Besides, if x ∈Ω∩ U^*, let us introduce the first time t_∂Ω such that γ(t_∂Ω) ∉Ω∩ U^* for the curve solution toγ'(t)=-∇Φ (γ(t))(with γ(0)=x). If t_∂Ω=∞, then γ belongs to U^* ∩Ω for all time, and since t↦Φ(γ(t)) is decreasing, necessarily, lim_t→ +∞γ(t)=x^*. If t_∂Ω< ∞, then, since t↦Φ(γ(t)) is decreasing and U^*=Φ^-1([0,ε)), necessarily, γ(t_∂Ω) ∈∂Ω∩ U^*. The curve γ is then defined on [t_∂Ω,∞) as above, for an initial condition in ∂Ω∩ U^*.We have thus shown that the functionγ is globally defined, piecewise C^∞,continuous, remains in ∂Ω if it enters ∂Ω, and satisfies lim_t→ +∞γ(t)=x^*.Recall that t_∂Ω=inf{t∈ [0,+∞),γ(t)∈∂Ω}∈ [0,∞]. One has-Φ(x) =∫_0^+∞d/dtΦ∘γ(t) dt=∫_0^t_∂Ω∇Φ (γ(t)) ·γ'(t) d t+ ∫_t_∂Ω^+∞∇_T Φ (γ(t)) ·γ'(t)dt =-(∫_0^t_∂Ω|∇Φ (γ(t)) |^2 dt + ∫_t_∂Ω^+∞|∇_T Φ (γ(t)) |^2 dt)=-(∫_0^t_∂Ω|∇Φ (γ(t))||γ'(t) |d t+ ∫_t_∂Ω^+∞|∇_T Φ (γ(t))||γ'(t) |dt)=- ∫_0^+∞ g(γ(t)) |γ'(t) |dt=-lim_t→ +∞ L(γ, (0,t)).Thanks to Lemma<ref>,d_a(x,x^*)≤ L(γ, (0,+∞))=Φ(x).In conclusion, Φ(x)=d_a(x^*,x) for all x∈ U^*. Let us mention a simple consequence of the previous proof that will be useful in the following. If x^*∈∂Ω is such that ∇_Tf(x^*)=0, there exists a neighborhood U^* of x^* such that for all x∈ U^*, there exists a piecewise C^∞ and continuous curve γ: ℝ_+→Ω such that d_a (x^*,x)=∫_0^+∞ g (γ(t)) |γ'(t) |dt,with γ(0)=x and lim_t→ +∞γ(t)=x^*. The curve γ is solution to (<ref>)–(<ref>). In addition, the set∂{ t∈ [0,∞),γ(t)∈∂Ω} either consists of one point or is empty.§.§ Curves realizing the Agmon distance In this section, it is proven that for any two points x∈Ω and y∈Ω, their exists a finite number of curves (γ_i)_i=1,…,N defined on the intervals (I_i)_i=1,…,N such that the sum of their lengths L(γ_i,I_i) equals the Agmon distance d_a(x,y). The precise statement is given in the following theorem. Assume that [H1] and [H3]hold. Let x,y ∈Ω. Then there exists a finite number of Lipschitz curves (γ_j)_j=1,…,N which are defined on possibly unbounded intervals I_j⊂ℝ, with values in Ω, such that for all j∈{1,…,N},the sets ∂{ t∈ I_j,γ_j(t)∈∂Ω} are finite and d_a(x,y)=∑_j=1^N L(γ_j,I_j).Additionally, by construction, the intervals (I_j)_j ∈{1, … , N} are either [0+∞), (-∞,0] or [0,1]. Moreover, if I_j=[0,+∞) or I_j=(-∞,0], then γ_j is continuous and piecewise C^∞ (see Lemma <ref> belowfor a more precise definition of the curves γ_j in this case). If I_j=[0,1], then γ_j ∈ A(γ_j(0),γ_j(1)). Finally the curves ((γ_1,I_1),…,(γ_N,I_N)) are ordered such that lim_t→ (inf I_1)^+γ_1(t)=x, lim_t→ (sup I_N)^-γ_N(t)=y,and for all k∈{1,…,N-1},lim_t→(sup I_k)^-γ_k(t)=lim_t→(inf I_k+1)^+γ_k+1(t). This section is entirelydedicated to the proof of Theorem <ref>. In the following, one denotes by {x_1,…,x_m}={x∈Ω, g(x)=0},where g is defined by (<ref>) (there is a finite number of zeros of the function g thanks to [H1]). §.§.§ Preliminary resultsLet us first consider the simple case when the curve realizing the Agmon distance does not meet zeros of the function g.Assume that [H1] and [H3]hold.Let (x,y)∈Ω×Ω. Let (γ_n)_n ≥0∈ A(x,y)^ℕ be a minimizing sequence of curves for d_a(x,y): lim_n→∞L(γ_n,(0,1))= d_a(x,y). In addition, assume that for each k ∈{1,…,m}, there exists a neighborhood V_k of x_k in Ω, such that: ∀ n ∈ℕ,∀ k ∈{1,…,m}, (γ_n)∩ V_k=∅.Then, there exists γ∈ A(x,y) such that L(γ,(0,1))=d_a(x,y).Let M be such that for all n, L(γ_n,(0,1))≤ M and let us definec:=inf_Ω∖ (V_1∪…∪ V_m) g>0.One defines for t∈ [0,1], ϕ_n(t)=L(γ_n,(0,t))+t/L(γ_n,(0,1))+1. The map ϕ_n is strictly increasing and continuous from [0,1] to [0,1]. Therefore it admits an inverse. Setting γ̃_n(u):=γ_n ∘ϕ_n^-1(u), one gets L(γ_n,(0,1))=L(γ̃_n,(0,1)) and |γ̃_n' | ( ϕ_n(t))=|γ_n'(t) |/ g( γ_n(t))|γ_n'(t)|+1(L(γ_n,(0,1))+1)≤|γ_n'(t) |/ c|γ_n'(t)|+1(L(γ_n,(0,1))+1)≤1/c(L(γ_n,(0,1))+1)≤1/c(M+1). Thus, up to replacing γ_n by γ̃_n, one may assumethat the Lipchitz constants of γ_n are bounded uniformly inn. In addition since for all t∈ [0,1], γ_n(t)∈Ω,the sequence (γ_n)_n ≥ 0 is relativelycompact in C^0([0,1], Ω). Thus, up to the extractionofa subsequence, there exists a Lipschitz curve γ such that lim_n→∞γ_n = γ uniformly on [0,1]. Moreover since(γ_n)_n≥ 0 is bounded in H^1([0,1],Ω), up to the extraction of a subsequence, (γ_n)_n≥ 0 converges weakly to γ in H^1([0,1],Ω). It is not difficult to see that for all t∈[0,1],lim inf_n→∞ g(γ_n(t))≥ g(γ(t)).Indeed, for t∈ [0,1], there are two cases: * If γ(t)∈Ω, then for n large enough, all the points γ_n(t) are in Ω and thus lim inf_n→∞ g(γ_n(t)) =lim_n→∞ g(γ_n(t))= |∇ f(γ(t))|= g(γ(t)), * If γ(t)∈∂Ω, since ℕ={n, γ_n(t)∈∂Ω}∪{n,γ_n(t)∈Ω}, one obtains that the set of limit points of (|∇ f(γ_n(t) ) | )_n ≥ 0 is included in {|∇ f(γ(t))|, |∇_T f(γ(t))|}. Therefore, from [H3], one has: lim inf_n→∞ g(γ_n(t))≥|∇_T f(γ(t))|=g(γ(t)).Then, one obtainsd_a(x,y)=lim_l→∞∫_0^1 g(γ_l(t))|γ_l'(t)| dt≥lim inf_n→∞lim inf_p→∞∫_0^1 g(γ_p(t))|γ_n'(t)| dt≥lim inf_n→∞∫_0^1 lim inf_p→∞ g(γ_p(t))|γ_n'(t)| dt≥lim inf_n→∞∫_0^1 g(γ(t))|γ_n'(t)| dt≥∫_0^1 g(γ(t))|γ ' (t)| dt.In the previous computation, one used Fatou Lemma and the lower semi continuity (for the weak convergence) of the convex functionalh∈ H^1([0,1],Ω) ↦∫_0^1 g(γ(t))| h' (t)| dt.Since [H3] holds, using Proposition <ref>, there existsa curve γ̃∈ A(x,y) such that L(γ,(0,1))≥ L(γ̃,(0,1)) and thus d_a(x,y)=L(γ̃,(0,1)).Let us now introduce asufficient condition so that a minimizing sequence ofcurves realizing the Agmon distance avoids a neighborhood of a zero of the function g. For x∈Ω, one introduces the following sets:∀k ∈{1,…,m} , A_k(x):={ z∈Ω,d_a(x,z)=d_a(x,x_k)+d_a(x_k,z)}.One notices that z∈ A_k(x) if and only if x∈ A_k(z). Assume that [H1] and [H3]hold. Let (x,y) ∈Ω^2 and assume that there exists k ∈{1,…,m} such that y∉ A_k(x). If (γ_n)_n ≥ 0∈ A(x,y)^ℕ is a minimizing sequence of curves for d_a(x,y), then there exists a neighborhood V_k of x_k in Ω and n_0∈ℕ, such that for all n≥ n_0,(γ_n)∩ V_k=∅.If y∉ A_k(x), for a k∈{1,…,m}, then d_a(x,y)<d_a(x,x_k)+d_a(x_k,y) and thus y≠ x_k and x ≠ x_k. Let us define ε:=d_a(x,x_k)+d_a(x_k,y)-d_a(x,y)>0,and V_k:=B_a(x_k, min(ε/3, d_a(x_k,y)/2)) where ∀ z∈Ω,∀ r>0, B_a(z,r):={ u ∈Ω,d_a(z,u) <r}. Notice that y∉ V_k. We now prove Proposition <ref> by contradiction. We assume that, up to the extraction of a subsequence, for all n∈ℕ, (γ_n)∩ V_k≠∅.Let usdefine, for all n∈ℕ, t_0^n:=inf{ t∈ [0,1],γ_n(t)∈ V_k}and t_1^n:=sup{ t∈ [0,1],γ_n(t)∈ V_k}.We have for all n∈ℕ, owing to the triangular inequality,L(γ_n,(0,t_0^n))≥ d_a(x,x_k)-ε/3,L(γ_n,(t_1^n,1))≥ d_a(x_k,y)-ε/3.Thus, for all n∈ℕ, it holds:L(γ_n(0,1)) ≥ L(γ_n,(0,t_0^n))+L(γ_n,(t_1^n,1))≥ d_a(x,x_k) +d_a(x_k,y)-2 ε/3= d_a(x,y)+ε/3.This contradicts the fact that lim_n →∞ L(γ_n,(0,1))= d_a(x,y).A direct corollary of Proposition <ref> andLemma <ref> is the following result: Assume that [H1] and [H3]hold.Let y ∈Ω and assume that y∉ A_j(x) for all j∈{1,…,m}. Then,there exists a curve γ∈ A(x,y)such that d_a(x,y)=L(γ,(0,1)). Notice that y∉ A_j(x) for all j∈{1,…,m} implies in particular that x and y are not zeros of the function g. This corollary will be used below to build the curves γ_j associated with intervals I_j=[0,1] in Theorem <ref>. The curves γ_j associated with intervals I_j=[0,+∞) or I_j=(-∞,0] willbe built using the following lemma, which is a direct consequence of Remarks <ref> and <ref>.Assume that[H1]and [H3]hold. Let k∈{1,…,m}. There exists a neighborhood V_k of x_k in Ω, such that for all y∈ V_k, there exists a continuous and piecewise C^∞ curve γ: (-∞,0]→ V_k satisfyingd_a(y,x_k)=L(γ,(-∞,0]), lim_t→ -∞γ(t)=x_k, γ(0)=y.If x_k∈Ω, γis with values in Ω and satisfies (<ref>). If x_k∈Ω,γ satisfies (<ref>)–(<ref>) and is such that ∂{ t∈ (-∞,0], γ(t)∈∂Ω} is either empty or a single point. Before proving Theorem <ref>, we finally need two additional preliminary lemmas. Assume that [H1] and [H3]hold.Let u∈Ω and w∈Ω. For any δ>0 small enough, there exists z_δ such that d_a(u,z_δ)=δ and d_a(w,u)=d_a(w,z_δ)+d_a(z_δ,u).Notice that d_a(u,z_δ)=δ is equivalent toz_δ∈∂ B_a(u,δ), where B_a is defined by (<ref>). We prove Lemma <ref> by contradiction. Assume that there exists δ∈ (0,d_a(u,w)/2) such that for all z ∈∂ B_a(u,δ), d_a(w,u)<d_a(w,z)+d_a(z,u). By compactness of ∂ B_a(u,δ), there exists a_δ>0 such that for all z ∈∂ B_a(u,δ), d_a(w,u) +a_δ≤ d_a(w,z)+d_a(z,u).Thus if γ∈ A(u,w), since there exists a time t_δ such that γ(t_δ)∈∂ B_a(u,δ), one hasL(γ,(0,1))=L(γ,(0,γ(t_δ)))+L(γ,(γ(t_δ),1))≥ d_a(u,γ(t_δ))+d_a(γ(t_δ),w)≥ d_a(u,w)+a_δ. This is impossible since by definition d_a(u,w)=inf_γ∈ A(u,w)L(γ,(0,1)).Assume that [H1] holds. Let (x,y)∈Ω^2 with x≠ y.Then, there exist N∈ℕ and a sequence (b_i)_i∈{0,…,N+1}∈Ω^N+2, b_0=x, b_N+1=y, (b_i)_i∈{1,…,N}∈{x_1,…,x_m }^N (with the convention {x_1,…,x_m }^0=∅) such that the following holds: * For all i∈{0,…,N}, b_i≠ b_i+1 and d_a(x,y)=∑_i=0^N d_a(b_i,b_i+1). * For all i∈{0,…,N} and for all z∈{x,y,x_1,…,x_m}∖{b_i,b_i+1},d_a(b_i,b_i+1)<d_a(b_i,z) + d_a(z,b_i+1).Since x≠ y, the following set E:={ (N,b), N∈ℕ,b=(b_i)_i∈{0,…,N+1}∈Ω^N+2,b_0=x, b_N+1=y, ∀ i∈{0,…,N}, b_i≠ b_i+1, (b_i)_i∈{1,…,N}∈{x_1,…,x_m}^N,(<ref>) holds}, is not empty since by assumption it contains (0,{x,y}). For (N,b)∈ E, one defines the cardinal of(N,b) by the number of different critical points b contains. The cardinal ofan element of E belongs to {0,…,m}. Let us now consider anelement (N,b)∈ Ewhich is maximal forthe cardinal. By construction, this element satisfies point 1 in Lemma <ref>. Let us now show that it also satisfies point 2 in Lemma <ref>. Notice that {b_0, … ,b_N+1}⊂{x,y,x_1,…,x_m}. Let i∈{0,…,N} and z∈{x,y,x_1,…,x_m}∖{b_i,b_i+1}. If z∈{x,y,x_1,…,x_m}∖{b_0, … ,b_N+1}, the equality d_a(b_i,b_i+1)=d_a(b_i,z) + d_a(z,b_i+1) cannot hold since b has been chosen maximal in E for the cardinal. Thus, by the triangular inequality d_a(b_i,b_i+1)<d_a(b_i,z) + d_a(z,b_i+1).If z∈{b_0, … ,b_N+1}∖{b_i,b_i+1}, let usprove that d_a(b_i,b_i+1)<d_a(b_i,z) + d_a(z,b_i+1) bycontradiction. By the triangular inequality, if the previousinequality does not hold, one has d_a(b_i,b_i+1)=d_a(b_i,z) +d_a(z,b_i+1) for some z∈{b_0, … ,b_N+1}∖{b_i,b_i+1}. Let us denote by j_0 ∈{0, …,i-1,i+2,…, N+1} theindex such that z=b_j_0. One has d_a(b_i,b_i+1)=d_a(b_i,b_j_0) +d_a(b_j_0,b_i+1). Let us assume without loss of generality thatj_0<i (the case j_0>i+1 is treated similarly). In this case, onehas, using the triangular inequality:d_a(x,y) =∑_j=0^N d_a(b_j,b_j+1)=∑_j=0^i-1 d_a(b_j,b_j+1)+d_a(b_i,b_j_0)+d_a(b_j_0,b_i+1)+∑_j=i+1^N d_a(b_j,b_j+1)=∑_j=0^j_0-1 d_a(b_j,b_j+1)+d_a(b_j_0,b_i+1)+∑_j=i+1^N d_a(b_j,b_j+1) +∑_j=j_0^i-1 d_a(b_j,b_j+1) +d_a(b_i,b_j_0)≥ d_a(x,y) +∑_j=j_0^i-1 d_a(b_j,b_j+1) +d_a(b_i,b_j_0).Thus, ∑_j=j_0^i-1 d_a(b_j,b_j+1) +d_a(b_i,b_j_0)=0 and b_j_0=b_i which is in contradiction with z∉{b_i, b_i+1}. Therefored_a(b_i,b_i+1)<d_a(b_i,b_j_0) + d_a(b_j_0,b_i+1). This concludes the proof ofLemma <ref>.§.§.§ Proof of Theorem <ref>Let us now prove Theorem <ref>.Recall that by assumption, the hypotheses[H1]and [H3]hold. Let x,y ∈Ω. If x=y, thenTheorem <ref> is proved by taking the constant curveγ(t)=x for all t∈ [0,1]. Let us deal with the casex≠ y. From Lemma <ref>, there exist N∈ℕ and a sequence (b_j)_j∈{0,…,N+1}⊂Ω^N+2 such that b_0=x, b_N+1=y, (b_j)_j∈{1,…,N}⊂{x_1,…,x_m }^N (with the convention {x_1,…,x_m }^0=∅)and for all k∈{0,…,N}, b_k≠ b_k+1 andd_a(x,y)=∑_k=0^N d_a (b_k,b_k+1).If N=0, then for all k∈{1,…,m} y∉A_k(x) and Theorem <ref> is then a consequence of Corollary <ref>. Let us now assume that N ≥ 1, namely that there existsk∈{1,…,m} such that y∈ A_k(x). Let us actually consider the case N≥ 2 (the case N=1 is treated similarly). Let k∈{1,…,N-1} and let us consider the term d_a(b_k,b_k+1) in (<ref>) (the first term d_a(x,b_1) and the last term d_a(b_N,y) in the sum are treated in a similar way). One can label the points {x_1,…,x_m} such that b_k=x_1 and b_k+1=x_2. Point 2 in Lemma <ref> implies that x_2∉ A_j(x_1) for all j∈{3,…,m}. From Lemma <ref>, for any δ>0 there exists z_1∈∂ B_a(x_1,δ) such that d_a(x_1,x_2)=d_a(x_1,z_1) + d_a(z_1,x_2) (where B_a is defined by (<ref>)). By taking δ small enough, this implies that z_1∉ A_1(x_2) and z_1∉{x_1,…,x_m}. Likewise, from Lemma <ref>, for any δ>0 there exists z_2∈∂ B_a(x_2,δ) such that d_a(z_1,x_2)=d_a(z_1,z_2) + d_a(z_2,x_2) and by taking δ small enough, this implies that z_2∉ A_2(z_1) and z_2∉{x_1,…,x_m}. Therefore one getsd_a(b_k,b_k+1)=d_a(x_1,x_2)=d_a(x_1,z_1)+d_a(z_1,z_2) + d_a(z_2,x_2).Taking δ small enough and using Lemma <ref>,there exists a continuous and piecewise C^∞ curve γ_1defined on (-∞,0] such that d_a(x_1,z_1)=L(γ_1,(-∞,0]), lim_t→ -∞γ_1(t)=x_1, γ_1(0)=z_1, and ∂{ t∈ (-∞,0],γ_1(t)∈∂Ω} is either empty or a single point. Similarly, there exists a continuous and piecewise C^∞ curve γ_2defined on [0,+∞) such that d_a(z_2,x_2)=L(γ_2,[0,+∞)), γ_2(0)=z_2, lim_t→ +∞γ_2(t)=x_2 and ∂{ t∈ [0,+∞),γ_2(t)∈∂Ω} is either empty or a single point.Let us show by contradiction that z_2∉ A_j(z_1) for all j∈{3,…,m}. On the one hand, ifz_2∈ A_j(z_1) for some j∈{3,…,m}, one has d_a(x_1,x_2)= d_a(x_1,z_1)+d_a(z_1,x_j)+ d_a(x_j,z_2)+d_a(z_2,x_2). On the other hand, x_1∉ A_j(x_2), and thus d_a(x_1,x_2)<d_a(x_1,x_j)+d_a(x_j,x_2)≤ d_a(x_1,z_1)+d_a(z_1,x_j)+ d_a(x_j,z_2)+d_a(z_2,x_2).This leads to a contradiction. Therefore z_2∉ A_j(z_1) for all j∈{3,…,m}. One also has by a similar reasoning that z_2∉ A_1(z_1). Indeed, If z_2∈ A_1(z_1), then one has on the one handd_a(z_1,x_2)=d_a(z_1,z_2)+d_a(z_2,x_2)=d_a(z_1,x_1)+d_a(x_1,z_2)+ d_a(z_2,x_2).On the other hand, since z_1∉ A_1(x_2), one hasd_a(z_1,x_2)<d_a(z_1,x_1)+d_a(x_1,x_2)≤ d_a(z_1,x_1)+d_a(x_1,z_2)+ d_a(z_2,x_2).This leads to a contradiction. In conclusion z_2∉ A_j(z_1) for all j∈{1,…,m}. Therefore, from Corollary <ref>, there exists a curve γ∈ A(z_1,z_2)such that d_a(z_1,z_2)=L(γ,(0,1)).In conclusion, we have built three curves γ, γ_1 and γ_2 such thatd_a(b_k,b_k+1)=L(γ_1,(-∞,0])+L(γ,(0,1))+L(γ_2,[0,+∞)). A similar reasoning for all the terms in the sum in (<ref>) concludes the proof of Theorem <ref>. A consequence of Theorem <ref> is the following.Let us assume that [H1] and [H3]hold. Let (x,y) ∈Ω. Let us denote by ((γ_1,I_1),…,(γ_N,I_N)) the curves given by Theorem <ref> ordered such that lim_t→ (inf I_1)^+γ_1(t)=x, lim_t→ (sup I_N)^-γ_N(t)=y,and which realize the Agmon distance between x and y.Let k_1≤ k_2 with (k_1,k_2)∈{1,…,N}^2 and let t_1∈I_k_1 and t_2∈I_k_2. If k_1=k_2, t_1 and t_2 are chosen such that t_1≤ t_2. Then,one has: * If k_1<k_2, d_a(γ_k_1(t_1),γ_k_2(t_2))= L(γ_k_1, ( t_1,sup I_k_1))+ ∑_k=k_1+1^k_2-1 L(γ_k, I_k)+L(γ_k_2, ( inf I_k_2,t_2)) ,where by convention, if k_2=k_1+1, ∑_k=k_1+1^k_2-1 L(γ_k, I_k)=0.* If k_1=k_2, d_a(γ_k_1(t_1),γ_k_2(t_2))= L(γ_k_1, ( t_1,t_2) ).In addition, the following equality holdsd_a(x,y)=d_a(x,γ_k_1(t_1))+d_a(γ_k_1(t_1),γ_k_2(t_2))+d_a(γ_k_2(t_2),y). The proof ofLemma <ref> is done easily reasoning by contradiction and using the triangular inequality on the Agmon distance. §.§.§ On the equality in (<ref>)We end up this section with some results in case of equality in the inequality (<ref>). We will prove in particular Proposition <ref> which has been used in Section <ref> above to give lower bounds on the Agmon distance.Let us assume that [H1] and [H3]hold.Let x,y ∈Ω with f(x)≤ f(y). Let us denote by ((γ_1,I_1),…,(γ_N,I_N)) thecurves given by Theorem <ref> ordered such that lim_t→ (inf I_1)^+γ_1(t)=x, lim_t→ (sup I_N)^-γ_N(t)=y,and which realize the Agmon distance between x and y. If it holds: d_a(x,y)= f(y)-f(x) ,then for all i∈{1,…,N}, there exist measurable functions λ_i : I_i→ℝ_+ such that for almost every t in{t∈ I_i,γ_i(t)∈Ω} γ_i'(t)=λ_i(t) ∇ f (γ_i(t) ),and such that for almost every t in int{t∈ I_i,γ_i(t)∈∂Ω}γ_i'(t)=λ_i(t) ∇_T f (γ_i(t) ).Moreover, for all i∈{1,…,N}, λ_i∈ L^∞(I_i,ℝ_+). Finally, if I_i is unbounded (i.e. I_i=[0,+∞) or I_i=(-∞,0]),it holds for almost every t∈ I_i,λ_i=1. Using Lemma <ref>, one gets using first the triangular inequality and then the Cauchy-Schwarz inequality f(y)-f(x) = ∑_k=1^N ( ∫_{ t ∈ I_k,γ_k(t)∈Ω}(∇ f)( γ_k)·γ_k' +∫_ int{ t ∈ I_k,γ_k(t)∈∂Ω}(∇_T f)( γ_k)·γ_k')≤∑_k=1^N ( ∫_{ t ∈ I_k,γ_k(t)∈Ω}|∇ f( γ_k)||γ_k' | +∫_ int{ t ∈ I_k,γ_k(t)∈∂Ω}|∇_T f ( γ_k)||γ_k'|)=∑_k=1^NL(γ_k,I_k)=d_a(x,y ).If d_a(x,y)=f(y)-f(x), then the previous inequality is necessarily an equality. Using the cases of equality in both the triangular inequality and the Cauchy-Schwarz inequalities, for k∈{1,…,N}, there exists a nonnegative functionλ_k:I_k→ℝ_+ such that (<ref>) and (<ref>) hold.Let i∈{1,…,N}.Let us first consider the case whenI_i=[0,1]. Then, by construction, the curve γ_i does not meet any critical points of the functions f and f|_∂Ω. This implies that inf_I_i |∇ f(γ_i)|>0 and inf_I_i |∇_T f(γ_i)|>0, and thus, since γ_i'_L^∞<∞, one concludes that λ_i∈ L^∞([0,1],ℝ_+).Let us now consider the case when I_i is not bounded. Using the construction of the curves (γ_k)_k=1,…,N,this implies that I_i is either (-∞,0] or [0,+∞) and γ_i is constructed using the gradient flow of the eikonal solution near a critical point x^* of f or of f|_∂Ω (see Lemma <ref>). Let us assume that x^* is a critical point of f|_∂Ω and I_i=[0,+∞) (the other cases are treated similarly). Let Φ be the solution to (<ref>) on the neighborhoodV^* of x^* in Ω (see Proposition <ref>). The curve γ_i satisfies by construction, Ran(γ_i)⊂ V^*,lim_t→∞γ_i(t)=x^*, and on(-∞, 0],γ_i'={ -∇Φ (γ_i)inΩ, -∇_T Φ (γ_i) on∂Ω. . In addition, by the previous reasoning, one also has on(-∞, 0],γ_i'={ λ_i ∇ f (γ_i)inΩ,λ_i∇_T f (γ_i) on∂Ω, .for some measurable function λ_i:[0,+∞)→ℝ_+.Furthermore, from the last point in Proposition <ref>,it holds on V^*∩Ω, |∇_T Φ| =|∇_T f|.Taking the norm in (<ref>) and (<ref>),and using the fact that Φ solves (<ref>)together with the equality (<ref>), one obtains that λ_i(t)=1 for almost everyt∈ I_i.This concludes the proof of Corollary <ref>. Let us define the notion of generalized integral curves. Let D⊂Ω be a C^∞ domain and X∈ C^∞(D,ℝ). Let N∈ℕ^* and for i∈{1,…,N}, letI_i⊂ℝ be an interval and γ_i: I_i→ D be Lipschitz and such thatlim_t→ (inf I_1)^+γ_1(t) andlim_t→(sup I_N)^-γ_N(t)existand for all k∈{1,…,N-1},lim_t→(sup I_k)^-γ_k(t)=lim_t→(inf I_k+1)^+γ_k+1(t).The set of curves {γ_1,…,γ_N} is a generalized integral curve of the vector field {∇ X inD∩Ω,∇_T X on D∩∂Ω, . if for all i∈{1,…,N}, there exist measurable functions λ_i : I_i→ℝ_+ such that for almost every t in{t∈ I_i,γ_i(t)∈ D∩Ω}:γ_i'(t)=λ_i(t) ∇ X (γ_i(t) ),and such that for almost every t in int{t∈ I_i,γ_i(t)∈∂Ω∩ D }: γ_i'(t)=λ_i(t) ∇_T X (γ_i(t) ).The notion of generalized integral curve has been introduced in the case of manifolds without boundary in<cit.>. As introduced in Definition <ref>, the set of curves {γ_1,…,γ_N} given by Corollary <ref> is a generalized integral curve of the vector field{∇ f inΩ,∇_T f on∂Ω. .Let us mention that in the case when Ω is a manifold without boundary, Corollary <ref> is exactly <cit.>.Let us end this section with the following proposition which which has been used in Section <ref>.Let us assume that [H1] and [H3] hold. Let us denote by {z_1,…,z_n}the local minima of f|_∂Ω ordered such that f(z_1)≤ f(z_2)≤…≤ f(z_n). Then, for all i<j, (i,j)∈{1,…,n}^2, one hasd_a(z_i,z_j)>f(z_j)-f(z_i).From the inequality (<ref>), one has d_a(z_i,z_j)≥ f(z_j)-f(z_i) for all i<j. Let us prove Proposition <ref> by contradiction. Assume that d_a(z_i,z_j)= f(z_j)-f(z_i) for some i<j. Denote by ((γ_1,I_1),…,(γ_m,I_m)) the curves given by Theorem <ref> ordered such that lim_t→ (inf I_1)^+γ_1(t)=z_i, lim_t→ (sup I_m)^-γ_m(t)=z_j,and which realize the Agmon distance between z_i and z_j. Since d_a(z_i,z_j)= f(z_j)-f(z_i), from Corollary <ref>, for all i∈{1,…,m}, there exist measurable functions λ_i : I_i→ℝ_+ such that for almost every t in{t∈ I_i,γ_i(t)∈Ω}, γ_i'(t)=λ_i(t) ∇ f (γ_i(t) ), and such that for almost every t in int{t∈ I_i, γ_i(t)∈∂Ω}, γ_i'(t)=λ_i(t) ∇_T f (γ_i(t) ). Let us recall that from Remark <ref>, I_1=(-∞, 0] and I_m=[0,+∞) since z_i and z_j are critical points of f|_∂Ω. Step 1. Let us show thatforall t∈ (-∞, 0], γ_1(t)∈∂Ω.On the one hand, from Remark <ref>, lim_t→ -∞γ_1(t)=z_i and on(-∞, 0],γ_1'={ ∇Φ (γ_1)inΩ∇_T Φ (γ_1) on∂Ω, .where Φ solves (<ref>). On the other hand, from Corollary <ref>, one has on(-∞, 0],γ_1'(t)={ ∇ f (γ_1)inΩ∇_T f (γ_1) on∂Ω. .Then, for all t≤ 0, one has d/dt( f (γ_1)(t)-Φ (γ_1(t))) =0. Therefore there exists C>0 such that forall t∈ (-∞, 0], γ_1(t)∈{x, f(x)-Φ(x)=C}. Since lim_t→ -∞γ_1(t)=z_i and (f-Φ)(z_i)=f(z_i), one gets that C=f(z_i) and thus forall t∈ (-∞, 0], γ_1(t)∈{x,f(x)-Φ(x)=f(z_i)}. From Corollary <ref> and Proposition <ref>, γ_1 lives in a neighborhood U^* of z_i such that (see Equation (<ref>)):{x,f(x)-Φ(x)=f(z_i)}⊂∂Ω.We thus get that for all t≤0, γ_1(t) ∈∂Ω, and then γ_1'(t)=∇_T f (γ_1(t))=∇_T Φ (γ_1(t)). Step 1 is proved.Step 2.We are going to show that forall t∈ I_k, γ_k(t)∈∂Ω. If it is not the case, from Step 1, there exist k∈{2,…,m} and t_k∈ I_k such that γ_k(t_k) ∈Ω. Let us define the first time, denoted by t^*, for which the curves ((γ_2,I_2),…,(γ_m,I_m)) leave ∂Ω.By construction of the curves γ_1,…,γ_m, there are two cases: either t^* is finite (and thus belongs to int(I_k) for k∈{1,…,m}) or, there exist j∈{2,…,m-1}, s<0 and z∈Ω such that g(z)=0, lim_t→ +∞γ_j(t)=z, lim_t→ -∞γ_j+1(t)=z and γ_j+1(-∞,s)⊂Ω in which case t^*=-∞. Let us assume that t^* is finite and belongs to int(I_k) (the other case is treated similarly).As in Step 1 of the proof of Proposition <ref>, let us now introduce a smooth tangential and normal system of coordinates around γ_k(t^*) in Ω, denoted by ϕ(x)=(x_T,x_N). The function ϕ is defined from a neighborhood ofγ_k(t^*) in Ω to ^d. Moreover, one has x_N≥ 0 and x_N(x)=0 if and only if x∈∂Ω. We may assume that the neighborhood V_α⊂^d on which ϕ is defined is such that ϕ(V_α)=U×[0,α] for α >0 and U ⊂^d-1. Since ∂_nf>0 on ∂Ω, α>0 can be chosen small enough such that ∇ f(x) · n(x)>0 for all x∈ V_α where n(x)=-∇ x_N(x)/|∇ x_N(x)|. Indeed, for x∈∂Ω, n(x) is nothing but the unit outward normal to ∂Ω.Now, by continuity of the curve γ_k, there exists >0 such that [t^*, t^*+]⊂ I_k and for all t∈ [t^*, t^*+], γ_k(t)∈ V_α. The mapping t∈ [t^*, t^*+]↦ x_N(γ_k(t)) is Lipschitz and satisfies: for almost every s∈ (t^*, t^*+), d/dsx_N(γ_k(s))=-|∇ x_N(γ_k(s))| γ_k'(s)· n(γ_k(s)).Then, for all t∈[t^*, t^*+], one has: d/dsx_N(γ_k(s))={ 0fora.e.s ∈ int { u∈ (t^*,t),γ_k(u)∈∂Ω}-|∇ x_N(γ_k(s))| λ_k(s) ∇ f(γ_k(s))· n(γ_k(s))fora.e.s ∈{ u∈ (t^*,t),γ_k(u)∈Ω}. .Since ∂{ u∈ (t^*,t),γ_k(u)∈∂Ω} is of Lebesgue measure zero (see Theorem <ref>) and since ∇ f · n >0 in V_α, one has from Lemma <ref>, for all t∈[t^*, t^*+]x_N(γ_k(t))=x_N(γ_k(t))-x_N(γ_k(t^*))=∫_t^*^td/dsx_N(γ_k(s))ds≤ 0.This implies that for all t∈[t^*, t^*+], x_N(γ_k(t))=0 and thusγ_k(t)∈∂Ω for all t∈[t^*, t^*+]. This contradicts the definition of t^*. Step 2 is proved.Step 3. End of the proof of Proposition <ref>.From the last two steps, for all t∈ [0,+∞), γ_m(t)∈∂Ω. From Corollary <ref>, one has γ_m'(t)=∇_Tf(γ_m(t)) for all t∈ [0,+∞) and therefore, the map t∈ [0,+∞)↦ f(γ_m(t)) is increasing (indeed, one has d/dtf(γ_m(t))=|∇_T f(γ_m(t))|^2). This is impossible since z_j is a local minimum of f|_∂Ω. This concludes the proof of Proposition <ref> by contradiction.§.§ Agmon distance in a neighborhood of the basin of attraction of a local minimum of f|_∂Ω and eikonal equation The aim of this section is to generalize the results of Section <ref> to relate the Agmon distance and the solution to an eikonal equation on a neighborhood of a basin of attraction B_z (see Definition <ref>) of a local minimum z of f|_Ω.Let first introduce asolution to the eikonal equation |∇ϕ|^2 =|∇ f |^2 defined globally on a neighborhood of the boundary ∂Ω. Let us assume that [H3] holds. There exists a neighborhood of ∂Ω in Ω, denoted V_∂Ω, such that there exists Φ∈ C^∞(V_∂Ω,ℝ) satisfying {|∇Φ|^2=|∇ f |^2 inΩ∩ V_∂Ω Φ = f on∂Ω ∂_n Φ =-∂_n f on∂Ω ..Moreover, one has the following uniqueness results: if Φ̃ is a C^∞ real valued function defined on a neighborhood Ṽ of ∂Ω satisfying (<ref>), then Φ̃=Φ on Ṽ∩ V_∂Ω.Let z∈∂Ω. Using <cit.> or <cit.>, thanks to [H3], there exists a neighborhood of z in Ω, denoted by 𝒱_z, such that there exists Φ∈ C^∞(𝒱_z,ℝ) satisfying {|∇Φ|^2=|∇ f |^2 inΩ∩𝒱_z Φ = f on∂Ω∩𝒱_z ∂_n Φ =-∂_n f on∂Ω∩𝒱_z. .Moreover, 𝒱_z can be chosen such that the following uniqueness result holds: if a function Φ̃∈ C^∞(𝒱_z,ℝ) satisfies the previous equalities, then Φ̃=Φ on 𝒱_z. Now, one concludes using the fact that ∂Ω is compact and can thus be covered by a finite number of these neighborhoods (𝒱_z)_z∈∂Ω.Let us mention another standard approach to prove Proposition <ref>, using the notion of viscosity solutions. Let us recall some resultsfrom <cit.>. For (x,y)∈Ω^2, one defines d̃(x,y):=inf_T>0,γ∫_0^T |∇ f(γ(t))| dt,where the infimum is taken over T>0 and over Lipschitz curves γ: [0,T]→Ω which satisfy γ(0)=x, γ(T)=y, |γ'|≤ 1. Then,v(x):=inf{ f(y)+d̃(x,y),y∈∂Ω} is Lipschitz and is a viscosity solution to{|∇ v | =|∇ f |inΩ v= f on∂Ω..Let us notice that this implies |∂_nv |=|∂_nf | on Ω.To prove Proposition <ref> using this result, one has to show that v is C^∞ near ∂Ω and ∂_nv =-∂_nf. This is a consequence of the characteristic method, see <cit.>. Let x^* be a local minimum of f|_∂Ω and let us denote by Φ̃ the solution to the eikonal equation (<ref>) introduced in Corollary <ref>, defined on a neighborhood V^* of x^*. Then, one has on V^* ∩ V_∂Ω:Φ̃=Φ-f(x^*)where Φ is the solution to (<ref>). Let us now introduce the function f_- which will be used in the sequel. Assume that [H3]holds. Let Φ∈ C^∞(V_∂Ω,) be the function introduced in Proposition <ref>. Let us define the function f_- ∈ C^∞(V_∂Ω,) byf_-=Φ-f/2. Then, f_-=0 on ∂Ω, and up to choosing a smaller neighborhood V_∂Ω of ∂Ω, the function f_- is positive in V_∂Ω∖∂Ω and |∇ f_-| >0 on V_∂Ω. Since ∂_n(Φ-f)=-2∂_n f<0 and Φ=f on ∂Ω,then, up to choosing a smaller neighborhood V_∂Ω of ∂Ω, one has Φ> f on V_∂Ω∖∂Ω and |∇ (Φ-f)| >0 on V_∂Ω.We are now in position to prove the main result of this section.Let us assume that [H1] and [H3]hold. Let Φ be the function given by Proposition <ref>.Denote by z a local minimum of f|_∂Ω and denote by B_z⊂∂Ω the associated basin of attraction (see Definition <ref>). Besides, let Γ_z⊂∂Ω be an open domain such that Γ_z⊂ B_z and z∈Γ_z. Then there exists a neighborhoodof Γ_z in Ω, denoted by V_Γ_z, such that ∂ V_Γ_z∩∂Ω⊂B_z and for all x∈ V_Γ_z,d_a(x,z)=Φ(x)-f(z). Notice that in this proposition, Γ_z can be chosenas large as neededin B_z.Let Φ be the function given by Proposition <ref>. The proof is divided into three steps. Step 1. Let us first define V_Γ_z.To this end let us denote by f_- and V_∂Ω respectively the function and the neighborhood of ∂Ωgiven by Proposition <ref>.For ε>0 small enough one defines V_ε= { y∈Ω,0≤f_- (y)≤ε}⊂ V_∂Ω.The parameter >0 can be chosen such that there is no critical point of f on ∂ V_∩Ω={ y∈Ω, f_-(y)=ε}. The set V_ is a neighborhood of ∂Ω in Ω (see Figure <ref> for a schematic representation). Let us now fix such a >0. Assumption [H3] together with the fact that∂_nΦ<0 on ∂Ω, imply that there exists a neighborhood V_Γ_z of Γ_z in Ω, such that V_Γ_z⊂ V_ε, ∂ V_Γ_z∩∂Ω⊂B_z and∂_nΦ>0, on ∂ V_Γ_z∩Ω. The set V_Γ_z isschematically represented on Figure <ref>.Step 2. Let us first prove that for all x∈ V_Γ_z, d_a(x,z)≥Φ(x)-f(z).For x∈ V_Γ_z,denote by ((γ_1,I_1),…,(γ_N,I_N)) the curves given by Theorem <ref> ordered such that lim_t→ (inf I_1)^+γ_1(t)=z, lim_t→ (sup I_N)^-γ_N(t)=x,and which realize the Agmon distance between x and z. One has to deal with the two following cases:* either ∀ k∈{1,…,N}, ∀ t∈ I_k, γ_k(t)∈ V_, * or ∃ k∈{1,…,N} and∃ t∈ I_k,γ_k(t)∈Ω∖ V_. In the first case, since Φ is defined on V_, it holdsΦ(x)-f(z)=Φ(x)-Φ(z)= ∑_j=1^N∫_I_j d /dtΦ∘γ_j(t) dt. Using Lemma <ref> and the fact that |∇Φ|=g on Ω∩ V_∂Ω and |∇_T Φ|=g on ∂Ω, it holds, for all j ∈{1, … ,N},∫_I_j d /dtΦ∘γ_j(t) dt ≤ L(γ_j,I_j)and thus Φ(x)-f(z)≤∑_j=1^N L(γ_j, I_j)=d_a(x,z).Let us now consider the second case. Let us introduce k_1 ∈{1,…,N} and t_1∈ I_k_1such that for all t<t_1, γ_k_1(t) ∈ V_, for all k ∈{1, …, k_1-1}, for all t∈ I_k, γ_k(t) ∈ V_ and such that there exists β>0 such that for all t∈ (t_1,t_1+β], γ_k_1(t) ∉ V_. The couple (k_1,t_1) thus represents the “first time” the curves γ_1,…,γ_N leave V_. Likewise, let us introduce k_2 ∈{1,…,N} and t_2∈ I_k_2such that for all t>t_2, γ_k_2(t) ∈ V_, for all k ∈{k_2+1, …, N}, for all t∈ I_k, γ_k(t) ∈ V_ and such that there exists β>0 such that for all t∈ [t_2-β,t_2), γ_k_2(t) ∉ V_. The couple (k_2,t_2) thus represents the “last time” the curves γ_1,…,γ_N leave Ω∖ V_. From Step 1, there is no critical point of f on ∂ V_∩Ω={ y∈Ω, f_-(y)=ε}. Therefore, by construction of the curves(γ_k)_k=1,…,N, the times t_1 and t_2 arefinite and belong respectively to int I_k_1 and int I_k_2. One has by continuity of γ_k_1 and γ_k_2, f_-(γ_k_1(t_1))= f_-(γ_k_2(t_2))=. Since Φ is defined on V_ε, using again Lemma <ref> and the fact that |∇Φ|=g on Ω and |∇_T Φ|=g, one has|Φ(γ_k_1(t_1))-Φ(z) | ≤∑_j=1^k_1-1 L(γ_j,I_j)+L(γ_k_1,(inf I_k_1,t_1)).In addition, using Lemma <ref>,∑_j=1^k_1-1 L(γ_j,I_j)+L(γ_k_1,(inf I_k_1,t_1))=d_a(z,γ_k_1(t_1)).Thus |Φ(γ_k_1(t_1))-Φ(z) |≤ d_a(z,γ_k_1(t_1)). By similar arguments, one obtains |Φ(x)-Φ(γ_k_2(t_2))|≤ d_a(γ_k_2(t_2),x).Thanks to the definition (<ref>) of f_- and using the fact that f_-(γ_k_1(t_1))= f_-(γ_k_2(t_2))=, one has | f(γ_k_2(t_2))-f(γ_k_1(t_1))|=|Φ(γ_k_2(t_2))-Φ(γ_k_1(t_1))|. In addition, using(<ref>) one obtainsd_a(γ_k_1(t_1),γ_k_2(t_2))≥| f(γ_k_2(t_2))-f(γ_k_1(t_1))|=|Φ(γ_k_2(t_2))-Φ(γ_k_1(t_1))|. Using Lemma <ref> and gathering these three last inequalities, one getsd_a(x,z) =d_a(z,γ_k_1(t_1))+d_a(γ_k_1(t_1),γ_k_2(t_2))+d_a(γ_k_2(t_2),x)≥|Φ(z)-Φ(γ_k_1(t_1))| + |Φ(γ_k_2(t_2))-Φ(γ_k_1(t_1))| +|Φ(x)-Φ(γ_k_2(t_2))|≥|Φ(z)-Φ(x)|≥Φ(x)-Φ(z)=Φ(x)-f(z). Step 3. Let us now show that for all x∈ V_Γ_z, d_a(x,z)≤Φ(x)-f(z).The proof of this inequality is very similar to the second step in the proof of Proposition <ref>. For x∈ V_Γ_z, let γ be defined by (<ref>)–(<ref>) (where Φ is defined by (<ref>)), with γ(0)=x. The function γ is with values in V_Γ_z since ∂_nΦ>0 on ∂ V_Γ_z∩Ω and ∂ V_Γ_z∩∂Ω⊂ B_z. Thus γ is defined on ℝ_+. Thanks to the definition (<ref>) of the vector field X, if there exists a time t_∂Ω such that γ(t_∂Ω) is in ∂Ω, then, for all t ≥ t_∂Ω, γ(t) ∈∂Ω. The function t∈ℝ_+↦γ(t) is continuous, piecewise C^∞ and satisfies lim_t→ +∞γ(t)=z.Then, as in the second step of the proof of Proposition <ref>, one has Φ(x)-f(z)=L(γ,(0,∞)).Using Lemma <ref>, one obtains that d_a(x,z)≤L(γ,(0,∞))=Φ(x)-f(z). This proves the inequality: for all x∈ V_Γ_z, d_a(x,z)≤Φ(x)-f(z). This concludes the proof of Proposition <ref>. The following corollary is similar to Corollary <ref> in the sense that is deals with the case of equality between the Agmon distance and the function Φ introduced in Proposition <ref>. Corollary <ref> will be needed inthe proof of Proposition <ref>.Let us assume that [H1] and [H3]hold.Let Φ be the function introduced in Proposition <ref> and, let f_- and V_∂Ω be respectively the function and the neighborhood of ∂Ωgiven by Proposition <ref>.Let V_α be defined by (<ref>),the parameter α >0 is chosen such that: (i) V_α⊂ V_Ω,(ii)there is no critical point of f on ∂ V_α∩Ω={ w∈Ω, f_-(w)= α}, (iii)_nf>0 on ∂ V_α∩Ω, (iv)_nf^-<0 on ∂ V_α∩Ω, and(v) |∇Φ|≠ 0 in V_α.Notice that it is possible to choose such an α>0 since _nf^-=∂_nΦ=-∂_nf<0 on Ω=V_0. Let x,y∈ V_α and denote by ((γ_1,I_1),…,(γ_N,I_N)) the curves given by Theorem <ref> ordered such that lim_t→ (inf I_1)^+γ_1(t)=x, lim_t→ (sup I_N)^-γ_N(t)=y and which realize the Agmon distance between x and y.Let us assume thatΦ(x)-Φ(y)=d_a(x,y).Then, for all i∈{1,…,N}, Im γ_i⊂ V_α and there exist measurable functions λ_i : I_i→ℝ_+ such that for almost every t in{t∈ I_i,γ_i(t)∈Ω}, one has γ_i'(t)=-λ_i(t) ∇Φ(γ_i(t) ), and such that for almost every t in int{t∈ I_i,γ_i(t)∈∂Ω}, one has γ_i'(t)=-λ_i(t) ∇_T Φ(γ_i(t) ). Moreover, if I_i is not bounded (namely I_i=(-∞,0] or I_i=[0,+∞)), λ_i(t)=1 for almost every t∈ I_i, and if I_i=[0,1], λ_i∈ L^∞([0,1],ℝ_+).According to Definition <ref>, the set of curves {γ_1,…,γ_N} introduced in Corollary <ref> is a generalized integral curve of the vector field{ -∇ΦinV_α∩Ω, -∇_T Φon∂Ω. . The proof of this statement is similar to the proof of Corollary <ref>.Let us first prove that for all i∈{1,…,N}, Im γ_i⊂ V_α. If it is not the case, then there exist k∈{1,…,N} andt∈ I_k such thatγ_k(t)∈Ω∖ V_α. Let the couples (t_1,k_1) and (t_2,k_2) be definedas in Step 2 of the proof ofProposition <ref>. Then, one has (see the second step of the proof of Proposition <ref>), d_a(x,γ_k_1(t_1))≥Φ(x)-Φ(γ_k_1(t_1)), d_a(γ_k_1(t_1),γ_k_2(t_2))≥Φ(γ_k_1(t_1))-Φ(γ_k_2(t_2)) and d_a(γ_k_2(t_2),y)≥Φ(γ_k_2(t_2))-Φ(y). Since one hasby assumption andfrom Lemma <ref>: Φ(x)-Φ(y)=d_a(x,y)=d_a(x,γ_k_1(t_1))+d_a(γ_k_1(t_1),γ_k_2(t_2))+d_a(γ_k_2(t_2),y),all the previous inequalities are equalities and in particular, it holds:d_a(γ_k_1(t_1),γ_k_2(t_2))= Φ(γ_k_1(t_1))-Φ(γ_k_2(t_2)) = f(γ_k_2(t_2)) - f(γ_k_1(t_1))≥ 0.Using Corollary <ref>, this implies that when restricting γ_k_1 to I_k_1∩ [t_1,∞) and γ_k_2 to I_k_2∩ (-∞,t_2] the set of curves {γ_k_1,…,γ_k_2} is a generalized integral curve of {∇ f inΩ ,∇_T f onΩ.see Definition <ref>. Let D=Ω∖ V_α (D=Ω∩ V_α={w∈Ω, f_-(w)=α} is C^∞since f^- is C^∞ and _nf^-<0 on ∂ V_α∩Ω= D which implies that there is no critical point of f_- on D). Then, fromCorollary <ref> andby definition of (t_1,k_1) (see the second step of the proof of Proposition <ref>), there exists >0 and a measurable function λ: [t_1, t_1+]→ℝ^+such that for all t ∈ [t_1, t_1+]:γ_k_1 '(t)=λ(t) ∇ f(γ_k_1 (t) )and for all t ∈ (t_1, t_1+]: γ_k_1(t) ∈ D. As in Step 2 of the proof of Proposition <ref>, let usintroduce a smooth tangential and normal system of coordinates around γ_k_1(t_1)∈ D in D, denoted by ϕ(x)=(x_T,x_N). The function ϕ is defined from a neighborhood ofγ_k_1(t_1) in D to ^d. Moreover, one has x_N≥ 0 and x_N(x)=0 if and only if x∈ D. We may assume that the neighborhood U_β⊂D on which ϕ is defined is such that ϕ(U_β)=U×[0,β] for β >0 and U ⊂^d-1. Since ∂_nf>0 on ∂ D, β>0 can be chosen small enough such that ∇ f(x) · n(x)>0 for all x∈ U_β where n(x)=-∇ x_N(x)/|∇ x_N(x)|. Indeed, for x∈∂D, n(x) is nothing but the unit outward normal to ∂ D.Now, by continuity of the curve γ_k_1, there exists μ>0 such that for all t∈ (t_1,t_1+μ], γ_k_1(t)∈ U_β. The same considerations as in Step 2 of the proof of Proposition <ref> can then be used to show that: x_N(γ(t))≤ 0, for all t∈ [t_1,t_1+μ] and thusγ_k_1(t)∉D for all t∈[t_1,t_1+μ]. This contradicts (<ref>). Thus, for all i∈{1,…,N}, Im γ_i⊂ V_α. Then, the announced result follows by the same arguments as those used in the proof of Corollary <ref> with f replaced by Φ together with the fact that Φsatisfies (<ref>) on V_α andfor all i∈{1,…,N}, Im γ_i⊂ V_α. § CONSTRUCTION OF THE QUASI-MODES AND PROOF OF THEOREM <REF>The aim of this section is to build the quasi-modes ũ and (ϕ̃_i)_i=1, … n satisfying the conditions stated in Section <ref>. Let us recall that span(ũ) (resp. span(ϕ̃_i, i=1,…,n)) is intended to be a good approximation (in the sense made precise in items 1 and 2 inProposition <ref>) of Ran(π_[0,√(h))(L^D,(0)_f,h(Ω))) (resp. Ran(π_[0,h^3/2)(Δ^D,(1)_f,h(Ω)))).As recalled in Proposition <ref>, it is known that the dimension of Ran(π_[0,h^3/2)(Δ^D,(1)_f,h(Ω))) is equal to the number of generalized critical points of index 1 (see <cit.>) which are in our setting, thanks to assumptions [H1], [H2] and [H3], the local minima (z_i)_i=1,…,n of f|_∂Ω. In addition, it is known that the 1-forms in Ran(π_[0,h^3/2)(Δ^D,(1)_f,h(Ω))) are localized in the limit h → 0 in small neighborhoods of the local minima (z_i)_i=1,…,n. For each local minimum z_i, we construct an associated quasi-mode ϕ̃_i, using an auxiliary Witten Laplacian on 1-forms with mixed tangential-normal boundary conditions. This Witten Laplacian is defined on a domain Ω̇_i ⊂Ω with suitable boundary conditions, so that its only small eigenvalue (namely in the interval [0,h^3/2)) is 0, thanks to a complex property (see <cit.>). The associated eigenform is localized near z_i, which can be proven thanks to Agmon estimates. Moreover, a precise estimate of this eigenform can be obtained thanks to a WKB expansion. The quasi-mode ϕ̃_i is then this eigenform multiplied by a suitable cut-off function. This section is organized as follows. In Section <ref>, we define a Witten Laplacian with mixed boundary conditions on a open domain Ω̇_i⊂Ω associated to each z_i, i∈{1,…,n}, and we study its spectrum.Section <ref> is dedicated to the construction of the quasi-modes ((ϕ̃_i)_i=1,…,n, ũ). In Section <ref>, we prove Agmon estimates on the eigenform associated with the smallest eigenvalue of the Witten Laplacian with mixed boundary conditions on Ω̇_i and in Section <ref> we compare this eigenform with a WKB approximation. We finally use this construction and these estimates to prove Theorem <ref> in Section <ref>.§.§ Geometric setting and definition of the Witten Laplacians with mixed boundary conditionsThis section is organized as follows. In Section <ref>, we discuss some general results on traces of differential forms and we introduce the Witten Laplacians with mixed tangential Dirichlet and normal Dirichlet boundary conditions on manifolds with boundary. In Section <ref>, the domain Ω̇_i⊂Ω associated with each z_i, i∈{1,…,n}, is defined. Finally, Section <ref> is dedicated to the study of the spectrum of the Witten Laplacian with mixed tangential Dirichlet boundary conditions and normal Dirichlet boundary conditions on Ω̇_i.§.§.§ Trace estimates for differential forms and Witten Laplacians on Lipschitz domain with mixed boundary conditionsIn this section, we first discuss some general results on traces of differential forms. This is crucial to then build the Witten Laplacians with mixed boundary conditions. In the following, Ω̇ refers to anysubmanifold Ω̇ of Ω with Lipschitz boundary. We will call such a submanifold a Lipschitz domain.We first recall that for any Lipschitz domain Ω̇, the trace application{Λ^pH^1 ( Ω̇)→Λ^pH^1/2 (Ω̇) G↦ G|_∂Ω̇.is a linear continuous and surjective application. We would like to present extensions of this result to less regular forms. Weak definition of traces For a Lipschitz domain Ω̇, let us introduce the functional spaces Λ^p H_d ( Ω̇):= {u∈Λ^p L^2 ( Ω̇), du∈Λ^p+1 L^2 ( Ω̇)}andΛ^p H_d^* ( Ω̇) := {u∈Λ^p L^2 ( Ω̇), d^*u∈Λ^p-1 L^2 ( Ω̇)} equipped with their natural graph norms. One recalls that for a differential form f in L^2 (Ω̇), the tangential and normal components are defined as follows:f= 𝐭 f+𝐧 fwith𝐭 f= 𝐢_n(n^♭∧ f)and𝐧 f= n^♭∧(𝐢_n f),where the superscript ♭ stands for the usual musical isomorphism: n^♭ is the 1-form associated with the outgoing unit normal vector n. Moreover,f^2_ L^2 (Ω̇)=𝐭 f^2_ L^2 (Ω̇)+ 𝐧 f^2_ L^2 (Ω̇) =n^♭∧ f^2_ L^2 (Ω̇)+ 𝐢_n f^2_ L^2 (Ω̇). TheGreen formula for differential forms (u,v)∈Λ^p H^1 ( Ω̇) ×Λ^p +1H^1 ( Ω̇) writes ⟨ du, v⟩_ L^2 ( Ω̇)- ⟨ u, d^*v⟩_ L^2 ( Ω̇) = ∫_∂Ω̇⟨n^♭∧ u,v⟩_ T^*_σΩ̇ dσ= ∫_∂Ω̇⟨n^♭∧ u, 𝐧 v⟩_ T^*_σΩ̇ dσ=∫_∂Ω̇⟨u,𝐢_n v⟩_ T^*_σΩ̇ dσ=∫_∂Ω̇⟨𝐭u,𝐢_n v⟩_ T^*_σΩ̇ dσ,where we used the standard relation (n^♭∧)^*=𝐢_n. Using this Green formula, the tangential (resp. normal) traces can be defined for forms in Λ H_d(Ω̇)(resp. Λ H_d^*(Ω̇)) by duality. Indeed, for anyu ∈Λ^p H_d ( Ω̇), n^♭∧ u∈Λ^p+1 H^-1/2 (Ω̇) is defined by∀ g∈Λ^p+1H^1/2 (Ω̇), ⟨ n^♭∧ u , g⟩_H^-1/2 (Ω̇),H^1/2 (Ω̇) =⟨ du, G⟩_ L^2 ( Ω̇)-⟨ u, d^*G⟩_ L^2 ( Ω̇),where G is any form in Λ^p+1H^1 ( Ω̇) whose trace in Λ^p+1H^1/2 (Ω̇) is g. This definition is independent of thechosen extension G (this indeed follows from the Green formula (<ref>)together with the density of Λ^p𝒞^∞ ( Ω̇)in Λ^p H_d ( Ω̇), seefor example <cit.>). Similarly, for any u ∈Λ^p H_d^* ( Ω̇), 𝐢_n u∈Λ^p-1H^-1/2 (Ω̇) is defined by∀ g∈Λ^p-1 H^1/2 (Ω̇), ⟨𝐢_n u , g⟩_H^-1/2 (Ω̇),H^1/2 (Ω̇) = ⟨ u, dG⟩_ L^2 ( Ω̇)-⟨ d^*u, G⟩_ L^2 ( Ω̇),where G is any extension of g in Λ^p-1H^1 ( Ω̇). Let Γ be any subset of ∂Ω̇. For u∈Λ^p H_d ( Ω̇), we will write 𝐭 u |_Γ=0 if n^♭∧ u|_Γ=0. Ifu∈Λ^p H_d ( Ω̇) and n^♭∧ u|_Γ∈Λ^p+1 L^2(Γ), the tangential trace on Γ is defined by𝐭 u|_Γ:=𝐢_n (n^♭∧ u) ∈Λ^p L^2(Γ) , so that 𝐭 u_ L^2(Γ) = n^♭∧ u_ L^2(Γ). Similarly, for u∈Λ^p H_d^* ( Ω̇), we will write 𝐧 u |_Γ=0 if 𝐢_nu|_Γ=0. If u∈Λ^p H_d^* ( Ω̇)and 𝐢_n u|_Γ∈Λ^p-1 L^2(Γ), the normal traceon Γ is defined by𝐧 u|_Γ:=n^♭∧ (𝐢_nu) ∈Λ^p L^2(Γ) , so that 𝐧 u_ L^2(Γ) =𝐢_nu_ L^2(Γ).Lastly, if u∈Λ^pH_d(Ω̇)∩Λ^pH_d^*(Ω̇) is such that n^♭∧ u|_Γ∈Λ^p+1L^2(Γ) and 𝐢_nu∈Λ^p-1L^2(Γ) then u admits a trace u|_Γin L^2(Γ)defined byu|_Γ := 𝐭 u|_Γ+ 𝐧 u|_Γ.This definition iscompatible with (<ref>) and such a differential form satisfiesu|_Γ^2_ L^2(Γ) =𝐭 u|_Γ^2_ L^2(Γ) + 𝐧 u|_Γ^2_ L^2(Γ) = n^♭∧ u_ L^2(Γ)^2 + 𝐢_nu_ L^2(Γ)^2 . All the above definitions coincide moreover with the usual ones when u belongs to Λ^ℓ H^1 ( Ω̇). Let us finally note for further references that if traces are in L^2(∂Ω̇), a direct consequence of the Green formula (<ref>) is the following: for every u,v ∈Λ L^2 ( Ω̇) such thatdu, d^*u,d^*du, dd^*u, dv, d^*v∈Λ L^2 ( Ω̇)and n^♭∧ d^*_f,hu, 𝐢_n d_f,hu,n^♭∧ v, 𝐢_n v∈Λ L^2(Ω̇),⟨ (d_f,hd^*_f,h+d^*_f,hd_f,h)u,v⟩_L^2(Ω̇) =⟨ d_f,hu,d_f,hv⟩_L^2(Ω̇)+ ⟨ d_f,h^*u,d_f,h^*v⟩_L^2(Ω̇) +h ∫_Ω̇⟨n^♭∧ d^*_f,h u,n^♭∧(𝐢_n v) ⟩_T^*_σΩ̇dσ -h ∫_Ω̇⟨n^♭∧ v,n^♭∧(𝐢_n d_f,hu) ⟩_ T^*_σΩ̇dσ . The Gaffney's inequality The following extension of Gaffney's inequality (see <cit.>) will be useful in the sequel (we refer to Section <ref> for the definitions of the Hilbert space Λ^p H^1_T(Ω̇) and Λ^p H^1_N(Ω̇)). Notice that in the following resultwe usethat Ω̇ is smooth (there are actually counterexamples for Lipschitz domains, see for example <cit.>).Let Ω̇ be a smooth domain. The equality{u∈Λ^p L^2(Ω̇) s.t. du, d^*u∈ L^2(Ω̇)and 𝐭 u=0on Ω̇}= Λ^p H^1_T(Ω̇)holds algebraically and topologically, the functional space in the left-hand side being equipped with the norm associated with the scalar productQ(u,v) :=⟨ u, v⟩_ L^2(Ω̇) +⟨ du,dv ⟩_ L^2(Ω̇)+⟨ d^*u, d^*v⟩_ L^2(Ω̇).In a similar way, the following equality holds algebraically and topologically: {u∈Λ^p L^2(Ω̇) s.t. du, d^*u∈ L^2(Ω̇)and 𝐧 u=0on Ω̇} = Λ^p H^1_N(Ω̇).Notice that in the defintion of the functional spaces above, the equality 𝐭 u=0 and 𝐧 u=0 hold in the weak sense defined above (see (<ref>) and (<ref>)). A direct consequence of this lemma is that a differential form in Λ H_d(Ω̇) ∩Λ H_d^*(Ω̇) such that 𝐭u=0 or 𝐧 u=0 on ∂Ω̇ admits a trace in Λ L^2(∂Ω̇).The statement of Gaffney's inequality in <cit.> reads as follows (see indeed Corollary 2.1.6 and Theorem 2.1.7 there):∃C>0, ∀u∈Λ^p H^1_T(Ω̇)∪Λ^p H^1_N(Ω̇),u^2_H^1(Ω̇)≤ C Q(u,u).Since it also holds that, for some C'>0 and any u∈Λ^p H^1(Ω̇),Q(u,u)≤ C' u^2_H^1(Ω̇), the scalar products ⟨·,·⟩_H^1and Q(·,·) are then equivalenton bothΛ^p H^1_T(Ω̇) and Λ^p H^1_N(Ω̇). The above lemma can be seen as a generalization of this result to the spaces {u∈Λ^p L^2(Ω̇) s.t. du, d^*u∈ L^2(Ω̇)and 𝐭 u=0on Ω̇} and {u∈Λ^p L^2(Ω̇) s.t. du, d^*u∈ L^2(Ω̇)and 𝐧 u=0on Ω̇}. We only prove the first equality in Lemma <ref>, the second one being similar. Let us defineH := {u∈Λ^p L^2 ( Ω̇) s.t. du, d^*u∈Λ L^2 (Ω̇) and n^♭∧u=0on Ω̇} which is a Hilbert space once equipped with the scalar product Q. FromGaffney's inequality (<ref>),Λ^p H^1_T(Ω̇) is a closed subset of H and to conclude, we just have to show that (Λ^p H^1_T(Ω̇))^⊥={0}, the orthogonal complement of H being taken with respect to the norm inherited from Q. Consider then u∈ H such that for any v∈Λ^pH^1_T(Ω̇),0 =Q(u,v) = ⟨ u, v⟩_ L^2(Ω̇) +⟨ du,dv ⟩_ L^2(Ω̇)+⟨ d^*u, d^*v⟩_ L^2(Ω̇).The above equality holds in particular for every v∈ D whereD={v ∈Λ^pH^2(Ω̇) , 𝐭 v|_Ω̇=𝐭 d^* v|_Ω̇=0}. Fix such a v. Since n^♭∧ u=0 on Ω̇, applying (<ref>) to u and d v∈Λ^p+1H^1(Ω̇)then leads to ⟨ du,dv ⟩_ L^2(Ω̇) = ⟨ u,d^*dv ⟩_ L^2(Ω̇).Applying also (<ref>) to u and d^*v∈Λ^p-1H^1(Ω̇) gives ⟨ d^*u,d^*v ⟩_ L^2(Ω̇)= ⟨ u,dd^*v ⟩_ L^2(Ω̇) -⟨𝐢_n u , d^*v|_Ω̇⟩_H^-1/2 (Ω̇),H^1/2 (Ω̇).Since Λ^p𝒞^∞ (Ω̇ ) is densely embedded in both Λ^p H_d ( Ω̇) and Λ^p H_d^* ( Ω̇) (see for example <cit.>), we have moreoverfor some sequence (u_k)_k∈ℕ of Λ^p𝒞^∞(Ω̇) forms:⟨𝐢_n u,d^*v|_Ω̇⟩_H^-1/2 (Ω̇),H^1/2 (Ω̇) = lim_k→+∞∫_Ω̇⟨𝐢_n u_k,d^*v ⟩_T^*_σΩ̇dσ= lim_k→+∞∫_Ω̇⟨𝐢_n u_k,n^♭∧(𝐢_n d^*v) ⟩_T^*_σΩ̇dσ= 0,where the second equality is a consequence of 𝐭 d^* v|_Ω̇=0. It consequently follows 0= ⟨ u, v⟩_ L^2(Ω̇) +⟨ u,d^*dv ⟩_ L^2(Ω̇)+⟨ u, dd^*v⟩_ L^2(Ω̇) = ⟨ u, (I+Δ_H^(p))v⟩_ L^2(Ω̇) ,where Δ_H^(p) denotes the Hodge Laplacian on Ω̇ with domain D defined by (<ref>).Since the unbounded operator (Δ_H^(p), D) isselfadjoint and nonnegative on Λ^p L^2(Ω̇), we have in particular(I+Δ_H^(p))= Λ^p L^2(Ω̇) and we deduce from (<ref>) that u=0, which completes the proof.The case of mixed normal-tangential Dirichlet boundary conditions Let Γ_T and Γ_N be two disjoint open subsets of ∂Ω̇ such that Γ_T∪Γ_N = Ω̇. The objective of this section is to consider differential forms such that 𝐭u=0 on Γ_T and 𝐧 u=0 on Γ_N, and to state results on the existence of a trace in L^2(∂Ω̇) for such differential forms, as well as subelliptic estimates. In general,a trace in L^2(∂Ω̇) does not exist in such a setting <cit.>: one needs a geometric assumption, namley that Γ_T and Γ_N meet at an angle strictly smaller than π. This means that the angle between Γ_T and Γ_N measured in Ω̇ is smaller than π. More precisely, see <cit.>, locally around any point x_0 ∈Γ_T∩Γ_N, one requires that there exists a local system of coordinates (x_1,x”,x_n) ∈×^d-2× on a neighborhood V_0 of x_0, and two Lipschitz functions φ̃:^n-1→ and ψ̃:^n-2→ such that Ω̇∩ V_0={x_n > φ̃(x_1,x”)}, Γ_T ∩ V_0={x_n=φ̃(x_1,x”)andx_1 > ψ̃(x”)} and Γ_N ∩ V_0={x_n=φ̃(x_1,x”)andx_1 < ψ̃(x”)} and∂_x_1φ̃(x_1,x”) ≥κonx_1>ψ̃(x”) ∂_x_1φ̃(x_1,x”)≤ -κonx_1<ψ̃(x”)for some positive κ. This is equivalent to the existence of a smooth vector field θ on ∂Ω̇ such that ⟨θ , n ⟩ < 0 on Γ_T and ⟨θ, n⟩ > 0 on Γ_N, which is one of the key ingredient of the proofs used in <cit.>. Let Γ be any open Lipschitz subset of ∂Ω̇. According to <cit.>, the space{u∈Λ^p𝒞^∞ (Ω̇ ),u≡ 0in a neighborhood of Ω̇∖Γ}is densely embedded in bothΛ^p H_d,Γ ( Ω̇) := {u∈Λ^p H_d ( Ω̇), (n^♭∧ u)⊂Γ}andΛ^p H_d^*,Γ ( Ω̇) := {u∈Λ^p H_d^* ( Ω̇), (𝐢_nu)⊂Γ}.Inaddition, according to <cit.>, for(u,v)∈Λ^p H_d ( Ω̇)×Λ^p+1 H_d^* ( Ω̇) satisfyingthe trace conditions 𝐢_n v ∈Λ^p L^2 (Ω̇), 𝐢_n v⊂Γand n^♭∧ u ∈Λ^p+1 L^2(Γ),orn^♭∧ u∈Λ^p+1 L^2 (Ω̇), (n^♭∧ u)⊂Γand𝐢_n v ∈Λ^p L^2(Γ),one has the following Green formula (compare with (<ref>)):⟨ du, v⟩_ L^2 ( Ω̇)-⟨ u, d^*v⟩_ L^2 ( Ω̇) =∫_Γ⟨n^♭∧ u,n^♭∧(𝐢_n v) ⟩_T^*_σΩ̇dσ= ∫_Γ⟨𝐢_n (n^♭∧ u),𝐢_n v ⟩_T^*_σΩ̇ d σ .One is now ready to state the following proposition implied by Theorems 1.1 and 1.2 of <cit.> (see also Theorems 4.1 and 4.2of <cit.>). Let us assume that Ω̇ is a Lipschitz domain. Let Γ_T and Γ_N be two disjoint Lipschitz open subsets of ∂Ω̇ such that Γ_T∪Γ_N = Ω̇ and such that Γ_T and Γ_N meet at an angle strictly smaller than π. Then, the following results hold:(i) Let u be a differential form such that u∈Λ^p L^2 ( Ω̇), du ∈L^2 ( Ω̇), d^*u∈L^2 ( Ω̇), 𝐭 u|_Γ_T=0and 𝐧 u|_Γ_N=0.Then u satisfies u∈Λ^p H^1/2 ( Ω̇)and𝐢_nu, n^♭∧ u ∈Λ^pL^2 (Ω̇)as well as the subelliptic estimate:u_ H^1/2 ( Ω̇)+ u|_Ω̇_ L^2 (Ω̇)≤ C(u_ L^2 ( Ω̇) +du_ L^2 ( Ω̇)+d^*u_ L^2 ( Ω̇)),where u|_Ω̇ is defined by (<ref>).(ii) The unbounded operators d_T^(p) ( Ω̇) and δ_N^(p) ( Ω̇)on Λ^p L^2 ( Ω̇) defined byd_T^(p) ( Ω̇) = d_f,h^(p)with domain D (d_T^(p) ( Ω̇))={u∈Λ^pL^2 ( Ω̇), d_f,hu ∈Λ^p+1L^2(Ω̇), 𝐭u|_Γ_T=0 },andδ_N^(p) ( Ω̇) =( d_f,h^(p))^*with domain D(δ_N^(p) ( Ω̇))={u∈Λ^p L^2 ( Ω̇), d^*_f,hu∈Λ^p-1L^2( Ω̇), 𝐧u|_Γ_N=0 } ,are closed, densely defined, and adjoint one of each other.Note that in the point (i) of Proposition <ref>, d and d^* can be replaced by d_f,h and d^*_f,h owing to the relations d_f,h=hd+df∧ and d^*_f,h=hd^*+𝐢_∇ f. Moreover, the point (ii) is actually proven in <cit.> for d and d^* but remains true for d_f,h and d^*_f,h since (df∧)^*=𝐢_∇ f on L^2 ( Ω̇).The mixed Witten Laplacian Δ^M_f,h(Ω̇) We are now in position to define the mixed Witten Laplacian Δ^M_f,h(Ω̇) (the upperscript M stands for mixed boundary conditions) with tangential Dirichlet boundary conditions on Γ_T and normal Dirichlet boundary conditions on Γ_N (see <cit.> for more results on such operators). The operator Δ_f,h^M,(p) ( Ω̇)on L^2 (Ω̇) is defined byΔ_f,h^M,(p) ( Ω̇) := d_T^(p-1) ( Ω̇)∘δ_N^(p) ( Ω̇)+δ_N^(p+1) ( Ω̇)∘ d_T^(p) ( Ω̇),inthe sense of composition of unbounded operators, where d_T and δ_N have been introduced in Proposition <ref>. Notice thatfor any u∈Λ^p H_d ( Ω̇) such that 𝐭 u|_Γ_T=0,one hasdu∈Λ^p+1H_d( Ω̇)and𝐭 du|_Γ_T=0. The latter is easy to check whenu∈Λ^p H^2 ( Ω̇) and can be proved here using (<ref>) together with the density of {u∈Λ^p𝒞^∞ (Ω̇ ),u≡ 0in a neighborhood of Ω̇∖Γ_T} intoΛ^p H_d,Ω̇∖Γ_T ( Ω̇). Likewise, one has d_f,hd_f,h=0 in the distributional sense and𝐭 d_f,hu|_Γ_T=0 for u∈Λ^p H_d ( Ω̇) such that 𝐭 u|_Γ_T=0. This implies in particular { d_T⊂ d_Tand d_T^2=0,δ_N⊂δ_Nandδ_N^2=0. .Owing to this last relation and to Proposition <ref>, a result due to Gaffney(see e.g. the proof of <cit.>) states that Δ_f,h^M,(p) ( Ω̇) is a densely defined nonnegative selfadjointoperator on L^2 ( Ω̇) (with domain defined below in (<ref>)).The domain D( 𝒬_f,h^M,(p) ( Ω̇) ) of the closed quadratic form 𝒬_f,h^M,(p) ( Ω̇) associated withΔ_f,h^M,(p) ( Ω̇) is given byD ( 𝒬^M,(p)_f,h ( Ω̇) ) = D (d_T^(p) ( Ω̇) )∩ D (δ_N^(p) ( Ω̇))={v∈Λ^p L^2 ( Ω̇), dv ∈L^2 ( Ω̇), d^*v∈L^2 ( Ω̇), 𝐭 v|_Γ_T=0and 𝐧 v|_Γ_N=0} and for any u,v∈ D( 𝒬^M,(p)_f,h ( Ω̇)), 𝒬_f,h^M,(p) ( Ω̇)(u,v)=⟨ d_T u,d_Tv ⟩_L^2 +⟨δ_Nu,δ_Nv ⟩_L^2.This is proven in <cit.>. The domain D (Δ_f,h^M,(p) ( Ω̇)) is explicitly given by: D (Δ_f,h^M,(p) ( Ω̇)) ={ u∈ L^2 ( Ω̇) s.t.d_f,h u,d^*_f,hu,d^*_f,hd_f,h u, d_f,hd^*_f,h u ∈L^2 ( Ω̇), 𝐭u|_Γ_T=0, 𝐭d^*_f,hu|_Γ_T=0, 𝐧u|_Γ_N=0, 𝐧d_f,hu|_Γ_N=0 }.The traces 𝐭d^*_f,hu and 𝐧d_f,h u are a priori defined inH^-1/2 ( Ω̇) but actually belong to L^2 (Ω̇). Indeed, we have 𝐧d_f,hu|_Γ_N=0 by definition of D (Δ_f,h^M,(p) ( Ω̇) ) and 𝐭 d_f,hu|_Γ_T=0 by (<ref>), sod_f,hu is in D ( 𝒬^M,(p+1)_f,h ( Ω̇) ) and therefore has a trace in L^2 ( Ω̇) according to Proposition <ref>. This argument also holds for d^*_f,hu∈ D ( 𝒬^M,(p-1)_f,h ( Ω̇) ).We end up this section with the following lemma which will be frequently used in the sequel. Let us assume that the assumptions of Proposition <ref> are satisfied.Let us moreoverassume that Γ_T is C^∞ and that there exist two disjoint C^∞ open subsets Γ_N,1 and Γ_N,2of Ω̇ such that Γ_N=Γ_N,1∪Γ_N,2.Then, the following formula holds: for any u∈ D ( 𝒬^M,(p)_f,h ( Ω̇)),𝒬^M,(p)_f,h ( Ω̇)(u,u) = d_f,hu^2_L^2 ( Ω̇)+ d_f,h^*u^2_L^2 ( Ω̇)= h^2 du^2_L^2 ( Ω̇)+ h^2 d^*u^2_L^2 ( Ω̇)+ |∇ f|u^2_L^2 ( Ω̇) +h⟨(ℒ_∇ f+ℒ_∇ f^*)u,u⟩_L^2 ( Ω̇)-h(∫_Γ_T-∫_Γ_N) ⟨ u,u ⟩_T^*_σΩ̇_n fdσ,where ℒ stands for the Lie derivative.Notice that the boundary integral terms are well defined since u|_∂Ω̇∈ L^2(Ω̇) thanks to point (i) in Proposition <ref>. From the proof of Lemma <ref>, it will be clear that (<ref>) actually holds ifΓ_T and Γ_N areonly piecewise smooth. In the following, we will only need the result for Γ_Tsmooth and Γ_N the union of two smooth pieces and this is why we present this result in this setting. For u∈D ( 𝒬^M,(p)_f,h ( Ω̇)), one first gets by straightforward computations,d_f,hu^2_L^2 ( Ω̇)+ d_f,h^*u^2_L^2 ( Ω̇) = h^2 du^2_L^2 ( Ω̇)+ h^2 d^*u^2_L^2 ( Ω̇) + d f∧ u^2_L^2 ( Ω̇)+ 𝐢_∇ f u^2_L^2 ( Ω̇) + h⟨ d f∧ u, du ⟩_L^2 ( Ω̇) + h ⟨ d u, d f∧ u ⟩_L^2 ( Ω̇) +h⟨ d^* u, 𝐢_∇ f u ⟩_L^2 ( Ω̇) + h ⟨𝐢_∇ f u, d^*u ⟩_L^2 ( Ω̇)= h^2 du^2_L^2 ( Ω̇)+ h^2 d^*u^2_L^2 ( Ω̇)+ |∇ f|u^2_L^2 ( Ω̇)+h⟨ (ℒ_∇ f+ℒ^*_∇ f) u, u ⟩_L^2 ( Ω̇)+ h⟨ d f∧ u, du ⟩_L^2 ( Ω̇) -h ⟨ d^*(d f∧ u), u ⟩_L^2 ( Ω̇) -h⟨ d 𝐢_∇ f u , u ⟩_L^2 ( Ω̇)+ h⟨𝐢_∇ f u, d^*u ⟩_L^2 ( Ω̇) ,where the last equality holds thanks to therelations(df∧)^*=𝐢_∇ f,ℒ_∇ f=d∘𝐢_∇ f+𝐢_∇ f∘ dand𝐢_∇ f(df∧ u) + df∧(𝐢_∇ fu) =|∇ f|^2u.To get the boundary integral terms in (<ref>) one uses (<ref>), which gives here, since u∈D ( 𝒬^M,(p)_f,h ( Ω̇)) anddf∧ u, 𝐢_∇ f u∈Λ H_d ( Ω̇)∩Λ H_d^* ( Ω̇):⟨ d f∧ u, du ⟩_L^2 ( Ω̇) - ⟨ d^*(d f∧ u), u ⟩_L^2 ( Ω̇) = ∫_Γ_N⟨n^♭∧ u ,n^♭∧𝐢_n (df∧ u)⟩_ T^*_σΩ̇dσ= ∫_Γ_N,1⟨n^♭∧ u ,n^♭∧𝐢_n (df∧ u)⟩_ T^*_σΩ̇dσ +∫_Γ_N,2⟨n^♭∧ u ,n^♭∧𝐢_n (df∧ u)⟩_ T^*_σΩ̇dσ,where the last equality follows from (<ref>). Likewise, one has:⟨𝐢_∇ f u, d^*u ⟩_L^2 ( Ω̇) -⟨ d 𝐢_∇ f u , u ⟩_L^2 ( Ω̇)=-∫_Γ_T⟨n^♭∧𝐢_∇ f u ,n^♭∧𝐢_n u ⟩_ T^*_σΩ̇dσ.Since u∈D ( 𝒬^M,(p)_f,h ( Ω̇)), and Γ_T, Γ_N,1, and Γ_N,2 are smooth open subsets of Ω̇,Lemma <ref> impliesthatu is in Λ ^p H^1 outside (Γ_T∩Γ_N) ∪ (Γ_N,1∩Γ_N,2), bya localization argument. Therefore, u admits a boundary trace defined a.e. on Ω̇ and belonging to L^2_loc(Ω̇∖(Γ_T∩Γ_N) ∪ (Γ_N,1∩Γ_N,2) ). But this trace has to be u|_Ω̇ as defined by (<ref>) and is hence in L^2 (Ω̇) owing to item (i) inProposition <ref>. Let us now conclude the proof of Lemma <ref>. Let usconsider (<ref>). For j∈{1,2} and >0, one defines Γ_N,j^ε:= {x∈Γ_N,j, d^Ω̇(x,Γ_N,j)>ε}. One then has for j∈{1,2}: ∫_Γ_N,j⟨n^♭∧ u ,n^♭∧𝐢_n (df∧ u)⟩_ T^*_σΩ̇dσ = lim_ε→ 0^+∫_Γ_N,j^ε⟨n^♭∧ u ,n^♭∧𝐢_n (df∧ u)⟩_ T^*_σΩ̇dσ=lim_ε→ 0^+∫_Γ_N,j^ε⟨u ,𝐢_n(n^♭∧𝐢_n (df∧ u))⟩_ T^*_σΩ̇dσ=lim_ε→ 0^+∫_Γ_N,j^ε⟨u , 𝐢_n (df∧ u)⟩_ T^*_σΩ̇dσ=lim_ε→ 0^+∫_Γ_N,j^ε(_nf ⟨u , u⟩_ T^*_σΩ̇- ⟨u , df∧𝐢_nu⟩ _ T^*_σΩ̇ )dσ=∫_Γ_N,j_nf ⟨u , u⟩_ T^*_σΩ̇dσ,where we used the usual trace properties for H^1 forms on Γ_N,j^ε, the fact that 𝐢_nu =0 at the second to last lineand the Lebesgue dominated convergence theorem at the last line. From (<ref>), one then has:∫_Γ_N⟨n^♭∧ u ,n^♭∧𝐢_n (df∧ u)⟩_ T^*_σΩ̇dσ= ∫_Γ_N_nf ⟨u , u⟩_ T^*_σΩ̇dσ.The fact that ∫_Γ_T⟨n^♭∧𝐢_∇ f u ,n^♭∧𝐢_n u ⟩_ T^*_σΩ̇dσ=∫_Γ_T_nf ⟨u , u⟩_ T^*_σΩ̇dσ is provedsimilarly.This concludes the proof of Lemma <ref>. §.§.§ Construction of the domain Ω̇_iIn this section, we assume [H1], [H2] and [H3]. Let us consider z_i∈{z_1,…,z_n} a local minimum of f|_Ω. The objective of this section is to build the domain Ω̇_i on which theWitten Laplacian with mixed tangential-normal Dirichletboundary conditions will be defined. This auxiliary operatorissuch that z_i remainsthe only generalized critical point.Let us recall that x_0 ∈Ω is the minimum of f on Ω. LetΩ_0 be a small smooth open neighborhoodof x_0such that the∂_n f < 0 on Γ_0=Ω_0,n being the outward normal derivative to Ω∖Ω_0. Let Γ_1,i denote a subset of B_z_i, as large as we want in B_z_i, and such that z_i∈Γ_1,i. The basic idea is to define Ω̇=Ω∖Ω_0 and to consider a Witten Laplacian on Ω̇, with tangential Dirichlet boundary conditions on Γ_0∪Γ_1,i and with normal Dirichlet boundary conditions on Ω∖Γ_1,i. This would indeed yield an operator on a domain Ω̇ with a single generalized critical point, namely z_i.There is however a technical difficulty in this approach, related to the fact that differential forms with mixed normal and tangential Dirichlet boundary conditions are singular at the boundary between the domains where tangential and normal boundary conditions are applied, as explained in Section <ref>. With the previous construction, Γ_1,i and Ω∖Γ_1,i meet at an angle π. We therefore need to define a domain Ω̇_i stricly included in Ω∖Ω_0, with boundary Ω̇_i=Γ_0∪Γ_1,i∪Γ_2,i where Γ_0=Ω_0 as defined above, Γ_1,i∩Γ_2,i=∅, Γ_1,i⊂ B_z_i is as large as we want in B_z_i and Γ_2,i meets Γ_1,i at an angle strictly smaller than π (see (<ref>) above for a proper definition). We will then consider a Witten Laplacian with tangential Dirichlet boundary conditions on Γ_0∩Γ_1,i and normal Dirichlet boundary conditions on Γ_2,i. Moreover, in order not to introduce new generalized critical point on Γ_2,i, we would like to keep the property ∂_n f >0 on Γ_2,i (where n denotes the outward normal derivative to Ω̇_i). The aim of this section is indeed to define such a domain Ω̇_i.A system of coordinates on a neighborhood of ∂Ω. Let us consider the function f_- defined on a neighborhood V_∂Ω of ∂Ω, as introduced in Proposition <ref>. Recall that f_-(x)=0 for x ∈∂Ω and that V_∂Ω can be chosen such that f_->0 on V_∂Ω∖∂Ω and |∇ f_-|≠ 0 on V_∂Ω. Let us now consider ε >0 such that V_ε={y ∈Ω, 0 ≤ f_-(y) ≤ε}⊂ V_∂Ω.For any x ∈ V_ε, the dynamics{γ_x'(t) =-∇ f_-/|∇ f_-|^2(γ_x(t)) γ_x(0) =x .is such that γ_x(t_x)∈∂Ω, wheret_x=inf{t,γ_x(t) ∉ intV_ε}.This is indeed a consequence of the fact that d/dt f_-(γ_x(t)) =-1 < 0 on [0,t_x). Notice that this also implies thatt_x describes [0,] when x describes V_. The applicationΓ:{ V_ε →∂Ω× [-ε,0]x↦ (γ_x(t_x),-t_x) .defines a C^∞ diffeomorphism.The inverse application of Γ is (x',x_d) ∈∂Ω× [-ε,0] ↦γ_x'(x_d).Let us assume that the hypothesis [H3] holds. Let us define the following system of coordinates for x ∈ V_ε:∀ x ∈ V_ε, (x'(x),x_d(x))= (γ_x(t_x),-t_x) ∈∂Ω× [-ε,0].Notice that, by construction (since d/dt f_-(γ_x(t)) =-1), x_d(x)=-f_-(x).Thus, in this system of coordinates, {x_d=0} = ∂Ω and {x_d<0} = Ω∩ V_ε. We will sometimes need to use a local system of coordinates in ∂Ω, that we will then denote by the same notation x'. By using the same procedure as above, (x',x_d) then defines a local system of coordinates. Let us make this precise. For y∈∂Ω, let us consider x':V_y →^d-1 a smooth local system of coordinates in ∂Ω, in a neighborhood V_y ⊂∂Ω of y. These coordinates are then extended in a neighborhood of V_y in Ω, as constant along the integral curves of γ'(t)=∇ f_-/|∇ f_-|^2(γ(t)), for t ∈ [0,ε].The function x↦ (x',x_d) (where, we recall, x_d(x)=-f_-(x)) thus defines a smooth system of coordinates in a neighborhood W_y of y in Ω. In this system of coordinates, the metric tensor G writes:G(x',x_d)=G_dd(x',x_d)dx_d^2+∑_i,j=1^d-1G_ij(x',x_d)dx_idx_j,where x'=(x_1, …, x_d-1). In particular if ψ: V_y→ℝ is a Lipschitz function which only depends on x', it holds a.e. on V_y:|∇ψ(x',x_d)|=|∇ (ψ|_Σ_x_d)(x')|,where ∀ a>0, Σ_a:={x ∈ V_, x_d(x)=a} is endowed with the Riemannian structure induced by the Riemannian structure in Ω.Definitions of the functions Ψ_i, f_+,i and f_-,i. Let us assume that the hypotheses [H1] and [H3] hold. Let us consider z_i a local minimum of f|_∂Ω as introduced in hypothesis [H2]. Let us define onΩ the following Lipschitz functions Ψ_i(x):=d_a(x,z_i),f_+,i:=Ψ_i+ f-f(z_i)/2 andf_-,i:=Ψ_i-( f-f(z_i))/2. Owing to Ψ_i(x)=d_a(x,z_i)≥|f(x)-f(z_i)| for all x∈Ω, the functions f_±,i are non negative and f=f(z_i)+f_+,i-f_-,i and Ψ_i=f_+,i+f_-,i on Ω.Let Γ_1,i⊂ B_z_i be an open smooth d-1 dimensional manifold with boundary such that z_i∈Γ_1,i and Γ_1,i⊂ B_z_i.From Proposition <ref>, there exists a neighborhoodof Γ_1,i in Ω, denoted V_Γ_1,i, such that ∂ V_Γ_1,i∩∂Ω⊂B_z_i and for all x∈ V_Γ_1,i,Ψ_i(x)=Φ(x)-f(z_i)where Φ is the solution to the eikonal equation in a neighborhood of the boundary (see Proposition <ref>).Notice that it implies that on V_Γ_1,i the function f_-,i coincides with the function f_- defined in Proposition <ref> on V_∂Ω∩ V_Γ_1,i. Moreover, it implies that the functions f_±,i are C^∞on V_Γ_1,i and one has:onV_Γ_1,i∩∂Ω, f_+,i= f -f(z_i), f_-,i= 0 , _n f_+,i = 0 , and _n f_-,i = -_n f,where n is the unit outward normal to Ω. Therefore, as in Proposition <ref>, up to choosing a smaller neighborhood V_Γ_1,i of Γ_1,i in Ω, the function f_-,i is positive on V_Γ_1,i∖∂Ω and such that|∇ f_-,i| ≠0inV_Γ_1,i. Besides, since |∇Ψ_i| =|∇ f | in V_Γ_1,i, one has∇ f_+,i·∇ f_-,i=0inV_Γ_1,i,and thus|∇Ψ_i|^2 =|∇ f|^2= |∇ f_+,i|^2 + |∇ f_-,i|^2 inV_Γ_1,i.In the following, we will assume in addition that V_Γ_1,i is sufficiently small so that the system of coordinates(x',x_d) introduced in Definition <ref> is well defined on V_Γ_1,i. A consequence of (<ref>) is that d/dt f_+,i(γ_x(t))=0, where γ_x satisfies (<ref>). Thus, in the system of coordinates (x',x_d), the functions f_+,i, Ψ_i and f write:f_+,i(x',x_d)=f_+,i(x',0),Ψ_i(x',x_d)=f_+,i(x',0) -x_dandf(x',x_d)=f(z_i)+f_+,i(x',0)+x_d.Notice that by construction ∀ x ∈ V_Γ_1,i, |∇ f_+,i|(x)=0x'(x)=x'(z_i).Indeed, f_+,i(x',x_d)=f_+,i(x',0) and x' ↦ f_+,i(x',0)=f(x',0)-f(z_i) has a single critical point at x'(z_i).Strongly stable domain in B_z_i. In order to build an appropriate domain Ω̇_i, we will need to define Γ_1,i⊂ B_z_i as a strongly stable domain, as defined now. A smooth open set A⊂Ω is called strongly stable if ∀σ∈∂ A, ⟨∇ (f|_Ω)(σ), n_σ( A )⟩_T_σΩ >0,where n_σ( A)∈ T_σΩ denotes the outward normal to A at σ∈∂ A. Notice that ∇ (f|_Ω )=∇_T f=∇ f_+,i (this is due to the fact that on B_z_i, one has f-f(z_i)=Ψ_i and thus ∇_Tf=∇_TΨ_i). Thus, the strong stability condition appearing in Definition <ref> is equivalent to ∀σ∈∂ A, ∂_n_σ( A ) f_+,i(σ) >0.The name "stable" is justified by the following: if A⊂Ω is strongly stable, then for any curve satisfying for all t> 0, γ'(t)=-∇(f|_Ω)(γ(t)) with γ(0)∈A, one has for all t≥ 0, γ(t)∈A.The following proposition will be needed to get the existence of an arbitrary large and strongly stable domain in B_z_i. Let us assume that the hypotheses [H1] and [H2] hold.For all compact setsK⊂ B_z_i there exists a C^∞ open domain A which is strongly stable in the sense of Definition <ref>, simply connected and such that K⊂ A and A⊂ B_z_i.For the ease of notation, we drop the subscript i in the proof. One will first construct the set A. Then it will be proven that A has thestated properties. For a>0, let us defineL_a:=f|_∂Ω^-1 ( [f(z),f(z)+a) ) ∩ B_z.For a fixed a>0 small enough L_a is a C^∞ simply connected open set (which contains z) with boundary the level set f|_∂Ω^-1({f(z)+a}). The domain L_a is C^∞ since f is C^∞. Let us define for x∈ B_z the curves γ_x by γ_x'(t)=∇ f|_∂Ω(γ_x(t)), γ_x(0)=x.For any x∈∂ L_a, for all t>0, γ_x(-t)∈ L_a since t≥ 0↦ f|_∂Ω(γ_x(-t)) is decreasing (d/dtf|_∂Ω(γ_x(-t))=-|∇ f|_∂Ω(γ_x(-t))|^2 and f|_∂Ω(γ_x(0))=a). Let us now define for T> 0A_T:={γ_x(t), x∈∂ L_a,t∈ [0,T)}∪ L_a ⊂ B_z.One clearly has A_T⊂ A_T' if T<T'. One claims that A_T is a C^∞simply connected open set which satisfies ∀σ∈∂ A_T, ∂_n_σ( A_T ) f|_Ω(σ) >0. Let us first prove that A_T is C^∞. One has ∂ A_T={γ_x(T),x∈∂ L_a}. The boundary of A_T is thus a C^∞ homotopy of ∂ L_a where the homotopy function is H(t,x)=γ_x(t). Additionally since this homotopy is with values in B_z and since L_a is simply connected (because L_a can be asymptotically retracted on z in the sense that for all x∈ L_a, lim_t→ -∞H(t,x)=z), A_T is simply connected. Let us prove thatA_T is open. Let us denote by d_∂Ω the geodesic distance in ∂Ω. Let x_0∈A_T∖L_a. There exists a time t_0∈ (0,T) such that γ_x_0(-t_0)∈ L_a. Let us define _0=d_∂Ω(γ_x_0(-t_0),∂ L_a)/2>0. Since the mapping y↦γ_y(-t_0) is C^∞, there exists _1>0 such that if d_∂Ω(x,y)≤_1 then d_∂Ω(γ_y(-t_0),γ_x_0(-t_0))≤_0/2 and thus γ_y(-t_0)∈ L_a. Moreover, since B_z∖L_a is open, it can be assumed, taking maybe _1>0 smaller, that B_∂Ω(x_0,_1)⊂ B_z∖L_a. Then, by continuity, for ally∈ B_∂Ω(x_0,_1), there exists t_0(y)∈ (0,t_0)⊂ (0,T) such that γ_y(-t_0(y))∈ L_a, which implies that y∈ A_T∖L_a. Thus A_T∖L_a is open. In addition, since L_a is open and since L_a⊂ A_T, one has that int(A_T)=int(A_T∖L_a)∪L_a=(A_T∖L_a)∪L_a=A_T. Therefore the set A_T is open.Let us now prove that A_T is strongly stable (see Definition <ref>). By construction, A_T is stable for the dynamics γ'=-∇ f|_Ω(γ) and thusone has ∀σ∈∂ A_T, ∂_n_σ( A_T ) f|_Ω(σ) ≥ 0.Let us defined now the functionΥ :x∈ B_z∖L_a ↦ (x',t)∈ L_a×ℝ_+s.t γ_x'(t)=x.Notice that Υ is a C^∞ diffeomorphism from B_z onto its range, and let us denote F:=Υ^-1 its inverse function (F(x',t)=γ_x'(t)). Assume that there exists x∈A_T such that ∂_n_x( A_T ) f|_Ω(x)=0 and let (x',T)=Υ(x). This implies that ∇ f|_Ω(x)∈ T_x∂ A_T and thus ∂ _t F(x',T)∈T_x∂ A_T. Furthermore (d_x'F(.,T))=T_x∂ A_T and thus d_(x',T)F is not invertible which contradicts the fact that F is a diffeomorphism. It remains to prove that for any compact set K⊂ B_z, there existsT>0 such that K⊂ A_T. One has B_z=⋃_T>0 A_T.Indeed, if x∈L_a, x∈ A_T for all T>0 and if x∈ B_z∖L_a, lim_t→∞γ_x(-t)= z and thus there exists s>0 such that γ_x(-s)∈∂ L_a which implies that x∈ A_s. Let K⊂ B_z be a compact set. Then K⊂⋃_T>0 A_T and thus by compactness there exists a sequence (T_j)_j=1,…,N⊂ℝ^N, with0<T_1<…<T_m such that K ⊂⋃_j=1^m A_T_j=A_T_m. This concludes the proof.Construction of the domain Ω̇_i. In this section, we introduce the domain Ω̇_i (associated with z_i) on whichthe auxiliary Witten Laplacian with mixed tangential-normal Dirichlet boundary conditions is constructed. Let us assume that the hypotheses [H1], [H2] and [H3] hold. Let us fix a neighborhood Ω_0 of x_0 (the global minimum of f in Ω) such that ∂_n f < 0on Γ_0 := ∂Ω_0 where n denotes the outward normal to Ω∖Ω_0 on Γ_0. Let us consider a critical point z_i of f|_∂Ω. Then there exists a smooth open subset Γ_1,i of B_z_i containing z_i and arbitrarily largein B_z_i, a neighborhood V_Γ_1,i of Γ_1,i in Ω such that V_Γ_1,i∩∂Ω⊂ B_z_i and a Lipschitz subset Ω̇_i of Ω∖Ω_0 which are such that the following properties are satisfied: * Following Proposition <ref>,∀ x ∈ V_Γ_1,i,d_a(x,z_i)=Φ(x) - f(z_i)where Φ is the solution to the eikonal equation (<ref>) ;* The system of coordinates (x',x_d) is defined on V_Γ_1,i, see Definition <ref> ;* ∂Ω̇_i is composed of two connected components: Γ_0 andΓ_1,i∪Γ_2,i, where Γ_2,i is an open subset of Ω̇_i, Γ_1,i∩Γ_2,i=∅, andΓ_2,iis such that there exist two disjoint C^∞ open subsets Γ_2,i^1 and Γ_2,i^2of Ω̇_i such thatΓ_2,i=Γ_2,i^1∪Γ_2,i^2; * Γ_1,i and Γ_2,i meetat an angle smaller than π, see (<ref>) for a precise definition ;* It holds,∀ x ∈∂Ω̇_i ∖((Γ_2,i^1∩Γ_2,i^2)∪ (Γ_1,i∩Γ_2,i) ∪Γ_0 ),∂_n f(x) > 0where n is the outward normal to Ω̇_i ;* It holds,∀ x ∈Γ_2,i∩ V_Γ_1,i,∂_n f_+,i(x) > 0; * Moreover, for all δ > 0,Ω̇_i (and Ω_0) can be chosen such thatsup{d_e(x,y), x ∈Γ_2,i, y ∈ B_z_i^c }≤δandsup{d_e(x_0,x), x ∈Γ_0 }≤δwhere, we recall, d_e denotes the geodesic distance for the Euclidean metric on Ω.We refer toFigure <ref> for a schematic representation of Ω̇_i.The domain Ω̇_i ⊂Ω is built as follows. First, let us fix a neighborhood Ω_0 of x_0 such that (<ref>) is satisfied and ∂_n f < 0on Γ_0 = ∂Ω_0 where n denotes the outward normal to Ω∖Ω_0 on Γ_0 (this can be done for example by considering Ω_0={x, f(x) < f(x_0) + η} for some positive η). Second, let us consider a smooth subset Γ_1,i of B_z_i which canbe as large as needed in B_z_i, and which is strongly stable (see Proposition <ref> for the existence of such a set):⟨∇ f|_∂Ω , n(Γ_1,i) ⟩_T Ω > 0on ∂Γ_1,i and thus _n(Γ_1,i)f_+,i>0on ∂Γ_1,i,where n(Γ_1,i) denotes the outward normal derivative of Γ_1,i (see Definition <ref> and (<ref>)). Once Γ_1,i is fixed, the existence of a neighborhood V_Γ_1,i of Γ_1,i in Ω such that V_Γ_1,i∩∂Ω⊂ B_z_i and such that items 1 and 2 are fulfilled is a direct consequence of Proposition <ref>. Let us now consider the system of coordinates (x',x_d) introduced in Definition <ref>. Let V_∂Γ_1,i⊂∂Ω denotes a neighborhood of ∂Γ_1,i in ∂Ω andV^+_∂Γ_1,i =V_∂Γ_1,i∩Γ_1,i^c.The domain Ω̇_i is then defined as follows:Ω̇_i = Ω∖ (Ω_0∪{x=(x',x_d),x'∈ V^+_∂Γ_1,i such thatx_d(x) ∈ [-φ(x'),0] })where φ:V^+_∂Γ_1,i→_+ is a smooth function such that∃ε > 0,∀ x' ∈∂Γ_1,i,φ(x') ≥ε,see Figure <ref> for a schematic representation. Notice that by construction, Ω̇_i is a connected Lipschitz subset of Ω and, denoting by Γ_2,i=∂Ω̇_i ∖ (Γ_1,i∪Γ_0), item 3 is satified. For each point z ∈∂Γ_1,i, there is a small neighborhood 𝒱 of z such that 𝒱∩Γ_2,i⊂{x=(x',x_d), x' ∈∂Γ_1,i andx_d(x) ∈ (-η,0] },for some η∈ (0,). By choosing Γ_1,i sufficiently large in B_z_i, and φ such that maxφ is sufficiently small, (<ref>) is satisfied. This concludes the verification of item 7. For each pointy ∈∂Γ_1,i, it is possible to construct locally a normal system of coodinate x'=x_T=(x_T,1,x_T,2, …, x_T,d-1) in a neighborhood V_y of y in ∂Ω, such that Γ_1,i∩ V_y = {x ∈ V_y, x_T,1(x) ≤ 0}, V^+_∂Γ_1,i∩ V_y = {x ∈ V_y, x_T,1(x) ≥ 0}and ∂Γ_1,i∩ V_y = {x ∈ V_y, x_T,1(x) = 0}. As explained after Definition <ref>, by extending this system of coordinate inside Ω as constant along the curve associated with the vector field ∇ f_-,i/|∇ f_-,i|^2, x↦ (x'(x),x_d(x)) then defines a local system of coordinates in a neighborhood W_y of y in Ω. For all x ∈∂Γ_1,i, the vector n_z(Γ_1)=∇ x_T,1(x)/|∇ x_T,1(x)| is the outward normal vector to Γ_1,i on ∂Γ_1,i. By a compactness argument, one gets that ∂Γ_1,i⊂∪_k=1^K 𝒱_y_k for afinite number of points y_k ∈∂Γ_1,i. See Figure <ref> for a schematic representation of the function φ in this system of coordinates.Let us now look at the boundary of Ω̇_i in a neighborhood of ∂Γ_1,i (see Figure <ref>). For σ∈∂Ω̇_i, let us denote by n_σ(Ω̇_i) the unit outward normal to Ω̇_i. Let us show that for all z ∈∂Γ_1,i,lim_σ→ z n_σ(Ω̇_i)=n_z(Γ_1,i)where the limit is taken for σ∈Γ_2,i. Let us prove (<ref>). For any point z ∈∂Γ_1,i, there is a small neighborhood 𝒱 of z in Ω such that the system of coordinates (x_T,x_d) introduced above is well defined. In this system of coordinates,∂Ω̇_i ∩𝒱∩Γ_2,i⊂{x ∈𝒱, x_T,1(x)=0andx_d(x) ∈ [-φ(x'(x_T(x))),0] }.Moreover, the outward normal to Ω̇_i on this subset is n(Ω̇_i)=∇ x_T,1/|∇ x_T,1| and thus, by construction, for all z ∈∂Γ_1,i, (<ref>) holds. As a consequence of (<ref>), the two submanifolds Γ_1,i and Γ_2,i meet at an angle smaller than π (see (<ref>) and Figure <ref>). This shows that item 4 is satisfied. Moreover, using (<ref>),one has: for all z ∈∂Γ_1,i,lim_σ→ z∇ f_+,i (σ) · n_σ(Ω̇_i)= ∇ f_+,i (z) · n_z(Γ_1,i)>0,and lim_σ→ z∇ f (σ) · n_σ(Ω̇_i)= ∇ f (z) · n_z(Γ_1,i)>0,where the limits are taken for σ∈Γ_2,i. Thus, up to choosing φ with maxφ sufficiently small, it is possible to build Ω̇_i such that (see Figure <ref>)∀ x ∈Γ_2,i such thatx'(x) ∈∂Γ_1,i,∂_n f_+,i(x) > 0and ∂_n f(x) > 0,where n here denotes the outward normal to Ω̇_i. Finally, by using (<ref>) and since ∂_n f > 0 on ∂Ω, up to choosing φ with maxφ sufficiently small, it is possible to build Ω̇_i such that (see Figure <ref>)∀ x ∈∂Ω̇_i ∖((Γ_2,i^1∩Γ_2,i^2)∪ (Γ_1,i∩Γ_2,i) ∪Γ_0 ), ∂_n f(x) > 0where n again denotes the outward normal to Ω̇_i. This is item 5, and this concludes the proof of Proposition <ref>.Let us assume that the hypotheses [H1], [H2] and [H3] hold. Let us consider a critical point z_i of f|_∂Ω. In the following, we denote by𝒮_M,i:={Ω̇_i, Γ_0,Γ_1,i,Γ_2,i,V_Γ_1,i}an ensemble of sets satisfying the requirements of Proposition <ref>. In the following, in order to reduce the amount of notation, the index i will sometimes be omitted. Thus, we will denotez=z_i,Γ_1=Γ_1,i,Γ_2=Γ_2,i,Ω̇=Ω̇_i,V_Γ_1=V_Γ_1,i, Ψ=Ψ_i,f_+=f_+,iand f_- =f_-,i.We shall warn the reader whenever the index i is dropped. §.§.§ On the spectrum of the Witten Laplacian Δ^M_f,h(Ω̇_i)Troughout this section, one assumes [H1], [H2] and [H3]. In this section, we introduce a Witten Laplacian with mixed tangential and normal Dirichlet boundary conditions, associated with the critical point z_i. Let 𝒮_M,i:={Ω̇_i,Γ_0, Γ_1,i,Γ_2,i,V_Γ_1,i} be an ensemble of sets associated with z_i, see Definition <ref>. Let us now consider the Witten Laplacian Δ^M_f,h on Ω̇_i with homogeneous Dirichlettangential boundary conditions onΓ_T=Γ_0∪Γ_1,i and homogeneous Dirichletnormal boundary conditions onΓ_N=Γ_2,i as defined at the end of Section <ref> (see (<ref>)–(<ref>)). The main result of this section concerns the spectrum of the operator Δ_f,h^M ( Ω̇_i ).Let us assume that the hypotheses [H1], [H2] and [H3] hold.Let Δ_f,h^M,(p) ( Ω̇_i ) be the unboundednonnegative selfadjointoperator onL^2 ( Ω̇_i ) defined by (<ref>) and with domain (<ref>) with Γ_T=Γ_1,i∪Γ_0 and Γ_N=Γ_2,i. One has: (i) The operator Δ_f,h^M,(p) ( Ω̇_i ) has compact resolvent.(ii) For any eigenvalue λ_p of Δ_f,h^M,(p) ( Ω̇_i ) and associated eigenform u_p∈ D (Δ_f,h^M,(p) ( Ω̇_i )), one has d_f,hu_p∈ D (Δ_f,h^M,(p+1)) and d^*_f,hu_p∈ D (Δ_f,h^M,(p-1)), withd_f,hΔ_f,h^M,(p)u_p=Δ_f,h^M,(p+1)d_f,hu_p = λ_pd_f,hu_pandd^*_f,hΔ_f,h^M,(p) ( Ω̇_i )u_p=Δ_f,h^M,(p-1) ( Ω̇_i )d^*_f,hu_p = λ_pd^*_f,hu_p.If in addition λ_p≠ 0, either d_f,hu_p or d^*_f,hu_p is nonzero.(iii) There exist c>0 and h_0>0 such that for any p∈{0,…,n} and h∈ (0,h_0),π_[0,ch^3/2)(Δ_f,h^M,(p) ( Ω̇_i )) =δ_1,pand(Δ_f,h^M,(1) ( Ω̇_i ))∩ [0,ch^3/2) = {0},where δ is the Kronecker delta:δ_1,p=1 iff p=1.Since the criticial point z_i is fixed, for the ease of notation, we drop the subscript i in the proof. The point (i) follows from the compactness of the embedding H^1/2 ( Ω̇)↪L^2 ( Ω̇) (since additionally D ( 𝒬^M_f,h ( Ω̇) )↪ H^1/2 ( Ω̇) is continuous according to Proposition <ref>).The point (ii) is then a straightforwardconsequence of the characterization of the domain of Δ_f,h^M,(p) ( Ω̇) together with (<ref>). The statement in the case λ_p≠ 0follows from0 ≠ λ_p⟨ u_p,u_p⟩_L^2 ( Ω̇)= ⟨Δ_f,h^M,(p) ( Ω̇)u_p,u_p⟩_L^2 ( Ω̇) = ⟨ d_f,h u,d_f,hu_p⟩_L^2 ( Ω̇) +⟨ d^*_f,hu_p,d^*_f,hu_p⟩_L^2 ( Ω̇) .Let us now give the proof of the last point (iii), which is a consequence of (ii) together with ideas from <cit.>, since z_i is the only generalized critical point of f in Ω̇ for Δ_f,h^D,(1)(Ω̇). Let us first provethat for some c>0, one has for any p∈{0,…,n} and h small enough,π_[0,ch^3/2)(Δ_f,h^M,(p) ( Ω̇)) =δ_1,p.Pick up u∈ D ( 𝒬^M,(p)_f,h ( Ω̇)). From the Green formula (<ref>) and from the fact that ℒ_∇ f+ℒ_∇ f^* is a 0^th order differential operator, there exists C_0>0 such thatfor all u∈D ( 𝒬^M,(p)_f,h ( Ω̇)) and all smooth cut-off function χsupported in Ω̇ (whose support avoids Ω̇):d_f,hχ u^2_L^2 ( Ω̇)+ d_f,h^*χ u^2_L^2 ( Ω̇)≥(inf_ suppχ|∇ f |^2 -hC_0)χ u^2. Thus, since f has no critical point in Ω̇,there exists some C>0 independent of u∈D ( 𝒬^M,(p)_f,h ( Ω̇)) such thatfor any smooth cut-off function χsupported in Ω̇(whose support avoids Ω̇)and h small enough,𝒬^M,(p)_f,h ( Ω̇)(χ u)≥Cχ u^2. Note in addition that owing to _nf> 0 on Γ_2∖ (Γ_2,i^1∩Γ_2,i^2) and _nf<0 on Γ_0,the boundary terms in the Green formula (<ref>) are non negative, for any smooth cut-off function χ supported in a neighborhood of any point in Γ_2∪Γ_0 (whose support avoids some neighborhood of Γ_1). Thus, thesame considerations show thatfor h small enough, for such functions χ,taking maybe C smaller, ∀ u ∈D ( 𝒬^M,(p)_f,h ( Ω̇))𝒬^M,(p)_f,h( Ω̇)(χ u)≥Cχ u^2.According to the analysis done in <cit.>, the same estimate also holdsfor χ supported in a sufficiently small neighborhood of some point x≠ z, x∈Γ_1 (whose support avoids a neighborhood of {z}∪Γ_1). Thisis related to the fact that Γ_1 does not contain any generalized critical point of f in the tangential sense except z. Let us now show that such an estimate is also valid near Γ_1. In order to prove it, one recalls that f=f(z)+f_+-f_- a.eon Ω and|∇ f|^2=|∇ f_-|^2 +|∇ f_+|^2 a.enear Γ_1,where f_± are smooth and satisfy the following relations on B_z:f_+= f -f(z), f_-= 0, _n f_+ = 0and _n f_- = -_n f.Hence, for any χ supported in a sufficiently small neighborhood of Γ_1,one deduces fromthe relation 𝒬^M,(p)_-f_-,h ( Ω̇)(χ u)≥ 0, the Green formula (<ref>), and the fact that ℒ_-∇ f_-+ℒ_-∇ f_-^* is a 0^th order differential operator, that there exists C_1>0 independent of u ∈D ( 𝒬^M,(p)_f,h ( Ω̇)) such that:h(∫_Γ_1-∫_Γ_2)⟨χ u , χ u ⟩_ T^*_σΩ̇_n f_-d σ ≥ -h^2 dχ u^2_L^2 ( Ω̇)- h^2 d^*χ u^2_L^2 ( Ω̇) -|∇ f_-|χ u^2_L^2 ( Ω̇) -C_1hχ u^2_L^2 ( Ω̇).Using again the Green formula (<ref>),the relation f-f(z)=f_+-f_- with _nf_+=0 on Γ_1, and the fact that ℒ_∇ f+ℒ_∇ f^* is a 0^th order differential operator then leads to the existence of C_2>0 independent of u ∈D ( 𝒬^M,(p)_f,h ( Ω̇)) s.t.𝒬^M,(p)_f,h ( Ω̇)(χ u) ≥|∇ f_+|χ u^2_L^2 ( Ω̇) -C_2hχ u^2_L^2 ( Ω̇)+h∫_Γ_2⟨χ u|χ u ⟩_ T^*_σΩ̇_n f_+d σ.Since f_+ has no critical point around Γ_1 (see (<ref>)), one has then for h small enough, taking maybe C smaller:𝒬^M,(p)_f,h ( Ω̇)(χ u) ≥ Cχ u^2_L^2 ( Ω̇)+h∫_Γ_2⟨χ u|χ u ⟩_ T^*_σΩ̇_n f_+d σ.Let us recall that due to our construction of Γ_2 near ∂Γ_1, one has (see (<ref>) in Proposition <ref>): _n f_+(σ)> 0 for σ∈Γ_2 sufficiently close to Γ_1.This implies that for χ supported near Γ_1 with sufficiently small support and h small enough:𝒬^M,(p)_f,h ( Ω̇)(χ u) ≥ Cχ u^2_L^2 ( Ω̇).Lastly, since z is a generalized critical point with index 1 in the tangential sense,it follows from <cit.> thatfor χ supported ina neighborhood ofzand h small enough, the spectrum of the Friedrichs extension associated with the quadratic form{v∈Λ^p H^1(χ);𝐭v|_Γ_1=v|_χ∖Γ_1=0}∋ v ↦d_f,h v_ L^2^2+ d_f,h^* v_ L^2^2does not meet[0,h^3/2) if p≠ 1, and consists of exactly one eigenvalue in[0,h^3/2) which is actually exponentially small – i.e. of the size O(e^-C_3/h) –if p=1. Denote by ψ_1∈Λ^1 H^1(χ) some normalized eigenvalue associated with this exponentially small eigenvalue. Using the IMS localization formula(see for example <cit.>)∀(χ_k)_k∈{1,…,K}∈(C^∞ ( Ω̇ ))^K s.t. ∑_k=1^Kχ_k^2=1 𝒬^M,(p)_f,h (Ω̇) (u) = ∑_k=1^K(𝒬^M,(p)_f,h ( Ω̇)(χ_ku) -h^2|∇χ_k | u_L^2 ( Ω̇)^2), the previous analysis then shows that choosing χ_1∈ C^∞ (Ω̇)supported ina neighborhood of z with χ_1=1 near z, one getsfor some C,C'>0 independent of u∈ D ( 𝒬^M,(p)_f,h ( Ω̇)) and h small enough:𝒬^M,(p)_f,h ( Ω̇)(u)≥𝒬^M,(p)_f,h ( Ω̇)(χ_1u) + C (1-χ_1^2)^1/2 u_L^2 ( Ω̇)^2 -C'h^2u_L^2 ( Ω̇)^2.If p≠ 1, one deduces immediately from (<ref>) that for h small enough,𝒬^M,(p)_f,h ( Ω̇)(u)≥h^3/2χ_1u_L^2 ( Ω̇)^2+ C (1-χ_1^2)^1/2 u_L^2 ( Ω̇)^2 -C'h^2u_L^2 ( Ω̇)^2,and then that for some c>0 and h small enough: 𝒬^M,(p)_f,h ( Ω̇)(u)≥c h^3/2 u_L^2 ( Ω̇)^2.If p=1, one obtains from (<ref>) the same conclusion for any u such that ∫_Ω̇ uχ_1ψ_1=0 and therefore Δ_f,h^M,(p) ( Ω̇) has no eigenvalue in [0,ch^3/2) if p≠1 and at most one if p=1. To end up the proof, it is sufficient to remark that ψ̃_1, the extension of ψ_1 to Ω̇ by 0 outside χ_1, belongs to D (𝒬^M,(1)_f,h ( Ω̇)) and satisfies for h small:𝒬^M,(1)_f,h ( Ω̇)(ψ̃_1)=d_f,hψ_1_L^2 ( Ω̇)+ d_f,h^*ψ_1_L^2 ( Ω̇) = O(e^-C_3/h) < ch^3/2.This proves thatthere exists c>0 and h_0>0 such that for allp∈{0,…,n} and h∈ (0,h_0),π_[0,ch^3/2)(Δ_f,h^M,(p) ( Ω̇_i )) =δ_1,p.Then, the fact that there exists c>0 and h_0>0 such thatfor all h∈ (0,h_0),(Δ_f,h^M,(1) ( Ω̇_i ))∩ [0,ch^3/2) = {0},is a direct consequence of item (ii) in Proposition <ref> together with (<ref>). This concludes the proof ofProposition <ref>.Following Proposition <ref>, let us introduce an L^2-normalized eigenform u^(1)_h,i of Δ_f,h^M,(1)(Ω̇_i) associated with the eigenvalue 0:Δ_f,h^M,(1) ( Ω̇_i ) u^(1)_h,i=0 in Ω̇_iand u^(1)_h,i_L^2 ( Ω̇_i )=1.Using standard elliptic regularity results, one can check that u^(1)_h,i is actually in C^∞( Ω̇_i ) and is smooth in a neighborhood of any regular point of Ω̇_i.Notice that thanks to item (iii) in Proposition <ref>, u^(1)_h,i is unique up to a multiplication by ±1: this multiplicative constant will be fixed in Proposition <ref> below. The quasi-mode ϕ̃_i will be built using a suitable truncation of u^(1)_h,i. §.§ Definition of the quasi-modesTroughout this section, one assumes [H1], [H2] and [H3]. In this section, weconstruct the function ũ and a family of 1-forms (ϕ̃_i)_i=1,…,n which will satisfy the estimates stated in Section <ref>. For each i∈{1,…,n}, the 1-form ϕ̃_i will be constructed by a suitable truncation of an eigenfunction u_h,i associated with the eigenvalue 0 of the mixed WittenLaplacian attached with z_i∈{z_1,…,z_n}, as defined in Section <ref>.We recall thatΣ_i is an open set included in ∂Ω containing z_i which is such that Σ_i ⊂ B_z_i (see Definition <ref>). §.§.§ Definition of the quasi-mode ũLet us consider the global minimum x_0 introduced in the hypothesis[H2]. Let χ∈ C^∞_c(Ω) such that {x ∈Ω | χ(x)=1} is a neighborhood of x_0 and such that0≤χ≤ 1 (in particular χ(x_0)=1). The quasi-mode ũ is defined byũ := χ/√(∫_Ωχ^2 e^- 2f/h). The function ũ belongs to C^∞_c(Ω) and therefore ũ∈H^1_0(e^-2/h f(x) dx). The function χ will be chosen such that supp(|∇χ|) is as close as needed to ∂Ω, as will be made precise in Section <ref>.Let us first prove that ũ satisfies item 2(b) in Proposition <ref>.Let us assume that the hypotheses [H1] and [H2]hold. Then for anyδ>0, there exist h_0>0, C>0, andthere exists χ∈ C^∞_c(Ω) such that the set {x ∈Ω|χ(x)=1} is a neighborhood of x_0,0≤χ≤ 1 and for all h∈ (0,h_0) ∫_Ω|∇ũ|^2e^-2f/h≤ C h^-d/2e^-2(f(z_1)-f(x_0))-δ/h,where ũ is defined in Definition <ref>. There exists a constant C such that∫_Ω|∇ũ(x)|^2e^-2f(x)/hdx ≤ C ∫_ supp∇χ e^-2f(x)/h dx/∫_Ωχ^2(y)e^-2f(y)/hdy.Since supp∇χ can be chosen arbitrarly close to ∂Ω and since z_1 is the minimum of V on ∂Ω, by continuity of f, for anyδ>0 there exists χ∈ C^∞(Ω) such that {x ∈Ω|χ(x)=1} is a neighborhood of x_0,0≤χ≤ 1 and ∫_ supp∇χ e^-2f(x)/h dx ≤ Ce^-2f(z_1)+2δ/h.Moreover, since x_0 is the global minimum of f in Ω, one gets, using Laplace's method∫_Ωχ^2(y)e^-2f(y)/h dy = (π h)^d/2/√( detHessf (x_0) )e^-2f(x_0) /h(1+O(h)).This yields the desired estimate.Notice that item 2(b) in Proposition <ref> is a direct consequence ofLemma <ref>. §.§.§ Definition of the quasi-mode ϕ̃_i attached to z_i Let z_i be a local minimum of f|_Ω. Let us recall that Σ_i is an open subset of Ω such that z_i∈Σ_i and Σ_i⊂ B_z_i.Let 𝒮_M,i:={Ω̇_i,Γ_0, Γ_1,i,Γ_2,i,V_Γ_1,i} be an ensemble of sets associated with z_i, see Definition <ref>. Thanks to Propositions <ref> and <ref>,the set Γ_1,i can be taken such that Σ_i ⊂Γ_1,i.We recall that Section <ref> was dedicated to the construction of a domain Ω̇_i⊂Ω and a mixed WittenLaplacian Δ^M,(1)_f,h ( Ω̇_i ) (see (<ref>)) associated with this ensemble of sets 𝒮_M,i. Proposition<ref> gives the spectral properties of the operator Δ^M_f,h ( Ω̇_i ). In the following, we consider a normalized eigenform u^(1)_h,i∈ D(Δ_f,h^M,(1) ( Ω̇_i )) associated with the first eigenvalue 0, i.e. such that (<ref>) holdsThe quasi-mode ϕ̃_i is defined as the following truncation of u^(1)_h,i.Let us assume that the hypotheses [H1], [H2] and [H3] hold.Let χ_i ∈ C^∞(Ω) be such that:* χ_i ∈ C^∞_c( Ω̇_i ∪Γ_1,i) (and thus χ_i=0 on a neighborhood of Γ_2,i∪Γ_0 andon a neighborhood of ∂Ω∖Γ_1,i),* χ_i=1 on a neighborhood of Σ_i in Ω̇_i,* 0≤χ_i≤ 1.One defines 𝒱_i:={x∈Ω | χ_i(x)=1 }. The quasi-mode ϕ̃_i is defined on Ω by:ϕ̃_i :=χ_i u^(1)_h,i/√(∫_Ω|χ_i(x) u^(1)_h,i(x) |^2 dx ). The support of χ_i on ∂Ω is represented on Figure <ref> and the support of χ_i in Ω is represented in Figure <ref>.The sets Γ_1,i, Γ_2,i and, the function χ_i will be chosen such that supp(|∇χ_i|) is as close as needed from B_z_i^c ⊂∂Ω or from x_0, as will be made precise at the end of Section <ref>. This is possible thanks to item 7 in Proposition <ref>. UsingLemma <ref> and the fact that tχ_i u_h,i^(1)=0 on ∂Ω, one easily shows thatϕ̃_i ∈Λ^1 H^1_T(Ω). Using now the regularity of u_h,i^(1), one can check that ϕ̃_i is actually a C^∞_c(Ω∪Γ_1,i) function. We will show in Section <ref> that the family of forms (ũ,ϕ̃_1, … ,ϕ̃_n) satisfy the estimates stated in Sections <ref> and <ref>. This requires some preliminary results on the eigenforms (u^(1)_h,i)_i ∈{1, … n} that are provided in Section <ref> and <ref>.§.§ Agmon estimates on u^(1)_h,i Throughout this section, one assumes [H1], [H2] and [H3]. In all this section, we consider, for a fixed critical point z_i, an ensemble of sets 𝒮_M,i associated with z_i (see Definition <ref>) and an L^2-normalized eigenformu^(1)_h,i of Δ^M,(1)_f,h(Ω̇_i) associated with the eingevalue 0, as introduced at the end of Section <ref>.The aim of this section is to prove the following proposition.Let us assume that the hypotheses [H1], [H2] and [H3] hold. Any L^2-normalized eigenform u^(1)_h,i ofΔ_f,h^M,(1) ( Ω̇_i )associated with the eigenvalue 0 satisfies:∃N ∈ℕ,e^Ψ_i/ h u^(1)_h,i_L^2 ( Ω̇_i ) +d(e^Ψ_i/ h u^(1)_h,i) _L^2 ( Ω̇_i )+d^* (e^Ψ_i/ h u^(1)_h,i) _L^2 ( Ω̇_i )=O(h^-N)where, we recall, Ψ_i(x)=d_a(x,z_i) (see Definition <ref>). For the ease of notation, we drop the subscript i in the remaining of this section.The proof is inspired by the first part of the proof of <cit.> where the authors consider a Witten Laplacian with mixed tangential – full Dirichlet boundary conditions in a local system of coordinates in a neighborhood of z. The proof actually only requires thatu^(1)_h is an eigenform ofΔ_f,h^M,(1) ( Ω̇)associated with an eigenvalue λ_h=O (h). It crucially relies on the following Agmon-type energy equality.Let us assume that the hypotheses [H1], [H2] and [H3] hold. Let φ be a real-valued Lipschitz function on Ω̇. Then, for any u∈ D ( 𝒬_f,h^M,(1) ( Ω̇)), one has: 𝒬^M,(1)_f,h ( Ω̇)( u,e^2φ/h u) =h^2 d e^φ/h u^2_ L^2 ( Ω̇)+ h^2 d^* e^φ/h u^2_ L^2 ( Ω̇) + ⟨ (|∇ f|^2-|∇φ|^2+hℒ_∇ f+hℒ_∇ f^*)e^φ/h u, e^φ/hu⟩_ L^2 ( Ω̇) +h(-∫_Γ_0∪Γ_1 +∫_Γ_2) ⟨ u, u ⟩_ T_σ^*Ω̇ e^2 /hφ_n fdσ.Moreover, when u∈ D (Δ_f,h^M,(1) ( Ω̇)), the left-handside equals ⟨e^2φ/hΔ_f,h^M,(1) ( Ω̇) u, u⟩_L^2 ( Ω̇). This result is standard for manifolds without boundary or for bounded manifolds and quadratic forms with full normal or tangential boundary conditions (see e.g. <cit.>). We extend it here to our setting. Note first that u∈ D (𝒬^M,(1)_f,h ( Ω̇)) implies e^2φ/h u∈ D (𝒬^M,(1)_f,h ( Ω̇)), sincefor u∈ D ( 𝒬^M,(1)_f,h ( Ω̇) ),n^♭∧ e^2φ/h u=e^2φ/h n^♭∧ u and 𝐢_n e^2φ/hu=e^2φ/h𝐢_n u. One then gets by straightforward computations:𝒬^M,(1)_f,h ( Ω̇) ( u,e^2φ/h u)= ⟨ d_f,h u, d_f,h(e^2φ/h u)⟩ + ⟨ d_f,h^* u, d_f,h^*(e^2φ/h u)⟩= ⟨ e^φ/hd_f,h u, d_f,h(e^φ/h u)⟩ + ⟨ e^φ/hd_f,h u, dφ∧(e^φ/h u)⟩ + ⟨ e^φ/hd_f,h^* u, d_f,h^*(e^φ/h u)⟩ -⟨ e^φ/hd_f,h^* u, 𝐢_∇φ(e^φ/h u)⟩= ⟨ d_f,h(e^φ/h u), dφ∧(e^φ/h u)⟩ - ⟨ dφ∧(e^φ/h u), d_f,h(e^φ/h u)⟩ + d_f,h(e^φ/h u)^2 - d φ∧ (e^φ/h u)^2+d_f,h^*(e^φ/h u)^2 -𝐢_∇φ e^φ/h u^2 + ⟨𝐢_∇φ(e^φ/h u), d_f,h^*(e^φ/h u)⟩- ⟨ d_f,h^*(e^φ/h u), 𝐢_∇φ(e^φ/h u)⟩.Let us set ũ:=e^φ/h u∈D (Q^M,(1)_f,h ( Ω̇)). The formulas stated in (<ref>) lead to:𝒬^M,(1)_f,h ( Ω̇) ( u,e^2φ/h u)=𝒬^M,(1)_f,h ( Ω̇) (ũ) - ⟨|∇φ|^2ũ,ũ⟩ - ⟨ dφ∧ũ, d_f,hũ⟩+⟨ d_f,hũ, dφ∧ũ⟩ +⟨𝐢_∇φũ,d_f,h^*ũ⟩ - ⟨ d_f,h^*ũ, 𝐢_∇φũ⟩and hence to𝒬^M,(1)_f,h ( Ω̇) ( u,e^2φ/h u) =𝒬^M,(1)_f,h ( Ω̇) (ũ) - ⟨|∇φ|^2ũ,ũ⟩ .One concludes by applying Lemma <ref>. We are now in position to prove Proposition <ref>. (of Proposition <ref>)Following the proof of <cit.>, one proves the result in two steps. First, the Agmon estimate along Γ_1⊂Ω is proven by applying Lemma <ref> with a function φ closeto f_+ (recall that on Γ_1, Ψ=f_+). The Agmon estimate in Ω̇ is then obtained using again Lemma <ref> with φ close to Ψ, and the Agmon estimate along Γ_1. In order to separate the analysis along Γ_1 and elsewhere, one introduces two smooth cut-off functions χ_0 and χ_1 on Ω such that: χ_1:= √(1-χ_0^2) , χ_0=1on V_Γ_1 with χ_0⊂ V_Γ_1',for a set V_Γ_1'⊂Ω such that for some ε>0, (see Figure <ref>) (i) (V_Γ_1+B(0,ε))∩Ω⊂ V_Γ_1',(ii) Γ_1':=V_Γ_1'∩Ω is smooth and (Γ_1 + B(0,ε)) ∩∂Ω⊂Γ_1' and (Γ_1' + B(0,ε)) ∩∂Ω⊂ B_z,(iii) Ψ=d_a(z,·) is a smooth solution to the followingeikonal equation in V_Γ_1' (see Proposition <ref>):{ |∇Ψ |^2= |∇ f |^2inV_Γ_1' Ψ = f-f(z) on Γ_1' _nΨ = -_nf on Γ_1' . . It is possible to choose V_Γ_1' such that all the properties stated previously on V_Γ_1 also hold on V_Γ_1' (in particular (<ref>), (<ref>)and the properties stated in Proposition <ref>). We recall that one has by (<ref>):|∇Ψ|^2 ≤ |∇ f|^2a.e. in Ω.Thus |∇ f_±| = |∇(Ψ± (f-f(z))/2)| ≤ |∇ f|a.e. in Ω.Thanks to the relations f-f(z)=f_+-f_- andΨ=f_++f_-, together with the equality|∇Ψ |^2=|∇ f |^2 a.e inV_Γ_1', one has∇ f_-·∇ f_+ =0 a.ein V_Γ_1',|∇Ψ|^2 = |∇ f|^2 = |∇ f_+ |^2+ |∇ f_- |^2 a.ein V_Γ_1'.Let now u^(1)_h∈ D (Δ_f,h^M,(1) ( Ω̇) ) satisfyΔ_f,h^M,(1) ( Ω̇)u^(1)_h=0and u^(1)_h_L^2 ( Ω̇)=1. Step 1: Agmon estimate in Γ_1. In this part, we are going to prove the estimate (<ref>)with Ψ replaced by f_+ namely:e^f_+/hu^(1)_h_L^2 ( Ω̇) +d(e^f_+/hu^(1)_h)_L^2 ( Ω̇)+d^*(e^f_+/hu^(1)_h)_L^2 ( Ω̇) = O(h^-N_0)for some integer N_0. By the trace result (<ref>), this will give the estimatee^f-f(z)/hu^(1)_h_ L^2(Γ_1)= O(h^-N_0),which is the first step to prove (<ref>).To get these results, the idea is to apply Lemma <ref> to a convenient φ comparable with f_+ and such that |∇φ|≤ |∇ f_+|. This kind of estimate is classic and the ideas behind the computations presented below, which follow <cit.>, originate from the article <cit.> where similar estimates were obtained in the case of manifolds without boundary.The presence of a boundary leads here to some technicalities which can somehow hide the reasoning andwe refer for example to <cit.> fora presentation of semi-classical Agmon estimates in manifolds without boundary.We recall from the work <cit.> that if one just wants to get an error of the sizeO(e^ε/h)with ε>0 arbitrarily small, the choice φ=(1-ε)f_+ is sufficient, but it does not yield an error of the size O(h^-N). To get such an error term, a good choice for φ is the following. Let φ:Ω̇→ be the following Lipschitz function: φ= {[c] f_+-Chlnf_+/hif f_+ > Ch,f_+-Chln Cif f_+≤ Ch , .for some constant C> 1 that will be fixed at the end of this step. Define the level sets:Ω_-={x∈Ω̇ s.t.f_+(x) ≤ Ch}andΩ_+=Ω̇∖Ω_-.Then ∇φ = ∇ f_+ a.e. in Ω_- and∇φ=∇ f_+(1-Ch/f_+) a.e. in Ω_+.This implies in particular the two following inequalities valid a.e. on Ω_+ and that will be used in the sequel:|∇ f_+|^2- |∇φ |^2 = |∇ f_+|^2(2Ch/f_+-C^2h^2/f_+^2)≥Ch |∇ f_+|^2/f_+ on Ω_+ |∇ f|^2-|∇φ |^2 ≥ |∇ f|^2-|∇ f_+|^2(1-Ch/f_+) ≥ Ch |∇ f |^2/f_+ on Ω_+.The last inequality in (<ref>) is a consequence of the inequality |∇ f_+|^2 ≤ |∇ f|^2. This implies in particular that |∇φ| ≤ |∇ f| a.e. in Ω̇. Note lastly that there exists a constant K>0 depending on f_+ and f,such that|∇ f |^2/f_+≥ Kin Ω̇and|∇ f_+|^2/f_+≥ Kin V_Γ_1',the last inequality being a consequence of the facts that f_+(x',x_d)=f_+(x',0) andx' ↦ f(x',0)=f(z)+f_+(x',0) is a Morse function with z as only critical point.Now, using the fact that Δ^M,(1)_f,h(Ω̇) u^(1)_h=0 and the IMS localisation formula (<ref>), one gets 0 =𝒬^M,(1)_f,h ( Ω̇)(u^(1)_h,e^2φ/hu^(1)_h)= ∑_k∈{0,1}[ 𝒬^M,(1)_f,h ( Ω̇)(χ_ku^(1)_h,e^2φ/hχ_ku^(1)_h) -h^2|∇χ_k | e^φ/hu^(1)_h_ L^2 ( Ω̇)^2 ].Settingũ^(1)_h:=e^φ/hu^(1)_h,and applying (<ref>) to χ_k u^(1)_h, k∈{0,1}, one obtains:C_1h^2∑_k∈{0,1}χ_kũ^(1)_h_ L^2 ( Ω̇)^2=C_1h^2ũ^(1)_h_ L^2 ( Ω̇)^2≥∑_k∈{0,1}[ hdχ_kũ^(1)_h^2_L^2 ( Ω̇) +hd^*χ_kũ^(1)_h^2_ L^2 ( Ω̇) + ⟨(|∇ f|^2-|∇φ|^2)χ_kũ^(1)_h,χ_kũ^(1)_h⟩ _ L^2 ( Ω̇)+h⟨(ℒ_∇ f+ℒ_∇ f^*)χ_kũ^(1)_h,χ_kũ^(1)_h⟩_ L^2 ( Ω̇)] + h(∫_Γ_2-∫_Γ_1)⟨χ_0ũ^(1)_h,χ_0ũ^(1)_h⟩_n fdσ,where C_1=max(∇χ_0^2_∞,∇χ_1^2_∞).Note that one has used thatχ_0=0 on Γ_0, χ_1=0 on Γ_1 and (-∫_Γ_0+∫_Γ_2)⟨χ_1ũ^(1)_h,χ_1ũ^(1)_h⟩_n fdσ≥ 0,which follows from _nf>0 on Γ_2 and from _nf<0 on Γ_0.Now, since ℒ_∇ f+ℒ_∇ f^* is a 0^th order differential operator, and |ũ^(1)_h(x)|≤ e^C |u^(1)_h(x)| a.e. on Ω_-, one obtains that for some constants C_2 (independent of C) and C_3(C) dependingon C,C_3(C)h ≥∑_k∈{0,1}[ hdχ_kũ^(1)_h^2_ L^2 ( Ω̇) +hd^*χ_kũ^(1)_h^2_ L^2 ( Ω̇)+ ⟨(|∇ f|^2-|∇φ|^2 )χ_kũ^(1)_h,χ_kũ^(1)_h⟩ _ L^2 ( Ω̇) -C_2hχ_kũ^(1)_h^2_ L^2(Ω_+)] +h(∫_Γ_2-∫_Γ_1)⟨χ_0ũ^(1)_h,χ_0ũ^(1)_h⟩_n f dσ.Let us first consider the case k=1. Using |∇φ|≤ |∇ f| and (<ref>)–(<ref>), one gets:⟨(|∇ f|^2-|∇φ|^2 )χ_1ũ^(1)_h,χ_1ũ^(1)_h⟩_ L^2 ( Ω̇) -C_2hχ_1ũ^(1)_h^2_ L^2(Ω_+)≥⟨(|∇ f|^2-|∇φ|^2 )χ_1ũ^(1)_h,χ_1ũ^(1)_h⟩_ L^2(Ω_+) -C_2hχ_1ũ^(1)_h^2_ L^2(Ω_+)≥⟨(Ch |∇ f|^2/f_+ -C_2h) χ_1ũ^(1)_h,χ_1ũ^(1)_h⟩_ L^2(Ω_+)≥(KC-C_2)h χ_1ũ^(1)_h^2_ L^2(Ω_+) .Let us then consider the case k=0. In this case, one deduces from χ_0⊂ V_Γ_1' where |∇ f|^2=|∇ f_+|^2 +|∇ f_-|^2, from |∇φ|^2=|∇ f_+|^2 on Ω_-, and from (<ref>)–(<ref>) the inequality:⟨(|∇ f|^2-|∇φ|^2 )χ_0ũ^(1)_h,χ_0ũ^(1)_h⟩_ L^2 ( Ω̇) -C_2hχ_0ũ^(1)_h^2_ L^2(Ω_+)=|∇ f_-|χ_0ũ^(1)_h^2_ L^2 ( Ω̇) +⟨(|∇ f_+|^2-|∇φ|^2_=0)χ_0ũ^(1)_h,χ_0ũ^(1)_h⟩_ L^2(Ω_-) + ⟨(|∇ f_+|^2-|∇φ|^2 -C_2h )χ_0ũ^(1)_h,χ_0ũ^(1)_h⟩_ L^2(Ω_+)≥ |∇ f_-|χ_0ũ^(1)_h^2_ L^2 ( Ω̇) + (KC-C_2)h χ_0ũ^(1)_h^2_ L^2(Ω_+)≥ (1+ 2C_4(C)h) |∇ f_-|χ_0ũ^(1)_h^2_ L^2 ( Ω̇) - (KC-C_2)hχ_0ũ^(1)_h^2_ L^2(Ω_-) ,where C_4(C):= KC-C_2/2∇ f_-^2_L^∞(V_Γ_1')(see (<ref>)) and C has been chosen large enough to ensure that KC-C_2>0.In order to get a lower bound for the boundary term in (<ref>), one uses the fact that the mixed Witten Laplacian Δ^M,(1)_f̂, ĥ ( Ω̇) associated with f̂=-χ̃_0f_- where χ̃_0∈ C^∞(Ω,[0,1]), χ̃_0=1 on supp χ_0, supp χ̃_0⊂ (V_Γ_1'+B(0,α))∩Ω for α>0 such that f_- is smooth on supp χ̃_0 and ĥ= h/1+C_4(C)h, is nonnegative. Starting from the inequality (1+C_4(C)h) 𝒬_f̂,ĥ^M,(1) ( Ω̇)(χ_0ũ^(1)_h,χ_0ũ^(1)_h)≥ 0and then applying Lemma <ref> to χ_0ũ^(1)_h∈ D(𝒬^M,(1)_f̂,ĥ ( Ω̇))lead to (since χ_0=0 on Γ_0):h (∫_Γ_1-∫_Γ_2) ⟨χ_0ũ^(1)_h,χ_0ũ^(1)_h⟩_nf_-dσ≥ -(1+C_4(C)h) |∇ f_-|χ_0ũ^(1)_h_L^2(Ω̇)^2-h^2/1+C_4(C)h(dχ_0ũ^(1)_h^2_L^2(Ω̇) +d^*χ_0ũ^(1)_h_L^2(Ω̇)^2) - h C_5χ_0ũ^(1)_h_L^2(Ω̇)^2,where C_5 is some positive constant independent of C (it only depends on f_-). Injectingthe estimates (<ref>)–(<ref>) in (<ref>) andusing f=-f_-+f_++f(z) on V_Γ_1' with _nf_+=0 on Γ_1, then leads to:C_3h ≥∑_k∈{0,1}C_4h^3/1+C_4h( dχ_kũ^(1)_h^2_ L^2(Ω̇) +d^*χ_kũ^(1)_h^2_ L^2(Ω̇)) +(KC-C_2)h χ_1ũ^(1)_h^2_ L^2(Ω_+)+ C_4h |∇ f_-|χ_0ũ^(1)_h^2_ L^2 (Ω̇) - (KC-C_2)hχ_0ũ^(1)_h^2_ L^2(Ω_-) - h C_5χ_0ũ^(1)_h^2_ L^2 (Ω̇) +h∫_Γ_2⟨χ_0 u|χ_0 u ⟩_ T^*_σΩ̇_n f_+ d σ.In the last computation, one has used that 1 ≥C_4h/1+C_4h, 1-1/1+C_4h=C_4h/1+C_4h. It follows moreover from (<ref>) that_n f_+ > 0 on χ_0∩Γ_2. Then,since |ũ^(1)_h(x) |≤ e^C |u^(1)_h(x)| a.e. on Ω_-, there exists C_6(C,C_2,K) such thatC_6h ≥∑_k∈{0,1}C_4h^3/1+C_4( dχ_kũ^(1)_h^2_ L^2(Ω̇) +d^*χ_kũ^(1)_h^2_ L^2(Ω̇)) +2C_4∇ f_-^2_L^∞(V'_Γ_1)h χ_1ũ^(1)_h^2_ L^2(Ω̇)+ C_4h |∇ f_-|χ_0ũ^(1)_h^2_ L^2 (Ω̇) - h C_5χ_0ũ^(1)_h^2_ L^2 (Ω̇).One has used that 1+C_4≥ 1+C_4h, for h≤ 1 and KC-C_2 =2C_4∇ f_-^2_L^∞(V'_Γ_1).Additionally, since |∇ f_-|≥ c>0 on V_Γ_1' (see (<ref>)), lim_C →∞ C_4(C)= +∞ and C_5 is independent of C, one can then choose C such that c^2C_4-C_5>0. This implies the existence of a constant C_7>0 such that for h_0>0 small enough and for all h∈ (0,h_0],ũ^(1)_h_L^2 ( Ω̇) +dũ^(1)_h_L^2 ( Ω̇)+d^*ũ^(1)_h_L^2 ( Ω̇) ≤ C_7/h.Since φ-f_+≥ - C_8 h ln1/h for some constant C_8, there exists N_0>0 such that:e^f_+/hu^(1)_h_L^2 ( Ω̇) +d(e^f_+/hu^(1)_h)_L^2 ( Ω̇)+d^*(e^f_+/hu^(1)_h)_L^2 ( Ω̇) = O(h^-N_0).One has in particular, owing tothe trace result (<ref>) stated inProposition <ref> and since f_+=f-f(z) on Γ_1,e^f-f(z)/hu^(1)_h_ L^2(Γ_1)= O(h^-N_0). Step 2: Agmon estimate in Ω̇. One follows the same approach as in step 1 but with the functionφ= {[c]Ψ-ChlnΨ/h, if Ψ > Ch,Ψ-Chln C,if Ψ≤ Ch, .where the constant C> 1 will be fixed later on, and with thelevel sets:Ω_-={x∈Ω̇ s.t. Ψ(x) ≤ Ch}andΩ_+=Ω̇∖Ω_-. Applying formula (<ref>) then leads to (note that |ũ^(1)_h|≤ e^C |u_h| on Ω_-, ∂_nf <0 on Γ_0 and ∂_nf >0 on Γ_2):C_2(C) h(1+∫_Γ_1⟨ũ^(1)_h,ũ^(1)_h⟩_ T_σ^*Ω dσ) ≥hdũ^(1)_h^2_ L^2(Ω̇) +hd^*ũ^(1)_h^2_ L^2(Ω̇) +⟨ (|∇ f|^2-|∇φ|^2)ũ^(1)_h,ũ^(1)_h⟩_L^2(Ω̇)-C_1hũ^(1)_h_ L^2(Ω_+)^2,whereũ^(1)_h:=e^φ/h u^(1)_h, the constant C_1 is independent of C, whereas C_2 is a constant depending on C. Besides, due to the relationsΨ= f-f(z)on Γ_1 ande^φ/h ≤e^Ψ/h on Ω̇,the trace estimate obtained in (<ref>) implies∫_Γ_1⟨ũ^(1)_h,ũ^(1)_h⟩_ T_σ^*Ω dσ =O (h^-2N_0).Injecting (<ref>) in (<ref>) then giveshdũ^(1)_h^2_ L^2(Ω̇) +hd^*ũ^(1)_h^2_ L^2(Ω̇)+⟨ (|∇ f|^2-|∇φ|^2)ũ^(1)_h,ũ^(1)_h⟩ _ L^2(Ω̇)-C_1hũ^(1)_h_ L^2(Ω_+)^2 = O( h^1-2N_0).Since moreover |∇Ψ|^2≤ |∇ f|^2 (see (<ref>)) and f has no critival point in Ω̇, one gets:|∇ f|^2-|∇φ |^2≥|∇ f|^2-|∇Ψ|^2(1-Ch/Ψ)≥ Ch |∇ f |^2/Ψ≥CC_3h on Ω_+where C_3>0 is independent of C. Since |∇f|^2≥ |∇Ψ|^2=|∇φ|^2 a.e. on Ω_-, adding the term (CC_3-C_1)hũ^(1)_h^2_ L^2(Ω_-) to (<ref>) leads tohdũ^(1)_h^2_ L^2(Ω̇) +hd^*ũ^(1)_h^2_ L^2(Ω̇) + (CC_3-C_1)hũ^(1)_h^2_ L^2 ( Ω̇) = O(h^1-2N_0)Now, since φ - Ψ≥ -C_4 h ln1/h, taking C>C_1/C_3,there exists N_1>0 such that:e^Ψ/hu^(1)_h_ L^2 ( Ω̇) +d(e^Ψ/hu^(1)_h)_ L^2 ( Ω̇)+d^*(e^Ψ/hu^(1)_h)_ L^2 ( Ω̇) = O(h^-N_1).This concludes the proof of (<ref>).§.§ Comparison of the eigenform u^(1)_h,i and its WKB approximationThroughout this section, one assumes [H1], [H2] and [H3]. In all this section, we consider, for a fixed critical point z_i, an ensemble of sets 𝒮_M,i associated with z_i (see Definition <ref>) and an L^2-normalized eigenformu^(1)_h,i of Δ^M,(1)_f,h(Ω̇_i) associated with the eigenvalue 0, as introduced at the end of Section <ref>.For the ease of notation, we drop the subscript i in all this section.§.§.§ Construction of the WKB expansion of u^(1)_h Let z be a local minimum of f|_Ω.Before going through a rigorous construction of a WKB expansionu^(1)_z,wkb of u^(1)_h in a neighborhood of z, let us explain formally howweproceed. Let us recall that the 1-form u^(1)_h satisfies:{ Δ_f,h^(1)u^(1)_h=0in Ω̇,t u^(1)_h=0and t d^*_f,hu^(1)_h =0on Γ_1, .plus additional boundary conditions on Γ_0∪Γ_2 that we do not recall since the objective is to approximate u^(1)_h in a neighborhood of Σ in Ω̇ (where we recall Σ is an open subset of Ω containing z and such that Σ⊂Γ_1, see (<ref>)). The behavior of u^(1)_h in a neighborhood of Γ_1 exhibited in Proposition <ref> suggests to take u^(1)_z,wkb of the form u^(1)_z,wkb(x,h)=a^(1)(x,h)e^-d_a(x ,z) /h where a^(1) is expanded in powers of h: a^(1)(x,h)=∑_k≥ 0 a_k^(1)(x)h^k and to look for 1-forms (a_k^(1))_k≥ 0 so that u^(1)_z,wkb is a nontrivial 1-form satisfying (compare with (<ref>)):{ Δ_f,h^(1)u^(1)_z,wkb=O(h^∞)e^-d_a(· ,z) /h in Ω̇,t u^(1)_z,wkb =0and t d^*_f,hu^(1)_z,wkb =O(h^∞)e^-d_a(· ,z) /h on Γ_1, .where the meaning of O(h^∞) is formally h^sO(h^∞)=o_h(1) for any s∈ℝ. The boundary conditions in (<ref>) ensures that when cutting suitably a solution to (<ref>) near Γ_1, the resulting 1-form belongs to the form domain of Δ^M,(1)_f,h(Ω̇) (this is needed if one wants to approximate u^(1)_h on Ω̇).Instead of directly trying to solve (<ref>), the construction of u^(1)_z,wkb can be simply done as follows (see <cit.>). Using the complex property, one considersu^(1)_z,wkb=d_f,hu^(0)_z,wkb where the function u^(0)_z,wkb=a^(0)(·,h) e^-d_a(. ,z) /h where a^(0)(x,h)=∑_k≥ 0 a_k^(0)(x)h^k for a non trivial family of functions (a_k)_k≥ 0 such that: { Δ_f,h^(0)u^(0)_z,wkb=O(h^∞)e^-d_a(· ,z) /h in Ω̇, u^(0)_z,wkb =e^-1/h f on Γ_1. . This implies the boundary condition: a^(0)=1 on Γ_1.Then, if u^(0)_z,wkb=a^(0)e^-d_a(. ,z) /h is a solution to (<ref>),we set:u^(1)_z,wkb=d_f,hu^(0)_z,wkb.One can easily check that the 1-form u^(1)_z,wkb then satisfies (<ref>) and the extra boundary condition t d^*_f,hu^(1)_z,wkb = O(h^∞)e^-f-f(z) /h on Γ_1. Indeed, it holds:d_f,h u^(0)_z,wkb=e^-d_a(· ,z) /h( d(f-d_a(·,z))a^(0)+ h d a^(0)),which implies t u^(1)_z,wkb =0 since a^(0)=1 andf-d_a(·,z)=f(z)on Γ_1.In addition, one hasd_f,h^* d_f,h u^(0)_z,wkb=Δ_f,h^(0)u^(0)_z,wkb=O(h^∞)e^-d_a(· ,z) /h which impliest d^*_f,hu^(1)_z,wkb =O(h^∞)e^-d_a(· ,z) /h andΔ_f,h^(1) d_f,h u^(0)_z,wkb=d_f,hΔ_f,h^(0)u^(0)_z,wkb=O(h^∞)e^-d_a(· ,z) /h.Thus, the 1-form u^(1)_z,wkb satisfies (<ref>).Expanding in powers of h the function e^d_a(x ,z) /h Δ_f,h^(0)( (∑_k≥ 0 a_k^(0)(x)h^k ) e^-d_a(x ,z) /h), u^(0)_z,wkb is a solution to (<ref>) if it holds:|∇ d_a(x, z)| =|∇ f(x)|, forx∈Ω̇,which is satisfied at least in a neighborhood of z (see Proposition <ref>) and if (a_k^(0))_k≥ 0 satisfies the following transport equations, defined recursively by:{[ (ΔΦ-Δ f +2∇Φ·∇)a_0^(0)=0 Ω̇; (ΔΦ-Δ f +2∇Φ·∇)a_k+1^(0)=Δ a_k^(0) Ω̇,∀ k≥ 0, ].with boundary conditions{[a_0^(0)=1Γ_1; a_k^(0) =0 Γ_1,∀ k≥ 1 ].. The equation (<ref>) together with the boundary condition (<ref>) justify a posteriori the choice of the function d_a(·,z) in the exponential for the ansatz on u^(1)_z,wkb.Let us mention that∂_n f > 0 on Γ_1 implies that there exists a non trivial solution u^(0)_z,wkb to (<ref>) in a neighborhood of Γ_1 since in that case the transport equations (<ref>) are non degenerate. Let us now justify rigorously the construction of the WKB expansion u^(1)_z,wkb of u^(1)_h, which is, in view of (<ref>) and (<ref>),possible near Γ_1.A preliminary construction. Let Φ be the solution to the eikonal equation (<ref>) on a neighborhood V_∂Ω of the boundary ∂Ω. Let us introduce the formal transport operator T:=ΔΦ-Δ f +2∇Φ·∇.Let us consider the solutions to the following transport equations, defined recursively by{[Ta_0=0V_∂Ω;Ta_k+1=Δ a_k V_∂Ω, ∀ k≥ 0, ].with boundary conditions {[ a_0=1∂Ω;a_k =0 ∂Ω, ∀ k≥ 1. ].For a fixed k, the transport equation can be solved locally around each z∈∂Ω thanks to the condition ∂_nΦ=-∂_nf<0 on ∂Ω and thus on a neighborhood of ∂Ω (independent of k) using a compactness argument. Therefore, up to choosing a smaller neighborhood V_∂Ω of ∂Ω in Ω, there exists a unique sequence ofC^∞(V_∂Ω) functions (a_k)_k≥ 0 solution to (<ref>)-(<ref>).There exists a function a=a(x,h)(called a resummation of the formal symbol∑_k=0^+∞ a_kh^k) C^∞ and uniformly bounded together with all its derivatives such that a(x,h)=1 on ∂Ω and a(x,h)∼∑_k=0^+∞ a_k(x)h^k.This means that a-∑_k=0^+∞ a_kh^k is O(h^∞) in the following sense: for all compact K in V_∂Ω, for all α∈ℕ^d, for all N ∈ℕ,∂^α_x ( a- ∑_k=0^N a_k(x) h^k )_L^∞(K)≤ C_K,α,N h^N+1.Such a construction is standard and can be found in <cit.> or in<cit.>, where it is done using a Borel summation. Moreover a is unique up to a term oforder O(h^∞). Let us now define on V_∂Ω: u_wkb^(0)(x,h):=a(x,h)e^-Φ/h. By construction of the sequence (a_k)_k≥ 0, the function u_wkb^(0) solves {Δ_f,h^(0)u^(0)_wkb =O(h^∞) e^-Φ/h inV_∂Ω,u^(0)_wkb=e^-Φ/h=e^-f/h on ∂Ω, .where O(h^∞) is defined in (<ref>).Indeed, using (<ref>), |∇ f |^2=|∇Φ|^2 on V_∂Ω, and the equations (<ref>) satisfied by (a_k)_k ≥ 0,e^Φ/hΔ_f,h^(0)u^(0)_wkb =-h^2Δ a(x,h)+h[a(x,h)ΔΦ +2 ∇Φ·∇ a(x,h)] -a(x,h) |∇Φ|^2 + a(x,h) |∇ f |^2 -h a(x,h)Δ f∼ hTa_0 + h^2 ∑_k=0^+∞ h^k(Ta_k+1-Δ a_k) =O(h^∞).In addition, it holds u^(0)_wkb=e^-Φ/hon ∂Ω since a(x,h)=1 on ∂Ω. Let us now define on V_∂Ω:u^(1)_wkb:=d_f,h u^(0)_wkb . The 1-form u^(1)_wkbsatisfies:{Δ_f,h^(1) u^(1)_wkb =O(h^∞)e^-Φ/h in V_∂Ω , 𝐭u^(1)_wkb=0 on ∂Ω , 𝐭d_f,h^*u^(1)_wkb=O (h^∞)e^-Φ/h on ∂Ω, . where O(h^∞) is defined in (<ref>). Indeed, one has 𝐭u^(1)_wkb=𝐭d_f,h u^(0)_wkb=d_f,h𝐭 u^(0)_wkb=d_f,h𝐭(a(x,h) e^-Φ/h)=d_f,h𝐭 e^-f/h = d_f,h e^-f/h = 0since a(x,h)=1 and Φ=f on∂Ω. Moreover 𝐭d_f,h^*u^(1)_wkb=𝐭d_f,h^*d_f,hu^(0)_wkb= 𝐭Δ_f,h^(0)u^(0)_wkb= O (h^∞) e^-Φ/h. Finally, Δ_f,h^(1)u^(1)_wkb= Δ_f,h^(1)d_f,hu^(0)_wkb=d_f,hΔ_f,h^(0)u^(0)_wkb=(hd+df ∧) O (h^∞) e^-Φ/h=O (h^∞) e^-Φ/h. WKB expansion of u^(1)_h. Let z be a local minimum of f|_Ω.Let us now define the WKB expansion of u^(1)_h on V_∂Ω by:u^(1)_z,wkb:=e^f(z)/h u^(1)_wkb= e^f(z)/h d_f,hu^(0)_wkb = d_f,h(a(·,h) e^-Φ-f(z)/h).One recalls (see Proposition <ref>) that for any smooth open domain Γsuch that Γ⊂Γ_1 and z∈Γ, there exists a neighborhoodof Γ in Ω, denoted by V_Γ⊂ V_∂Ω∩ (Γ_1∪Ω̇), such that for all x∈ V_Γ,Ψ(x)=d_a(x,z)=Φ(x)-f(z).Let us assume that the hypotheses [H1] and [H3] hold. Let us consider z a local minimum of f|_∂Ω as introduced in hypothesis [H2]. The 1-form u^(1)_z,wkbsatisfies {Δ_f,h^(1) u^(1)_z,wkb =O(h^∞)e^-Φ-f(z)/hin V_∂Ω , 𝐭u^(1)_z,wkb=0 on ∂Ω , 𝐭d_f,h^*u^(1)_z,wkb =O (h^∞)e^-Φ-f(z)/hon ∂Ω, .where O(h^∞) is defined in (<ref>). For any χ∈ C^∞_c(V_Γ) such that χ=1 on a neighborhood of z, it holds: in the limit h→ 0,∫_Ω|χ(x) u^(1)_z,wkb(x)|^2dx=C_z,wkb^2h^d+1/2 (1+O(h)), where C_z,wkb:= π^d-1/4√(2∂_nf(z))/( detHess f|_∂Ω(z) )^1/4.Furthermore, there exists C>0 such that for h small enough,‖χu^(1)_z,wkb‖_H^1(Ω)≤ Ch^-1,and 𝒬_f,h^M,(1) ( Ω̇)(χ u^(1)_z,wkb)=O(h^∞).Equation (<ref>) is easily obtained from (<ref>). Let us now prove (<ref>) and (<ref>). Notice that one can write (using (<ref>)) u^(1)_z,wkb =d_f,h[e^-Φ-f(z)/h∑_k=0^+∞ a_kh^k]=e^-f/h(hd)e^f/h[ e^-Φ-f(z)/h∑_k=0^+∞ a_kh^k] =e^-Φ-f(z)/h e^-f-Φ/h(hd)e^f-Φ/h[∑_k=0^+∞ a_kh^k]= e^-Φ-f(z)/h( d(f-Φ)∧ a_0 )+e^-Φ-f(z)/h[h d∑_k=0^+∞ a_kh^k + d(f-Φ)∧∑_k=1^+∞ a_kh^k ]. Recall that the function χ is supported in V_Γ and the function x↦Φ(x)-f(z) has a unique minimum on V_Γ which is z sinceΦ(x)-f(z)=d_a(x,z) ≥ 0 on V_Γ. Therefore, in the limit h→ 0:√(∫_Ω|χ(x) u^(1)_z,wkb(x)|^2dx)=√(∫_Ω| e^-Φ-f(z)/hχ d(f-Φ)∧ a_0|^2) (1+O(h)). Additionaly, since χ(z)=1 and | d(f-Φ)(z)| ^2=|∇ (f-Φ)(z)|^2= |∇_T (f-Φ)(z)|^2+(2∂_nf(z))^2=(2∂_nf(z))^2, one getsusing Laplace's method∫_Ω| e^-Φ-f(z)/hχ d(f-Φ)∧ a_0|^2=(π h)^d+1/22∂_nf(z)/π√( detHess f|_∂Ω(z) ) (1+O(h)). Let us give more details on how to obtain (<ref>). Recall that on supp(χ), Φ-f(z)=f_++f_-, f-Φ=f-Ψ-f(z)=-2f_- and, on supp(χ) ∩∂Ω, ∂_nf = -∂_nf_- =|∇f_-|. Thus, using the coordinate set introduced in Definition <ref> and the co-area formula dx=dσ_Σ_η/|∇ f_-| dη (see for example <cit.>)∫_Ω| e^-Φ(x)-f(z)/hχ(x) d(f-Φ) (x) ∧ a_0 (x)|^2dx =4∫_-α^0e^-2η/h∫_Σ_ηe^-2f_+/hχ^2a_0^2|∇ f_-| dσ_Σ_ηdη=4∫_-α^0e^-2η/h∫_∂Ωe^-2f_+(x',0)/hχ^2(x',η)a_0^2(x',η)|∇ f_-|(x',η) j(x',η) dσ_∂Ω(x') dηwhere Σ_η={x, f_-(x)=-η}, σ_Σ_η is the Lebesgue measure on Σ_η. In the last equality, j(x',η) is the Jacobian of the parametrization of Σ_η by x' ∈∂Ω. Using the Laplace formula, for any η∈ [-η_0,0] with η_0>0 sufficiently small so that χ^2(z,η)≠ 0 for all η∈ [-η_0,0], one has∫_∂Ωe^-2f_+(x',0)/hχ^2(x',η) a_0^2(x',η)|∇ f_-|(x',η) j(x',η) dσ_∂Ω(x')=(π h)^d-1/2 ( f_+(z))^-1/2χ^2(z,η)|∇ f_-|(z,η) j(z,η) a_0^2(z,η)(1+O(h))where O(h) is a function of η and h with L^∞ norm in η∈ [0,η_0] bounded from above by a constant times h (thanks to the regularity of the involved terms), for sufficiently small h. Thus, using again Laplace's method: ∫_Ω| e^-Φ-f(z)/hχ d(f-Φ)∧ a_0|^2=4 ∫_-α^0e^-2η/h (π h)^d-1/2 ( f_+(z))^-1/2χ^2(z,η) a_0^2(z,η) |∇ f_-|(z,η) j(z,η) dη (1+O(h))=2h (π h)^d-1/2 ( f_+(z))^-1/2χ^2(z,0)a_0^2(z,0)|∇ f_-|(z,0) j(z,0) (1+O(h))Since χ(z,0)=1, a_0(z,0)=1, f_+(z)= f|_∂Ω(z) and j(z,0)=1, this concludes the proof of (<ref>), and thus of (<ref>)-(<ref>).Now, writingu^(1)_z,wkb = e^-Φ-f(z)/h[d(f-Φ)∧ a(·,h) +h da(·,h)],and noticing that Φ-f(z)≥ 0 on suppχ, one has ‖χu^(1)_z,wkb‖_H^1( Ω)≤ Ch^-1.It remains to prove the last statement. Using the fact that supp[Δ_f,h^(1),χ]⊂ supp χ, Φ-f(z)= Ψ≥ c'>0 on supp∇χ and (<ref>), one getsΔ_f,h^(1)(χ u^(1)_z,wkb) =χΔ_f,h^(1)( u^(1)_z,wkb)+[Δ_f,h^(1),χ](u^(1)_z,wkb) =O(h^∞)+ O(e^-c/h)=O(h^∞).Therefore, from the Cauchy-Schwartz inequality, one has ⟨χ u^(1)_z,wkb, Δ_f,h^(1)(χ u^(1)_z,wkb)⟩_L^2(Ω̇)=O(h^∞). The fact that 𝒬_f,h^M,(1) ( Ω̇)(χ u^(1)_z,wkb)=O(h^∞) then follows from an integration by parts and the boundary conditions in (<ref>).§.§.§ A first estimate of the accuracy of the WKB approximation Recall that z∈{z_1,…,z_n} is a local minimum of f|_∂Ω and that u^(1)_h is a L^2-normalized eigenform of Δ^M,(1)_f,h(Ω̇) associated with the eigenvalue 0. The objective of this section is to prove that u^(1)_h is accurately approximated by the function u^(1)_z,wkb defined in the previous section. The computations below are inspired by those made in <cit.> where the authors were adapting <cit.> to manifolds with boundary. The novelty is that we compare the two 1-forms in a neighborhood of B_z, instead of a neighborhood of z. Take two smooth open sets Γ_St⊂Γ_St'⊂Γ_1 which are strongly stable (see Definition <ref> and Proposition <ref>) and such that, for some positive ε, (Γ_St+B(0,ε)) ∩∂Ω̇⊂Γ_St' and (Γ_St'+B(0,ε)) ∩∂Ω̇⊂Γ_1, see Figure <ref>. The fact that Γ_St and Γ_St' are strongly stable and thus that V_Γ_St and V_Γ_St' are stable under the dynamics (<ref>) -see below-will actually be needed only to get refined estimates in Section <ref>. Let us now consider the system of coordinate (x',x_d)(see Definition <ref>) which is well defined on V_Γ_1 by assumption (see item 2 in Proposition <ref>). Let us introducethe Lipschitz sets V_Γ_St and V_Γ_St'V_Γ_St = {(x',x_d)∈Γ_St×(-a,0) } andV_Γ_St' = {(x',x_d)∈Γ_St'×(-a',0) }where 0< a < a' are small enough so that V_Γ_St⊂ V_Γ_St'⊂ V_Γ_1. By construction, there exists ε > 0 such that V_Γ_St + B(0,ε) ⊂ V_Γ_St' and V_Γ_St' + B(0,ε)⊂ V_Γ_1∩ (Ω̇∪Γ_1) (see again Figure <ref> for a schematic representation of these sets). In addition V_Γ_St∩Γ_1=Γ_St and V_Γ_St'∩Γ_1=Γ_St'. Moreover, a and a' can be chosen sufficiently small so that the sets V_Γ_St and V_Γ_St' are stable under the dynamics x'(t)= {-∇Φ (x(t)) on Ω-∇_T Φ (x(t))on ∂Ω . . This stability is a consequence of two facts. First, for x(t) solution to (<ref>), d/dt f_-(x(t))'=-|∇ f_-(x(t))|^2 (since ∇Φ·∇ f_-=|∇ f_-|^2 on V_Γ_1, thanks to (<ref>), and ∇_T Φ·∇ f_-=0 in ∂Ω) so that ∀ t≥ 0,x_d(x(0))≤ x_d(x(t))≤ 0. Second, by construction, for sufficiently small a and a',∀ x ∈∂ V_Γ_St such thatx'(x) ∈∂Γ_St,∇Φ(x) · n_x(V_Γ_St)>0(where n(V_Γ_St) is the unit ouward normal to V_Γ_St). Indeed for any z ∈∂Γ_St, lim_σ→ z n_σ(V_Γ_St)=n_z(Γ_St) (where the limit is taken for σ∈∂ V_Γ_St with x'(σ) ∈∂Γ_St), see (<ref>) for a proof, and sinceΓ_St is chosen strongly stable, for z ∈∂Γ_St, ∇Φ(z) · n_z(V_Γ_St)=(∇ f_+ + ∇ f_-) · n_z(Γ_St)=∇ f_+ · n_z(Γ_St)=∇ f|_∂Ω· n_z(Γ_St) > 0. The argument is of course the same for V_Γ_St'.Let us now introduce two smooth cut-off functions 0≤χ≤η∈𝒞^∞_c(Ω̇∪Γ_1) satisfyingχ=1 in a neighborhood of V_Γ_St, χ⊂ V_Γ_St'andη=1 in a neighborhood of V_Γ_St',η⊂ V_Γ_1∩ (Ω̇∪Γ_1).Notice that by construction, η=0 on Γ_2. In the following, we moreover assume that χ and η are tensor products in the system of coordinates (x',x_d) (this will actually be needed only to get refined estimates in Section <ref>):χ(x',x_d)=χ_1(x')χ_d(x_d)and η(x',x_d)=η_1(x')η_d(x_d).Let κ∈{χ,η}. Owing to Lemma <ref>, the 1-formκ u^(1)_h belongs to Λ^1H_T^1(Ω̇). The first a priori estimate on κ(u^(1)_h-c(h)u^(1)_z,wkb)is the following:Let us assume that the hypotheses [H1], [H2] and [H3] hold. For κ∈{χ,η}, one hasκ(u^(1)_h-c_z(h)u^(1)_z,wkb)_ H^1(Ω̇)=O (h^∞)where c_z(h)^-1=⟨u^(1)_h, χ u^(1)_z,wkb⟩_L^2(Ω̇). The 1-form u^(1)_h can be chosen such that c_z(h)>0. Additionally, when h→ 0c_z(h) =C_z,wkb^-1h^-d+1/4(1+O(h^∞)),where C_z,wkb is defined by (<ref>).Notice that |c_z(h)|^-1 is equivalent (in the limit h → 0) to κ u^(1)_z,wkb_L^2(Ω̇) (see (<ref>)), and can thus be simply understood as a normalizing factor.Let us first consider the case κ=χ, the other case is considered at the end of the proof.One definesk(h):= ⟨u^(1)_h, χ u^(1)_z,wkb⟩_L^2(Ω̇)∈ℝ.If k(h)<0, then one changes u^(1)_h to -u^(1)_h so that one can suppose without loss of generality thatk(h) ≥ 0. For h small enough, one has (from Proposition <ref>, item (iii))π_[0,ch^3/2)( Δ_f,h^M, (1)(Ω̇) )( χ u^(1)_z,wkb)= k(h) u^(1)_h.Let us define α_h:=χ(u^(1)_z,wkb- k(h) u^(1)_h). Thus, the following identity holds for h small enoughα_h=k(h) (1-χ)u^(1)_h+π_[h^3/2, +∞]( Δ_f,h^M,(1) ( Ω̇) ) ( χ u^(1)_z,wkb).Notice that, using Cauchy-Schwarz inequality and Lemma <ref>, there exist C>0 andh_0>0 such that for all h∈ (0,h_0)| k(h)|≤ Ch^d+1/4. Therefore, using Lemma <ref>, Proposition <ref> and Lemma <ref> we get‖α_h ‖^2_L^2(Ω̇) ≤ 2k(h)^2 ‖(1-χ)u^(1)_h‖^2_L^2(Ω̇)+ 2 ‖π_[ch^3/2, +∞] ( Δ_f,h^M,(1) ( Ω̇) ) ( χ u^(1)_z,wkb) ‖^2_L^2(Ω̇)≤ Ch^d+1/2‖(1-χ)u^(1)_he^Ψ/he^ -Ψ/h‖^2_L^2(Ω̇) + C h^-3/2𝒬_f,h^M,(1) ( Ω̇)(χ u^(1)_z,wkb)≤ Ch^d+1/2 h^-N_0 e^-c/h+ C h^-3/2𝒬_f,h^M,(1) ( Ω̇)(χ u^(1)_z,wkb)= O(h^∞), with c:= inf_ supp( 1-χ)Ψ>0 (since χ=1 near z) and the integer N_0 is given by Proposition<ref>.Moreover, since d_f,h =hd+df∧ and d_f,h^* =hd^*+i_∇ f, one obtainsusing the triangular inequality, the Gaffney inequality (<ref>) (since α_h ∈ H^1_T(Ω̇)), the fact that 𝒬_f,h^M,(1) ( Ω̇)(χ u^(1)_h)=O(e^-c/h) (from the Agmon estimate (<ref>)) and 𝒬_f,h^M,(1) ( Ω̇)(χ u^(1)_z,wkb)=O(h^∞),‖α_h‖_H^1(Ω̇)^2≤ C( ‖ dα_h‖^2_L^2(Ω̇)+‖ d^*α_h‖^2_L^2(Ω̇)+‖α_h‖^2_L^2(Ω̇) )≤ Ch^-2(𝒬_f,h^M,(1) ( Ω̇)(α_h) + ‖α_h ‖^2_L^2(Ω̇)) = O(h^∞).Moreover since ‖χ u^(1)_h ‖_L^2(Ω̇) =1+O(e^-c/h) (from the Agmon estimate (<ref>)), by considering ‖χ (u^(1)_z,wkb-k(h) u^(1)_h )‖_L^2(Ω̇) =O(h^∞), one gets:k(h) ^2= ‖χ u^(1)_z,wkb‖^2_L^2(Ω̇) + O(h^∞)/2-‖χ u^(1)_h‖^2_L^2(Ω̇)= C^2_z,wkb h^d+1/2 +O(h^∞)/1+O(e^-c/h),with C_z,wkbgiven by (<ref>) inLemma <ref>. Therefore, since k(h) ≥ 0,k(h)=C_z,wkb h^d+1/4(1+O(h^∞)). This concludes the proof of (<ref>) for κ=χ, by choosingc_z(h):=k(h)^-1. Let us now deal with the case κ=η. There exists c>0 such that, for h sufficiently small, ‖η(u^(1)_z,wkb- k(h) u^(1)_h) ‖_H^1(Ω̇) ≤‖α_h ‖_H^1(Ω̇)+‖ (η-χ)(u^(1)_z,wkb- k(h) u^(1)_h) ‖_H^1(Ω̇)≤ O(h^∞) +‖ (η-χ)u^(1)_z,wkb‖_H^1(Ω̇) + | k(h)|‖ (η-χ) u^(1)_h ‖_H^1(Ω̇)≤ O(h^∞) + e^-c/h.The last inequality is the consequence of two facts. First, ‖ (η-χ) u^(1)_h ‖_H^1(Ω̇)=e^-c/h thanks to Proposition <ref> and (<ref>) together with the fact that χ= η near z. Second, a direct computation shows that ‖ (η-χ)u^(1)_z,wkb‖_H^1(Ω̇)≤ Ch^-1e^-inf_ supp(η-χ)Ψ/h≤ e^-c/h . This concludes the proof of Proposition <ref>.The estimate we obtained in Proposition <ref> is sufficient to get the result of Theorem <ref>. The more precise estimates obtained inSection <ref> are only needed to prove Theorem <ref>.§.§.§ A more accurate comparison on the WKB approximation The objective of this section is to combine the techniques used to obtain the Agmon estimates of Proposition <ref> and the first estimate of the accuracy of the WKB approximation of Proposition <ref> in order to obtain a more precise estimate of the latter. Let us start with estimates which are simple consequences of Propositions <ref> and <ref>. Notice that,for κ∈{χ,η}, one obviously gets from Proposition <ref>the following relation in Λ^1H^1(Ω̇):∃ N_0∈ℕ , e^Ψ/hκ(u^(1)_h-c_z(h)u^(1)_z,wkb) = O (h^-N_0).For the term involving u^(1)_z,wkb, this is due to Ψ(x)=Φ(x)-f(z) on κ and the estimate (<ref>) on c_z(h). Let us now set w_h:=κ(u^(1)_h-c_z(h)u^(1)_z,wkb).The 1-form w_h is in C^∞_c(Ω̇∪Γ_1) andsatisfies in Ω̇:Δ_f,h^(1)w_h =κΔ_f,h^(1)( u^(1)_h-c_z(h)u^(1)_z,wkb) +[Δ_f,h^(1),κ](u^(1)_h-c_z(h)u^(1)_z,wkb)=-c_z(h)κΔ_f,h^(1)u^(1)_z,wkb +[Δ^(1)_f,h,κ] (u^(1)_h-c_z(h)u^(1)_z,wkb) = (r_1+r_1') e^-Ψ/h ,where, owing to (<ref>) and (<ref>):r_1:= - e^Ψ/h c_z(h)κΔ_f,h^(1)u^(1)_z,wkb=O(h^∞)in Λ^1 L^2(Ω) and, from (<ref>):r_1':= e^Ψ/h[Δ^(1)_f,h,κ] (u^(1)_h-c_z(h)u^(1)_z,wkb) = O (h^-N_0)inΛ^1 L^2(Ω)andr_1'⊂∇κ. Additionally, one gets similarly on the boundary Γ_1:𝐭w_h|_Γ_1=0and 𝐭d_f,h^*w_h|_Γ_1 = (r_2 +r'_2)e^-Ψ/h =(r_2 +r'_2)e^-f-f(z)/h,where owing to (<ref>) and (<ref>):r_2 :=𝐭 e^Ψ/hκd^*_f,h (u^(1)_h-c_z(h)u^(1)_z,wkb) |_Γ_1 =-𝐭 e^Ψ/h κc_z(h) d^*_f,h u^(1)_z,wkb|_Γ_1=O(h^∞)inL^2(Ω) andr_2':= 𝐭 e^Ψ/h h i_∇κ (u^(1)_h-c_z(h)u^(1)_z,wkb)|_Γ_1= O (h^-N_0)in L^2(Ω̇)with r_2'⊂Γ_1∩∇κ.We are now in position to prove the following proposition. Let us assume that the hypotheses [H1], [H2] and [H3] hold. One has the following estimate in the limit h→0:e^Ψ/h(u^(1)_h-c_z(h)u^(1)_z,wkb)_ H^1(V_Γ_St) =O(h^∞),where c_z(h) is defined by (<ref>) and where, we recall, Ψ(x)=d_a(x,z) and V_Γ_St is defined by (<ref>).As for the proof of Proposition <ref>,one first proves an estimate along the boundary Γ_1before propagating it in V_Γ_St.Step 1. Comparison in Γ_1. Let us consider w_h defined by (<ref>) and the cut-off function κ=η defined in (<ref>).Like in the first step of the proof ofProposition <ref>, we are going to prove an estimate of the form (<ref>) with Ψ replaced by f_+. More precisely, we want to show that e^f_+/hw_h_H^1(V_Γ_St')=e^f_+/h(u^(1)_h-c_z(h)u^(1)_z,wkb)_ H^1(V_Γ_St')= 𝒪(h^∞),which implies in particular the following estimate along the boundary, since f_+=f-f(z) in Γ_1,e^f-f(z)/h(u^(1)_h-c_z(h)u^(1)_z,wkb) _ H^1/2(Γ_St') =𝒪(h^∞).In the following, we denote (see Figure <ref> for a schematic representation of the set V_η) V_η=η. In the system of coordinates (x',x_d), x ∈ V_η if and only if x'(x) ∈η_1 and x_d(x) ∈η_d. We recall that V_η is a compact set of Ω̇∪Γ_1. As for the proof ofProposition <ref>, weintroduce the setsΩ_-={x∈ V_η s.t.f_+(x) ≤ Ch} and Ω_+=V_η∖Ω_-,and define the Lipschitz function φ:V_η→ byφ= { f_+-Chlnf_+/hiff_+ > Ch,f_+-Chln C iff_+≤ Ch , .for some constant C> 1 that will be fixed at the end of this step. Notice for further purposes that lim_h → 0φ - f_+_L^∞(V_η)=0. We recall that in the system of coordinates (x',x_d), φ is independent of x_d.The reasoning below is based on <cit.>, see also <cit.> for a presentation in the case without boundary. According to (<ref>), we want to get an error of the form 𝒪(h^N) with N arbitrary. We are not going to work withthe above phase function φ as we did in the proof of Proposition <ref>, but with a phase function φ_N alsodependingon some arbitrary N∈. Let us definew̃_h =e^φ_N/hw_h= e^φ_N/hη(u^(1)_h-c_z(h)u^(1)_z,wkb).Combining the integration by parts formula (<ref>) (with u=w_h and φ=φ_N) with the Green formula (<ref>) (with u=w_h and v=e^2φ_N/hw_h)leads to the estimatee^φ_N/hΔ^(1)_f,hw_h_ L^2(V_η) w̃_̃h̃_ L^2(V_η) + he^φ_N/h𝐭d^*_f,hw_h_ L^2(Γ_1)w̃_h_ L^2(Γ_1)≥hdw̃_h^2_ L^2(V_η)+hd^*w̃_h^2_ L^2(V_η) -h∫_Γ_1⟨w̃_h,w̃_h⟩_ T_σ^*Ω_n fdσ + ⟨ (|∇ f|^2-|∇φ_N|^2+hℒ_∇ f+ hℒ^*_∇ f)w̃_h,w̃_h⟩_L^2(V_η). Let us explain formally how the function φ_N is chosen.Roughly speaking, using similar arguments as in the proof of Proposition <ref>, it is natural to choose φ_N=φ+Nhln1/hand to try to prove thatthe left-hand side of (<ref>) is bounded from above by𝒪(h^-N_1)w̃_̃h̃_ H^1(V_η) for some N_1∈ℕ independent of N. This would indeed lead to an estimate of the form w̃_̃h̃_ H^1(V_η)=𝒪(h^-N_1) (for some maybe larger N_1) and finally to the desired estimate on w_h since w̃_h_H^1(V_η)≃ h^-Ne^f_+/h w_h_H^1(V_η). To get this upper bound,a trace theorem and(<ref>)–(<ref>) yield the following estimate from (<ref>):e^φ_N-Ψ/h_L^∞(V_η) O(h^∞) + e^φ_N-Ψ/h_L^∞( supp (∇η)) O(h^-N_1) ≥w̃_h_H^1(V_η)≃ h^-Ne^f_+/h w_h_H^1(V_η)for someN_1∈ℕ independent of N. It can be checked that e^φ_N-Ψ/h_L^∞(V_η)= e^φ_N-Ψ/h_L^∞( supp (∇η)) = O(h^-N) so that the first term is well controlled, but the second one is of order O(h^-N-N_1). These relations suggesta choice of φ_N satisfying φ_N≤f_+ ≤Ψ on ∇η so thate^φ_N-Ψ/h_L^∞( supp (∇η)) = O(1). This would yield the desired estimate e^f_+/h w_h_H^1(V_η)=O(h^N-N_1). Let us now enter the rigorous proof. The above considerations (see also <cit.>) lead to define, for any N∈,φ_N=min{φ+Nhln1/h ,ψ},where the Lipschitz function ψ:V_η→ is defined by the following relation, for some ε∈ (0,1) that will be specified below:ψ(x',x_d)=ψ(x',0)=min{φ(y',0)+(1-ε) d^Ω_a(x',y'),y'∈∇η_1} .Here, d^Ω_a(x',y') denotes the Agmon distance associated with f|_Ω between x' and y' along the boundary (see Definition <ref>), i.e. the distance induced by the metric |∇(f|_Ω)|^2 ds^2, where ds^2 denotes the restriction of the Euclidean metric to the boundary Ω .Step 1-a: Preliminary estimates on φ_N. Let us first show that there exists ε∈ (0,1) such that for any h∈ (0,h_0(N,ε)) with h_0=h_0(N,ε) small enough,φ_N=φ+Nhln1/h < ψ inV_η∩{x'∈Γ'_St}. The proof of (<ref>) is as follows. From (<ref>) applied to d^Ω_a,∀ (x',y')∈Γ_St'×∇η_1 , f_+(x',0)< f_+(y',0)+ d^Ω_a(x',y').The inequality above is strict since if f_+(x',0)= f_+(y',0)+ d^Ω_a(x',y') for some (x',y') ∈Γ_St'×∇η_1, then there exists a generalized integral curve (in the sense of Definition <ref>) of -∇ (f|_Ω)=-∇ f_+joining x'∈Γ_St' to y∈∇η_1 (this isa consequence of Corollary <ref> applied to the Agmon distance d^Ω_a on Ω rather than the Agmon distance d_a in Ω). But since Γ'_St is strongly stable, any integral curveof -∇ f_+remains in Γ'_St, and thus cannot reach y' which is not in Γ'_St (see (<ref>)). From the strict inequality (<ref>), there exists _0>0 such that for all ∈ [0,_0),∀ (x',y')∈Γ_St'×∇η_1 , f_+(x',0) ≤ f_+(y',0)+ (1-)d^Ω_a(x',y'),and thus, considering the limit h → 0 (see (<ref>)) and the infimum over y'∈∇η_1 of the right-hand side, there exists >0 used to define ψ (see (<ref>)) and such that, for sufficiently small h,∀ x'∈Γ_St' , f_+(x',0)< ψ(x',0).Moreover, since lim_h → 0φ+Nhln1/h -f_+_L^∞(V_η)=0 (thanks to (<ref>)), one obtains for h small enough, ∀ x' ∈Γ_St', φ(x',0) + N h ln1/h< ψ(x',0), and by definition of φ_N, φ_N(x',0)= φ (x',0)+Nhln1/h which leads to (<ref>) for x=(x',0), with x' ∈Γ'_St. The fact that φ_N and ψ donot depend on x_d in the system of coordinates (x',x_d) concludes the proof of (<ref>). Let us now prove that∃ M< 1/1-ε, ∀ x ∈ V_η, |∇ψ(x) |≤M(1-ε)|∇ f_+(x) |.The triangular inequality applied to d^Ω_a leads to the relation (since ψ(x',x_d) does not depend on x_d)∀ x,y ∈ V_η,|ψ(x',x_d)-ψ(y',y_d)| ≤ (1-ε)d^Ω_a(x',y').where we denote (x',x_d) (resp. (y',y_d)) the coordinates of x (resp. y) in the system of coordinates (<ref>). Let us first show that (<ref>) implies thatfor a.e. x' ∈ V_η∩∂Ω, |∇ (ψ|_Ω) (x')|≤ (1-ε)|∇ (f|_Ω) (x')| = (1-ε)|∇ (f_+|_Ω )(x') | .Indeed, let us consider a local parametrization in ^d-1 of a neighborhood in ∂Ω of a point x' ∈∂Ω. In this local chart, let us consider y_α=x'+α∇ (ψ|_Ω)/|∇ (ψ|_Ω)|(x'). One has, in the limit α→ 0,ψ(y'_α,0)-ψ(x',0)=α |∇ (ψ |_Ω) (x')| + o(α)and likewise using the inequality (<ref>) applied to d^Ω_a (see also <cit.>)d^Ω_a(x,'y'_α)-d^Ω_a(x',x')≤α |∇ (f|_Ω) (x')| + o(α).By considering the limit α→ 0, one thus deduces (<ref>) from (<ref>).Now, one can check that, uniformly in x' ∈ V_η∩∂Ω,lim_x_d → 0 |∇ψ(x',x_d)|=|∇ (ψ|_∂Ω) (x')|and lim_x_d → 0 |∇ f_+(x',x_d)|=|∇ ( f_+|_∂Ω )(x')|.Indeed, using the fact that ψ does not depend on x_d, one first has almost everywhere (see (<ref>)) |∇ψ(x',x_d)|=|∇ (ψ|_Σ_x_d)(x')| where ∀ a>0, Σ_a={x ∈ V_η, x_d(x)=a} is endowed with the Riemannian structure induced by the Riemannian structure in Ω. Now, let us consider the smooth diffeomorphism Γ_x_d: Σ_x_d→∂Ω such that for all x=(x',x_d) ∈Σ_x_d, Γ_x_d(x)=(x',0) ∈∂Ω. The result (<ref>) on ψ is then a consequence of the fact that ψ|_∂Ω∘Γ_x_d = ψ|_Σ_x_d and lim_x_d → 0Γ_x_d -Id_W^1,∞(Σ_x_d) = 0 so that the Jacobian associated to the change of metric from Σ_x_d to ∂Ω converges to Id, uniformly on Σ_x_d. The same reasoning show that (<ref>) also holds for f_+ since f_+ does not depend on x_d.By combining (<ref>) and (<ref>), one obtains (<ref>) for some M>1. Moreover, M can be chosen as close to 1 as needed, up to modifying η (and thus V_Γ_St⊂ V_Γ'_St⊂ V_η) such that for all x ∈ V_η, Γ_x_d -Id_W^1,∞(Σ_x_d) is as close to 0 as needed.Let us finally mention the following inequalities, valid for h∈ (0,h_0)with h_0=h_0(N,ε)>0 small enough:φ_N≤f_++Nhln1/h≤Ψ+Nhln1/h inV_ηφ_N=ψ≤φ≤f_+≤Ψ inV_η∩{x' ∈∇η_1}and since Ψ=f_++f_->f_+ on {x_d∈η_d'}, one hasφ_N≤f_++Nhln1/h≤Ψ inV_η∩{ x_d∈η_d' }. Step 1-b: Proof of (<ref>). We are now readyto prove(<ref>). Controlling the left-hand side of (<ref>) using the relations (<ref>)–(<ref>) gives(r_1 +r_1')e^φ_N- Ψ/h_ L^2(V_η) w̃_̃h̃_ L^2(V_η) +(r_2+r_2')e^φ_N- Ψ/h_ L^2(Γ_1)w̃_h_ L^2(Γ_1)≥hdw̃_h^2_L^2(V_η)+hd^*w̃_h^2_L^2(V_η) -h∫_Γ_1⟨w̃_h,w̃_h⟩_ T_σ^*Ω_n f dσ + ⟨ (|∇ f|^2-|∇φ_N|^2+hℒ_∇ f+ hℒ^*_∇ f)w̃_h,w̃_h⟩_L^2(V_η) ,where, since φ_N- Ψ≤ Nhln1/h (by (<ref>)) and r_i=𝒪(h^∞) for i∈{1,2} (by (<ref>) and (<ref>)),r_1e^φ_N- Ψ/h_ L^2(V_η) + r_2e^φ_N- Ψ/h_ L^2(Γ_1) =𝒪(h^∞),and, since φ_N≤Ψ on ∇η (by (<ref>)–(<ref>)) and r_i'⊂∇η for i∈{1,2} (by (<ref>) and (<ref>)),r_1'e^φ_N- Ψ/h_ L^2(V_η) + r_2'e^φ_N- Ψ/h_ L^2(Γ_1)=𝒪(h^-N_0).This leads to the existence ofC_1=C_1(N)>0 such that for h small enough:C_1h^-N_0w̃_̃h̃_ H^1(V_η) ≥hdw̃_h^2_ L^2(V_η)+hd^*w̃_h^2_ L^2(V_η) -h∫_Γ_1⟨w̃_h,w̃_h⟩_ T_σ^*Ω_n f dσ+ ⟨ (|∇ f|^2-|∇φ_N|^2+hℒ_∇ f+ hℒ^*_∇ f)w̃_h,w̃_h⟩_L^2(V_η).Since φ_N≤φ+Nhln1/h, φ≤ Ch on Ω_- and w_h_ H^1(V_η)=𝒪 (h^∞) (see (<ref>))w̃_h_ L^2(Ω_-)≤ e^C h^-Nw_h_ L^2(Ω_-)≤ C_2(C,N).Thus, since ℒ_∇ f+ ℒ^*_∇ f is a 0^th order differential operator, we get the existence ofC_3>0 independent of (C,N) and of C_4=C_4(C,N) such that: C_4(h^-N_0w̃_h_ H^1(V_η)+1)≥hdw̃_h^2_ L^2(V_η)+hd^*w̃_h^2_ L^2(V_η) -h∫_Γ_1⟨w̃_h,w̃_h⟩_ T_σ^*Ω_n f dσ + ⟨ |∇ f_-|^2w̃_h,w̃_h⟩_ L^2(V_η)+ ⟨ (|∇ f_+|^2-|∇φ_N|^2-C_3h) w̃_h,w̃_h⟩_ L^2(Ω_+).Moreover, by definition of φ_N, a.e. in Ω_+, ∇φ_N=∇ψ 1_{φ_N=ψ} + ∇ f_+(1-Ch/f_+) 1_{φ_N<ψ}. Now, * On {φ_N=ψ}, since by (<ref>) {φ_N=ψ}avoids a neighborhood of {(z,x_d),x_d∈ supp η_d}={x∈ V_η,|∇ f_+(x)|=0} (see (<ref>)), we get|∇ f_+|^2-|∇φ_N|^2≥(1-M^2(1-ε)^2)|∇ f_+|^2≥c_ε>0,where (<ref>) have been used;* On {φ_N<ψ}∩Ω_+, we get like in the proof ofProposition <ref> (see (<ref>) and (<ref>)),|∇ f_+|^2-|∇φ_N|^2≥ KCh. Choosing C> max (1,C_3/K), weobtain that for h small enough: C_4(h^-N_0 w̃_h_ H^1(V_η)+1) ≥hdw̃_h^2_ L^2(V_η)+hd^*w̃_h^2_ L^2(V_η)-h∫_Γ_1⟨w̃_h,w̃_h⟩_ T_σ^*Ω_n fdσ+ ⟨ |∇ f_-|^2w̃_h,w̃_h⟩_ L^2(V_η) + (KC-C_3)h(w̃_h^2_ L^2(V_η) -w̃_h^2_ L^2(Ω_-)).We can now control from below the r.h.s. of the above estimate exactly as we didat the end of the first step of Proposition <ref>: defining C_5(C):= KC-C_3/2|∇ f_-|^2_L^∞(V_η) (see (<ref>)), one gets the inequality(KC-C_3)hw̃_h^2_ L^2(V_η) + ⟨ |∇ f_-|^2w̃_h,w̃_h⟩_ L^2(V_η) ≥(1+2C_5h) ⟨ |∇ f_-|^2w̃_h,w̃_h⟩_ L^2(V_η)and from Lemma <ref> applied withu=w̃_h, f=-η̃f_- where η̃∈ C^∞(Ω,[0,1]), η̃=1 on supp η, supp η̃⊂ ( supp η+B(0,α))∩Ω for α>0 such that f_- is smooth on supp η̃and h/1+C_5h instead of h, one gets the following lower bound: -h ∫_Γ_1⟨w̃_h,w̃_h⟩_nf dσ = h ∫_Γ_1⟨w̃_h,w̃_h⟩_nf_-dσ≥-(1+C_5h) |∇ f_-|w̃_h_L^2(V_η)^2-h^2/1+C_5h(dw̃_h^2_L^2(V_η) +d^*w̃_h_L^2(V_η)^2) -C_6h w̃_h_L^2(V_η)^2,where C_6 is some positive constant independent of C (it only depends on f_-). Injectingthe estimates (<ref>) and (<ref>) in (<ref>) then leads to:C_4(h^-N_0 w̃_h_ H^1(V_η)+1)≥C_5h^3/1+C_5h( dw̃_h^2_ L^2(V_η) +d^*w̃_h^2_ L^2(V_η))+ C_5h |∇ f_-|w̃_h^2_ L^2(V_η) - (KC-C_3)hw̃_h^2_ L^2(Ω_-)-C_6h w̃_h^2_ L^2(V_η).Then, since |∇ f_-|≥ c>0 on V_η (see (<ref>)), lim_C→∞ C_5(C)= +∞. Therefore, since C_6 is independent of C, one can choose C such that c^2C_5-C_6>0, which implies, remembering also w̃_h_ L^2(Ω_-)≤ C_2(C,N) (see (<ref>)),the existence of a constant C_7>0 and a constant h_0>0 such that, for every h∈ (0,h_0),w̃_h^2_L^2(V_η) +dw̃_h^2_L^2(V_η)+d^*w̃_h^2_L^2(V_η)≤C_7/h^3(h^-N_0w̃_h_ H^1(V_η)+1).According to Gaffney's inequality (<ref>), this finally leads to the existence of a positive constant C_8 such thatw̃_h_ H^1(V_η) ≤C_8 h^-N_0-3.Moreover, according to (<ref>), we have φ_N=φ+ Nhln1/h in V_η∩{x'∈Γ'_St}and then φ_N-Nhln1/h-f_+≥- C_9 hln1/h (with a constant C_9 independent of N) in V_Γ_St'⊂ V_η∩{x'∈Γ'_St}. Therefore, there exists N_1 independent of N such that for h small enough,e^f_+/hw_h_ H^1(V_Γ_St')≤C_N h^N-N_1,from which (<ref>) and(<ref>) follow since N is arbitrary.Step 2: Comparison in V_Γ_St. We work now with the cut-off function χ defined in (<ref>). Recall that ηχ=χ. Similarly as in the previous step, let us define the sets Ω_-={x∈ V_Γ_St' s.t. Ψ(x)≤ Ch} and Ω_+=V_Γ_St'∖Ω_-,and the function φ_N= min{φ+Nhln1/h,ψ}, where φ and ψ are respectively defined byφ= {[c]Ψ-ChlnΨ/hif Ψ> Ch Ψ-Chln Cif Ψ≤ Ch, .andψ(x)=min{φ(y)+(1-ε) d_a(x,y),y∈∇χ}.The constant C>1 will be chosen at the end of the proof. Following the proof of (<ref>), there exists ε∈ (0,1) such that for any h∈ (0,h_0) with h_0=h_0(N,ε) small enough,φ_N=φ+Nhln1/h <ψ V_Γ_St.Indeed, using the fact that Ψ(x)=d_a(x,z) and a triangular inequality,∀(x,y)∈V_Γ_St×∇χ, Ψ(x)< Ψ(y)+ d_a(x,y).The inequality is strict since if Ψ(x)= Ψ(y)+ d_a(x,y) for some (x,y) ∈V_Γ_St×∇χ, then Φ(x)-Φ(y)=d_a(x,y) and from Corollary <ref>, up to modifying η such that V_η⊂ V_α (see Corollary <ref> for the definition of V_α) there exists a generalized integral curve (in the sense of Definition <ref>) of{ -∇ΦonV_α∩Ω-∇_TΦon Ω.joining x∈V_Γ_St toy∉V_Γ_St. This contradicts the fact that V_Γ_St is stable for (<ref>). The end of the proof of (<ref>) then follows exactly the same lines of the proof of (<ref>).Moreover, owing to the properties of d_a, one has analogously to (<ref>) the following estimate valid a.e. in V_Γ_St':|∇ψ|≤ (1-ε)|∇ f |= (1-ε)|∇Ψ|. Let us finally mention the following inequalities, valid for h∈ (0,h_0)with h_0=h_0(N,ε)>0 small enough: φ_N≤Ψ+ Nhln1/h inV_Γ_St' andφ_N=ψ≤φ≤Ψ on ∇χ.We are now in position to prove (<ref>). Let us definew̃_̃h̃= e^φ_N/hw_h= e^φ_N/hχ(u^(1)_h-c_z(h)u^(1)_z,wkb). Using the relations (<ref>)–(<ref>) and the integration by parts formulae (<ref>) and (<ref>), there exists C_1>0 (only depending on f) such that(r_1+r_1')e^φ_N- Ψ/h_ L^2(V_Γ_St')w̃_̃h̃_ L^2(V_Γ_St') +(r_2+r_2')e^φ_N- Ψ/h_ L^2(Γ_1)w̃_h_ L^2(Γ_1)+C_1 h∫_Γ_1⟨w̃_h,w̃_h⟩_ T_σ^*Ω dσ≥hdw̃_h^2_ L^2(V_Γ_St')+hd^*w̃_h^2_ L^2(V_Γ_St') + ⟨ (|∇ f|^2-|∇φ_N|^2-C_1h)w̃_h,w̃_h⟩_L^2 (Ω_+) -C_1hw̃_h^2_ L^2(Ω_-),where we have used the fact that almost everywhere on Ω_-, |∇φ_N| is either equal to |∇φ|=|∇Ψ|=|∇ f| or to |∇ψ| ≤ (1-ε) |∇ f|. Moreover, since φ_N- Ψ≤ Nhln1/h (see (<ref>)), one has from (<ref>) and (<ref>)r_1e^φ_N- Ψ/h_ L^2(V_Γ_St') + r_2e^φ_N- Ψ/h_ L^2(Γ_1) =𝒪(h^∞)and, since φ_N≤Ψ on ∇χ (see (<ref>)), one gets from (<ref>) and (<ref>)r_1'e^φ_N- Ψ/h_ L^2(V_Γ_St') + r_2'e^φ_N- Ψ/h_ L^2(Γ_1)=𝒪(h^-N_0).Additionally, since Ψ=f-f(z) on Γ_1 and φ_N- Ψ≤ Nhln1/h (see (<ref>)), we deduce from the relation (<ref>) obtained in the first step the following estimate:w̃_h_ L^2(Γ_1)=e^φ_N/hχ(u^(1)_h-c_z(h)u^(1)_z,wkb)_ L^2(Γ_St') = 𝒪(h^∞) .Consequently, using in addition the relation w̃_h_ L^2(Ω_-)≤ e^C h^-Nw_h_ L^2(Ω_-)≤ C_2(C,N),(since φ_N≤φ+Nh ln1/h, φ≤ Ch on Ω_- and w_h_L^2(V_Γ_St')=O(h^∞) from (<ref>)), we deduce from (<ref>) the existence of some positive constant C_3=C_3(C,C_1,N) such thatC_3(h^-N_0w̃_h_ L^2 (V_Γ_St')+ 1 )≥hdw̃_h^2_ L^2(V_Γ_St')+hd^*w̃_h^2_ L^2(V_Γ_St') +⟨ (|∇ f|^2-|∇φ_N|^2-C_1h)w̃_h, w̃_h⟩_L^2(Ω_+) .Lastly, one has a.e. in Ω_+, ∇φ_N=∇ψ 1_{φ_N=ψ} + ∇Ψ(1-Ch/Ψ) 1_{φ_N<ψ}, and thus * on {φ_N=ψ}, from (<ref>),|∇ f|^2-|∇φ_N|^2≥ (2ε-ε^2)|∇ f|^2≥ c_ε>0, * on {φ_N<ψ}∩Ω_+, there exists C_4>0 independent ofC such that,|∇ f|^2-|∇φ_N|^2≥ Ch|∇ f|^2/Ψ≥ C_4C h. Taking C>C_1/C_4 and adding(CC_4-C_1)hw̃_h^2_ L^2(Ω_-) to (<ref>) then leads toC_5( h^-N_0w̃_h_ L^2 (V_Γ_St')+ 1 )≥hdw̃_h^2_ L^2(V_Γ_St') +hd^*w̃_h^2_ L^2(V_Γ_St') + (CC_4-C_1)hw̃_h^2_ L^2(V_Γ_St'),for a constant C_5 depending on C and N. Using Gaffney's inequality (<ref>), we consequently get the existence of C_6>0such thatw̃_h_ H^1(V_Γ_St')≤ C_6 h^-N_0-3/2.Now, sinceφ_N-Nhln1/h- Ψ≥ -C_7 hln1/h in V_Γ_St (with a constant C_7 independent of N, from the definition of φ and the fact that φ_N-Nhln1/h=φ in V_Γ_St, see (<ref>)), we also get the existence of N_2 independent of N such that for h small enough,e^Ψ/hw_h_ H^1(V_Γ_St)≤C_N h^N-N_2,for some constant C_N >0, which concludes the proof of Proposition <ref>.§.§ Proof of Theorem <ref>The aim of this section is to conclude the proof of Theorem <ref> by checking that the function ũ and the family of 1-forms (ϕ̃_i )_i=1,…,n introduced in Section <ref>satisfy the estimates appearing in Proposition <ref> rewritten in the flat space (see Section <ref>).In all this section, we assume in addition to thehypotheses [H1], [H2] and [H3], that (<ref>) and (<ref>) hold.From Sections <ref> and <ref>, it only remains to prove (<ref>), (<ref>), (<ref>) and (<ref>). Let us start with a lemma about the normalisation term appearing in (<ref>). Let us assume that the hypotheses [H1], [H2] and [H3] hold. Let us define Θ_i:=√(∫_Ω|χ_i(x) u^(1)_h,i(x) |^2 dx). There exist c>0 and h_0>0 such that for all h∈ (0,h_0),Θ_i^2=1+O(e^- c/h).On the one hand, one has the upper boundχ_i u^(1)_h,i_L^2(Ω)=χ_i u^(1)_h,i_L^2(Ω̇_i)≤ u^(1)_h,i_L^2(Ω̇_i) =1. On the other hand, the triangular inequality yields the lower boundχ_i u^(1)_h,i_L^2(Ω̇_i) ≥u^(1)_h,i_L^2(Ω̇_i) - (1-χ_i) u^(1)_h,i_L^2(Ω̇_i) = 1- (1-χ_i) u^(1)_h,i_L^2(Ω̇_i) . Thanks to Proposition <ref>, there exist N∈ℕ and c>0 independent of h such that(1-χ_i) u^(1)_h,i_L^2(Ω̇_i)^2 = ∫_Ω̇_i|(1-χ_i(x)) u^(1)_h,i(x) e^1/h d_a(x,z_i) e^ -1/h d_a(x,z_i)|^2dx ≤∫_Ω̇_i∖𝒱_i| u^(1)_h,i(x) e^1/h d_a(x,z_i) e^ -1/h d_a(x,z_i)|^2dx ≤ C h^-N e^-inf_Ω̇_i∖𝒱_i2/h d_a(.,z_i)≤ C e^- c/h,where, we recall 𝒱_i={x ∈Ω,χ_i=1}. This concludes the proof of Lemma <ref>. We are now in position to check the estimates stated in Section <ref>.Step 1. Study of the term ( 1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω) ) )ϕ̃_i_H^1(Ω). We recall that from (<ref>), ϕ̃_i belongs to Λ^1H^1_T(Ω) and then we get from Lemma <ref> that there exist c>0 and h_0>0 such that for all h∈ (0,h_0),(1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i_L^2(Ω)^2 ≤ c h^-3/2( d_f,hϕ̃_i_L^2(Ω)^2+ d_f,h^* ϕ̃_i _L^2(Ω)^2 )= c h^-3/2( d_f,hϕ̃_i_L^2 ( Ω̇_i )^2+d_f,h^* ϕ̃_i_L^2 ( Ω̇_i )^2). Moreover, from Proposition <ref> (items (ii) et (iii))d_f,hϕ̃_i =Θ_i^-1( χ_i d_f,hu^(1)_h,i + hdχ_i ∧u^(1)_h,i)=Θ_i^-1 h dχ_i ∧u^(1)_h,i,andd_f,h^* ϕ̃_i =Θ_i^-1( χ_i d_f,h^* u^(1)_h,i - hu^(1)_h,i·∇χ_i)=-Θ_i^-1hu^(1)_h,i·∇χ_i.As a consequence, using Lemma <ref> and Proposition <ref>, one gets for someN∈ℕ and for some c>0 which may change from one occurrence to another,(1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i_L^2(Ω)^2 ≤ c h^-3/2(hdχ_i ∧u^(1)_h,i_L^2 ( Ω̇_i )^2+ h u^(1)_h,i·∇χ_i_L^2 ( Ω̇_i )^2) ≤ ch^1/2∫_ supp∇χ_i| u_h,i^(1)(x)e^1/h d_a(x,z_i) e^ -1/h d_a(x,z_i)| ^2 dx≤ c h^1/2-N e^-inf_supp∇χ_i2/h d_a(· ,z_i).The function χ_i can be chosen such that the set {x ∈Ω̇_i,∇χ_i(x)≠ 0} is asclose as needed to Γ_2,i and to Γ_0 (see Figure <ref>).Therefore by continuity of the Agmon distance,using (<ref>)–(<ref>),for any δ>0, one can choose χ_i satisfying the three conditions stated inDefinition <ref> and such that inf_z∈ supp∇χ_i d_a(z,z_i)≥min(d_a(x_0,z_i),inf_z∈ B_z_i^c d_a(z,z_i) )-δ.From (<ref>), there exists r>0 such that inf_z∈ B_z_i^c d_a(z,z_i) ≥max[f(z_n)-f(z_i),f(z_i)-f(z_1)] +r.In addition, using (<ref>), there exists r'>0 such thatd_a(z_i,x_0) ≥ f(z_i)-f(x_0)≥ f(z_1)-f(x_0)≥ f(z_n) - f(z_1) + r'≥max[f(z_n)-f(z_i),f(z_i)-f(z_1)]+r'.Therefore, choosing χ_i such that δ < min(r,r'),there exists '>0 such thatinf_z∈ supp∇χ_i d_a(z,z_i) ≥max[f(z_n)-f(z_i),f(z_i)-f(z_1)] +'.Using the estimate (<ref>) in (<ref>), there exist _1>0, c>0, N∈ℕ and h_0>0, such that for every h∈ (0,h_0)(1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i_L^2(Ω)^2≤ ch^-N e^- 2/h(max[f(z_n)-f(z_i),f(z_i)-f(z_1)]+ ')≤ e^ -2/h(max[f(z_n)-f(z_i),f(z_i)-f(z_1)]+ _1).This last inequality leads to the desired estimate in the L^2(Ω)-norm. In order to get the same upper bound in the H^1(Ω)-norm, notice now that one has (1-π_[0,h^3/2 )( Δ^D,(2)_f,h(Ω) )) d_f,hϕ̃_i=d_f,h(1-π_[0,h^3/2 )( Δ^(D,1)_f,h(Ω))) ϕ̃_i= hd (1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i + df∧(1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i.Therefore it holds h d (1-π_[0,h^3/2 )(Δ^D,(1)_f,h(Ω))) ϕ̃_i =(1-π_[0,h^3/2 )(Δ^D,(2)_f,h(Ω) )) d_f,h ϕ̃_i - df∧(1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i.Let us introduce K_i:=max[f(z_n)-f(z_i),f(z_i)-f(z_1)]. From (<ref>), there exist C>0 and h_0>0, such that for all h∈ (0,h_0)‖ df∧(1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i ‖_L^2(Ω)^2≤ C e^- 2/h (K_i+ _1). Moreover, using (<ref>) and (<ref>) there exist > 0, C>0 and h_0>0, such that for all h∈ (0,h_0),‖(1-π_[0,h^3/2 ) ( Δ^D,(2)_f,h(Ω) ) ) d_f,hϕ̃_i ‖_L^2(Ω)^2≤‖ d_f,hϕ̃_i ‖_L^2(Ω)^2 = Θ_i^-2 hdχ_i ∧u^(1)_h,i_L^2 ( Ω̇_i )^2≤ C e^- 2/h (K_i+ ) .Thus one gets: there exist > 0, C>0 and h_0>0, such that for all h∈ (0,h_0),h^2 ‖ d (1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i‖_L^2(Ω)^2≤ C e^- 2/h (K_i+ ) .Similarly, there exist >0, C>0 and h_0>0, such that for all h∈ (0,h_0)h^2 ‖ d^* (1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω))) ϕ̃_i‖_L^2(Ω)^2≤ C e^- 2/h (K_i+ ).As a consequence, using (<ref>), there exist >0, C>0 and h_0>0, such that for all h∈ (0,h_0)‖(1-π_[0,h^3/2 )( Δ^D,(1)_f,h(Ω)))ϕ̃_i‖_H^1(Ω)^2 ≤ C e^- 2/h (K_i+ ).This concludes the proof of (<ref>).Step 2. Study of the terms ∫_Ωϕ̃_i ·ϕ̃_j for (i,j)∈{1,…,n}^2. Let (i,j)∈{1,…,n}^2 be such that i<j. One then has f(z_i)≤ f(z_j).From Proposition <ref>, it holdsd_a(z_i,z_j)>f(z_j)-f(z_i). Now, according to Proposition <ref> and Lemma <ref> and to the triangular inequality for d_a,there exist >0, N∈ℕ and h_0>0 such that for all h∈ (0,h_0),|∫_Ωϕ̃_i(x) ·ϕ̃_j(x) dx | ≤ e^ -d_a(z_j,z_i)/h∫_ suppχ_i∩ suppχ_j e^d_a(x,z_i)/h|ϕ̃_i(x)| e^d_a(x,z_j)/h|ϕ̃_j(x) |dx ≤Θ_i^-1Θ_j^-1‖ e^d_a(.,z_i)/hχ_i u^(1)_h,i‖_L^2(Ω̇_i)‖ e^d_a(.,z_j)/hχ_j u^(1)_h,j‖_L^2(Ω̇_j) e^ -d_a(z_i,z_j)/h≤ Ch^-N e^ - 1/h (f(z_j)-f(z_i)+) .This concludes the proof of (<ref>).Step 3. Study of the terms ∫_Σ_iϕ̃_j · n e^-f/h for (i,j)∈{1,…,n}^2. By construction, for all (i,j)∈{1,…,n}^2 such that i≠ j, one has∫_Σ_iϕ̃_j · n e^- 1/hf=0.Now, let us compute the term ∫_Σ_iϕ̃_i · ne^- 1/hf. Let u^(1)_z_i,wkb be the WKB expansion defined by (<ref>). Following the beginning of Section <ref>, let us consider * a neighborhood V_Γ_St,i of Σ_i in Ω, which is stable under the dynamics (<ref>) and such that, for some > 0, V_Γ_St,i + B(0,) ⊂V_Γ_1,i∩ (Γ_1,i∪Ω̇_i)* and a cut-off function χ_wkb,i∈ C^∞_c(Ω̇_i ∪Γ_1,i) with χ_wkb,i≡ 1 on a neighborhood of V_Γ_St,i and such that χ_wkb,i⊂ V_Γ_1,i∩ (Ω̇_i ∪Γ_1,i).Using Proposition <ref>, there exists c_z_i(h)∈ℝ_+^* such thatχ_wkb,i(u^(1)_h,i-c_z_i(h)u^(1)_z_i,wkb)_ H^1(Ω̇_i)=O (h^∞).Let us now introduceϕ̃_z_i,wkb:=c_z_i(h) χ_wkb,i u_z_i,wkb^(1) so that∫_Σ_iϕ̃_i · n e^- 1/hf= ∫_Σ_iϕ̃_z_i,wkb· n e^- 1/hf+∫_Σ_i(ϕ̃_i-ϕ̃_z_i,wkb) · ne^- 1/hf .Let us first deal with the term ∫_Σ_iϕ̃_z_i,wkb· n e^- 1/hf. Using Laplace's method (the computation is similar to (<ref>)), one gets when h→ 0 (since Φ=f and ∂_nΦ=-∂_nf on ∂Ω, see (<ref>) and (<ref>))∫_Σ_iχ_wkb,iu_z_i,wkb^(1)· ne^- 1/hf =∫_Σ_ie^-Φ-f(z_i)/ha_0 ∂_n(f-Φ) e^- 1/hf(1+O(h))=2e^f(z_i)/h∫_Σ_ie^-2f/ha_0 ∂_nf (1+O(h))= 2 ∂_nf(z_i) π ^d-1/2/√(detHessf|_∂Ω (z_i))h ^d-1/2 e^- 1/hf(z_i)(1+O(h)). Then one obtains when h→ 0∫_Σ_iϕ̃_z_i,wkb· ne^- 1/hf=c_z_i(h) 2 ∂_nf(z_i) π ^d-1/2/√(detHessf|_∂Ω (z_i))h ^d-1/2 e^- 1/hf(z_i) (1+O(h)).We recall from Proposition <ref> that in the limit h→ 0:c_z_i(h)= C_z_i,wkb^-1h^-d+1/4(1+O(h^∞)),where the constant C_z_i,wkb is defined by (<ref>). Therefore, in the limit h→ 0∫_Σ_iϕ̃_z_i,wkb· ne^- 1/hf=π ^d-1/4√(2 ∂_n f(z_i) )/ (detHessf|_∂Ω (z_i) )^1/4h^d-3/4 e^- 1/hf(z_i) (1+O(h)). Let us now deal with the term ∫_Σ_i(ϕ̃_i-ϕ̃_z_i,wkb) · ne^- 1/hf. One obtains usingLemmata <ref> and <ref>, that there exist C>0, h_0>0 and η>0 such that for all h∈ (0,h_0) |∫_Σ_i(ϕ̃_i-ϕ̃_z_i,wkb) · ne^- 1/hf| =|∫_Σ_i(u^(1)_h,i/Θ_i-c_z_i(h)u_z_i,wkb^(1)) · ne^- 1/hf|= 1/Θ_i|∫_Σ_i(u^(1)_h,i-Θ_ic_z_i(h) u_z_i,wkb^(1)) · ne^- 1/hf|≤e^- 1/hf(z_i)/Θ_i∫_Σ_i| (u^(1)_h,i- c_z_i(h) u_z_i,wkb^(1)) · n | +e^- 1/hf(z_i) |Θ_i-1|/Θ_i |c_z_i(h)| ∫_Σ_i| u_z_i,wkb^(1)· n|≤ C e^- 1/hf(z_i)‖χ_wkb,i(u^(1)_h,i- c_z_i(h)u_z_i,wkb^(1))‖_H^1(Ω̇_i) +C e^- 1/hf(z_i) e^-η/h h^-d+1/4‖χ_wkb,i u_z_i,wkb^(1)‖_H^1(Ω̇_i).Therefore, one obtains using Proposition <ref> and (<ref>)e^1/hf(z_i)|∫_Σ_i(ϕ̃_i-ϕ̃_z_i,wkb) · ne^- 1/hf|= O(h^∞) + C e^-η/h h^-d+5/4 = O(h^∞).In conclusion, we have when h→ 0∫_Σ_iϕ̃_i · n e^- 1/hf= ∫_Σ_iϕ̃_z_i,wkb· ne^- 1/hf(1+O(h^∞)),which gives the expected estimate ∫_Σ_i ϕ̃_j · n e^- 1/hf = B_i h^me^- 1/hf(z_i)( 1+ O(h )) ifi=j, 0 ifi≠ j,where m=d-3/4 and B_i= π ^d-1/4√(2∂_n f(z_i) )/ (detHessf|_∂Ω (z_i) )^1/4 .This concludes the proof of (<ref>).Step 4. Study of the term∫_Ω∇ũ·ϕ̃_ie^- 1/hf. First one has the equality by Definition <ref>, ∫_Ω∇ũ·ϕ̃_ie^- 1/hf = ∫_Ω∇χ·ϕ̃_ie^- 1/hf/√(∫_Ωχ^2 e^-2/hf), where ∇χ·ϕ̃_i= i_∇χϕ̃_i=ϕ̃_i(∇χ).The denominator of the right-hand side iseasily computed thanks to Laplace's method:√(∫_Ωχ^2 e^- 2/hf)= (πh)^d/4/ (detHessf (x_0) )^1/4e^- f(x_0)/h (1+O(h) ).Using an integration by parts and the fact that d^*(u^(1)_h,ie^-f/h )=0 in L^2(Ω̇_i) (see Proposition <ref> items (ii) and (iii)) which is valid for all h small enough, one obtains ∫_Ω∇χ·ϕ̃_ie^-f/h = - ∫_Ω∇(1- χ) ·χ_iu^(1)_h,i/Θ_i e^-f/h=∫_Ω(1-χ)∇χ_i · u^(1)_h,i/Θ_i e^-f/h- ∫_∂Ω(1-χ)ϕ̃_i · ne^-f/h.Using the fact that χ = 0 on ∂Ω, one then obtains:∫_∂Ω(1-χ)ϕ̃_i · ne^-f/h = ∫_∂Ω∩ suppχ_iϕ̃_i · ne^- f/h= ∫_Σ_i ϕ̃_i · n e^-f/h + ∫_(∂Ω∩ suppχ_i) ∖Σ_iϕ̃_i · ne^-f/h . Using (<ref>), in the limit h→ 0:∫_Ω∇χ·ϕ̃_ie^-f/h =- B_i h^me^-1/h f(z_i)(1+ O(h)) - ∫_(∂Ω∩ suppχ_i) ∖Σ_iϕ̃_i · ne^-f/h+∫_Ω(1-χ)∇χ_i · u^(1)_h,i/Θ_i e^-f/h.Let us now prove that the two last terms in (<ref>) are negligible compared to the first one. On the compact set (∂Ω∩ suppχ_i)∖Σ_i onehas f(z)>f(z_i) since z_i∈Σ_i is the only global minimum of f on B_z_i and χ_i ∩∂Ω⊂Γ_1,i⊂ B_z_i. Then, using Proposition <ref> and (<ref>), there exist >0, h_0>0, C>0 and N∈ℕ such that for all h∈ (0,h_0) | ∫_(∂Ω∩ suppχ_i) ∖Σ_iϕ̃_i · ne^-f/h| ≤ e^-f(z_i)+/h| ∫_(∂Ω∩ suppχ_i) ∖Σ_iϕ̃_i· n |≤ C e^-f(z_i)+/h| ∫_(∂Ω∩ suppχ_i) ∖Σ_iϕ̃_i· n e^d_a(.,z_i)/h|≤ C e^-f(z_i)+/h/Θ_i ‖χ_iu^(1)_h,ie^d_a(.,z_i)/h‖_H^1(Ω̇_i)≤ C e^ -f(z_i)+/h h^-N≤ C e^- f(z_i)+/2/h. Let us now deal with the last term of (<ref>). The support of (1-χ)∇χ_i is included in the support of ∇χ_i anddoes not contain x_0 since χ≡ 1 around x_0. The function χ_i can be chosen such that the set {x ∈Ω̇_i, |∇χ_i| ≠ 0and χ≠ 1} is as close as one wants from Γ_2,i (see Figure <ref>). Therefore, by continuity of the Agmon distance, using (<ref>), for any δ > 0, one can choose χ_i satisfying the three conditions stated in Definition <ref> and such thatinf_ supp(1-χ)∇χ_i ( d_a( · ,z_i)+f)≥inf_ B_z_i^c( d_a(· ,z_i)+f)-δ.Thus, using the Cauchy-Schwarz inequality and Proposition <ref>, there exists N∈ℕ such that∫_ supp(1-χ)∇χ_i|(1-χ) ∇χ_i ·u^(1)_h,i e^-f/h| ≤C‖ u^(1)_h,i e^d_a(·,z_i)/h‖_L^2(Ω̇_i)e^-1/hinf_ supp(1-χ) ∇χ_i (d_a(·,z_i)+f)≤C h^-Ne^- 1/hinf_B_z_i^c (d_a(·,z_i)+f-δ). Besides, from assumption (<ref>)inf_z∈ B_z_i^c[ d_a(z,z_i)+f(z)]> f(z_i).Indeed,the inequality (<ref>) implies that there exists r>0 such that for all z∈ B_z_i^c, d_a(z,z_i)≥ f(z_i)-f(z_1)+r and therefore for all z∈ B_z_i^c one obtainsd_a(z,z_i)+f(z) ≥ f(z_i)+ (f(z)-f(z_1)) +r ≥ f(z_i)+r.Therefore, taking χ_i such that δ < r/2, one has, when h→ 0∫_ supp(1-χ)∇χ_i|(1-χ) ∇χ_i ·u^(1)_h,i e^-f/h| = O(e^-f(z_i)+c/h)for some constant c>0.In conclusion, for all i∈{1,…,n}, one has in the limit h → 0,∫_Ω∇ũ·ϕ̃_ie^- 1/hf dx =C_i h^pe^-1/h(f(z_i)- f(x_0))(1+ O(h)) ,with {C_i=-B_i(detHessf (x_0) )^1/4/π^d/4=- π ^-1/4√(2∂_n f(z_i) )(detHessf (x_0) )^1/4/ (detHessf|_∂Ω (z_i) )^1/4, p=m-d/4=-3/4, .where B_i and m have both been defined in (<ref>). This concludes the proof of (<ref>), and thus the proof of Theorem <ref>.§ CONSEQUENCES AND GENERALIZATIONS OF THEOREM <REF> §.§ Proofsof Proposition <ref>, Proposition <ref>, Corollary <ref> and Corollary <ref>§.§.§ Proof of Proposition <ref> Assume that [H1], [H2] and [H3] hold. From Lemma <ref> and since the function ũ is non negative in Ω, there exists h_0>0 such that for all h∈ (0,h_0)u_h=π_h^(0)ũ/‖π_h^(0)ũ‖_L^2_w,where u_h is the eigenfunction associated with the smallest eigenvalue of -L^(D),(0)_f,h(Ω) (see Proposition <ref>)and ũ is introduced in Definition <ref>. Then, there exists h_0>0 such that for all h∈ (0,h_0),∫_Ωu_he^-2/h f =∫_Ωπ_h^(0)ũ/‖π_h^(0)ũ‖_L^2_we^-2/h f=1/‖π_h^(0)ũ‖_L^2_w∫_Ω[ũ + (π_h^(0) -1) ũ] e^-2/h f . From the definition of χ (see Definition <ref>) and using Laplace's method, one obtains (in the limit h→0) ∫_Ωχ^2 e^-2/h f=h^d/2π ^d/2/√( detHessf (x_0) )e^-2/h f(x_0) (1+O(h) ) and likewise ∫_Ωχ e^- 2/hf=h^d/2π ^d/2/√( detHessf (x_0) )e^-2/h f(x_0) (1+O(h) ).In addition, using Lemma <ref>, one has ‖π_h^(0)ũ‖_L^2_w=1+O(e^-c/h). Therefore, it holds when h→ 0,1/‖π_h^(0)ũ‖_L^2_w∫_Ωũ e^- 2/h f=∫_Ωχ e^- 2/h f /√(∫_Ωχ^2 e^- 2f/h) (1+O(e^-c/h))=h^d/4π ^d/4/ ( detHessf (x_0) )^1/4e^-1/h f(x_0) (1+O(h) ).Moreover, from Lemma <ref>, there exist c>0, h_0>0 and C>0 such that for h∈ (0,h_0)1/‖π_h^(0)ũ‖_L^2_w|∫_Ω(π^(0)_h -1) ũ e^-2/h f| ≤C ‖ (1-π_h^(0) )ũ‖_L^2_w√(∫_Ω e^-2/h f)≤C e^-c/h e^-1/h f(x_0)Thus, one has when h→ 0,∫_Ω u_he^- 2/h f = π ^d/4/( detHessf (x_0) )^1/4 h^d/4 e^-1/hf(x_0) (1+O(h) ).This proves Proposition <ref>.§.§.§ Proof of Proposition <ref>The aim of this section is to prove (<ref>).To this end, we first state in Proposition <ref> some estimates thatthe quasi-modes constructedin Section <ref> satisfy under hypotheses [H1], [H2] and [H3]. Let us emphasize that these estimates are weaker than those obtained in Section <ref> where in addition to [H1]-[H2]-[H3], the hypotheses (<ref>) and (<ref>) were also assumed. Then, we prove that the estimates of Proposition <ref> imply (<ref>). Let us assume that the hypotheses [H1], [H2] and [H3] hold. Then there exist n+1 quasi-modes ( ( ψ̃_i)_i=1,…,n, ũ) which satisfy the following estimates: * ∀ i∈{1,…,n}, ψ̃_i∈Λ^1 H^1_w,T(Ω) and ũ∈Λ^0 H^1_w,T(Ω). The function ũ is non negative in Ω. Moreover∀ i∈{1,…,n},ψ̃_i _L^2_w= ũ _L^2_w = 1.* (a) There exists_1>0,for all i∈{1,…,n}, in the limit h→ 0:(1-π_h^(1)) ψ̃_i _H^1_w^2=O(e^-_1/h).(b) For any δ>0, ∇ũ_L^2_w^2 = O (e^-2/h(f(z_1)-f(x_0) - δ)). * There exists _0>0 such that ∀ (i,j) ∈{1,…,n}^2, i<j, in the limit h→ 0: ψ̃_i, ψ̃_j_L^2_w=O ( e^-_0/h ).* There exist _0>0, such that for all i∈{1,…,n},in the limit h→ 0:∇ũ,ψ̃_i_L^2_w=C_ih^pe^-1/h(f(z_i)- f(x_0)) (1+ O(h ) )+O( e^-1/h(f(z_1 )- f(x_0)+_0)),where the constants p and(C_i)_i=1,…,n are given by (<ref>).Thanks to the hypotheses [H1], [H2] and [H3], one can introduce then+1 quasi-modes ( ( ϕ̃_i)_i=1,…,n, ũ) built in Section <ref>. Recall that ψ̃_i=e^1/h fϕ̃_i for i∈{1,…,n}.Then, one easily obtains that( ( ψ̃_i)_i=1,…,n, ũ) satisfy the estimates stated in Proposition <ref>, following exactly the computations made on ( ( ϕ̃_i)_i=1,…,n, ũ) in Section <ref>: 2(a) follows from (<ref>), 2(b) is a consequence of Lemma <ref>, 3 follows from (<ref>) and 4 is a consequence of (<ref>)-(<ref>)-(<ref>) (in (<ref>), one uses that for δ>0 small enough, there exists c>0 such thatinf_B_z_i^c (d_a(.,z_i)+f-δ)≥ f(z_1)+c) . Let us now prove that the estimates stated in Proposition <ref> imply (<ref>), which will conclude the proof of Proposition <ref>.From (<ref>) together with the assumption ‖u_h ‖_L^2_w=1, it holds λ_h =-⟨ L^D,(0)_f,h(Ω)u_h,u_h ⟩_L^2_w=h/2‖∇u_h ‖^2_L^2_wwhere u_h is the eigenfunction associated with the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>). Recall that ∇ u_h ∈π_h^(1) (see (<ref>)). In addition, let us recall that from items 1, 2(a) and 3 in Proposition <ref> and using the proof of Lemma <ref>, there exists h_0 such that for all h ∈ (0,h_0),span( π_h^(1)ψ̃_i,i=1, …,n)=π_h^(1).Let us denote by (ψ_i)_i=1, …,n the orthornormal basis of π_h^(1) resulting from the Gram-Schmidt orthonormalisation procedure applied to the set (π_h^(1)ψ̃_i)_i=1, …,n (see Lemma <ref>) so that‖∇u_h ‖^2_L^2_w = ∑_j=1^n| ⟨∇ u_h , ψ_j_L^2_w |^2.We now want to estimate the terms ⟨∇ u_h , ψ_j_L^2_w.Using2(b) in Proposition <ref>and using the proof of Lemma <ref>, one has that for h small enough π_h^(0)ũ≠ 0 and therefore, since moreover ũ is non negative in Ω,u_h=π_h^(0)ũ/π_h^(0)ũ_L^2_w. Thus one has (see (<ref>)), for j ∈{1, …, n}, ∇ u_h , ψ_j _L^2_w =Z^-1_j/π_h^(0)ũ_L^2_w[ ∇ũ,ψ̃_j_L^2_w +∇ũ, ( π_h^(1)-1 ) ψ̃_j_L^2_w] +Z^-1_j/π_h^(0)ũ_L^2_w[∑_i=1^j-1κ_ji (∇ũ,ψ̃_i_L^2_w + ∇ũ, ( π_h^(1)-1 ) ψ̃_i_L^2_w) ] where (κ_ji)_1≤ i<j ≤ n and (Z_j)_1≤ j ≤ n are defined in Lemma <ref>. Now, using the items 1, 2(a) and 3 of Proposition <ref> and the proof of Lemma <ref>, oneobtains that there exist _0>0 and h_0>0 such that∀ h∈ (0,h_0), ∀ (i,j) ∈{1,…,n}^2, it holdsκ_ji=O (e^-ε_0/h )andZ_i=1+O ( e^-_0/h),Injecting (<ref>) and the estimates 2 and 4 of Proposition <ref> into (<ref>) leads to the existence of '>0 and h_0>0 such that∀ h∈ (0,h_0),∇ u_h,ψ_j_L^2_w=C_ih^pe^-1/h(f(z_j)- f(x_0)) (1+ O(h ) )+O(e^-1/h(f(z_1 )- f(x_0)+')),wherethe constants p and(C_i)_i=1,…,n are given by (<ref>). Using (<ref>) in (<ref>) and (<ref>), there exists '>0 such that in the limit h→ 0λ_h =h/2∑_j=1^nC_i^2h^2pe^-2/h(f(z_i)- f(x_0)) (1+ O(h ) ) + O( e^-2/h(f(z_1 )- f(x_0)+')).Therefore, the estimate (<ref>)holds and Proposition <ref> is proved.§.§.§ Proof of Corollary <ref> According to (<ref>) onehas for i∈{1,…,n}:_ν_h [X_τ_Ω∈Σ_i]=- h/2λ_h∫_Σ_i (∂_n u_h)(z) e^-2/hf(z)σ(dz)/∫_Ω u_h(y) e^-2/hf(y)dy.Let us assume that [H1], [H2] and [H3] together with the inequalities (<ref>) and (<ref>)hold. From Propositions <ref> and <ref>, one obtains when h→ 0λ_h 2/h∫_Ω u_he^- 2/hf dx =2 π ^d-2/4 (detHessf (x_0) )^ 1/4h^d-6/4×∑_k=1^n_0∂_nf(z_k)/√( detHessf|_∂Ω (z_k) )e^-1/h(2f(z_1)- f(x_0)) (1+O(h) ).Then, using in addition Theorem <ref> to estimate ∫_Σ_i (∂_n u_h) e^-2/hfdσ (i∈{1,…,n}), one proves Corollary <ref>.§.§.§ Proof of Corollary <ref>Before starting the proof of Corollary <ref>, let us notice that under the assumptions stated inCorollary <ref>, for all i∈{1,…,n} and for any test function F∈ C^∞(∂Ω) satisfying suppF ⊂ B_z_i and z_i∈ int ( suppF),when h→ 0,_ν_h [ F(X_τ_Ω )]= ∂_nf(z_i)/√( detHessf_|∂Ω (z_i) ) ( ∑_k=1^n_0∂_nf(z_k) /√( detHessf_|∂Ω (z_k) ))^-1e^-2/h (f(z_i)-f(z_1)) ( F(z_i) + O(h) ).The strategy for the proof of Corollary <ref> is to first extend (<ref>) to a deterministic initial condition, and then to deduce the result of Corollary <ref>. To this end, let i_0∈{2,…, n} be as in (<ref>), j∈{1,…,i_0} and α∈ℝ be such thatf(x_0)<α<2f(z_1)-f(z_j).Notice that we can assume without loss of generality (up to increasing α if α is smaller than f(x_0)+f(z_j)-f(z_1), see (<ref>)) thatf(x_0)+f(z_j)-f(z_1)<α< 2 f(z_1)-f(z_j).For such an α, let us define K_α:=f^-1( (-∞, α]) ∩Ω.For i∈{1,…,j}, we would like to show that (<ref>) holdswhenX_0= x∈ K_α, for any test function F∈ C^∞(∂Ω) satisfying suppF ⊂ B_z_i and z_i∈ int ( suppF). Let usintroduce the principal eigenfunction ũ_h of -L^D,(0)_f,h(Ω):{-L^ (0)_f,hũ_h=λ_h ũ_honΩ, ũ_h = 0 on∂Ω,.withũ_h>0 on Ω and normalized such that∫_Ωũ_h^2dx=1.Notice that u_h solution to (<ref>) only differs from ũ_h by a multiplicative constant so that, from Proposition <ref>ν_h(dx)=Z_h(Ω)^-1ũ_h(x) e^-2/hf(x)dx,where, for any set O⊂Ω, Z_h(O):=∫_Oũ_he^- 2/h f .For F∈ C^∞(∂Ω), let us definew_h(x)=𝔼_x [ F (X_τ_Ω )]for allx∈Ω. The function w_h is such that: ∀ h >0 and x ∈Ω,| w_h(x)|≤F_L^∞.Moreover, a standard Feynman-Kac formula shows that w_h satisfies{ L^(0)_f,h w_h=0onΩ, w_h = F on∂Ω, .where, we recall, the differential operator L^(0)_f,h is defined by (<ref>). Our objective is to compare w_h(x) with _ν_h[F(X_τ_Ω)].By the Markov property, using (<ref>), we have _ν_h[F(X_τ_Ω)] =(∫_Ωũ_he^-2/hf )^-1 (∫_Ωw_h ũ_he^-2/hf ) = Z_h^-1(Ω) (∫_Ω∖ K_αw_h ũ_he^-2/hf) + Z_h^-1(Ω) (∫_K_α w_h ũ_he^-2/hf ).In order to estimate the first term in (<ref>), we need a leveling property for ũ_h, which is stated in <cit.>. Let us assume that [H1], [H2] and [H3] hold and let us consider ũ_h the principal eigenfunction of L^D,(0)_f,h(Ω) (see (<ref>)) with normalization (<ref>). Then, for any compact set K⊂Ω, lim_h→ 0ũ_h - (∫_Ωdx )^-1/2_L^∞(K)=0.Notice that the reason why we consider a smooth test function F rather than 1_Σ_i is that we would like to apply the results in <cit.>. A direct consequence of Lemma <ref> is the following limit,lim_h→ 0 hln(Z_h(Ω))=-2f(x_0).Indeed,from the normalization of ũ_h, we get Z_h(Ω)≤e^-2/hf(x_0), and from Lemma <ref> we have, for h small enough and for r>0 such that the open ball B(x_0,2r) is included in Ω,Z_h(Ω) ≥Z_h(B(x_0,r))≥1/2(∫_Ωdx )^-1/2∫_B(x_0,r)e^-2f/h.Since lim_h→ 0 hln(∫_B(x_0,r) e^-2f/hdx)=-2f(x_0), we get (<ref>).Let us now consider i∈{1,…,j}. For the first term in (<ref>), using (<ref>) and (<ref>), we have for any δ>0, for h small enough,Z_h(Ω)^-1| ∫_Ω∖ K_αw_h ũ_he^-2/hf| ≤F_L^∞e^δ/he^-2/h(inf_Ω∖ K_α f-f(x_0)) =F_L^∞e^δ/he^-2/h(α -f(x_0))and thus, thanks to (<ref>) and the fact that f(z_i)≤ f(z_j), by choosing δ small enough, there exists c>0 such that, for all h small enough,Z_h(Ω)^-1| ∫_Ω∖ K_αw_h ũ_he^-2/hf|≤F_L^∞e^-2/h(f(z_i)-f(z_1) + c) .In order to estimate the second term in (<ref>), we needa leveling property for w_h.Let us assume that [H1], [H2] and [H3] hold, as well as (<ref>). Let us consider w_h solution to (<ref>). Then there exists C>0 such that for any δ>0, for any h small enough, for all x,y∈ K_α, | w_h(x)-w_h(y)|≤ Ce^δ/h e^-2/h(f(z_1)-α ) .From <cit.>, it is known that for any δ >0, for any h small enough and for all x,y∈ K_α, | w_h(x)-w_h(y)|≤ Ce^δ/h e^-2/h V_Ω(K_α) ,where V_Ω(K_α) is defined by,V_Ω(K_α)=inf_x∈ K_α inf_T> 0 inf_γ∈ Abs(T,x,∂Ω) 1/4∫_0^T |γ̇+∇ f(γ)|^2 dtwhere Abs(T, x,∂Ω) is the set of absolutely continuous functions γ:[0,T]→Ω satisfying γ(0)=x and γ(T)∈∂Ω. For any γ∈ Abs(T, x,∂Ω), we have ∫_0^T |γ̇+∇ f(γ)|^2 dt-∫_0^T |γ̇-∇ f(γ)|^2 dt=4∫_0^Tγ̇·∇ f(γ) dt=4 ( f(γ(T))-f(x) ),and therefore, it holds ∫_0^T |γ̇+∇ f(γ)|^2 dt≥ 4 ( f(γ(T))-f(x) )≥ 4 ( f(z_1)-f(x)).Finally we obtain V_Ω(K_α)≥ f(z_1)-max_x∈ K_α f(x)=f(z_1)-α.This concludes the proof of Lemma <ref>.We are now in position to estimate the second term in (<ref>). Using Lemma <ref>, we get, for any δ >0, in the limit h → 0, uniformly in y_0∈ K_α, Z_h^-1(Ω) (∫_K_α w_h ũ_he^-2/hfdx ) = w_h(y_0)Z_h(K_α)/Z_h(Ω)+ O ( e^δ/h e^-2/h(f(z_1)-α ) )Z_h(K_α)/Z_h(Ω).Therefore, by choosing δ > 0 small enough, thanks to (<ref>) and the fact that f(z_i)≤ f(z_j) (since we recall that i∈{1,…,j}),there exists c>0 such that, in the limit h → 0,Z_h^-1(Ω) (∫_K_α w_h ũ_he^-2/hfdx ) =w_h(y_0)Z_h(K_α)/Z_h(Ω)+ O (e^-2/h (f(z_i)-f(z_1)+c ))Z_h(K_α)/Z_h(Ω).In addition we have Z_h(K_α)/Z_h(Ω)=1+O (e^-c/h ) for some c>0 independent of h. Indeed Z_h(K_α)/Z_h(Ω)=1-Z_h(Ω∖ K_α)/Z_h(Ω) and using (<ref>), we getfor any δ>0,Z_h(Ω∖ K_α)/Z_h(Ω)≤ e^δ/he^-2/h(min_Ω∖ K_α f-f(x_0) ) =O (e^-c/h ),for some c>0 independent of h by choosing δ small enough. Gathering the results (<ref>)–(<ref>)–(<ref>) in (<ref>), for all i∈{1,…,j}, there exists c>0 independent of h such that, in the limit h→ 0, it holds: uniformly in y_0∈ K_α,_ν_h[ F (X_τ_Ω )]=w_h(y_0) ( 1+O (e^-c/h ) ) + O (e^-2/h (f(z_i)-f(z_1)+c ) ).Let i∈{1,…,j} and let us assumethat suppF ⊂ B_z_i and z_i∈ int ( suppF). Then, combining the last estimate with (<ref>) implies that uniformly in x∈ f^-1( (-∞, α ]) ∩Ω, in the limit h→ 0: _x [ F(X_τ_Ω )]= ∂_nf(z_i)/√( detHessf_|∂Ω (z_i) ) ( ∑_k=1^n_0∂_nf(z_k) /√( detHessf_|∂Ω (z_k) ))^-1e^-2/h (f(z_i)-f(z_1)) ( F(z_i) + O(h) ).LetΣ_i ⊂∂Ω containing z_i andsuch that Σ_i ⊂ B_z_i. Then, there exit F,G∈ C^∞(∂Ω) such that suppF ⊂ B_z_i, z_i∈ int ( suppF), suppG ⊂ B_z_i, z_i∈ int ( suppG), F≤ 1_Σ_i≤ G and F(z_i)=G(z_i)=1. From the inequality _x [ F(X_τ_Ω )]≤_x [ X_τ_Ω∈Σ_i]≤_x [ G(X_τ_Ω )],together with (<ref>) applied to F and G, one gets, for all i∈{1,…,j},in the limit h→ 0:_x [ X_τ_Ω∈Σ_i]=∂_nf(z_i)/√( detHessf_|∂Ω (z_i) ) ( ∑_k=1^n_0∂_nf(z_k) /√( detHessf_|∂Ω (z_k) ))^-1e^-2/h (f(z_i)-f(z_1)) ( 1 + O(h) ). This concludes the proof of Corollary <ref>.§.§ Proofs of Theorem <ref> and Corollary <ref> In this section, we prove Theorem <ref>. The proof is similar to the one made for Theorem <ref>: the estimates of Proposition <ref> and the construction of the quasi-mode associated with z_j_0 are modified. The proof of Theorem <ref> is organized as follows. In Section <ref>, we give the estimates required for the n+1 quasi-modes. Then, in Section <ref>, we prove that these estimates implyTheorem <ref>. In Section <ref>, the construction of the quasi-modes is given and we check that they satisfy the estimatesstated in Section <ref>.§.§.§ Statement of the assumptions required for the quasi-modesLet su recall that, for p∈{0,1}, the orthogonal projector π_[0,√(h)/2)( -L^D,(p)_f,h(Ω)) is still denoted by π^(p)_h, see (<ref>). The next proposition gives the assumptions needed on the quasi-modes (ψ̃_i)_i=1,…,n whose span approximates π_h^(1), and ũ whose span approximates π_h^(0), in order to prove Theorem <ref>. It is the equivalent of Proposition <ref> in the more general setting of Theorem <ref>. Let us assume that the hypotheses [H1], [H2] and [H3] hold. Let Σ_i denotes an open set included in ∂Ω containing z_i (i∈{1,…,n}) and such that Σ_i ⊂ B_z_i.Let k_0∈{1,…,n} and f^* such thatf(z_k_0) ≤ f^*≤ f(z_k_0+1),(with the convention f(z_k_0+1)=+∞ if k_0=n). Finally, let Σ⊂∂Ω be a smooth open set such that inf_Σ f=f^* andΣ⊂ B_z_j_0, for some j_0∈{1,…,k_0}.Let us assumethat there exist n quasi-modes (ψ̃_i)_i=1,…,n and a family of quasi-modes (ũ=ũ_δ)_δ>0 satisfying the following conditions: * ∀ i∈{1,…,n}, ψ̃_i∈Λ^1 H^1_w,T(Ω) and ũ∈Λ^0 H^1_w,T(Ω). The function ũ is non negative in Ω. Moreover, ∀ i∈{1,…,n},ψ̃_i _L^2_w= ũ _L^2_w = 1. * (a) There exists_1>0 such thatfor all i∈{1,…,k_0}, it holds in the limit h→ 0:(1-π_h^(1)) ψ̃_i _H^1_w^2=O(e^-2/h( max[f^*-f(z_i), f(z_i)-f(z_1)] +_1 )), and for all i∈{k_0+1,…,n},(1-π_h^(1)) ψ̃_i _H^1_w^2=O (e^-2/h(f^*-f(z_1) +_1 )).(b) For any δ>0, in the limit h→ 0: ∇ũ_L^2_w^2 = O (e^-2/h(f(z_1)-f(x_0) - δ)). * There exists _0>0, ∀ (i,j) ∈{1,…,n}^2,in the limit h→ 0,if i<j≤ k_0:ψ̃_i, ψ̃_j_L^2_w=O ( e^-1/h(f(z_j)- f(z_i)+ε_0) ),and, ifk_0<j, i<j:ψ̃_i, ψ̃_j_L^2_w=O ( e^-1/h(f^*- f(z_1)+ε_0)). * (a)There exists '>0 and there exist constants (C_i)_i=1,…,n andp which do not depend on h such that, in the limit h→ 0,for i∈{1,…,k_0}:∇ũ,ψ̃_i_L^2_w=C_ih^pe^-1/h(f(z_i)- f(x_0)) (1+ O(h ) ),and for i∈{k_0+1,…,n}:∇ũ,ψ̃_i_L^2_w=C_ih^pe^-1/h(f(z_i)- f(x_0)) (1+ O(h ) )+O( e^-1/h(f^*- f(x_0)+')).(b) There exist constants (B_i)_i=1,…,n andm which do not depend on h such that for all (i,j) ∈{1,…,n}^2, in the limit h→ 0: ∫_Σ_i ψ̃_j · ne^- 2/h fdσ = B_i h^m e^-1/h f(z_i)(1+ O(h )) ifi=j0 ifi≠ j. (c) There exist C^*and p^* independent of h such that for alli∈{1,…,n}, in the limit h→ 0:∫_Σ ψ̃_i · ne^- 2/h f dσ=δ_j_0,iC^*h^q^*e^-1/h(2f^*-f(z_j_0)) ( 1+ O(h)). Let u_h be the eigenfunction associated with the smallest eigenvalue of -L^D,(0)_f,h(Ω) (see Proposition <ref>) which satisfies (<ref>).Then, one has: * For all i∈{1,…,k_0}, in the limit h→ 0∫_Σ_i(∂_nu_h) e^-2/hf dσ=C_iB_ih^p+me^-1/h(2f(z_i)- f(x_0)) (1+O(h) ).Moreover, if f(z_k_0)<f(z_k_0+1), there exists >0 such that for all i ∈{k_0+1,…,n} in the limit h→ 0∫_Σ_i(∂_nu_h) e^-2/hfdσ= ( ∫_Σ_k_0(∂_nu_h)e^-2/hfdσ ) O ( e^-/h). * In the limit h→ 0:∫_Σ (∂_nu_h) e^-2/hf dσ =C^*C_j_0h^q^*+pe^-1/h(2f^*- f(x_0)) (1+O(h) ).The asymptotic estimates (<ref>) and (<ref>) in Theorem <ref> areconsequences of this proposition and of the construction of someappropriate quasi-modes ((ψ̃_i)_i=1,…,n, ũ), see Section <ref>, which willshow that (B_i)_i=1,…,n, m, (C_i)_i=1,…,n, p are given by (<ref>)-(<ref>) and C^*, q^* will be given in Lemma <ref> below. Moreover, the other asymptotic estimates (<ref>) and (<ref>) in Theorem <ref> areconsequencesof the asymptotic estimates (<ref>) and (<ref>) together with Proposition <ref>,Proposition <ref> and (<ref>). §.§.§ Proof of Proposition <ref>The proof ofProposition <ref> follows closely the same steps as the proof ofProposition <ref>. We only highlight the main differences. In all this section, we assume that the hypotheses [H1], [H2] and [H3] hold. Let f^*∈ℝ, k_0∈{1,…,n},j_0∈{1,…,k_0}, (Σ_i)_i∈{1,…,n} and Σ be as stated in Proposition <ref>. In addition, let us assume the existence of n+1 quasi-modes (ũ, ( ψ̃_i)_i=1,…,n) satisfying all the conditions of Proposition <ref>. In the following, ε denotes a positive constant independent of h, smaller than min (ε_1, ε_0, '), and whose precise value may vary (a finite number of times) from one occurrence to the other.Let us recall a result relating ũ with u_h on the one hand, and span(ψ̃_j,j=1,…,n) with π_h^(1) on theother hand.The following lemma is a direct consequence of Lemma <ref>, Lemma <ref> and the assumptions 1, 2 and 3 of Proposition <ref>. Let us assume that the assumptions of Proposition <ref> hold. Then,there exist c>0 and h_0>0 such that for h ∈ (0,h_0),π_h^(0)ũ_L^2_w=1+O ( e^-c/h). In addition, there exists h_0>0 such that for all h ∈ (0,h_0),span( π_h^(1)ψ̃_i,i=1, …,n)=π_h^(1).A direct consequence of Lemma <ref> and the fact that ũ is non negative in Ω is that it holds for h small enough: u_h=π_h^(0)ũ/π_h^(0)ũ_L^2_w.Let us denote by (ψ_i)_i=1,…,n the orthonormal basis of π_h^(1) resulting from the Gram-Schmidt orthonormalization procedure on the set (π_h^(1)ψ̃_i)_i=1, …,n (see Lemma <ref>).Then, since ∇ u_h ∈ Ran(π_h^(1))=span(ψ_j,j=1,…,n) (see (<ref>))andψ_j,ψ_i_L^2_w=δ_i,j,one has for any Γ⊂∂Ω∫_Γ (∂_nu_h)e^-2/h fdσ= ∑_j=1^n ⟨∇ u_h , ψ_j_L^2_w∫_Γψ_j· ne^- 2/hf dσ.Let (κ_ji)_(i,j) ∈{1,…,n}^2, i<j and (Z_j)_j∈{1,…,n} be the matrix and vector obtained through the Gram-Schimdt othonormalization procedure, see Lemma <ref>. The strategy toprove Proposition <ref>consists inestimatingprecisely the following terms in the limit h→ 0: κ_ji , Z_j,⟨∇ u_h , ψ_j_L^2_w and (∫_Γψ_j · ne^- 2/hf dσ)_Γ∈{Σ, Σ_1,…,Σ_n} for (i,j)∈{1,…,n}^2, i<j. Then, they will be used to obtain a precise estimateof the right-hand-side of (<ref>) when h→ 0. Step 1. Estimates on the terms (κ_ji)_(i,j) ∈{1,…,n}^2, i<j and (Z_i)_i∈{1,…,n}.Let us assume that the assumptions of Proposition <ref> hold. Then, there exist >0 and h_0>0 such that for all (i,j) ∈{1,…,n}^2 with i<j and all h∈ (0,h_0), if j≤ k_0:π_h^(1)ψ̃_i,π_h^(1)ψ̃_j _L^2_w=O(e^-1/h(f(z_j) - f(z_i)+)),and if j>k_0:π_h^(1)ψ̃_i,π_h^(1)ψ̃_j _L^2_w=O(e^-1/h(f^* - f(z_1)+)).The proof follows the same lines of the proof of Lemma <ref>. If i<j and j≤ k_0, fromassumption 2(a) in Proposition <ref> and since f^*≥ f(z_1), one gets(1-π_h^(1))ψ̃_i, (1-π_h^(1)) ψ̃_j _L^2_w ≤‖ (1-π_h^(1))ψ̃_i‖_L^2_w‖ (1-π_h^(1))ψ̃_j‖_L^2_w≤ O(e^-1/h(f^*- f(z_i)+ f(z_j)-f(z_1) +))= O(e^-1/h(f(z_j) - f(z_i)+)).If i<j and k_0<j, fromassumptions 1 and 2(a) in Proposition <ref>, one gets(1-π_h^(1))ψ̃_i, (1-π_h^(1)) π_h^(1)ψ̃_j _L^2_w≤‖ψ̃_i‖_L^2_w‖ (1-π_h^(1))ψ̃_j‖_L^2_w≤O(e^-1/h(f^* - f(z_1)+)).Lemma <ref> is then a consequence of (<ref>) together with assumption 3 in Proposition <ref>.Let us assume that the assumptions of Proposition <ref> hold. Then, there exist >0 and h_0>0 such that for all (i,j) ∈{1,…,n}^2 with i<j and all h∈ (0,h_0), if j≤ k_0:κ_ji=O(e^-1/h(f(z_j) - f(z_i)+)),and if j>k_0:κ_ji=O(e^-1/h(f^* - f(z_1)+)).In addition, there exist c>0 and h_0>0 such that for all j ∈{1,…,n} and h∈ (0,h_0),Z_j=1+O ( e^-c/h).If i<j and j≤ k_0, the estimates on κ_ji and Z_j are proved by induction as in the proof of Lemma <ref>. Let us now deal with the case i<j and k_0<j. For j=k_0+1, it follows from (<ref>) that for all i<k_0+1,κ_(k_0+1) i =- ∑_k=i^k_0∑_l=1^k1/Z_k^2π_h^(1)ψ̃_k_0+1, π_h^(1)ψ̃_l_L^2_w κ_klκ_ki,where we use the notation κ_ii=1 for every i∈{1,…,n}. Since 1≤ k≤ k_0, one hasZ_k^-1=1+O ( e^-c/h). In addition, since 1≤ l≤ k≤ k_0 and 1≤ i≤ k≤ k_0, one has κ_klκ_ki=O( 1 ). From Lemma <ref>, one has for 1≤ l<k_0+1, π_h^(1)ψ̃_k_0+1, π_h^(1)ψ̃_l_L^2_w=O(e^-1/h(f^* - f(z_1)+)). Therefore, one obtains for all i<k_0+1, κ_(k_0+1) i=O(e^-1/h(f^* - f(z_1)+)).The fact that Z_k_0+1=1+O ( e^-c/h), comes from the fact that the terms (κ_(k_0+1) i)_i ∈{1,… ,k_0} are exponentially small and the fact that ‖π_h^(1)ψ̃_k_0+1‖_L^2_w = 1+O ( e^-c/h).In order to proveby induction the estimates on κ_ji for i<j and j>k_0, let us now assume that for some k∈{k_0+1,…,n} and for allj∈{k_0+1,…,k}, i∈{1,…,j-1},κ_ji= O (e^-1/h(f^*-f(z_1)+)) andZ_j=1+O ( e^-c/h).It follows from (<ref>), for q∈{1, … ,k},κ_(k+1) q =- ∑_j=q^k ∑_l=1^j1/Z_j^2π_h^(1)ψ̃_k+1, π_h^(1)ψ̃_l_L^2_w κ_jlκ_jq,where we used the notation κ_ii=1. Since 1≤ j≤ k, one has Z_j^-1=1+O ( e^-c/h). In addition, since 1≤ l≤ j≤ k and 1≤ q≤ j≤ k, one has κ_jlκ_jq=O( 1 ). From Lemma <ref>, one has for 1≤ l<k+1 and k> k_0, π_h^(1)ψ̃_k+1, π_h^(1)ψ̃_l_L^2_w=O(e^-1/h(f^* - f(z_1)+)). Therefore, one obtains for all 1≤ q<k+1, κ_(k+1) q=O(e^-1/h(f^* - f(z_1)+)).The fact that Z_k+1=1+O ( e^-c/h), comes from the fact that the (κ_(k+1) q)_q ∈{1,… ,k} are exponentially small and the fact that ‖π_h^(1)ψ̃_k+1‖_L^2_w = 1+O ( e^-c/h). This concludes the proof by induction.Step 2. Estimates on the interaction terms (⟨∇ u_h , ψ_j_L^2_w)_j∈{1,…,n}.Let us assume that the assumptions of Proposition <ref> hold.Then,for j∈{1,…,k_0}, in the limit h→ 0:∇ u_h,ψ_j_L^2_w=C_jh^pe^-1/h(f(z_j)- f(x_0)) (1+ O(h ) ),and for j∈{k_0+1,…,n},there exists '>0 such that in the limit h→ 0:∇ u_h, ψ_j_L^2_w=C_jh^pe^-1/h(f(z_j)- f(x_0)) (1+ O(h ) )+O( e^-1/h(f^*- f(x_0)+')).For j∈{1,…,k_0}, the proof of the estimate of ∇ u_h,ψ_j_L^2_w is exactly the same as for Lemma <ref>. Let j∈{k_0+1,…,n}. Using (<ref>)–(<ref>)–(<ref>)–(<ref>), one has∇ u_h , ψ_j _L^2_w=a_j+b_j+c_j,where a_j, b_j and c_j are defined by a_j=Z^-1_j/π_h^(0)ũ_L^2_w( ∇ũ,ψ̃_j_L^2_w +∇ũ, ( π_h^(1)-1 ) ψ̃_j_L^2_w),b_j= Z^-1_j/π_h^(0)ũ_L^2_w∑_i=1^k_0κ_ji (∇ũ,ψ̃_i_L^2_w + ∇ũ, ( π_h^(1)-1 ) ψ̃_i_L^2_w), and c_j=Z^-1_j/π_h^(0)ũ_L^2_w∑_i=k_0+1^j-1κ_ji (∇ũ,ψ̃_i_L^2_w + ∇ũ, ( π_h^(1)-1 ) ψ̃_i_L^2_w),with the convention ∑_i=k_0+1^k_0=0. FromLemmata <ref> and <ref>, one hasZ_j^-1/π_h^(0)ũ_L^2_w=1+O ( e^-c/h). Using assumptions 2 and 4(a) in Proposition <ref> and Lemma <ref>, there exists δ_0>0 such that for all δ∈ (0,δ_0),a_j =C_j h^pe^-1/h(f(z_j)- f(x_0)) ( 1 + O(h ) )+ O(e^-1/h(f^*-f(x_0) + ) ) + O(e^-1/h(f^*- f(x_0) -δ + )),b_j = ∑_i=1^k_0O (e^-1/h(f^*-f(z_1) + +f(z_i)- f(x_0)-δ) )+O(e^-1/h(f^*-f(z_1) + +f(z_1)- f(x_0) -δ+ f(z_i)-f(z_1) + ))=O(e^-1/h( f^*-f(x_0) -δ+ )), c_j =∑_i=k_0+1^j-1O (e^-1/h(f^*-f(z_1) + )) (O(e^-1/h(f(z_i)- f(x_0)-δ) ) + O (e^-1/h(f^*-f(x_0) + ) ) ) +O (e^-1/h(f^*-f(z_1) + +f(z_1)- f(x_0) -δ+ f^*-f(z_1) + ) )=O(e^-1/h( f^*-f(x_0) -δ+ ) ).Therefore, choosing δ∈ (0,ε), there exists '>0 such that∇ u_h , ψ_j _L^2_w =C_j h^pe^-1/h(f(z_j)- f(x_0)) ( 1+ O(h ) ) + O (e^-1/h(f^*- f(x_0) + ')). This concludes the proof of Lemma <ref>. Step 3. Estimates on the boundary terms (∫_Γψ_j · ne^- 2/hf dσ )_j∈{1,…,n}, Γ∈{Σ, Σ_1,…,Σ_n}. Let us assume that the assumptions of Proposition <ref> hold. Then, there exists >0 such that in the limit h→ 0 one has: * Ifk∈{1,…,k_0} andj∈{1,…,k_0},∫_Σ_k ψ_j · ne^- 2/h f dσ =δ_j,kB_k h^me^-1/h f(z_k) (1+ O(h )) + O(e^-1/h (2f(z_k)-f(z_j)+))+1_j>kO(e^-1/h (f(z_k)+)). * Ifk∈{1,…,k_0} andj∈{k_0+1,…,n}, ∫_Σ_k ψ_j · ne^- 2/h f dσ= O ( e^- 1/h( f(z_k)+)).* Ifk∈{k_0+1,…,n} andj∈{1,…,k_0}, ∫_Σ_k ψ_j · ne^- 2/h f dσ=O(e^-1/h (2f(z_k_0)-f(z_j)+)). * Ifk∈{k_0+1,…,n} and j∈{k_0+1,…,n},∫_Σ_k ψ_j · n e^- 2/h fdσ=δ_j,kO( h^me^-1/h f(z_k)) +O ( e^- 1/h( f(z_k_0+1)+)). For all (j,k)∈{1,…,n}^2, using (<ref>)–(<ref>) and Lemmata <ref> and <ref> together with assumption 4(b) in Proposition <ref>, one has in the limit h→ 0:∫_Σ_k ψ_j · n e^- 2/h fdσ =Z^-1_j [ ∫_Σ_k ψ̃_j · n e^- 2/h f dσ + ∫_Σ_k( π_h^(1)-1) ψ̃_j · ne^- 2/h fdσ] +Z^-1_j∑_i=1^j-1κ_ji (∫_Σ_k ψ̃_i · n e^- 2/h f dσ+ ∫_Σ_k( π_h^(1)-1) ψ̃_i · ne^- 2/h fdσ) = δ_j,kB_k h^me^-1/h f(z_k) (1+ O(h )) +1/h(1-π_h^(1)) ψ̃_j_H^1_wO ( e^- 1/h f(z_k)) +∑_i=1^j-1κ_jiδ_i,kO( h^me^-1/h f(z_k)) + κ_ji/h(1-π_h^(1)) ψ̃_i_H^1_w O ( e^- 1/h f(z_k)).Let us know deal separately with the two cases k∈{1,…,k_0} and k∈{k_0+1,…,n}. In the following, we use assumption 2(a) in Proposition <ref> and Lemma <ref> toestimate(<ref>).Case 1: k∈{1,…,k_0}. If j∈{1,…,k_0}, from (<ref>), one gets in the limit h→ 0:∫_Σ_k ψ_j · n e^- 2/h fdσ =δ_j,kB_k h^me^-1/h f(z_k) (1+ O(h )) +O ( e^- 1/h(f^*-f(z_j)+ f(z_k)+)) +∑_i=1^j-1δ_i,kO(e^-1/h (f(z_k)+))+∑_i=1^j-1O ( e^- 1/h(f(z_j)-f(z_i)+f^*-f(z_i)+ f(z_k)+)).Since f^*≥ f(z_k) for all k∈{1,…,k_0} and since there exists i<j such that δ_i,k=1 if and only if j>k, there exists >0 such that for all (k,j)∈{1,…,k_0}^2 and for all h small enough, ∫_Σ_k ψ_j · ne^- 2/h fdσ=δ_j,kB_k h^me^-1/h f(z_k) (1+ O(h )) + O(e^-1/h (2f(z_k)-f(z_j)+))+1_j>kO(e^-1/h (f(z_k)+)). If j∈{k_0+1,…,n}, from(<ref>), one gets ∫_Σ_k ψ_j · n e^- 2/h fdσ =O ( e^- 1/h(f^*-f(z_1)+ f(z_k)+))+∑_i=1^j-1 O( e^-1/h ( f(z_k)+)). Case 2:k∈{k_0+1,…,n}.If j∈{1,…,k_0}, from (<ref>) andsincef(z_k)≥ f^*≥ f(z_k_0), one has∫_Σ_k ψ_j · n e^- 2/h f dσ =O ( e^- 1/h (f^*-f(z_j)++f(z_k))) +∑_i=1^j-1 O ( e^- 1/h (f^*+f(z_k)+f(z_j)-2f(z_i)+))=O ( e^- 1/h (2f(z_k_0)-f(z_j)+)).If j∈{k_0+1,…,n}, from(<ref>), one gets∫_Σ_k ψ_j · n e^- 2/h fdσ =δ_j,kO( h^me^-1/h f(z_k)) +O ( e^- 1/h( f(z_k)+)),which leads to the desired estimate since f(z_k)≥ f(z_k_0+1). This concludes the proof of Lemma <ref>. Let us assume that the assumptions of Proposition <ref> hold.Then, for j∈{1,…,k_0} one has when h→ 0:∫_Σψ_j · ne^- 2/h fdσ=δ_j_0,jC^* h^q^* e^-1/h(2f^*-f(z_j_0))( 1+ O(h) ) +O ( e^- 1/h (2f^*-f(z_j)+))and for j∈{k_0+1,…, n} one has ∫_Σψ_j · ne^- 2/h fdσ =O ( e^- 1/h (2f^*-f(z_1)+)). Let j∈{1,…,n}. Using (<ref>)–(<ref>) and Lemmata <ref> and <ref> together with assumption 4(c) in Proposition <ref>, one has∫_Σψ_j · n e^- 2/h fdσ =Z^-1_j[ ∫_Σψ̃_j · n e^- 2/h fdσ + ∫_Σ ( π_h^(1)-1) ψ̃_j · ne^- 2/h fdσ] +Z^-1_j∑_i=1^j-1κ_ji (∫_Σψ̃_i · n e^- 2/h f dσ + ∫_Σ ( π_h^(1)-1) ψ̃_i · ne^- 2/h fdσ) = δ_j_0,jC^*h^q^*e^-1/h(2f^*-f(z_j_0)) ( 1+ O(h))+1/h(1-π_h^(1)) ψ̃_j_H^1_wO(e^- 1/h f^*) +∑_i=1^j-1δ_j_0,iκ_ji O(h^q^* e^-1/h(2f^*-f(z_j_0)))+ κ_ji/h(1-π_h^(1)) ψ̃_i_H^1_wO( e^- 1/h f^*). Let us first deal with the case j∈{1,…,k_0}. Using assumption 2(a)in Proposition <ref> and Lemma <ref>, one gets from (<ref>)∫_Σψ_j · n e^- 2/h fdσ =δ_j_0,jC^* h^q^* e^-1/h(2f^*-f(z_j_0))( 1+ O(h) ) +O ( e^- 1/h (f^*-f(z_j)++f^*)) + ∑_i=1^j-1O ( e^- 1/h ( 2f^*+f(z_j)-2f(z_i)+))=δ_j_0,jC^* h^q^* e^-1/h(2f^*-f(z_j_0))( 1+ O(h) ) +O ( e^- 1/h (2f^*-f(z_j)+)). Let us now deal with the case j∈{k_0+1,…,n}. In that case, one obtains from (<ref>), assumption 2(a)in Proposition <ref>, Lemma <ref> together with the fact that f^*≥ f(z_i) for all i∈{1,…,k_0} and f^*≥ f(z_k_0)≥ f(z_j_0),∫_Σψ_j · n e^- 2/h fdσ = O ( e^- 1/h (f^*-f(z_1)++f^*))+ ∑_i=1^j-1δ_j_0,iO ( e^-1/h(f^*-f(z_1)+ 2f^*-f(z_j_0 ) +)) + ∑_i=1^k_0O ( e^- 1/h (2f^*-f(z_1)+f^*-f(z_i)+)) + ∑_i=k_0+1^j-1O ( e^- 1/h (2f^*-2f(z_1)++f^*)) =O ( e^- 1/h (2f^*-f(z_1)+)).This concludes the proof of Lemma <ref>. Step 4. Estimates on the boundary terms (∫_Γ (∂_nu_h)e^- 2/hfdσ )_Γ∈{Σ, Σ_1,…,Σ_n}. We are now in position to conclude the proof of Proposition <ref>. Let us assume that the assumptions of Proposition <ref> hold. The proof is divided into two cases.Case 1: Γ=Σ_k in (<ref>) for some k∈{1,…,n}. If k∈{1,…,k_0}, from Lemmata <ref> and <ref> and the fact that f^*≥ f(z_k_0)≥ f(z_k), one obtains that there exists >0 such that for all j∈{1,…,n}, in the limit h→ 0∇ u_h , ψ_j _L^2_w∫_Σ_k ψ_j · ne^- 2/h fdσ =δ_jkB_kC_k h^m+pe^-1/h (2f(z_k)-f(x_0)) (1+ O(h )) +O (e^-1/h(2f(z_k)- f(x_0) + )). Therefore, from (<ref>), one gets for all k∈{1,…,k_0},in the limit h→ 0∫_Σ_k(∂_nu_h)e^- 2/h f dσ= B_kC_k h^m+pe^-1/h (2f(z_k)-f(x_0)) (1+ O(h )).If k∈{k_0+1,…,n}. From Lemmata <ref> and<ref>, one has for j∈{1,…,k_0}:∇ u_h , ψ_j _L^2_w∫_Σ_k ψ_j · ne^- 2/h fdσ=O ( h^p e^-1/h(2f(z_k_0)- f(x_0)+ )).and for j∈{k_0+1,…,n} (since f(z_j)≥ f(z_k_0+1) and f(z_k_0)≤ f^*≤(z_k_0+1)):∇ u_h , ψ_j _L^2_w∫_Σ_k ψ_j · ne^- 2/h fdσ =δ_jk O ( h^p+me^-1/h(2f(z_k)- f(x_0))) + O (e^-1/h(2f(z_k_0)- f(x_0) + )). Therefore, if one assumes that f(z_k_0+1)>f(z_k_0), from (<ref>), one gets for all k∈{k_0+1,…,n} and for all h small enough∫_Σ_k(∂_nu_h ) e^- 2/h f dσ= O (e^-1/h(2f(z_k_0)- f(x_0) + )).Case 2: Γ=Σ in (<ref>).From (<ref>) and using Lemmata <ref> and <ref>, one has ∫_Σ∂_nu_h e^- 2/h fdσ =C_j_0C^* h^q^*+p e^-1/h(2f^*-f(x_0))(1+O(h)) +∑_j=1, j≠ j_0^k_0O ( h^pe^-1/h(- f(x_0)+2f^*+))+ ∑_j=k_0+1^nO ( h^pe^-1/h(f(z_j)- f(x_0)+2f^*-f(z_1)+))+ O (e^-1/h(f^*- f(x_0) +2f^*-f(z_1)+))= C_j_0C^* h^q^*e^-1/h(2f^*-f(x_0))(1+O(h)), which is the desired result. Proposition <ref> is proved. §.§.§ Construction of the quasi-modes which satisfy the estimates of Proposition <ref> In this section, we first construct the quasi-modes (ψ̃_i)_i∈{1,…,n} and the family of quasi-modes (ũ=ũ_δ)_δ>0. Then, we prove that they satisfy the estimates stated in Proposition <ref>. In all this section,one assumes that the hypotheses [H1], [H2] and [H3] hold.Let(Σ_i)_i∈{1,…,n} and Σ be as in Proposition <ref>.Construction of the quasi-modes. The n+1 quasi-modes ( ( ψ̃_i)_i∈{1,…,n}, ũ) are constructed as in Section <ref> except that one addsan extra condition on the set Γ_1,j_0 used to define ψ̃_j_0 (where we recall that j_0∈{1,…,n} is such that Σ⊂B_z_j_0). Let us be more precise on this point. Let us recall thatfor all i∈{1,…,n}∖{j_0}, the 1-form ψ̃_i is defined as:ψ̃_i=e^1/hfϕ̃_i∈Λ^1H^1_w,T(Ω),where the n-1 quasi-modes ( ϕ̃_i)_i∈{1,…,n}∖{j_0} are built in Section <ref> (see Definition <ref>). Let us also recall thatthe function ũ is the one introduced in Definition <ref>. The construction ofψ̃_j_0 requires to take into account the set Σ in addition to the set Σ_j_0 when defining the cut off function χ_j_0 in Definition <ref>. More precisely, and thanks to Proposition <ref>,the set Γ_1,j_0 can be taken such thatΣ_j_0∪Σ⊂Γ_1,j_0. Then,withΓ_1,j_0 satisfyingthe previous condition, the cut-off fonction χ_j_0 and the 1-form ϕ̃_j_0∈Λ^1H^1_T(Ω) are defined exactly as in Definition <ref> for i=j_0. Finally, one sets: ψ̃_j_0=e^1/hfϕ̃_j_0∈Λ^1H^1_w,T(Ω).The quasi-modes satisfy the estimates stated in Proposition <ref>. Using in addition to [H1]-[H2]-[H3] the hypotheses (<ref>) and (<ref>), one easily obtains that( ( ψ̃_i)_i=1,…,n, ũ) satisfy the estimates 1, 2, 3, 4(a) and 4(b) stated in Proposition <ref>, following exactly the computations made on ( ( ϕ̃_i)_i=1,…,n, ũ) in Section <ref>: 2(a) follows from (<ref>)-(<ref>)-(<ref>)-(<ref>), 2(b) is proven in Lemma <ref>, 3 follows from (<ref>)-(<ref>), 4(b) is proven in Step 3 in Section <ref>and 4(a) is a consequence of (<ref>)-(<ref>)-(<ref>)-(<ref>). In particular, one gets that the constants (B_i)_i=1,…,n, m, (C_i)_i=1,…,n and p in Proposition <ref> are given by (<ref>)-(<ref>).The following lemmadeals with the assumption 4(c) inProposition <ref> which requires to use Proposition <ref>.Let us assume that the hypotheses [H1], [H2] and [H3] hold. Let j∈{1,…,n}. Then, when h→ 0, one has ∫_Σψ̃_j· n e^-2/hf dσ=δ_j_0,jB^* √(2 ) ( detHessf|_∂Ω (z_j_0) )^1/4/π^d-1/4√(∂_nf(z_j_0))h^p^*-d+1/4e^-1/h(2f^*-f(z_j_0))( 1+ O(h) ), where B^* and p^* are defined by (<ref>). By construction, if j≠ j_0 then ψ̃_j≡ 0 on B_z_j_0. Let us deal with the case j=j_0. Using (<ref>), one has∫_Σψ̃_j_0· n e^-2/hfdσ=∫_Σϕ̃_j_0· n e^- 1/hfdσ.Let u^(1)_z_j_0,wkb be the WKB expansion defined by (<ref>). Following the beginning of Section <ref>, let us consider * a neighborhood V_Γ_St,j_0 of Σ in Ω, which is stable under the dynamics (<ref>) and such that, for some > 0, V_Γ_St,j_0 + B(0,) ⊂ V_Γ_1,j_0∩ (Ω̇_j_0∪Γ_1,j_0),* and a cut-off function χ_wkb,j_0∈ C^∞_c(Ω̇_j_0∪Γ_1,j_0) with χ_wkb,j_0≡ 1 on a neighborhood of V_Γ_St,j_0 such that χ_wkb,j_0⊂ V_Γ_1,j_0∩ (Ω̇_j_0∪Γ_1,j_0).Using Proposition <ref>, there exists c_z_j_0(h)∈ℝ_+^* such thate^1/hd_a(·, z_j_0)(u^(1)_h,j_0-c_z_j_0(h)u^(1)_z_j_0,wkb)_ H^1(V_Γ_St,j_0)=O (h^∞).Let us now introduceϕ̃_z_j_0,wkb:=c_z_j_0(h) χ_wkb,j_0 u_z_j_0,wkb^(1)so that∫_Σϕ̃_j_0· n e^- 1/hf dσ = ∫_Σϕ̃_z_j_0,wkb· n e^- 1/hf dσ +∫_Σ(ϕ̃_j_0-ϕ̃_z_j_0,wkb) · ne^- 1/hf dσ.Let us first deal with the term ∫_Σϕ̃_z_j_0,wkb· n e^- 1/hf in (<ref>). Using (<ref>), one has (since Φ=f, ∂_nΦ=-∂_nf and a_0=1 on ∂Ω, see (<ref>)) when h→ 0,∫_Σϕ̃_z_j_0,wkb· ne^- 1/hf dσ = c_z_j_0(h) ∫_Σχ_wkb,j_0 u_z_j_0,wkb^(1)· ne^- 1/hf=2c_z_j_0(h)∫_Σ∂_nf e^- 1/h(2f-f(z_j_0)) ( 1+ O(h) )=2 c_z_j_0(h) B^*h^p^* e^-1/h(2f^*-f(z_j_0))( 1+ O(h) ).Then using (<ref>), one obtains in the limit h→ 0:∫_Σϕ̃_z_j_0,wkb· ne^- 1/hf dσ= B^* √(2 )( detHessf|_∂Ω (z_j_0) )^1/4/π^d-1/4√(∂_nf(z_j_0))h^p^*-d+1/4e^-1/h(2f^*-f(z_j_0))( 1+ O(h) ).Let us now estimate the term ∫_Σ(ϕ̃_j_0-ϕ̃_z_j_0,wkb) · ne^- 1/hf in (<ref>). Since d_a(·,z_j_0)=f-f(z_j_0)=Φ-f(z_j_0) on Σ, one obtains usingLemma <ref>: there exist C>0, h_0>0 and η>0 such that for all h∈ (0,h_0), |∫_Σ(ϕ̃_j_0-ϕ̃_z_j_0,wkb) · ne^- 1/hf dσ| =|∫_Σ(u^(1)_h,j_0/Θ_j_0-c_z_j_0(h)u_z_j_0,wkb^(1)) · ne^- 1/hf dσ|≤e^- 1/h(2f^*-f(z_j_0))/Θ_j_0∫_Σ|(u^(1)_h,j_0-c_z_j_0(h) u_z_j_0, wkb^(1)) e^d_a(·,z_j_0)/hdσ| +e^- 1/h(2f^*-f(z_j_0)) |Θ_j_0-1|/Θ_j_0| c_z_j_0(h)|∫_Σ| u_z_j_0,wkb^(1)e^Φ-f(z_j_0)/h dσ|≤ C e^- 1/h(2f^*-f(z_j_0)) e^d_a(·,z_j_0)/h(u^(1)_h,j_0-c_z_j_0(h)u^(1)_z_j_0,wkb)_ H^1(V_Γ_St,j_0) +C e^- 1/h(2f^*-f(z_j_0)) | c_z_j_0(h)|e^-η/h‖χ_wkb,j_0u_z_j_0,wkb^(1) e^Φ-f(z_j_0)/h‖_H^1(Ω̇_j_0) . Since it holdsu_z_j_0,wkb^(1) e^Φ-f(z_j_0)/h= d_f-(Φ-f(z_j_0)),ha(·,h)=hda(·,h)+∇(f-Φ)∧ a(·,h) (see (<ref>)), there exists C>0 such that for all h small enough, ‖χ_wkb,j_0u_z_j_0,wkb^(1) e^Φ-f(z_j_0)/h‖_H^1(Ω̇_j_0)≤ C.Then, one obtains using Proposition <ref> and (<ref>):e^1/h(2f^*-f(z_j_0))|∫_Σ(ϕ̃_j_0-ϕ̃_z_j_0,wkb) · ne^- 1/hf|= O(h^∞) + C e^-η/h h^-d+1/4 = O(h^∞).Injecting the estimates (<ref>)–(<ref>) in(<ref>) and using (<ref>) imply that in the limit h→ 0: ∫_Σψ̃_j_0· n e^-2/hfdσ =B^* √(2 ) ( detHessf|_∂Ω (z_j_0) )^1/4/π^d-1/4√(∂_nf(z_j_0)) h^p^*-d+1/4 e^-1/h(2f^*-f(z_j_0))( 1+ O(h) ).This proves Lemma <ref>. In conclusion, the n quasi-modes (ψ̃_i)_i=1,…,n and the family of quasi-modes (ũ=ũ_δ)_δ>0 satisfy all the conditions of Proposition <ref>. This concludes the proof of Theorem <ref>. §.§.§ Proof of Corollary <ref> Let us assume that the hypotheses [H1]-[H2]-[H3] hold and let us assume that f|_∂Ω has only two local minima z_1 and z_2 such that f(z_1)<f(z_2). Let Σ⊂∂Ω be a smooth open set such that Σ⊂ B_z_1 and f^*:=inf_Σ f.In addition, let us assume that (<ref>) and (<ref>) hold and let us assume that f^*=f(z_2). Then, the inequalities (<ref>) and (<ref>) are exactly (<ref>) and (<ref>) (in the case n=2 with j_0=1 and k_0=2). Therefore, (<ref>) holds. It remains to compute the prefactor in (<ref>). To this end, we need the constants B^* and p^*in (<ref>). Let us assume that there is only one minimizer z^* off on Σ. This impliesthatz^*∈∂Σ since z_1 is the only critical point of f|_∂Ω in B_z_1. Furthermore, we assumethat z^* is a non degenerate minimumof f_|∂Σ with ∂_n(∂Σ) f(z^*)<0 where n(∂Σ) is the unit outward normal to ∂Σ⊂∂Ω. Then, using Laplace's method, in the limit h → 0:∫_Σ∂_nfe^-2/h f dσ=-∂_n f(z^*)(π h)^d/2/2π∂_n(∂Σ) f(z^*) √( detHessf_|∂Σ (z^*) ) e^-2/h f^* (1+O(h)),with by convention, detHessf_|∂Σ (z^*) =1 if d=2. This specifies the constants B^* and p^* appearing in (<ref>). This endsthe proof of Corollary <ref>.§ ACKNOWLEDGEMENTS The authors would like to thank Guy Barles, Bernard Helffer, François Laudenbach, Francis Nier and Benoît Perthame for fruitful discussions on preliminary versions of this work. This work is supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492. § MAIN NOTATION USED IN THIS WORK 2 ∙ (τ_Ω,X_τ_Ω), p. page.tau ∙ _x, p. page.px ∙ ν_h, p. page.nuh ∙ L_f,h^(0) and L_f,h^D,(0)(Ω), p. page.LOFH∙ λ_h, p. page.lambdah (see also p. page.uhlambdah)∙ u_h, p. page.uh (see also p. page.uhlambdah) ∙ {k_0,1,…,k_0,n}, p. page.koi∙g, p. page.g∙L (γ,I), p. page.L∙d_a, p. page.dagmon∙x_0, p. page.z1zn∙{z_1,…,z_n}, p. page.z1zn∙Hypotheses [H1], [H2], and [H3], p. page.z1zn ∙n_0, p. page.n0 ∙ B_z_i and B_z_i^c, p. page.Bzi∙Σ_i, p. page.Sigmai∙ f^*,p. page.fstar∙ B^* and p^*,p. page.fstar ∙z^*,p. page.zstar∙ Λ^pC^∞(Ω), p. page.cinfty∙ Λ^pC^∞_T(Ω) and Λ^pC^∞_N(Ω), p. page.cinftyt∙ Λ^pL^2_w( Ω) andΛ^pH^q_w(Ω), p. page.wsobolevq∙Λ^pH^q_w,T( Ω) and Λ^pH^q_w,N(Ω), p. page.wsobolevqt∙ Λ^pL^2(Ω) and Λ^pH^q(Ω), p. page.psl2∙ Λ^pH^q_T(Ω) and Λ^pH^q_N(Ω), p. page.psl2∙ ‖ .‖_H^q_w and ⟨, ⟩_L^2_w,p. page.psl2w ∙‖ .‖_H^q and ⟨, ⟩_L^2,p. page.psl2∙ Δ^(p)_f,h,p. page.wlaplacien∙ Δ^D,(p)_f,h(Ω) andΔ^N,(p)_f,h(Ω),p. page.wlaplaciend ∙ L^D,(p)_f,h(Ω),p. page.generatord∙ π_E(A),p. page.piE∙For p∈{0,1},π^(p)_h, p. page.pihp ∙ ũ,p. page.pre (constructionpage.tildeu)∙ p, m, C_i, and B_i,p. page.pre (explicit valuesp. page.bim and page.ci)∙ψ̃_i,p. page.pre∙ ϕ̃_i,p. page.tildephi0(constructionp. page.tildephii) ∙ κ_ji,p. page.kappaji∙ Z_j andψ_j p. page.zjpsij ∙A(x,y), p. page.axy ∙d_a^Ω, p. page.dapaomega∙{x_1,…,x_m}, p. page.x1xm ∙V_Ω, p. page.vpaomega∙f_-, p. page.fmoins∙ Ω̇, p. page.omegapoint ∙ Λ^pH_d(Ω̇) and Λ^pH_d^*(Ω̇), p. page.omegapoint ∙Γ_T and Γ_N, p. page.gammatn (see alsop. page.gammatbis)∙Λ^pH_d,Γ(Ω̇) and Λ^pH_d^*,Γ(Ω̇), p. page.Hdgamma ∙ d^(p)_T(Ω̇) and δ^(p)_N(Ω̇), p. page.dtp∙ Δ^M,(p)_f,h(Ω̇), p. page.deltam∙ 𝒬^M,(p)_f,h(Ω̇), p. page.qm ∙ x_d, p. page.xd ∙Ψ_i, f_+,i, and f_-,i, p. page.fplus ∙ Σ_a (a>0), p. page.sigmaa∙Γ_1,i, p. page.gamma1i∙V_Γ_1,i, p. page.vgamma1i ∙Ω̇_̇i̇, p. page.omegaipoint ∙ Ω_0, Γ_0, p. page.gamma0 ∙Γ_2,i, p. page.gamma2i∙𝒮_M,i, p. page.SMi ∙ When the index i∈{1,…,n} is dropped (p. page.nota0): z=z_i,Γ_1=Γ_1,i, Γ_2=Γ_2,i, Ω̇=Ω̇_i, V_Γ_1=V_Γ_1,i, Ψ=Ψ_i, f_+=f_+,iand f_- =f_-,i. ∙u_wkb^(0), p. page.uowkb ∙u_wkb^(1), p. page.u1wkb ∙u_z,wkb^(1), p. page.u1zwkb ∙C_z,wkb, p. page.czwkb∙ V_Γ_st and V_Γ_st', p. page.vgammast∙ κ, p. page.kappa∙c_z(h), p. page.czh∙Θ_i, p. page.thetai ∙ ϕ̃_z_i,wkb, p. page.tildephiwkb∙K_α, p. page.kalpha∙ũ_h, p. page.tildeuh ∙Z_h(O), p. page.zho∙ w_h, p. page.whh∙ Σand j_0, p. page.jo∙ k_0, p. page.ko∙ C^*and q^*, p. page.cstarqstar ∙χ_j_0 andϕ̃_j_0, p. page.phijoplain | http://arxiv.org/abs/1706.08728v3 | {
"authors": [
"Giacomo Di Gesù",
"Tony Lelièvre",
"Dorian Le Peutrec",
"Boris Nectoux"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170627085100",
"title": "Sharp asymptotics of the first exit point density"
} |
[pages=1-last]ALMrate_submit.pdf | http://arxiv.org/abs/1706.08800v1 | {
"authors": [
"Ying Cui",
"Defeng Sun",
"Kim-Chuan Toh"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170627115931",
"title": "On the R-superlinear convergence of the KKT residues generated by the augmented Lagrangian method for convex composite conic programming"
} |
[email protected]@yahoo.comDepartment of Physics, Jadavpur University,188 Raja Subodh Chandra Mallik Road, Kolkata 700032, India Thermally stable quantum states with multipartite entanglements led by frustrationare found in the antiferromagnetic spin-1/2 Heisenberg hexagon.The model has been solved exactly to obtain all analytic expressions ofeigenvalues and eigenfunctions.Detection and characterizationsfor various types of entanglements have been carried out in terms of concurrence andentanglement witnesses based on several thermodynamic observables.Variations of entanglement properties with respect totemperature and frustration are discussed.Even though the frustration opposes the bipartite entanglement, itfavors the multipartite entanglement.Entangled states exhibit robustness against the thermal effectsin the presence of frustrationand they are found to survive at any temperature.03.65.Ud,03.65.Yz,03.67.Mn,03.67.-a,75.10.Jm,75.50.EeThermally stable multipartite entanglements in the frustrated Heisenberg hexagon Asim Kumar Ghosh today ================================================================================§ INTRODUCTIONThe field of quantum information processing experiences a mammoth growthin the last two decades <cit.>.Entanglement emerges as the most useful quantity among thequantum correlations through an extensive investigations in this field.Vast amounts of works involve in detection,characterization, distillation and quantificationof entanglements associated in various quantum systems.Nowadays, there are plethora of ideas which pave the way to realize the more secure and faster information processing tools as well as themore stable and efficient quantum communication networks.These technological innovations include cryptography <cit.>,dense coding <cit.>, teleportation <cit.> and many more.Interacting spin models consist of both small clusters and long chainswhere the spins are interacting through the exchangeinteractions can serve as the platforms to verify the outcomeof those proposals <cit.>.Thermal stability of the entangled state,on the other hand, is the main concern to make those protocols operational at room temperature. Besides those achievements, quantum correlations exhibit dramaticchanges in their values when the system undergoes a quantum phasetransition (QPT) <cit.>. Again, value of those correlations can be obtained exactly for the spinmodels as well as the locations of QPTs can be identified more precisely.In addition, real materials are also available those could serve as the macroscopicrealizations of any specific spin models.One of the example of such material is polyoxovanadate compound,(NHEt)_3 [V_8^ IVV_4^ VAs_8O_40(H_2O)]·H_2O <cit.>.The magnetic properties of this compound are faithfully explained by consideringa four-spin cluster, in which four spin-1/2 degrees are arranged on the vertices of a square and they are interacting with the nearest one withisotropic antiferromagnetic (AFM) Heisenberg exchange couplings.QPT occurs at a definite point for this model in the presence ofdiagonal exchange interaction <cit.>.Experimental evidence suggests thatentanglement can affect the macroscopic properties of solids.The observed values of specific heat and magnetic susceptibilityfor the compound LiHO_xY_1-xF_4 predict that those can be explainedif entanglement of the relevant quantum states are considered explicitly <cit.>.Thermodynamic observables of macroscopic system, likeinternal energy <cit.>, susceptibility <cit.> andstructure factor <cit.> serve as the entanglement witness (EW),since the measurement of those quantities eventually leads to the detection ofentanglement. For example, the magnetic structure of deuterated copper nitrate Cu(NO_3)_22.5D_2O has been considered as composed ofuncoupled spin-1/2 bond alternating AFM Heisenberg chainsand for this material susceptibility acts as EW <cit.>.A rigorous study of entanglement properties forHeisenberg spin chains in the thermodynamic limitis a theoretical challenge since the eigenvalues and eigenfunctions are not knownexactly in every case. In many cases, exact results areobtainable whenever the spin chain is mapped onto a spinlessfermionic models <cit.>.Bipartite and multipartite entanglement properties of variousspin models have been investigated at finite temperatures<cit.>. But the analytic derivations of all entanglement properties for an isolated cluster containingfew spins is possible so far as it is exactly diagonalizable.Moreover, spin cluster with higher values of exchangestrength can enhance the stability of entangled state at room temperature which is facing areal challenge nowadays. A spin-cluster material, copper carboxylate {Cu_2(O_2CH)_4}{Cu(O_2CH)_2(2-methylpyridine)_2} is found recently which supports entanglement above room temperature <cit.>.In this article, a cluster of six spins with all-to-alltwo-spin AFM exchange interactions isconsidered which gives rise to quantum states with multipartite entanglement those can withstand thermal agitations.The model has been solved exactly to obtain all analytic expressions ofeigenvalues and eigenfunctions.Symmetry of each eigenstate is studied by exploiting the six-foldrotational invariance of the Hamiltonian.Bipartite entanglements have been characterized with the help ofconcurrence (CN) while the multipartite entanglements are studied byintroducingseveral EWs. The model Hamiltonian is introduced in the Sec. <ref>along with the characterization of frustration embedded in it.The properties of thermal CNhave been discussed in the section <ref>.Detection of bipartite and multipartite entanglements in terms ofEWs based on susceptibility, fidelity and internal energy is presented in the Sec. <ref> while Sec.<ref> holds a comprehensive discussion on the results.§ THE SPIN-1/2 AFM J_1-J_2-J_3 HEISENBERG HEXAGONSpin-1/2 AFM Heisenberg Hamiltonian on thehexagonal cluster is defined byH = H_ NF+H_ F,H_ NF = J_1 ∑_i=1^6 S⃗_i ·S⃗_i+1+J_3∑_i=1,2,3S⃗_i·S⃗_i+3,H_ F = J_2∑_i=1^6 S⃗_i ·S⃗_i+2,S⃗_i+6=S⃗_i. S⃗_i is the spin-1/2 operator at the position i. In this model, every spin is interacting with all other spinsvia the AFM exchange interactions.As a result, three topologically different exchange couplings, say,nearest neighbor (NN), next nearest neighbor(NNN) and further neighbor (FN) or diagonal exchanges appearwhose strengths are J_1, J_2 and J_3, respectively. Geometrical view of this spin model is shown in the Fig. <ref>(a).Frustration appears in a magnetic system when all the AFM bondsare not energetically minimized in the classical ground state simultaneously.In this model, J_2 is frustrating, whileJ_1 and J_3 are non-frustrating.With this view, the total Hamiltonian, H, (Eq. <ref>) is decomposed intotwo parts, non-frustrated (H_ NF) and frustrated (H_ F).Frustration does not appear in this system if J_2 is assumednegative (ferromagnetic).For J_2< (J_1 + J_3/2),the classical ground state of this model is a doublet, where each stateis connected to other by flipping the spins in every site. One of such state, |𝒢⟩, is shown in the Fig. <ref>(c),in which the adjacent spins are antiparallel. In this case, energy minimization for both H_ NF and H_ F cannot be takenplace simultaneously in the ground state.Energy minimization of an AFM bond occurs when the spin alignments around this bondis antiparallel. As a result, energy corresponding to H_ NF with respect tothe ground state |⟩ is minimized butthat of H_ F is maximized with respect tothe same |⟩. The frustration of this model can becharacterized by using the quantity, frustration degree (ℱ)which is defined as <cit.>,ℱ= avg⟨ |H_ F|⟩/|⟨|H_ NF|⟩|,where “avg” denotes the averaging over all possible ground states. In this model, ℱ=J_2/(J_1+J_3/2).For the frustrated systemℱ>0, while itis non-frustrated when ℱ≤ 0.The higher value of ℱ corresponds to the stronger frustration.The variation ofℱ in the J_2-J_3 parameter spaceis shown in the Fig. <ref> (a).ℱ is found to increase (decrease) with the increase of J_2 (J_3). The maximum value ofℱ is unity which appears at the point J_3=0 over the line J_2=J_1 inthe parameter space.This particular point is labeled by the letter M in theparameter space (Fig. <ref> (d)).Therefore, the system is maximally frustrated at the point M.On the other hand, the minimum value of ℱ is zero for this AFM modelwhich is observedover the line J_2=0, where the system is said to be non-frustrated. Thus, effects of magnetic frustration on the entanglement propertiescan be studied with this model.The Hamiltonian, Eq. <ref>, commutes with total spin operator, S_ T,as well as the z-component of the total spin, S^z_ T.As a result, the Hamiltonian may be spanned in thedifferent subspaces of S^z_ T to obtain analytic expressions of eigenvalues andeigenfunctions. The exact analytic expressions of all 64 eigenstates (Ψ_n, n=1,2,3,⋯, 64) and corresponding energy eigenvalues (E_n) are available in the Appendix <ref>.Those states essentially comprise to five singlets(S_ T=0), nine triplets (S_ T=1),five quintets (S_ T=2) and one septet (S_ T=3).Five distinct singlets are denoted by the eigenstates Ψ_38, Ψ_39,Ψ_40, Ψ_41 and Ψ_42 in the Appendix <ref>. Among the five, two singlets, Ψ_42 and Ψ_38 can be expressed by two distinct combinations ofdimer states which are known as resonating valence bond (RVB) states.Those two particular singlets are defined by Ψ_ RVB and Ψ_ RVB'.The arrangements of dimer states in Ψ_ RVB (Ψ_42) and Ψ_ RVB' (Ψ_38) are shown in the Fig. <ref> (e). Ground state is always a total spin singlet.All the five singlets participate in four different manners to constitute the ground state in the whole parameter space.Thus, depending on the combinations of singlets in the ground states, J_2-J_3 parameter space is decomposed into four segments.Ψ_ RVB and Ψ_ RVB' are the ground states (non-degenerate) in the regions,R_1 (J_1+J_3>2J_2) andR_2 (J_1+J_3<2J_2), respectively. Ψ_ RVB and Ψ_ RVB' form the doubly degenerate ground state over the line, L(J_1+ J_3 = 2 J_2), junction of the two regions, R_1 and R_2.And all the five singlets constitute the ground state(five-folddegenerate) at the point P (J_1= J_3 = J_2).Positions of R_1, R_2, L and P on the parameterspace are shown in the Fig. <ref> (d). The area of R_1is three times larger than that of R_2.A first order QPT occursacross the line L as well as at the point P,where the ground state cross over takes place. In addition, the Hamiltonian possesses another useful symmetry, where it is invariant under the rotation by60^∘, (Fig. <ref>(b)). For the counter clockwise rotation by 60^∘,a rotational operator, R̂, can be defined as R̂⟨S_1S_2S_3S_4S_5S_6|=⟨S_2S_3S_4S_5S_6S_1|, where⟨S_1S_2S_3S_4S_5S_6|=⟨S_1^z|⊗⟨S_2^z|⊗⟨S_3^z|⊗⟨S_4^z|⊗⟨S_5^z|⊗⟨S_6^z|, in which⟨S_i^z| is the spin state at site i. So, R̂^(n) be the successive R̂ operation byn times, such that R̂^(6) is the identityoperation which leaves any stateunchanged. Each eigenstate (Ψ) of the Hamiltonianhas some definite rotational property,which can be characterized in terms of an eigenvalue equation, likeR̂^(n)|Ψ⟩=λ_r|Ψ⟩, where λ_r's are the eigenvalues of the rotational operator R̂^(n).λ_r can assume the value either +1 or -1for the minimum number (p) of R̂ operations on a definite state.Obviously, for the same state λ_r is always +1 for 2p number of R̂ operations.The states with λ_r=+1 for p number of R̂ operations have even parity (symmetric)while those with λ_r=-1 have odd parity (antisymmetric).It is found that, every eigenstate has definite valuesof both p and λ_r, and thus has definite parity.36 states have even parity while the remaining 28 states have odd parity.Values of p and λ_r for all eigenstates are shownin the Tab. I. It is observed thatp takes upeither 1 or 3 and nevertakes up 2, 4 and 5.For Ψ_ RVB, λ_r=-1 and p=1, while, for Ψ_ RVB', λ_r=1 and p=1. Thus,Ψ_ RVB' does not change signunder any number of R̂ operations, while Ψ_ RVB changes signfor odd numbers of R̂ operations. So, Ψ_ RVB isantisymmetric, whereas, Ψ_ RVB' is symmetric under the rotation by60^∘.§ THERMAL CONCURRENCEFor the Heisenberg hexagon, thermal state density matrix has been written down asρ(T)=1/Z∑_n=1^64e^-β E_nρ^n;ρ^n=⟨Ψ_n||Ψ_n⟩,where Z is the partition function of the system. β^-1=k_ BT, where k_ B and T are theBoltzmann constant and temperature, respectively.Eigenvalues, E_n and the corresponding eigenstates, Ψ_n areshown in the Appendix <ref>. Similarly, the reduced thermal state density matrix ρ_ij(T)can be written as,ρ_ij(T)=1/Z∑_n=1^64e^-β E_nρ_ij^n, where the reduced density matrix, ρ_ij^n is obtainedfrom ρ^nby tracing out the remaining four spin degrees of freedom, those arenot located at the sites i and j.CN is one of the simplest measure to quantify the entanglementbetween two qubits when they sit at two different sites in the surrounding ofother interacting spins and that can be derived from theexpression of ρ_ij(T).At T=0, ρ_ij(T) reduces to ρ^ G_ij, whereρ^ G=⟨Ψ_ G||Ψ_ G⟩ and Ψ_ Gis the ground state. Ψ_ G becomes equal toΨ_42 and Ψ_38 for the regions R_1 and R_2, respectively.On the line L, ρ^ G=(⟨Ψ_38||Ψ_38⟩+⟨Ψ_42||Ψ_42⟩)/2.Similarly, at P ρ^ G=(⟨Ψ_38||Ψ_38⟩+⟨Ψ_39||Ψ_39⟩+ ⟨Ψ_40||Ψ_40⟩+⟨Ψ_41||Ψ_41⟩+⟨Ψ_42||Ψ_42⟩ )/5.Depending on thepositions of the sites i and j, only threedifferent types ofρ^ G_ij can be constructed. They are ρ^ G_ NN, ρ^ G_ NNN and ρ^ G_ FN, when the sitesi and j are NN, NNN and FN, respectively.For example, there is six distinct pairs ofNN sites for different values ofi and j ({ij}), say, {12}, {23}, {34}, {45}, {56} and {61}.ρ^ G_ij is same for all these six NN pairs by virtueof the rotational symmetry of hexagon. So, theyare abbreviated as ρ^ G_ NN.The similar argument holds true for other combinations, NNN and FN.NNN corresponds to six distinct pairs while FN corresponds to only three pairs. The general form of two-qubit ρ^ G_ij in the space of S^z diagonal basis states, {⟨↑↑|, ⟨↑↓| , ⟨↓↑|, ⟨↓↓|}, looks like, ρ^ G_ij= [ [ a 0 0 f; 0 b_1 z 0; 0 z^* b_2 0; f^* 0 0 d ]].By expressingρ^ G_ij in this formone can define the spin reversed reduced density matrix as,ρ^ G_ij=(σ_y⊗σ_y)(ρ^ G)^^*_ij (σ_y⊗σ_y),where σ_y is the Pauli matrix. Then concurrence between the sitesi and j (CN_ij) is given byCN_ij= max{λ_1-λ_2-λ_3-λ_4,0},where λ_is are the square roots of the eigenvalues of the non-Hermitian matrixρ^ G_ij ρ^ G_ij, in descending order <cit.>.Since S^z_ T is the good quantum number,the element fin ρ^ G_ij (Eq. <ref>) vanishes.As a result, the expression of concurrence looks simpler, whichis given by <cit.> CN_ij=2max(0,|z|-√(ad)).CN_ij measures the pairwise entanglement between two spins at sites i and j,which varies from CN_ij=0 for a separable state toCN_ij=1 for a maximally entangled state. Variations of CN_ NN and CN_ FN for four differentlocations in the parameter space are shown in Fig. <ref> (b) and (c), respectively.CN_ NNN is zero everywhere which means that concurrence between NNN sites does not survive.CN_ NN is found to obey the relation CN_ NN=-1/2[4E_ G/N+1],for J_2=J_3=0 where E_ G is the ground state energy ofS=1/2 AFM Heisenberg chain with N sites and periodic boundarycondition <cit.>. Similar types of relations for CN_ NNN andCN_ FN are not found. CN_ NN=0.434 over the line J_2=J_3which is also maximum. This particular lineis marked by the dashed line OPin the parameter space (Fig. <ref> (d)). CN_ NN vanishes in theregion R_2.CN_ NN suffers a jump over the line L,which is the signature of a first-order QPT.In R_1, for fixed value of the frustrating bond (J_2),CN_ NN increases with J_3up to the line OP, where it acquires the maximum value. With further increase of J_3, it begins to decrease. On the other hand,CN_ FN is zero throughout the region R_2 in addition to theportion of R_1 whereJ_3≤(0.87J_2+0.14). In the region R_1, for fixed J_2, CN_ FNincreases with the increase of J_3 but for fixed J_3, itdecreases with increasing J_2. The maximum value of CN_ FN is observed over the lineJ_3=J_1 barring the point P. There is no effect of frustration onCN_ FN in the locations R_2 and L.On the other hand, they tend to decrease with the increase of J_2 in R_1. The thermal state concurrence (TCN)has been derived from ρ_ij(T) by using Eqs. (<ref>).The variations of TCN_ NN with respect tok_ BT/J_1 for the line L including the point P has been displayed in Fig. <ref> (d). TCN decreases with temperatureandexactly vanishes at the critical temperature T^ij_ c.Non-zero values for T^ NN_ c andT^ FN_ c have been observed while T^ NNN_ cis always zero. Variations of T^ NN_ c andT^ FN_ c have been shown in Fig. <ref> (a) and (b), respectively.For a fixed J_3, both T^ NN_ c and T^ FN_ c decreasevery fast with J_2 whereas for fixed J_2, they both increase slowly with J_3.The variations ofboth T^ NN_ c andT^ FN_ c with respect toJ_2 indicate that frustration opposes the bipartite entanglement in this system.§ ENTANGLEMENT WITNESSES: SUSCEPTIBILITY, FIDELITY AND INTERNAL ENERGYIn 1996, Horodeckiet. al. formulate the necessary and sufficientconditions for separability of a bipartite system <cit.>. This formulation leads tothe existence of a particular EW which is essentially a measure of violation of Bellinequality <cit.>.For a magnetic system, it has been shown that magnetic susceptibility can serveas an EW which can be applied without complete knowledge of the Hamiltonian <cit.>.For an isolatedN-spin cluster, which is SU(2) invariant and translationally symmetric,the condition of untangled states has been put forward interm of an inequality <cit.>.For the Heisenberg Hamiltonian, which is isotropic in the spin space,the magnetic susceptibility along a particular direction, α, (α=x,y,z) is given by χ_α=(gμ_ B)^2/k_ BT(⟨ M^2_α⟩-⟨ M_α⟩ ^2), where M_α=∑_i=1^N S^i_α is the magnetization along thedirection α, g is the g-factor and μ_B is the Bohr magneton. Thus,χ_α=(gμ_ B)^2/k_ BT(∑_i,j=1^N⟨ S^i_α S^j_α⟩-⟨∑_i=1^N S^i_α⟩^2). Since the Hamiltonian is isotropic in the spin space,χ=χ_x=χ_y=χ_z, or, χ=1/3(χ_x+χ_y+χ_z), and⟨∑_i=1^N S^i_α⟩=0, χ can be expressed asχ=(gμ_ B)^2/k_ BT(N/4+2/3∑_i<j⟨S⃗_i. S⃗_j⟩).The second term in the expression of χ,i. e.,the sum within the expectation value in Eq. <ref>, acts as the all-to-all spin interaction term.Alternately, in this particular case, this sum is equivalent to theHamiltonian (Eq. <ref>), at the point P when J_1=1, say, H_ P. As a result,∑_i<j⟨S⃗_i. S⃗_j⟩=⟨ H_ P⟩corresponds to the ground state energy at the point P for J_1=1.Due to AFM spin interaction the ground state expectation value ofH_ P is always negative. So, ⟨ H_ P⟩makes a negative contribution to χ.And the maximum negative value of ⟨ H_ P⟩is equal to the ground state energy of H_ P itself.It has been discussed in the next section that minimum energy of theseparable states is negative and equivalent to the ground-stateenergy of the corresponding classical Hamiltonian.For N=6, ⟨ H_ P⟩=-3/4. For any general separable states,⟨ H_ P⟩ always makelesser contribution to χ in comparison totheseparable state of minimum energy.Therefore, for a single cluster of N=6 spin the conditionof untangled states has been given by the inequalityχ≥(gμ_B)^2/k_BT.Curves describing the variation of χ/(g^2μ^2_ BJ_1) against k_ BT/J_1 arising from the above equality, Eq. <ref> and the same variation resultingfromEq. <ref> intersect at a critical temperature, T_ c, belowwhich the system is entangled. Thus, χ, (Eq. <ref>) acts as an EW.The variations of χ/(g^2μ^2_ BJ_1) against k_ BT/J_1 representingEqs. <ref> and <ref> have beenshown in Fig. <ref> (a). Eq. <ref> has been evaluated for N=6 where only NNinteractions are considered.Two curves intersects at T_ c≈ 1.43J_1/K_B.The variation of TCN_ NN with respect to k_ BT/J_1has been shown in Fig. <ref> (b), where only NN interactions are considered. This variation indicates that T_ c^ NN≈ 0.802 J_1/k_ B,where T_ c^ NN is that critical temperature beyond which the bipartite entanglement does not exist.By comparing the values of T_ c and T_ c^ NN,it is obvious that only multipartite entanglement is present in the system in the intermediate temperature range T_ c^ NN<T<T_ c. Thus, below T_ c^ NN, both bipartite and multipartite entanglements arepresent while they vanish above T_ c.The variation of k_ BT_ c/J_1 for the AFMHeisenberg hexagon has been shown in Fig. <ref> (a).T_ c has the maximum value at the point P whenJ_1=J_2=J_3,i. e., where all-to-all interactions of equal strength are present.The minimum value of T_ c appears when J_2=J_3=0, i. e., where only NN interactions are present.With the increase of both J_2 and J_3, T_ c increases steadily.But the rate of increase of T_ c with respect to J_2is more than that of J_3, which means thatfrustration enhances the multipartite entanglement in the system.In order to investigate the presence of six-qubit entanglement in theAFM Heisenberg hexagon, the state preparation fidelity, F isdefined as, F(ρ)=|Ψ_GHZ⟩ρ(T)⟨Ψ_GHZ|, where ⟨Ψ_GHZ|=1/√(2)(⟨↑↓↑↓↑↓|-⟨↓↑↓↑↓↑|)is the six-spin Greenberger-Horne-Zeilinger (GHZ) state <cit.>. The sufficient condition for the presence of six-particle entanglementin this six-qubit system is given by the inequality,F(ρ)>1/2<cit.>.For the hexagonal system with J_2=J_3=0, variation of F(ρ) againstk_ BT/J_1 has been shown in Fig. <ref> (b).The variation of ground state fidelity F in the parameter space is shown in Fig. <ref> (b). The value of Fis fixed over the line OP and that value of F is 0.458.The maximum value of F at zero temperature is 1/2 which is observedover another lineJ_3=J_1 except the point P.F, however, vanishes over the entire region R_2.The value of F just over the line L is fixed, and itsuffers a sudden jump, which is the manifestation of QPT. F decreases with the increase of T throughout the parameter space.Since F(ρ)≤1/2, the six-spin entanglement isabsent in the ground as well as the thermal statesat all temperatures.On the other hand, for S=1/2 AFMHeisenberg tetramer with NN interaction, F=2/3>1/2,which indicates the presence offour-particle entanglement in ground state <cit.>. In general,F increases with J_3 for fixed J_2 anddecrease with J_2 for fixed J_3. Therefore, frustration opposes thesix-spin entanglement in this case. Another kind of detection for EW has been introducedby Dowling and others based on a comparison betweenthe internal energy (U(T)) at finite temperature, T, andthe minimum separable energy (E_ sep) <cit.>.The entanglement gap energy, G_ E is defined byG_ E(T)=E_ sep-U(T), at non-zerotemperaturewhile that at zero temperature is given byG_ E(0)=E_ sep-E_ G, where E_ G is the ground state energy. U(T) is given by U(T)=-1/Z∂ Z/∂β. The multipartite entanglement would be present in thesystem at non-zero temperature, whenever G_ E(T)>0.With the increase of T, G_ E(T) decreasessince U(T) increases with T. Obviously, there would be a limiting valueofT above which G_ E(T)<0.This critical value of temperature, known as the entanglementgap temperature (T_ E) is define by, U(T_ E)=E_ sep.Therefore, below T_ Emultipartite entanglement is non-zero.Thus a thermal state is entangled if T<T_ E. To formalize this analysis, an EW, Z_ EW, a Hermitian operatoris introduced such that Tr[Z_ EWρ_ ent]<0, whenthere exists an entangled state, ρ_ ent. It is noted thatZ_ EW witnesses multipartite entanglement in ρ_ ent. Therefore, positive entanglement gap, G_ E(T)>0, defines theEW by the equation Z_ EW=H-IE_ sep,where I is the identity matrix on the Hilbert space.Hence, Tr[Z_ EWρ_ sep]=Tr[Hρ_ sep]-E_ sep≥ 0, when ρ_ sep is any separable state while E_sep is the lowest possibleenergy for a separable state. On the other hand, for the ground state, ρ_ G, Tr[Z_ EWρ_ G]=E_ G-E_ sep< 0 at T=0.Thus, Z_ EW serves as an EW. Generally variational methods are being employed to find thelowest possible energy for a separable state of spin chains. Otherwise, it has been noted that for AFM Heisenberg spin clusterwith all-to-all couplings of same strengths, a minimum energy separable state is given by that classicalspin configuration where the total spin vector is zero <cit.>. In order to find the separable state with minimum energyin this case, weintroduce the most general form of separable state, like, |ψ_ sep⟩=∏_j|S_j⟩, j=1,2,3,⋯, 6,where |S_j⟩=cosθ_j|↑⟩ +e^iϕ_jsinθ_j|↓⟩,0≤θ_j≤π, and 0≤ϕ_j≤ 2π. E_ sepis obtained by minimizing ⟨ψ_ sep|H|ψ_ sep⟩with respect to both θ_j and ϕ_j. By using simplex minimizingprocedure <cit.>, E_ sepis found to equal to -3/2 (J_1 - J_2) - 3/4 J_3, which essentially corresponds to θ_j=2π j/6 and ϕ_j=0.The symmetry in the Hamiltonian is responsible for thesymmetric solutions.The solutions always correspond to the classical spin configuration with the total spin vector is zero,although the condition of all-to-all couplings of same strength ismostly violated except the point P.The variations of E_ sep/J_1 and k_ BT_ E/J_1are shown in Figs. <ref> (c) and (d), respectively.Usually E_ sep is negative everywhere exceptthe extreme point, M, over the line J_1=J_2, where E_ sep becomes zero.At the point M, the value of frustration degree,is the maximum andT_ E tends to ∞, which is shown in the Fig. <ref> (d).The value of k_ BT_ c/J_1 at this point is 2.395. The bipartite entanglement vanishes over the same line including this point.Therefore, at this point multipartite entanglement survivesat all temperatures in the absence of bipartite entanglement.Besides this particular point, M, entangled states are found to exist at hightemperatures in the vicinity of the point.Thus, the entanglement in quantum states in this particular regionexhibits a robustness to the thermal noise.It appears from the expression of E_ sep that positive contributionto E_ sep only comes from the NNN frustrating bond, J_2.So, in the absence of NNN bond, E_ sep is always negative which gives rise tovery low T_ E. Therefore, the presence of frustration leadsto the high values of T_ E.This observation shows that the frustrationinduces the multipartite entanglement in this spincluster in such a manner that it does sustain against the thermal agitation. On the other extreme point P over the same line J_1=J_2, it is found that T_ E=T_ c= 2.862 J_1/K_ B.Since the bipartite entanglement vanishes over this lineonly multipartite entanglement survives in the system at P for 0<T<T_ E.The equality between T_ E and T_ c results from the fact that at this point the effective spin interactions are definedon a non-bipartite graph or lattice for which EW based on thermal energydetects only the multipartite entanglement. Now consider another point O (J_2=J_3=0) in the parameter space,where E_ sep=-1.5J_1, and the value ofT_ E is 0.802J_1/K_ B which is identical to that of T_ c^ NN.This is due to the fact that at this point the resulting spin interactions are definedon a bipartite graph or lattice and EW based on thermal energy in this casedetects only the bipartite entanglement.§ DISCUSSIONThe spin-1/2 AFM Heisenberg hexagon withall-to-all exchange couplings is considered toinvestigate the variety of entanglement properties. Four different locations, R_1, R_2, L and P have been identifiedwhere the nature of ground states are different whileQPT occurs over the line L including the point P.By exploiting its six-fold rotational symmetry three different kinds ofCNs, CN_ NN, CN_ NNN and CN_ FN are introduced and those givetotally different results.Both T_ c^ NN and T_ c^ FN decrease withthe increase of J_2 (Fig. <ref>) and ultimately vanish over the lines J_2/J_1=1 and J_3/J_1=0.87J_2/J_1+0.14, respectively.Those observations reveal the fact that the frustrationopposes the bipartite entanglement in this system.Multipartite entanglements of this model have been studied where susceptibility,fidelity and internal energy serve as the EWs. Multipartite entanglements surviveup to the temperature, T_ c which is always higher than the T_ c^ NN.Thus, the bipartite entanglement diminishes due to thermal agitation more quickly than the multipartite entanglement. Frustration leads to thehigher values of T_ c, so it favors the multipartite entanglements.Fidelity measurement indicates that this model exhibits no six-spin entanglementeven in zero temperature.Survival of multipartite entanglement at finite temperaturehas been studied in terms of internal energy as EW. Entanglement is found topersist at high temperatures in this system in the vicinity tothe point M,where value ofis the maximum.It appears that frustration is responsible for the robustness ofquantum entanglement against the thermal effect around this point. Existence of the multipartite entanglement at finite temperaturesis found in this system where the bipartite entanglement vanishes at non-zero temperatures. EW in terms of susceptibility can detect the existence of both bipartite and multipartiteentanglements collectively at finite temperatures.On the other hand, EW in terms of internal energy candetect bipartite and multipartite entanglements separately for the cases when the spin interactions are definedon bipartite and non-bipartite graphs or lattices, respectively.For this model, EW based on internal energy detects only the bipartite entanglement at the point O in the parameter space, andthat measures only the multipartite entanglement at the other point P.Therefore, at the point O,T_ c^ NN=T_ E.Similarly, T_ c becomes equal toT_ E only at the point P, whereonly multipartite entanglement survives and measured separately by the EWs based onsusceptibility and internal energy. It further appears that EW based oninternal energy detects the collective existence of both bipartiteand multipartite entanglements everywhere in the J_2-J_3 parameter spaceexcept the points O and P. Therefore, development ofmore effective EWs is necessary for precise detection of different types ofentanglements separately.Engineering of entangled quantum state at room temperature is a new challenge.So, the frustrated AFM spin models could shed light in this direction. The inelastic neutron scattering study on Cu_3WO_6reveals that spin-1/2 Cu^2+ ionsare arranged on the vertices ofhexagons in its crystalline state <cit.>.Dynamic structure factorpredicts the magnitudes of J_1, J_2 and J_3 are 78.5K, 50.4K and 40.0K, respectively.As they satisfy the relation J_1+J_3>2J_2, this system belongs tothe region R_1 having the RVB ground state, Ψ_42.Position of this compound in the J_2-J_3 parameterspace is identified by the point C.Estimations of various quantities forCu_3WO_6 yield following values. =0.51, CN_ NN=0.43,k_ BT^ NN_ c/J_1=0.50, F=0.44, k_ BT_ c/J_1=2.20,E_ sep/J_1=-0.92 andk_ BT_ E/J_1=1.33.Hence, this material is no more suitable to yield thermally stablemultipartite entanglement.Therefore, in our opinion synthesis of new AFM compoundswhose compositions as well as structures are very close to Cu_3WO_6or other one such that J_2 ≥ J_1+J_3 /2 becomes necessary for the production of thermally stable multipartite entanglement.§ ACKNOWLEDGMENTSMD acknowledges the UGC fellowship, no. 524067 (2014), India. AKG acknowledges a BRNS-sanctionedresearch project, no. 37(3)/14/16/2015, India.§ AUTHOR CONTRIBUTION STATEMENTMD did the analytical work and AKG did the numerical work. The manuscript was prepared jointly by both the authors.§ ENERGY EIGENVALUES AND EIGENSTATESIn this section, we provide the analytic expressions of all eigenvectors and corresponding eigenvalues of theHamiltonian, (Eq. <ref>). All energy eigenvalues with definite values of S_ T, S^z_ T, λ_r and phave been enlisted in the Tab. I.To express the eigenstates following notations have been used. ⟨ψ^3_n|=T^n-1⟨3|(n=1), ⟨3|=⟨↑↑↑↑↑↑|, ⟨ψ^2_n|=T^n-1⟨2|(n=1,2,3,4,5,6), ⟨2|=⟨↑↑↑↑↑↓|, ⟨ψ^1_n|_0=T^n-1⟨1|_0(n=1,2,3,4,5,6), ⟨1|_0=⟨↑↑↑↓↓↑|, ⟨ψ^1_n|_1=T^n-1⟨1|_1(n=1,2,3,4,5,6), ⟨1|_1=⟨↓↑↑↑↓↑|, ⟨ψ^1_n|_2=T^n-1⟨1|_2(n=1,2,3),⟨1|_2=⟨↓↑↑↓↑↑|, ⟨ψ^0_n|_0=T^n-1⟨0|_0(n=1,2,3,4,5,6), ⟨0|_0=⟨↑↑↑↓↓↓|, ⟨ψ^0_n|_1=T^n-1⟨0|_1(n=1,2,3,4,5,6), ⟨0|_1=⟨↑↑↓↓↑↓|, ⟨ψ^0_n|_2=T^n-1⟨0|_2(n=1,2,3,4,5,6), ⟨0|_2=⟨↑↓↑↓↓↑|, ⟨ψ^0_n|_3=T^n-1⟨0|_3(n=1,2),⟨0|_3=⟨↑↓↑↓↑↓|, ⟨ψ^-1_n|_0=T^n-1⟨-1|_0(n=1,2,3,4,5,6),⟨-1|_0=⟨↓↓↓↑↑↓|, ⟨ψ^-1_n|_1=T^n-1⟨-1|_1(n=1,2,3,4,5,6),⟨-1|_1=⟨↑↓↓↓↑↓|, ⟨ψ^-1_n|_2=T^n-1⟨-1|_2(n=1,2,3),⟨-1|_2=⟨↑↓↓↑↓↓|, ⟨ψ^-2_n|=T^n-1⟨-2|(n=1,2,3,4,5,6),⟨-2|=⟨↓↓↓↓↓↑|, ⟨ψ^-3_n|=T^n-1⟨-3|(n=1), ⟨-3|=⟨↓↓↓↓↓↓|.Here T is a unitary cyclic right shift operator such that T⟨abcdef|=⟨fabcde|, where⟨abcdef|=⟨a|⊗⟨b|⊗⟨c|⊗⟨d|⊗⟨e|⊗⟨f|. All the energy eigenstates are enlisted in the Tab. II.99AmicoL. Amico, R. Fazio, A. Osterloh and V. Vedral, Rev. Mod. 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Fehske, Phys. Rev. A68, 032318 (2003). Osterloh A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature (London)416, 608 (2002). Souza A. M. Souzaet. al., Phys. Rev. B.79, 054408 (2009).Sen A. Sen (De)et. al., Phys. Rev. Lett.101, 187202 (2008).HorodeckisM. Horodecki, P. Horodecki and R. Horodecki,Phys. Lett.A 223, 1 (1996).TerhalB. M. Terhal, Phys. Lett.A 271, 319 (2000).Wang X. Wang, Phys. Rev. A66, 044305 (2002). Sackett C. A. Sackettet. al., Nature (London)404, 256 (2000). Bennett C. H. Bennettet. al., Phys. Rev. Lett.76, 722 (1996).Nelder_Mead J. A. Nelder and R. Mead, Comput. J.7, 308 (1965).HaseM. Haseet. al., J. Phys. Soc. Jpn.65,372-375 (1996). | http://arxiv.org/abs/1706.08787v1 | {
"authors": [
"Moumita Deb",
"Asim Kumar Ghosh"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170627112758",
"title": "Thermally stable multipartite entanglements in the frustrated Heisenberg hexagon"
} |
Helmholtz-Institut Jena, 07743 Jena, Germany GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany[corresponding author ][email protected] Institut des NanoSciences de Paris, CNRS, Sorbonne Universités - UPMC Univ Paris 06, 75005, Paris, FranceGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyHelmholtz-Institut Jena, 07743 Jena, Germany GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyInstitute of Physics, Jan Kochanowski University, PL-25406 Kielce, PolandHelmholtz-Institut Jena, 07743 Jena, Germany deceased 16.12.2016 GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany I. Physikalisches Institut, Justus-Liebig-Universität Gießen, 35392 Gießen, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyHelmholtz-Institut Jena, 07743 Jena, Germany Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, 07737 Jena, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Institut für Kernphysik, Goethe-Universität, 60438 Frankfurt am Main, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany ExtreMe Matter Institute EMMI and Research Division, GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Institut für Kernphysik, Goethe-Universität, 60438 Frankfurt am Main, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyLaboratoire Kastler Brossel, Sorbonne Universités -UPMC Univ Paris 06, ENS-PSL Research University, Collège de France, CNRS, 75005 Paris, FranceDepartment of Mathematics and Physics, Kielce University of Technology,25-314 Kielce, PolandHelmholtz-Institut Jena, 07743 Jena, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Fakultät für Physik und Astronomie, Ruprecht-Karls-Universität, 69117 Heidelberg, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyHelmholtz-Institut Jena, 07743 Jena, Germany Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, 07737 Jena, GermanyLaboratoire de Physique des Lasers, CNRS, Université Paris 13, 93430 Villetaneuse, FranceHelmholtz-Institut Jena, 07743 Jena, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Institut für Kernphysik, Goethe-Universität, 60438 Frankfurt am Main, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyHelmholtz-Institut Jena, 07743 Jena, Germany Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, 07737 Jena, GermanyHelmholtz-Institut Jena, 07743 Jena, GermanyInstitut des Sciences de la Terre, UGA, CNRS, CS 40700, 38058 Grenoble, FranceGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyHelmholtz-Institut Jena, 07743 Jena, Germany GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, 07737 Jena, GermanyLaboratoire Kastler Brossel, Sorbonne Universités -UPMC Univ Paris 06, ENS-PSL Research University, Collège de France, CNRS, 75005 Paris, France Theiss Research, 7411 Eads Ave, La Jolla, CA 92037, United StatesHelmholtz-Institut Jena, 07743 Jena, GermanyHelmholtz-Institut Jena, 07743 Jena, Germany Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, 07737 Jena, GermanyHelmholtz-Institut Jena, 07743 Jena, GermanyHelmholtz-Institut Jena, 07743 Jena, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, GermanyEuropean Synchrotron Radiation Facility, 38043 Grenoble, FranceGSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Accurate spectroscopy of highly charged high-Z ions in a storage ring is demonstrated to be feasible by the use of specially adapted crystal optics. The method has been applied for the measurement of the 1s Lamb shift in hydrogen-like gold (Au^+78) in a storage ring through spectroscopy of the Lyman x rays. This measurement represents the first result obtained for a high-Z element using high-resolution wavelength-dispersive spectroscopy in the hard x-ray regime, paving the way for sensitivity to higher-order QED effects. 07.85.Fv, 07.85.Nc, 12.20.Fv, 31.30.J-, 32.30.RjWavelength-dispersive spectroscopy in the hard x-ray regime of a heavy highly-charged ion: The 1s Lamb shift in hydrogen-like gold H.F. Beyer December 30, 2023 ==================================================================================================================================The theory of Quantum Electrodynamics (QED) has been tested for light atoms with extraordinarily high accuracy <cit.>. Yet, in the recent years, measurements on muonic hydrogen (combined with the state-of-the-art QED calculations), have produced inconsistency with the results obtained from hydrogen spectroscopy <cit.>.The experimental verification of the QED predictions is still significantly less precise in the domain of extreme field strength as experienced by an electron bound to a nucleus with high atomic number Z. In contrast to low-Z ions, bound state QED corrections are still a challenge for theory since they have to be treated in all orders of αZ. Here, a very recent measurement of the hyperfine splitting in hydrogen- and lithium-like Bismuth has hints at a large disagreement with the QED predictions <cit.>.The QED corrections to the electronic binding energy, made up by the self energy and the vacuum polarization, are most important for the inner shells of high-Z systems since they approximately scale as Z^4/n^3 <cit.>, where n denotes the principal quantum number. Hydrogen and hydrogen-like ions are the most fundamental atomic systems where the QED effects can be calculated with high accuracy, thus offering a possibility of stringent experimental tests. From the experimental point of view it requires the preparation of heavy hydrogen-like ions where notably the 1s Lamb shift can be accessed via x-ray spectroscopy of an np → 1s Lyman transition from which the calculated Dirac energy plus the small QED contribution of the np level are subtracted. Such measurements have initially been performed at lower Z where ion intensities were sufficient for the use of high-resolution techniques with low detection efficiency <cit.>. With the advent of heavy-ion accelerators and storage rings the investigations could be extended to the highest nuclear charges up to Z=92 <cit.>. However the spectroscopy needed to be conducted with solid state Ge(i) detectors ensuring a high detection efficiency, although they soon faced their limits in spectral resolution. To circumvent the low resolving power of semiconductor detectors, they were replaced by specially adapted crystal spectrometers, as will be reported in this letter, and by calorimetric low-temperature detectors yielding promising results in first storage-ring experiments <cit.>. In the present experiment a pair of crystal spectrometers was used to measure the 1s Lamb shift of hydrogen-like gold accomplishing for the first time both, high-Z ions and high spectral resolution. Envisioned for a long time, the measurements have become feasible only recently because of the following developments:(i) adapted and optimized crystal-spectrometer optics, (ii) specially developed two-dimensionally position sensitive Ge(i) detectors for hard x rays with both energy and time resolution and (iii) a substantial increase of the ion-beam intensity in the Experimental Storage Ring (ESR) <cit.>.The experiment was performed at the accelerator and storage-ring facility of the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt, Germany <cit.>. Up to 10^8 of fully ionized gold atoms (Au^+79) per pulse with an initial kinetic energy of about 300 MeV per nucleon were injected into the ESR experimental storage ring (see Fig. <ref>). Here, they were stored, cooled, and decelerated to a final velocity of β = v_ion/c = 0.471 36(10). The relative momentum spread (Δ p/p) of the cooled ion beam is typically in the range of 10^-4 -10^-5. The cooled Au^+79 ions were then brought into interaction with the ESR internal gas target in the form of a supersonic gas jet overlapping with the circulating ion beam. A typical gas area density of ∼ 10^12 atoms/cm^2 guaranteed single collision conditions and a reasonably long ion beam storage times of several tens of seconds. In the present experiments, argon and krypton have been used as target gases. During each collision one ion has a chance to capture an electron from the target atom into an excited state, which then decays either directly or in a very rapid cascade to the 1s ground state of the newly formed hydrogen-like ion. About 1/3 of all down-charged ions decay (among other transitions in the cascade) via the Lyman-α_1 (2p_3/2→ 1s) transition, the accurate spectroscopy of which is the main goal of the present experiment.The Lyman-α_1 transition wavelength is measured by two twin spectrometers operated in the focussing compensated Laue (FOCAL) geometry <cit.>. This type of spectrometer is well suited to find the right compromise between superior spectral resolving power and sufficient detection efficiency in the situation of very limited source strength and the presence of strong Doppler effects. Since the radiation source moves with relativistic velocity relative to the resting detector assembly (the laboratory frame) the velocity and observation-angle dependent Doppler effect has to be taken into account. The velocity of the ion beam is set by the electron cooler, however it seems unfeasible to aim for a determination of the actual observation angle with comparable accuracy. The two identical crystal spectrometer arms are aligned perpendicular with respect to the ion beam at both sides of the interaction chamber on one common line of sight. Both spectrometers are used to measure the Lyman-α_1 transition independently of each other leading to two distinct results for the wavelength λ_1,2. In this special geometry the observation-angle dependency of the Doppler equation cancels out and the rest-frame transition wavelength λ_0 can be derived via λ_1+λ_2 = 2γ λ_0, with the velocity dependent Lorentz factor γ.The wavelengths λ_1,2 are measured with respect to a calibration line from an isotope enriched ^169Yb source. The strong and well known 63 120.44(4) eV γ transition <cit.> was selected as the main calibration line. The ion-beam velocity has been chosen such (β=0.471 36(10)), thatthe Doppler-shifted lab-frame energy of the Lyman-α_1 transition approximately coincides with this calibration energy thus avoiding systematic uncertainties due to large extrapolations. The wavelength comparison is made with respect to the dispersion plane defined by the crystals and detectors of the twin spectrometers.The actual crystal-optics layout of each FOCAL spectrometer arm is shown in the inset of Fig. <ref>. The emitted x-ray radiation is Bragg diffracted by the cylindrically bent silicon single crystal, with a bending radius of 2 m <cit.>. The diffracted x rays cross the polychromatic focus and are recorded in one of the position sensitive x-ray detectors. Due to the curvature of the crystal, the spatially wide x-ray radiation is focused to a narrow line at the edge of the Rowland circle whose diameter is equal to the crystal bending radius. The intensity of x rays emitted from the Au^78+ reaction products is too faint in order to allow the usage of a conventional crystal spectrometer geometry. For this purpose an asymmetric crystal cut has been applied with an angle deviation of χ=2^∘ from the symmetric Laue case, where the reflecting lattice planes are orientated perpendicular with respect to the principal crystal faces. This asymmetric cut leads to a broadening of the crystal reflection curve, thereby enhancing the efficiency by more than a factor of 20 <cit.>. The bent crystal is rotated by the angle χ to correct for the asymmetric cut, leading to symmetric but mirrored reflections above and below the optical axis.The position sensitive x-ray detectors are located close to the Rowland circle to make use of this focusing effect. Each spectrometer arm is equipped with one germanium microstrip detector consisting of an 11-mm-thick germanium single crystal with both anode and cathode segmented into many strips <cit.>. The cathode is divided into 128 56-mm-wide and 250-μm-high strips, whereas the anode is segmented into 48 1.2-mm-wide and 32-mm-high strips. The strips on the front and on the back side are oriented perpendicularly with respect to each other allowing a two dimensional position reconstruction if front and back side strips are combined for events with the same measured energy. The narrow strips on the front are orientated perpendicularly to the dispersive direction of the spectrometer allowing for a more accurate position determination.Both spectrometers are passively shielded by 15 mm thick lead plates and several thick tungsten diaphragms along the ray path to ensure that the majority of the detected photons stem from a diffractive process from the crystal. Additional background suppression was achieved by active shielding making use of the fact that the down-charged ions follow a different trajectory in the bending dipole magnets of the ESR, where they were recorded by a particle detector with efficiency close to 100 %.X-ray events in the germanium detectors have been taken into account only if a singly down-charged ion has been coincidently detected in the particle detector.Figure 2(a) shows the Lyman spectrum of H-like gold as measured by one of the two spectrometers by applying appropriate energy and time coincidence conditions to the data. In this way almost background free lines are revealed as can be seen in the figure. In three weeks of almost interruption free data taking about 1500 Lyman-α_1 photons per spectrometer arm could be collected. In addition to the Lyman-α, also the spatially resolved Lyman-β transitions could be recorded, clearly evidencing the high resolving power of FOCAL. The slight tilt of the lines over several horizontal strips is caused by an effect called Doppler slating. Due to the spatial extent of the 2D detector a certain observation angle interval is covered, leading to higher Doppler shifted (laboratory frame) transition energies in forward angles relative to the backward direction. Figure <ref>(b) shows the spectrum obtained by projection of the 2D image in Fig. <ref>(a) according to the tilt angle. Since the spectrometer is operated as a wavelength comparator only the relative distance Δ z_d between the main ^169Yb-γ calibration line and the Lyman-α_1 line matters. This distance was determined by fitting a 2D model function to the original (not projected) measurement data for the Lyman and the ^169Yb calibration data. The fitting results can be found in table <ref>. Here the minus sign indicates that the measured laboratory frame energy lies below the ^169Yb-γ line energy. Possible model dependencies and details of the line shape have also been addressed <cit.> by applying various fitting procedures resulting in negligible uncertainties.Besides the line spacing also the spectrometers dispersion D for both assemblies has to be measured. This was done by fitting in addition to the main ^169Yb-γ calibration line, the thulium Kβ_1,3 transitions, which are present in the calibration source spectra. The results are listed in table <ref>.By using Eq. (<ref>), with thus obtain a preliminary Lyman-α_1 transition energy of E_Ly-α_1^prel.=71 539.8(2.2) eV, which does not include any systematic effects so far.The systematic effects do not only increase the total uncertainty but may also shift the final value of the Lyman-α_1 transition energy. All possible contributions are discussed below and are listed in table <ref> with the corresponding estimated uncertainties.The first systematic effect was the temporal drift of the assembly during the three weeks of beam time. The drifts were monitored by the ^169Yb calibrations which were done every six hours. In total it was less than 100 μm for both spectrometers. With the help of the numerous calibrations the effect could be minimized, adding ±2.8 eV to the total uncertainty.If the moving x-ray source is shifted along the common line of sight between the two spectrometer arms, the FOCAL geometry corrects for that effect. However, if the source position (i.e. ion-beam–gas-target intersection region) is shifted out of that line (i.e. along the beam direction) this misalignment can not be corrected leading to a systematic deviation. For the actual position measurement of the gas-jet target a dedicated auxiliary experiment was performed in the aftermath of the beam time <cit.> andthe position of the gas-jet target was measured with an uncertainty of ±0.30 mm revealing an offset of 0.25 mm in the ion beam direction. The corrected gas-jet position represents our present best guess. However, because of the long time passed between the main and the auxiliary experiment, we need to increase the position uncertainty to ±1 mm in order to account for possible long-time changes (due to mechanical manipulations, venting and pumping, etc.). Fluctuations of this magnitude have previously been observed when checking the optical alignment of the gas-jet nozzles or when measuring the maximum overlap of the ion beam with the gas-jet. For the Lyman-α_1 transition energy it means a correction of 3.2 eV with an associated uncertainty of ±13 eV. Also the uncertainty in the ion-beam velocity has to be considered, which is mainly causedby an insufficiently accurate calibration of the high-voltage terminal of the electron cooler.Another correction to be added is due to the space charge of the electron beam. Details concerning the evaluation of these corrections and the associated uncertainties can be found in <cit.>. The influence on the total uncertainty of this systematic effect is ±4.3 eV.The last and strongest influence on the final value is given by the actual position of the germanium detector crystal inside the housing of the position sensitive x-ray detector. For its measurement, a dedicated beam time at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, has been conducted where an intense and narrow x-ray beam can be provided. The position sensitive detector was mounted on a movable table directly facing the x-ray beam. In small steps the detector was moved and the count rate of the strips as a function of the detector position was recorded. The location of the germanium crystal was thus measured relative to the outer fiducial marks, which were also used during the original experimental assembly. The findings from the ESRF measurement lead to a systematic energy decrease of 11.6 eV with an uncertainty of ±5.1 eV. Our final experimental value for the Lyman-α_1 transition energy including all statistical and systematic uncertainties (added quadratically) is given by E_Ly-α_1^exp.=71 531.5(15.0) eV.The experimental value for the 1s Lamb shift is obtained by subtracting the FOCAL value for the Lyman-α_1 transition energy from the theoretical value for the 2p_3/2 binding energy, which is sufficiently well known <cit.>. The difference between this value and the Dirac value for the 1s binding energy yields the 1s Lamb shift. With the theoretical value for the 2p_3/2 binding energy E_2p_3/2^theo.=-21 684.201(5) eV one obtains Δ E_1s^exp.=244.1(15.0) eV. In Table <ref> our result is compared to the experimental value obtained with a Ge(i) detector in an early experiment at the ESR electron cooler<cit.> and to the experimental result reported for the calorimetric low-temperature detectors which was gained in the same beam time <cit.> as our present experiment. In the last entry of the Table the theoretical value of Yerokhin and Shabaev <cit.> is given. Our present value of the Lamb shift is higher than the theoretical value and the other experimental results by about 2.5 standard deviations of the estimated experimental uncertainty.It is difficult at this stage to unambiguously pinpoint the reason behind this deviation. Without going into details of the other results which would be beyond the scope of this article, one can say that each of the measurements has been performed with different techniques, i.e. semiconductor detector at the electron cooler <cit.>, microcalorimeter at the gas jet target <cit.> and thus are prone to different systematic effects. It is important to emphasize that even though we have performed very thorough and extensive studies of the various possible systematic effects, since this is the first measurement of its kind at the storage ring, potentially underestimated or unknown systematic effects can not be fully excluded. Therefore more measurements are required in order to clarify this issue.In conclusion we performed a first measurement of the ground-state Lamb shift in a heavy H-like ion (Au^78+) using a high resolution crystal spectrometer in combination with a fast and dim source of hard x rays as present at a heavy-ion storage ring. The energy resolution corresponding to about 60 eV FWHM at 63 keV photon energy <cit.> surpasses the best semiconductor detectors by almost one order of magnitude. The achieved statistical uncertainty of 2.2 eV is groundbreaking for a crystal spectrometer operated in the region of hard x rays of H-like high-Z ions. Since storage rings are currently the only facilities routinely delivering high-Z hydrogen-like ions in large quantities, this measurement represents a very important milestone towards the challenging goal of achieving a sensitivity to higher-order QED effects as it is planned at the FAIR facility <cit.>. In a future run, particular effort has to be put into avoiding or reducing systematic uncertainties. The ion-beam velocity can already be determined with a much higher accuracy using a high-voltage divider from the Physikalisch-Technische Bundesanstalt (PTB) in the electron-cooler terminal, which will establish an absolutely calibrated velocity standard <cit.>. With a slightly modified assembly it will also be possible to measure the gas-target position relative to the detector-crystal position in situ, which will almost entirely eliminate these systematic uncertainties avoiding supplementary experiments alltogether.Furthermore, we would like to emphasize that this apparatus can also be applied for precision spectroscopy of heaviest helium-like ions (as well as other few-electron systems) which, taking into account the unprecedented resolution, would allow for resolving all the relevant fine structure levels for the first time. This is especially interesting in the light of the recent controversy with the comparison between the experimental and theoretical results for helium-like ions <cit.>. Laboratoire Kastler Brossel (LKB) is “Unité Mixte de Recherche de Sorbonne University-UPMC, de ENS-PSL Research University, du Collège de France et du CNRS n^∘ 8552”. Institut des NanoSciences de Paris (INSP) is “Unité Mixte de Recherche de Sorbonne University-UPMC et du CNRS n^∘ 7588”. This work has been partially supported by:the European Community FP7 - Capacities, contract ENSAR n^∘ 262010, the Allianz Program of the Helmholtz Association contract n^∘ EMMI HA-216 “Extremes of Density and Temperature: Cosmic Matter in the Laboratory, the Helmholtz-CAS Joint Research Group HCJRG-108 and by the German Ministry of Education and Research (BMBF) under contract 05P15RGFAA. | http://arxiv.org/abs/1706.08778v2 | {
"authors": [
"T. Gassner",
"M. Trassinelli",
"R. Heß",
"U. Spillmann",
"D. Banas",
"K. -H. Blumenhagen",
"F. Bosch",
"C. Brandau",
"W. Chen",
"C. Dimopoulou",
"E. Förster",
"R. Grisenti",
"A. Gumberidze",
"S. Hagmann",
"P. -M. Hillenbrand",
"P. Indelicato",
"P. Jagodzinski",
"T. Kämpfer",
"C. Kozhuharov",
"M. Lestinsky",
"D. Liesen",
"Y. Litvinov",
"R. Loetzsch",
"B. Manil",
"R. Märtin",
"F. Nolden",
"M. Petridis",
"M. Sanjari",
"K. Schulze",
"M. Schwemlein",
"A. Simionovici",
"M. Steck",
"T. Stöhlker",
"C. Szabo",
"S. Trotsenko",
"I. Uschmann",
"G. Weber",
"O. Wehrhan",
"N. Winckler",
"D. Winters",
"N. Winters",
"E. Ziegler",
"H. Beyer"
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"categories": [
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"published": "20170627110648",
"title": "Wavelength-dispersive spectroscopy in the hard x-ray regime of a heavy highly-charged ion: The 1s Lamb shift in hydrogen-like gold"
} |
=1 | http://arxiv.org/abs/1706.08537v2 | {
"authors": [
"Stefano Profumo",
"Tim Stefaniak",
"Laurel Stephenson Haskins"
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"published": "20170626180005",
"title": "The Not-So-Well Tempered Neutralino"
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Centralized and Distributed Sparsification for Low-Complexity Message Passing Algorithm in C-RAN Architectures Alessandro Brighente and Stefano Tomasin Department of Information Engineering, University of Padova via Gradenigo 6/B, 35131 Padova, Italy. Email: [email protected], [email protected] 16, 2017 ========================================================================================================================================================================================================================================================== C-RAN is a promising technology for fifth-generation (5G) cellular systems. However the burden imposed by the huge amount of data to be collected (in the uplink) from the RRH and processed at the BBU poses serious challenges. In order to reduce the computation effort of MMSE receiver at the BBU the Gaussian MP together with a suitable sparsification of the channel matrix can be used. In this paper we propose two sets of solutions, either centralized or distributed ones. In the centralized solutions, we propose different approaches to sparsify the channel matrix, in order to reduce the complexity of MP. However these approaches still require that all signals reaching the RRH are conveyed to the BBU, therefore the communication requirements among the backbone networkdevices are unaltered. In the decentralized solutions instead we aim at reducing both the complexity of MP at the BBU and the requirements on the RRH-BBU communication links by pre-processing the signals at the RRH and convey a reduced set of signals to the BBU.Cellular Systems; C-RAN; MP; Uplink.§ INTRODUCTIONThe fifth-generation (5G) of mobile communication systems has ambitious targets in terms (among others) of data rate, latency, number of supported users. Among the technologies envisioned to this end,C-RAN may provide the flexibility in the deployment and planning of the network, combined with powerful energy-efficient computational resources <cit.>. Indeed, since the signal processing of multiple cells is implemented in the centralized facility of the BBU, the computational resources are allocated on demand to the areas that have instantaneouslymore users, also with a better handling of inference and hand-off capabilities. On the other hand the need to process signals of many RRH poses significant challenges to the BBU. Various approaches have been proposed to reduce the huge amount of data that is exchanged in this centralized approach, including suitable quantization of either the received signal <cit.> or the log-likelihood ratios (LLRs) <cit.>. On the other hand, also the signal processing itself at the BBU is very challenging, since even a MMSE receiver requires the inversion of very large matrices. Similar problems are encountered in massive-MIMO systems with a huge number of users. About the reduction of signal processing burden in up-link detection, it has been proposedin <cit.> to cluster both users and RRH based on the distance of terminals from RRH thus parallelizing MMSE operations into small size matrix operations. A further step forward has been done in <cit.> where it is proposed to implement the MMSE receiver by the MP. By exploiting the Gaussian distribution of the noise, a simple solution is obtained where the complexity per unit network area remains constant with growing network sizes. In particular <cit.> combines MP with the sparsification approach of <cit.>, i.e., a first selection of users based on their distance from RRH reduces the size of the equivalent channel matrix before MP is applied.In this paper we leverage on the results of <cit.> to propose two sets of solutions, either centralized or distributed ones. In the centralized solutions, we propose different approaches to sparsify the channel matrix, in order to reduce the complexity of MP. However these approaches still require that all signals reaching the RRH are conveyed to the BBU, therefore the communication requirements among the backbone networkdevices are unaltered. In the decentralized solutions instead we aim at reducing both the complexity of MP at the BBU and the requirements on the RRH-BBU communication links by pre-processing the signals at the RRH and conveying a reduced set of signals to the BBU. The rest of the paper is organized as follows. We first introduce the system model in Section II. Then we propose the centralized sparsification techniques in Section III. The decentralized sparsification methods are discussed in Section IV. Numerical results are presented in Section V, before conclusions are obtained in Section VI. Notation: matrices and vectors are denoted in boldface. x^T and x^H denote the transpose and Hermitian of vector x, respectively.§ SYSTEM MODEL We consider the up-link of a cellular network with N_c cells, each one containing a BS equipped with N_a omnidirectional receive antennas (RRH). Each cell is populated by N_u MT uniformly distributed over the entire cell area, each one equipped with a single antenna and transmitting with power P. The overall network can be seen as a MIMO system, where the unit-power column vector x of size K=N_c N_u comprises the data signals of MT scaled by √(P) before transmission, whereas column vector y of size N = N_c N_a comprises all signals received by RRH. The MIMO channel model of the up-link from MT to the RRH can be written asy=√(P)Hx+w ,where H is the N × K channel matrix with entries [H]_i,j and w is the AWGN vector with iid complex Gaussian entries with zero-mean and variance N_0. The signals received by the RRH are forwarded to the BBU that aims at performing the MMSE receiver, i.e., computing x̂ = P^1/2H^H(PHH^H+N_0I)^-1y.§.§ Randomized Gaussian MP decoder TheMP algorithm can be used to solve the interference problem over sparse factor graphs <cit.>, therefore providing the solution of the MMSE receiver (<ref>). Since the received signal is affected by Gaussian noise we can use the GMP solution, and in particular we focus on the randomized RGMP of <cit.> which has been shown to have better convergence properties.In order to obtain the MMSE estimate of the transmitted signal x the proposed RGMP Algorithm exploits the knowledge of the statistical description of all the elements in (<ref>) and iteratively updates the values of mean and variance of all components of both x and y vectors. The Algorithm stops updating these values when a stopping criterion is satisfied and the MMSE estimate of x is returned. The computational complexity of the RGMP Algorithm is 𝒪(NK^2), hence itdepends on the number of users (growing quadratically with it) and receiving antennas of the system. In large systems, with many MT and RRH, the decoding process is therefore prohibitively complex. An approach to reduce the complexity is to reduce the number of non-zero entries in H over which the MP is run, i.e. applying MP on a sparsified version of H. Note that the sparsification on the one side will reduce the complexity, while on the other side provides an approximation of x̂, thus reducing the ASR (ASR) of the system. Different approaches will be analysed in the following sections: a centralised approach, where sparsification is performed at the BBU pool before RGMP decoding, and a distributed approach, where sparsification is applied as pre-coding operations at each BS. § CENTRALIZED SPARSIFICATION METHODSWith centralized sparsification methods the decoding process is entirely demanded to the central BBU pool. Then the signal received at the RRH, down-converted to base-band and converted to the digital domain, is entirely forwarded to the BBU. Hence no local processing is performed at the BS. Since no pre-processing operation is done at the BS in order to reduce the computational complexity of the decoding process, this latter task is demanded to the central BBU. We here introduce and discuss different approaches to sparsify the channel matrix by performing operations on its entries at the BBU.§.§ Sparsification based on the received power (CRPS) The first approach is based on the received power. In particular, we set to zero the channel matrix coefficients having power below a threshold value P_min.We thus obtain matrix Ĥ with entries [Ĥ]_i,j =[H]_i,j 0 . The neglected coefficients can be accounted for as additional noise into the system. In particular, defining the error matrix H̃=H-Ĥ the statistical power of noise and error N_0 becomesN̂_̂0̂= N_0+1/N∑_n=1^N∑_k=1^K|[H̃]_n,k|^2 RGMP is then run over channel Ĥ and considers as noise power N̂_̂0̂. §.§ Sparsification based on semi-orthogonality (MCOS) The second proposed approach is based on MT channels semi-orthogonality. Let us consider singularly each BS: we notice that MT having orthogonal channels do not interfere. Now, assuming that each MT signal is mainly detected by the antennas of its cell, we can ignore the contribution of the external MT since they will not significantly contribute to the computation of the MMSE. In formulas, let us consider the channel row vector h_k_1=[H]_n_1,k_1,[H]_n_2,k_1,...,[H]_n_N_a,k_1]from MT k_1 to all RRH belonging to a certain BS with indexes in the set 𝒜={n_1,n_2,...,n_N_a}. The orthogonality among channels toward the same BS is established by the internal product of the channels and we consider that two channels are semi-orthogonal if the product is below a threshold T_ prod, i.e.,|h_k_1h_k_2^H|^2< T_ prod.If MT k_1 outside the cell i is semi-orthogonal to all MT inside the cell, then entries of channel matrix H corresponding to the link between MT k_1 and all RRH of BS i are set to zero. §.§ Sparsification based on the correlation The idea is to reduce the number of rows of the channel matrix by selecting the subset 𝒮 of the antennas 𝒜(c) located in cell c. In order to chose a suitable subset and, hence, which rows to delete, we exploit the algorithms presented in <cit.>, i.e. correlation based sparsification (CBS) and mutual information based sparsification (MIBS). We denote by N_ac the number of antennas, and hence the number of rows of the channel matrix relative to c used for decoding.In formulas, we consider couples {n_1,n_2} of antennas and channel matrix rows g_n ∈𝒜(c)=[[H]_n,1,[H]_n,2,...,[H]_n,K], belonging to set 𝒜 of cell c and measure their correlation asc_n_1,n_2 = |g_n_1g_n_2^H|^2. For each cell the correlation between couples of antenna channels belonging to the considered cell is computed. Then the couple with highest correlation is selected and the antenna channel with lowest power is discarded. Its corresponding row in the channel matrix is hence set to zero. This procedure is repeated until we set to zero a number of rows equal to N_a-N_ac. A description of this method is provided in Algorithm 1. §.§ Sparsification based on the mutual information This antenna selection approach, MIBS, behaves similarly to Algorithm 1, except that correlation in step 1 is substituted by the normalized mutual information. The mutual information for a couple {n_1,n_2}∈𝒜(c) is computed asI(n_1,n_2) = log_2 ( h_n_1 ^ 2 h_n_2 ^ 2/h_n_1 ^ 2 h_n_2 ^ 2 + | h_n_1h_n_2^H |^2)whereas its normalized version is I_0(n_1,n_2) = I(n_1,n_2)/min{|log_2h_n_1^2|, |log_2h_n_2^2| }.In Algorithm 1 we replace (<ref>) with (<ref>).In both CBS and MIBS the noise power N_0 is not modified as in (<ref>), because, when deleting an antenna channel (and hence a channel matrix row), we assume that its information is contained in the other rows of the considered couple.§ DISTRIBUTED SPARSIFICATION METHODSThe centralized sparsification approach has the drawback that the entire received signal is forwarded from RRH to the central BBU. Since the requirements for a front-haul link are very stringent (multi-gigabit-per-second-capacity and few-milliseconds latency <cit.>) and this amount of data turns out to be prohibitively high for satisfying this requirements, we consider distributed sparsification solutions, which aim together at reducing both the decoding computational complexity and the amount of data flowing through the front-haul.In this section we will discuss sparsification applied as pre-coding at the BS of each cell before forwarding the received signals to theBBU. Let y_c be the received N_a-size column vector signal at the BS of cell c. If we consider a pre-coding N_r × N_a matrix B for cell c and we multiply it by the received signal we obtainBy_c = BH_N_r√(P)x̃+BH_N̅_̅r̅√(P)i+Bw,where x̃ is the vector containing signals coming from MT in set ℳ (as later discussed), H_N_r is the sub-channel matrix composed by the columns of H for users considered in ℳ, H_N̅_̅r̅ is the sub-channel matrix composed by the column of H for users ∉ℳ and is the vector containing signals coming from users ∉ℳ.Pre-coding matrix B can assume different forms and consider different number and types of users. In particular, we let G be the sub-channel matrix of users in ℳ. Then we set B = G^H, i.e. B assumes to form of the matched matrix to the considered channel. A second option provides that B is the zero-forcing matrix, i.e. B=G^H(GG^H)^-1. In the following we define different strategies to select ℳ. §.§ Selection based on the position (PSS) We first assume the knowledge of users location and, in particular, we know the cell each user belongs to. Then ℳ is the set of users located in cell c, with |ℳ|=N_u. Matrix B will hence be a N_u × N_a dimesnion matrix. Such a pre-coding operation hence reduces the number of rows of the sub-channel matrix of each cell from N_a (the number of antennas of the considered BS) to N_u. We notice that, with the pre-coding operation, noise vector entries are correlated and that the MP algorithm must be modified.Since noise power remains the same in all branches the noise level depends on n and becomes N_0 (n) = N_0∑_k=1^N_u|[B]_n,k|^2,with n ∈{1,...,N_u}, which takes into account correlation introduced by matrixB in each receiver branch. This new version of RGMP will be considered as default for henceforth presented methods. Note that this approach is sub-optimal respect to MMSE as the MP solution in this case neglects the correlation among the noise components. §.§ Selection based on received power (DRPS) In this approach MT are selected according to the received power. We select the N_p users with highest power reaching the BS of cell c, i.e. given the channel from user k to the BS in c, we compute the received power (<ref>) for each user in the cellular network, p(k) = ∑_n ∈𝒜(c)|[H]_n,k|^2and consider the N_p users with highest p(k) toward the BS of cell c. The channel matrix columns of this set of users will then compose the columns of matrix G for cell c. §.§ Selection based on mixed criterion (MSS) The third approach is a mix of the first two. In fact matrix G collects columns of both users located in cell c and the N_p most powerful users, i.e. with highest p(k), located outside cell c. § NUMERICAL RESULTS We here first present the ASR results obtained for all the sparsification methods introduced in previous sections and then discuss their computational complexity. Mostly the trade-off between ASR and computational complexity is analyzed. We consider a scenario with N_c=16 cells, each one equipped with a BS with N_a = 8 RRH. Each cell contains N_u=4 users and each user is allocated the same transmitting power P=1. Noise power is chosen to have a border cell SNR of 0 dB. In the following we assume that H is affected by both path loss (withcoefficient α=2) and Rayleigh fading, so that each entry is a zero-mean complex Gaussian random variable with variance equal to the inverse of the distance from the considered MT and the considered antenna of the BS. Channel matrix entries are i.i.d.The RGMP Algorithm is stopped when the mean of the transmitted signal does not change more than 1% in one iteration.Each method has been compared both in terms of sparsification level, i.e. the number of entries of the channel matrix H different from zero after sparsification, and channel ASR. All results have been compared with those of pure RGMP, i.e. without channel sparsification.§.§ Centralized sparsification We consider first the centralized sparsification. Fig. <ref> reports the mean ASR values vs. SNR for two parameter values of each centralized sparsification method and for RGMP without channel sparsification. ASR results for MIBS are analogous to the ones obtained with CBS, and are not reported here for brevity. With all the presented methods we can obtain good results in terms of ASR values, comparable or equal to that obtained with RGMP without channel sparsification. §.§ Distributed sparsificationDistributed sparsification has been implemented for both matched and zero forcing matrix B. Fig. <ref> reports mean ASR values vs. SNR obtained for the maximum and minimum considered users by distributed sparsification methods and for RGMP without channel sparsification. We denoted the different methods with their acronym followed by the number of considered users. We can see that the matched implementation of B outperforms the zero-forcing implementation is terms of mean ASR. Furthermore the matched implementation of all methods considering the maximum number of users, allows a better exploitation of the channel for low SNR values obtaining mean ASR values equal to the ones obtained with RGMP without channel sparsification.§.§ Computational complexity analysis We now analyse the computational complexity of the different approaches in terms of number of decoding operations after sparsification. This depends on the number of entries of Ĥ≠ 0 as eachrequires two sums over the total number of users K, operations that are repeated until the stopping criterion is satisfied. Hence the total number of operations isN_op = 2 Ks I,where s denotes the number of channel matrix entries different from 0, and I the number of message passing iterations needed to satisfy the stopping criterion.Fig. <ref> shows the ASR vs. the number of operations needed for the decoding process for the centralized sparsification methods with an SNR level of 0 dB. We notice that with semi-orthogonal-based sparsification we obtain the best performing system, with an achievable sum rate of 58 bit/s/Hz and a computational complexity of 9.2 · 10^5 operations. However notice that this implementation is not the best performing in terms of achievable sum rate, instead it is the best compromise between computational complexity and ASR. Notice that RGMP without channel sparsification obtains an ASR of 60 bit/s/Hz with a computational complexity of 2.1 · 10^6 operations. Hence the reduction of 1 · 10^6 operations comes with an ASR loss of 2 bit/s/Hz. Fig. <ref> reports the ASR vs. the number of operations needed for the decoding process for the centralized sparsification methods with an SNR level of 0 dB. We notice that the best compromise between ASR and computational complexity is obtained for MSS with matched matrix, which presents an ASR of approximately 63 bit/s/Hz with a computational complexity of 2 · 10^5 operations.Table I reports the obtained computational complexity and ASR values for the best performing parameter of each method when SNR value is 0 dB. A trade-off can be obtained, since we want to maximizethe ASR while maintaining a low computational complexity. We can hence state that all methods present a channel ASR comparable to the one obtained with pure RGMP, but generally need a significantly lower number of decoding operations. The best performing among all presented methods in terms of both computational complexity and ASR is MIBS sparsification when SNR value is 0 dB. This method needs less than half of the number of operations required by pure RGMP with an ASR loss of approximately 2 bit/s/Hz.§ CONCLUSIONS For a C-RAN system where signals coming from many RRH we have considered the problem of implementing a MMSE receiver at the BBU. In order to decrease the computational complexity a RGMP algorithm has been considered, and suitable sparsifications of the channel matrix have been introduced. We considered both centralized approaches, performed at the BBU and requiring a complete transfer of received signals from the RRHand decentralized solutions where a pre-processing is performed at the BS. This latter solution not only has been shown to be effective in terms of reduction of the computational complexity of the decoding process, but also of the amount of data flowing from the BS to the BBU, and hence of the front-haul network capacity as well as the centralization overhead. Numerical results have shown a variety of trade-off between complexity and performance (in terms of ASR) confirming that the proposed solutions are promising for an implementation of these approaches in 5G C-RAN systems. 9 CRAN C. Liu, K. Sundaresan, M. Jiang, S. Rangarajan and G. K. Chang, "The case for re-configurable backhaul in cloud-RAN based small cell networks," in Proc. IEEE INFOCOM 2013, pp. 1124-1132. Tomasin P. Baracca, S. Tomasin and N. Benvenuto, "Constellation Quantization in Constrained Backhaul Downlink Network MIMO," IEEE Trans. on Commun., vol. 60, no. 3, pp. 830-839, March 2012. Myamoto15 K. Miyamoto, S. Kuwano, J. Terada and A. Otaka, "Uplink Joint Reception with LLR Forwarding for Optical Transmission Bandwidth Reduction in MobileFronthaul," in Proc.2015 IEEE 81st Vehicular Technology Conference (VTC Spring), Glasgow, 2015.Fan C. Fan, Y. J. Zhang, and X. Yuan, "Dynamic nested clustering for parallel PHY-layer processing in cloud-RANs," IEEE Trans. Wireless Commun., vol. 15, no. 3, pp. 1881-1894, Mar. 2016.gmp C.Fan,X.Yuan,Y.J.A.Zhang Scalable Uplink Signal Detection in C-RANs via Randomized Gaussian Message Passing, arXiv:1511.09024, May2016.Kschischang F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, "Factor graphs and the sum-product algorithm," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498519, Feb. 2001.Molish A. F. Molish, M. Z. Win, Y. S. Choi, J. H. Winters, "Capacity of MIMO systems with antenna selection," IEEE Trans. Wireless Commun., vol. 4, no. 4, July 2005. Bartlet J. Bartlet, P. Rost, D. Wubben, J. Lessmann, B. Melis, G. Fettweis, "Fronthaul and Backhaul Requirements of Flexibly Centralized Radio Access", IEEE Wireless Communications, October 2015. | http://arxiv.org/abs/1706.08762v1 | {
"authors": [
"Alessandro Brighente",
"Stefano Tomasin"
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"cs.IT",
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"published": "20170627101024",
"title": "Centralized and Distributed Sparsification for Low-Complexity Message Passing Algorithm in C-RAN Architectures"
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α̅_sΛ_α_#1⟨#1⟩#1O(#1)σ_ effGPD_2 GeV MeV1 ⊗ 121⊗ 222 ⊗ 21/2DNQ_2Q^2_12.38.-t, 13.85.-t, 13.85.Dz, 14.80.BnAnalytical study of the rapidity distribution of the final state particles in deep inelastic scattering at small x is presented. We separate and analyse three sources of particle production: fragmentation of the quark-antiquark pair, accompanying coherent soft gluon radiation due to octet color exchange in the t-channel, and fragmentation of gluons that form parton distribution functions.Connection to Catani-Ciafaloni-Fiorani-Marchesini (CCFM) equations and the role of gluon reggezation are alsodiscussed. Rapidity distribution of particle multiplicity in DIS at small xB. Blok^1, Yu. Dokshitzer^2 and M. Strikman^3 ^1 Department of Physics, Technion – Israel Institute of Technology, Haifa, Israel^2 CNRS, LPTHE, University Pierre et Marie Curie, UMR 7589,Paris, France On leave of absence: St. Petersburg Nuclear Physics Institute, Gatchina, Russia ^3 Physics Department, Penn State University, University Park, PA, USADecember 30, 2023 ===========================================================================================================================================================================================================================================================================================================================================================================empty§ INTRODUCTION Perturbative QCD approach successfully describes internal structure of quark and gluon jets produced in e^+e^- annihilation into hadrons.The final states of hard lepton–hadron and hadron–hadron collisions have a more complicated structure. In this case, apart from jets formed by the partons originating from the underlying hard scattering, fragmentation of the initial state hadron(s) also contributes to the particle yield.Really, multiparticle production in hard processes is driven by radiation and successive cascading of relatively soft gluons. Soft bremsstrahlung is subject to coherence effects. Analysis of such effects in jets has led to “angular ordering” (AO) as means of organizing parton multiplication in terms of probabilistic time-like cascades <cit.>. For the space-like case the corresponding problem was addressed and solved by L. Gribov et al in <cit.> and independently by M. Ciafaloni <cit.>.It was shown that in order to formulate the probabilistic picture of the gluon radiation caused by fragmentation of a space-like parton ensemble, one has to impose AO as well, similar to the time-like jet evolution case. This result was further developed by S. Catani, F. Fiorani and G. Marchesini, and laid ground for the CCFM scheme for generating final states of small-x DIS processes in accord with the perturbative QCD <cit.>.In this letter we study pseudorapidity (angular) distribution of particles produced in small-x DIS processes.Similar to <cit.> where the inclusive energy spectra of final state particles were derived, we find three essential contributions to the answer. The first one (dn^(1)) originates from fragmentation of the struck quark and its partner antiquark (at small x it is a sea qq̅ pair that is hit by an incident lepton). This contribution dominates the particle yield at large momentum transfer q^2=-Q^2. Two more contributions, formally subleading but rather important, are due to the underlying space-like gluon cascade that produces the quark pair. One of them (dn^(2)) is driven by coherent soft gluon radiation caused by octet t-channel color exchange, and the last one (dn^(3)) — by fragmentation of relatively hard (energetic) gluons that determine the hadron structure functions.Section <ref> is devoted to derivation of corresponding analytic expressions for the spectrum of final particles. In Section<ref> we present numerical results for pseudorapidity distribution in DIS for different values of x and Q^2, both academic and realistic. § APPROXIMATION AND MAIN CONTRIBUTIONS Collinear approximation allows one to construct a probabilistic QCD cascade picture of multiparticle production and, in particular, to separate initial and final state radiation. Selecting and resuming contributions in which each power ofis accompanied by a logarithmic integration over parton transverse momentum, gives rise to the Leading LogarithmicApproximation (LLA).Because of the double-logarithmic nature of gluon radiation, the true perturbative expansion parameter for observables like particle multiplicity turns out to be not the QCD couplingitself, but rather √(). So, the next-to-leading logarithmic approximation (NLLA) effects are down by √() as compared with the LLA, etc.The general NLLA formulae describing the sub-jet structure of the DIS final state in terms of generating functionals were presented in the paper that has introduced the k_⊥–clustering algorithm for jets in DIS and hadron–hadron collisions <cit.>.This approximation, however, turns out to be insufficient to access the structure of the fragmentation of a target proton in DIS. This region provides afraction of the total particle yield, and therefore is formally of the NNLL nature. At the same time, this kinematical region widens and becomes important for small values of Bjorken x. In <cit.> it was demonstrated that the perturbative QCD analysis could be carried out, and NNLA expressions could be derived, if one looks upon ln 1/x as an additional enhancement, and selects specific NNLL contributions ln 1/x in each order of the pQCD expansion. Within this approach approximate analytic expressions for the inclusiveenergy spectrum of final particles has been derived. In this letter we extend the analysis of <cit.> to the case of thepseudorapidity (angular) distribution of particles produced in small-x DIS processes.We will treat the process in the Breit reference frame (μ=0, -q^2=Q^2, x=Q/2P≪ 1). In this frame proton fragmentation occupies positive pseudorapidities (target fragmentation region)η= - lnϑ/2,with ϑ the particle production angle with respect to the proton direction. The struck quark jet populates the region η <0 (current fragmentation region).DIS cross section at small x is dominated by space-like parton fluctuations that have the structure of a gluon ladder attached to the quark box as shown in Fig. <ref>. §.§ Quark box Fragmentation of the quark k' in Fig. <ref> gives rise to a jet with an opening angle Θ' ≃ k'_t/β' P. Here Θ' is the quark production angle, and k'_t and β'— its transverse momentum and longitudinal momentum fraction (for definition of the Sudakov decomposition of parton momenta see Appendix). This should be taken together with radiation off the virtual quark line k in the interval of Breit-frame angles Θ' ≤ϑ, thus forming a full quark jet in the target fragmentation region (proton hemisphere), similar to that in e^+e^-→ qq̅ annihilation with invariant annihilation energy s=Q^2. Similarly, the struck quark with momentum q+k produces the second jet populating the current fragmentation (photon hemisphere).Measuring preudorapidity of a final particle introduces certain competition at the level of collinear logarithmic enhancements.Consider a gluon with positive pseudorapidity. It can belong to the jet that the quark with momentum k' develops in the final state. In this case, logarithmic integration over the relative angle Θ_k', ℓ between the three-momenta of the quark, k⃗',and that of the gluon, ℓ⃗, runs in the kinematical region Θ_k', ℓ≪Θ' ≃ϑ.The energy of the quark k' is typically of the order of xP=Q/2. Thus, logarithmic integrations over the gluon energy ℓ and the relative angle (<ref>)produce the total quark jet multiplicity factor at a hardness scale Qϑ/2,k'_t≃k'_0·Θ' ∼Q/2 ϑ, _q(Qsinϑ/2).At the same time, the fact that the quark transverse momentum is fixed, corresponds to taking logarithmic derivative of the quark pdf D_h^q(x;μ^2 ). This gives rise to a contribution _q(Qsinϑ/2) ·d/dηD_h^q(x; Q^2sin^2 ϑ/2).Another logarithmic enhancement may originate from the integral over transverse momentum of the quark in the alternative region of production angles, namelyΘ' ≪ϑ≃Θ_k', ℓ .This integration gives rise to the quark pdf at the same scale Qϑ/2, while the multiplicity flow at a fixed angle ϑ is described by the derivative:D_h^q (x; Q^2sin^2 ϑ/2)·d/dη_q(Qsinϑ/2) . The two contributions (<ref>) combine into d/dη[ D_h^q(x; Q^2sin^2ϑ/2)_q(Qsinϑ/2) ]η>0.Analogous consideration applies to the radiation in the current fragmentation region, η<0 (with replacement of ϑ by π - ϑ).Finally, for entire pseudorapidity region we get an elegant expression D_h^q(x; Q^2)dn^(1)/dη = d/dη[ D_h^q(x; Q^2sin^2ϑ/2) _q(Qsinϑ/2) +D_h^q(x; Q^2cos^2ϑ/2)_q(Qcosϑ/2)] .Integrating over pseudorapidity,for accompanying particle multiplicity due to radiation off the quark box we obtainn^(1)=2 _q(Q) ,which expression coincides with the mean multiplicity in e^+e^-→ qq̅with invariant annihilation energy s=Q^2. §.§ Soft t-channel radiation If the quark box particle production is similar to that in e^+e^-→ qq̅, in small-x DIS there is an additional essential source of final particles that mimics agluon jet. It is due to coherent radiation of soft gluons ℓ with longitudinal momenta β_ℓ < x, and emission anglessmaller that the production angle of the quark k': ϑ <Θ'.Such soft gluons originate from coherent radiation off the s-channel partons at an anglelarger than their production angles. These are the partons (predominantly gluons) that are produced at an early stage of the parton system evolution, “below” the quark box in Fig. <ref>. Intensity of this coherent radiation is proportional to the color charge of the t-channel exchange, N_c.Substituting, as before,xP=Q/2 for the quark energy, the inequality (<ref>) translates into an upper limit for the quark transverse momentum in the box: k_t^2 ≈ k_t'^2 ≈ (β' P ·Θ')^2 ≃ (QΘ'/2)^2>(Qϑ/2)^2 .Integration over k_t^2 under this condition yields D_h^q (x; Q^2 ) - D_h^q (x; Q^2 sin^2 ϑ/2),0< ϑ≤π,where we have replaced ϑ/2 by sinϑ/2 to assure a smooth transition with the region of finite, η=1, and negative rapidities (large angles ϑ>1 where the collinear approximation is not applicable). As a result, the second contribution due to soft t-channel radiation takes the form D_h^q(x; Q^2)dn^(2)/dη=[ D_h^q (x; Q^2 ) - D_h^q (x; Q^2 sin^2 ϑ/2) ]·d/dη_g(Qsinϑ/2) .§.§ Fragmentation of structural gluons The soft t-channel radiation has to be combined with the fragmentation of the final state gluons that participate in formation of the quark pdf. We shall call them “structural gluons”.An inclusive gluon production cross section displayed in Fig. <ref> is given by a simple convolution of two parton distributions: ∫_μ^2^Q^2dk_t^2/k_t^2 (k_t^2)∫_x^1 dy/y D_h^g(y; k_t^2) [∫_x/y^1 dz/z Φ_g^g(z)D_g^q(x/zy; Q^2, k_t^2) ]·ϕ_g(ℓ).Hereis conveniently normalized coupling constant (see Appendix, Eq. (<ref>)), and Φ_g^g(z)— the DGLAP g→ g splitting function (<ref>). The first distribution D_h^g in (<ref>) is the customary gluon pdf. It stands for the probability to find, at a certain intermediate virtuality scale k_t^2 > μ^2, a gluon with the longitudinal momentum fraction y inside the target hadron, withμ the transverse momentum scale above which the pQCD approach can be applied. The second distribution D_g^q is a fundamental solution ofthe system of DGLAP evolution equations. It describes further evolution of the parton system, starting off from the gluon with the longitudinal momentum β_k=z· y (and an initial virtuality scale k_t^2) up to the hit quark x (Q^2). Finally, the factor ϕ_g(ℓ) encodes information about fragmentation of the gluon ℓ and depends on the observable under consideration. By virtue of the evolution equation for parton distributions, the z-convolution in the square brackets can be cast asderivative over the virtuality scale: [∫_x/y^1 dz/z Φ_g^g(z) D_g^q(x/zy; Q^2, k_t^2) ]= - [d/d ξ_k D_g^q(x/y; Q^2, k_t^2)]^(*) ,with ξ_k theevolution time parameter (<ref>).The quark distribution in the gluon, D_g^q, contains the Born contribution in which the target gluon coverts directly into a qq̅ pair without producing any s-channel gluons (quark box graph). Upon differentiation over the lower scale ξ_k, this contribution would have producedthe Φ_g^qkernel instead of the desired gluon–gluon splitting Φ_g^g. The superscript (*) stands as a reminder that the quark box term is subtracted from the derivative of D_g^q in (<ref>). §.§.§ Anomalous contribution In the kinematical region x≪ 1 structural gluons are, so to say, over-ordered. Indeed, climbing up the ladder, the longitudinal momenta of produced partons are decreasing, while their transverse momenta are strongly increasing. As a result, theemission angles are “double ordered”, and the gluons get separated by large rapidity intervals. When invariant pair energy ŝ between neighboring structural gluons becomes large, the t-channel gluon exchangereggeizes.This means that theelastic gluon exchange amplitude acquires a suppression factor ( ŝ)^α_g(t)-1,t = -k_t^2; α_g(-κ^2) -1 ≃ - ∫_μ^2^κ^2dq^2/q^2 (q^2) ,with α_g(t) denoting the gluon Regge trajectory. Physical meaning of the reggeization — suppression of the elastic amplitude due to vetoing particle production inside a large rapidity gap (the “fifth form factor”, <cit.>). In the inclusive cross section (pdf) this suppression is compensated, once again, by radiation of real gluons. In this case the gluons ℓ in Fig. <ref> are “soft” and “hard” at the same time. Namely, soft with respect to the structural gluons of preceding generations, but more energetic than the exchange line: β_k ≪β_ℓ≪β_p .In <cit.> such gluons were referred to as “anomalous”.In fact, this is nothing but radiation that compensates the so-called non-Sudakov form factor suppression in the language of the CCFM scheme of generating DIS events <cit.>. The origin of these gluons — coherent large-angle radiation off the external linespreceding the given cell. Their transverse momenta aresmaller than those flowing through the cell: ℓ_t^2 ≪ k_t^2≃ p_t'^2.The anomalous contribution reads ∫_ℓ_t^2^Q^2dk_t^2/k_t^2(k_t^2) ∫^1_β_ℓdβ_p/β_pD_h^g( β_p; k_t^2 )∫^β_ℓ_xdβ_k/β_k Φ_g^g(β_k/β_p) D_g^q( x/β_k; Q^2, k_t^2 ).It resembles the standard gluon ladder but with additional kinematical restrictions imposed on the parton energies (<ref>) and transverse momenta (<ref>) due to the presence of the anomalous gluon ℓ.In the small-x kinematics, the gluon splitting function can be approximated essentially asΦ_g^g(z)≃ 1/z. This allows us to substitute Φ_g^g(β_k/β_p) ⟹Φ_g^g(β_k/β_ℓ)·Φ_g^g(β_ℓ/β_p)and to use twice the evolution equation for parton distributions, to rewrite (<ref>) in a compact form ∫_ξ_ℓ^ξ_Q dξ_k [ d/dξ_kD_h^g( β_ℓ; k_t^2 ) ] ·[- d/dξ_k D_g^q( x/β_ℓ; Q^2, k_t^2 ) ]^(*) .Now we can simplify the k_t^2 integral in (<ref>). Integration by parts yields∫_ξ_ℓ^ξ_Q dξ_k D_h^g( β_ℓ; k_t^2)d^2/dξ_k^2 D_g^q( x/β_ℓ; Q^2, k_t^2 ) +D_h^g( β_ℓ; ℓ_t^2)[d/dξ_ℓ D_g^q( x/β_ℓ; Q^2, ℓ_t^2 )]^(*).The second line represents the surface term (k_t^2=ℓ_t^2) which cancels with the structural contribution (<ref>). (The second surface term, k_t^2=Q^2, vanishes.)Adding together (<ref>), (<ref>) and (<ref>) gives ∫_ℓ_t^2^Q^2dk_t^2/k_t^2 (k_t^2)D_h^g( β_ℓ; k_t^2 )d^2/dξ_Q^2 D_g^q( x/β_ℓ; Q^2, k_t^2 ) .Here we have replaced the second derivative over ξ_ℓ by that over ξ_Q since the parton distribution depends on the difference of evolution times, ξ_Q-ξ_ℓ.The expression (<ref>) has to be supplied with the final state factor ϕ_g(ℓ) that describes fragmentation of the gluon ℓ. This factor depends on the observable under consideration. In our case of multiplicity flow at a given pseudorapidity it reduces to the mean parton multiplicity in the gluon jet _g with the hardness parameter ℓ_t^2. The ratio of secondary parton multiplicities in gluon and quark jets is known to next-to-next-to leading order <cit.>, and numerically is close to the ratio of color factors N_c/C_F=9/4. For the sum of the structural and anomalous contributions we finally obtainD_h^q(x;Q^2) ·dn^(3)/dη=2 ∫_x^1 dβ_ℓ/β_ℓ(ℓ_t^2) _g(ℓ_t^2) ·∫_ℓ_t^2^Q^2dk_t^2/k_t^2 (k_t^2)D_h^g( β_ℓ; k_t^2 )d^2/dξ_Q^2 D_g^q( x/β_ℓ; Q^2, k_t^2 ) .Here the gluon radiation angle ϑ stays fixed, while its energy β_ℓ is integrated over. The virtuality scale ℓ_t^2 changes together with β_ℓ:ℓ_t = β_ℓ P ·ϑ = β_ℓ/x· xP ϑ≃β_ℓ/x· Q sinϑ/2.Note that for sufficiently large positive pseudorapiditiesη (small emission angles ϑ), the transverse momentum ℓ_t of the radiated gluon inside the integration region may become smaller than a critical value μ below which the pQCD approach is no longer applicable (the coupling may hit the “Landau pole”). A transverse momentum cutoffℓ_t>μ has to be introduced, and the third component (<ref>) becomes collinear sensitive. This happens for η≥log(Q/2μ).We chose a small value of μ in order to put maximal responsibility for particle production on the pQCD dynamics. We set μ^2 = 0.2 ^2 which value corresponds tothe initial scale of GRV parton distributions <cit.>.Since gluons with small transverse momentum ℓ_t∼μ do not cascade, collinear sensitivity of the answer turns out to be moderate. An uncertainty due to variation ofthe collinear cutoff is restricted to a narrow intervalof 0.5÷1 units in rapidity around η= log(Q/2μ), amounts to several percent and decreases with increase of the hardness of the process Q^2. §.§.§ Analytic estimate of the magnitude of the third component In oder to estimate relative weight ofthe third contribution, let us consider its share in the total particle yield. Introducing an integral over the full rapidity range (η>0) unties the β_ℓ and ℓ_t integrations by making them independent. Then, the convolution of the two successive parton distributions over the longitudinal momentum fraction β_ℓ, by virtue of the completeness relation, yields ∫_x^1 dβ_ℓ/β_ℓD_h^g( β_ℓ; k_t^2 )·D_g^q( x/β_ℓ; Q^2, k_t^2 ) ≃ D_h^q( x; Q^2) .Since the answer does not depend on an intermediate scale k_t^2, one immediately arrives atn^(3)=∫_μ^2^Q^2dℓ_t^2/ℓ_t^2 (ℓ_t^2)_g(ℓ_t^2) · (ξ_Q-ξ_ℓ) ×[. d^2 D_h^q(x; Q^2 )/dξ_Q^2/ D_h^q(x;Q^2)] .Given a sharp increase of the mean multiplicity factor _g with ℓ_t^2,'/∝√(),an estimate follows: dℓ_t^2/ℓ_t^2 (ℓ_t^2) ∼ξ_Q-ξ_ℓ=√().Invoking the known enhancement of the pdf scaling violation rate at small x, d/dξln D ∝√(ln x^-1), one finally obtains n^(3)/_g=ln x^-1.As envisaged in the introduction to this Section, this contribution is formally of NNLL nature, as it is proportional to the second power of √(). However, the fact that it is enhanced by the ln x factor, makes it legitimate to keep this term while neglecting other (non-enhanced) NNLL corrections.§ NUMERICAL RESULTS§.§ Ingredients The final result is given by the sum of three contributions:(<ref>), (<ref>) and (<ref>). These formulae contain three main ingredients.* _g(k^2)— multiplicity of partons in a gluon jet at hardness scale k^2. For this function we employed the analytic expression derived in the so called Modified Leading Logarithmic Approximation (MLLA). In our numerical calculations we used the parameter-free “limiting spectrum” approximation, that one obtains by setting the collinear cutoff Q_0=Λ_QCD. We chose Λ_QCD=320 , which value provided the best fit to LEP data <cit.>. In addition, for numerical analysis we use the leading order relation between parton multiplicities in quark and gluon jets _q=(C_F/C_A)_g.* D_g^q(x; Q^2,k^2)— the two-scale fundamental solution of the DGLAP evolution equations that describes distribution of sea quarks in the target gluon. This distribution is calculated numerically by inverting the corresponding LLA expression in the Mellin moment space N, known analytically, tothe x-space.* D_h^q(x; k^2) and D_h^g(x; k^2)— quark and gluon distributions in the target hadron. We employed the Gluck–Reya–Vogt (GRV) pdfs <cit.> at a low virtuality scale μ^2=0.2 ^2, and usedΛ_MS=230 to evolve them to arbitrary hardness scales k^2. This value of Λ_MS matches the above value Λ_ QCD=320that corresponds to the physical (“bremsstrahlung”, “MC”) scheme for the QCD coupling <cit.>. §.§ Figures We illustrate our results with several numerical examples. In order to demonstrate how does the pseudorapidity distribution evolves with Q^2 and x, we present here the curves for three momentum transfers, Q=10, 100 and 1000 for two values of the Bjorken variable x=10^-2 and x=10^-4.The upper curve In Figs. <ref>–<ref> shows the density of final state particles (charged hadrons) as a function ofthe Breit frame pseudorapidity. It is a sum of three contributions described above. *The leftmost curve describes fragmentation of the “quark box” given by (<ref>) (contribution 1). It is symmetric around η=0 and concentrated in the pseudorapidity interval |η|≤ln(Q/Λ_QCD).*The middle curve is due to accompanying soft gluons radiation in the target fragmentation region, (<ref>) (contribution 2).*The rightmost curve combines fragmentation of structural gluons and the anomalous contribution due to gluon reggeization, (<ref>) (contribution 3). It extends to large positive pseudorapidities and strongly increases with x decreasing. We also calculated the pseudorapidity distribution of the final particle multiplicity flow in DIS forQ^2=20^2 and x=10^-3,10^-4typical for HERA kinematics.Comparison of QCD expectations with the new HERA data <cit.> will be reported elsewhere. § CONCLUSIONS There are two sources of final state particles in DIS processes: fragmentation of partons forming pdfs, and accompanying coherent soft gluon radiation caused by t-channel color exchange. Both these sources are adequately embedded, in particular, in the CASCADE MC generator developed by G. Salam and J. Jung, which event generator incorporates both CCFM and BFKL physics <cit.>.Analytic pQCD analysis is an alternative to MC generation of events. In this letter we derived, in a compact analytic form, pQCD predictions for pseudorapidity distributions of final state particles produced in DIS processes at small Bjorken x. It would be highly desirable to reanalyzed the HERA data on hadron production<cit.> as a function dn/dy and compare them with QCD predictions.Itis straightforward to generalize and apply the above analysis to the double-differential distribution of final particles in pseudorapidity andenergy.A similar approach can be used to study the initial radiation in pp collisions at LHC. § NOTATIONSudakov decomposition:k =β_k P + k_t +α_k q',q'=q+xP,q'^2 =P^2 = 0.Coupling constant =N_c /π.In this normalization, the parton splitting functions that appear in the text are Φ_g^g(z)= 1-z/z + z/1-z + z(1-z) ,Φ_g^q(z)= 1/2N_c[ z^2 + (1-z)^2 ].Evolution time parameterdξ_k ≡(k_t^2) dk_t^2/k_t^2 .For one-loop coupling constant, ξ_k ≡ξ(k^2)=12N_c/11N_c -2n_fln1/(k^2)+99AOjet A.H. Mueller,Phys. Lett. 104 B (1981) 161; V.S. Fadin, Yad. Fiz. 37 (1983) 408.GDKTL.V. Gribov, Yu.L. Dokshitzer, S.I. Troian and V.A. Khoze, JETP Lett.45 (1987) 515[Pisma Zh. Eksp. Teor. Fiz.45 (1987) 405;Sov. Phys. JETP 67 (1988) 1303 [Zh. Eksp. Teor. Fiz. 94 (1988) 12].GDKT1 L.V. Gribov, Yu.L. Dokshitzer, V.A. Khoze and S.I. Troian, Phys. Lett. B 202 (1988) 276.Ciafaloni M. Ciafaloni,Nucl. Phys. B 296 (1988) 49. CCFM S. Catani, F. Fiorani and G. Marchesini, Phys. Lett. B 234 (1990) 339; Nucl.Phys.B 336 (1990) 18; G. Marchesini, Nucl. Phys. B 445 (1995) 49. CDW92 S. Catani, Yu.L. Dokshitzer and B.R. Webber, Phys. Lett.B 285 (1992) 291. FifthFF Yu.L. Dokshitzer and G. Marchesini,Phys. Lett. B631 (2005) 118. BookYu.L. Dokshitzer, V.A.K̇hoze, A.H. Mueller and S.I. Troian, “Basics of perturbative QCD,”Gif-sur-Yvette, France: Ed. Frontieres (1991) 274 p.DelphiP. Abreuet al. [DELPHI Collaboration],Phys. Lett. B449 (1999) 383. GRV M. Gluck, E. Reya and A. Vogt,Z. Phys. C53 (1992) 127.MCWS. Catani, B.R. Webber and G. Marchesini,Nucl. Phys. B349 (1991) 635.Salam1999 G.P. Salam, JHEP03 (1999) 009.CASCADE2001 H. Jung and G.P. Salam,Eur. Phys. J.C19 (2001) 351. H1new C. Alexaet al. [H1 Collaboration], Eur. Phys. J. C73 (2013) 4,2406. GaffMuel J.B. Gaffney and A.H. Mueller, Nucl. Phys. B250 (1985) 109. | http://arxiv.org/abs/1706.08585v2 | {
"authors": [
"B. Blok",
"Yu. Dokshitzer",
"M. Strikman"
],
"categories": [
"hep-ph",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170626203515",
"title": "Rapidity distribution of particle multiplicity in DIS at small x"
} |
Multi-spacecraft observations and transport simulations of solar energetic particles for the May 17th 2012 event M. Battarbee 1Currently at the Department of Physics, University of Helsinki, Finland J. Guo 2 S. Dalla 1 R. Wimmer-Schweingruber 2 B. Swalwell 1 D. J. Lawrence 3Received 26th June 2017 / Accepted 25th January 2018 =================================================================================================================================================================================================================================================================================================In this paper we characterize the set of polynomials f∈𝔽_q[X] satisfying the following property: there exists a positive integer d such that for any positive integer ℓ less or equal than the degree of f, there exists t_0 in 𝔽_q^d such that the polynomial f-t_0 has an irreducible factor of degree ℓ over 𝔽_q^d[X]. This result is then used to progress in the last step which is needed to remove the heuristic from one of the quasi-polynomial time algorithms for discrete logarithm problems (DLP) in small characteristic. Our characterization allows a construction of polynomials satisfying the wanted property. The method is general and can be used to tackle similar problems which involve factorization patterns of polynomials over finite fields.§ INTRODUCTION For a long time the discrete logarithm problem (DLP) over finite fields has been one of the most important primitives used for cryptographic protocols. The major breakthrough in recent years concerning DLPs in small characteristic consists of the heuristic quasi-polynomial time algorithms given in <cit.> (see also <cit.> for their origins).In this paper we focus on the algorithm in <cit.> which only relies on the field representation heuristic (see <cit.>). In fact, if that can be proved, this would show that DLP in small characteristic can indeed be solved in quasi-polynomial time. Our results characterize a class of polynomials which seem to be particularly suitable for performing the quasi-polynomial time DLP-algorithm described in <cit.> and show that if one wants to select polynomials satisfying the wanted property, these have to be chosen in this class (see Theorem <ref>). Our constructions involve some Galois theory over function fields, group theory and Chebotarev density theorem. Let us start with the motivating conjecture, which has to be proved in order to remove the remaining heuristic from the algorithm in <cit.>. For any finite field _q and any fixed positive integer ℓ≤ q+2, there exists an integer d=O(log(q)) and h_1,h_2∈_q^d[X] coprime of degree at most 2 such that h_1X^q+h_2 has an irreducible factor of degree ℓ. If this conjecture is true, then DLP in small characteristic can be solved in non-heuristic quasi-polynomial time as described in the algorithm presented in <cit.>. Such kind of requirement also appeared in <cit.>where it is observed that the choice h_1=1 and h_2=X^2-t_0 (for some well chosen t_0∈_q^d) seems to always satisfy the requirementsin odd characteristic and for d=2. This motivates us to formulate the following strongerLet 𝔽_q be a finite field of odd characteristic. There exists an integerd=O(log(q)) and h_1,h_2∈_q^d[X] coprime of degree at most 2 such that, for any positive integer ℓ≤(h_1)+qthere exists t_0∈_q^dsuch that h_1X^q+h_2-t_0 has an irreducible factor of degree ℓ. A polynomial satisfying Conjecture <ref> will allow to build extensions with the correct representation and of desired degree. Both these conjectures seem to be very hard. In this paper we make a step forward by showing a relaxed version of the stronger conjecture: in fact, we will fit the conjecture above in a general framework and will show a characterization of polynomials satisfying a weaker property than the one described in Conjecture <ref>. In particular we will be able to prove the followingLet 𝔽_q be a finite field of odd characteristic. There exists an integerd∈ and h_1,h_2∈_q^d[X] coprime of degree at most 2 such that, for any positive integer ℓ≤(h_1)+qthere exists t_0∈_q^dsuch that h_1X^q+h_2-t_0 has an irreducible factor of degree ℓ. Moreover, such polynomials can be constructed explicitly. More in general, we characterize completely (in any characteristic) polynomials f∈_q[X] having the property that there exists a d∈ such that for any ℓ≤(f), there exists t_0∈_q^d such that f-t_0 has an irreducible factor of degree ℓ in _q^d[X].On the theoretical side, our result shows the existence of such d for a certain class of polynomials, which is the first step in the attempt of giving an explicit bound. In practice, our methods are constructive and they allow to build new families of polynomials (see for example the constructions in subsection <ref>) which always satisfy the wanted requirements. Even though we can show the existence of such d for these families of polynomials, the wrinkle is that the required d might in principle be large (but in practice, if one follows our recipe, this seems to be never the case).In a nutshell, what we will do in this paper is to solve the geometric part of the problem connected with the two conjectures above and what remains to do to completely remove the heuristic is togive an explicit logarithmic bound for d for at least one polynomial in our families.The key idea of the method is the following. We look at the problem in a function field theoretical framework, explaining that the factorization conditions can be translated into group theoretical properties of the Galois closure of a certain extension L:K of global function fields. Then, we use the rigidity of group theory to determine the Galois group that can occur for the polynomials we are interested in. Finally, Chebotarev Density Theorem for global function fields will ensure that, for any fixed element γ in the Galois group, there exists an unramified place P of K for which the cycle decomposition of γ (when you look at its action on a certain set of homomorphisms) appears exactly as the splitting of P in L. The paper is structured as follows. In Section <ref> we recap the basic tools we need from algebraic number theory and group theory. In Section <ref> we characterize the monodromy groups of the class of polynomials we are interested in. In Section <ref> we specialize to the polynomial X^q+X^2 and compute its monodromy group, showing that for odd q, it is indeed the full symmetric group. In Section 5 we show other examples of polynomials of the wanted form that have symmetric monodromy group. §.§ NotationFor the entire paper p is a prime (even or odd) and q=p^a for some positive integer a. Let k:=_q be the finite field of order q. Let f∈ k[X]∖ k[X^p]. Let M_f be the splitting field of f-t over k(t), which is a separable extension of k(t). Let k̃ be the field of constants of M_f i.e. the integral closure of k in M_f. Let A_f=(M_f:k(t)) be the arithmetic monodromy group of f andG_f=(M_f:k̃(t))⊴ A_f be the geometric monodromy group of f. Let 𝒮_n be the symmetric group of degree n. Notice that if F_1,F_2 are subfields of a larger field F, we denote by F_1F_2 the the compositum of F_1 and F_2. Let G be a group acting on a set Y. For any y∈ Y we denote by _G(y) the stabilizer of y in G. § D-UNIVERSAL POLYNOMIALS AND GALOIS THEORY OVER FUNCTION FIELDSIn this section we define the notion of universal polynomial and state the basic results from global function field theory we will be using in the rest of the paper.Let f∈_q[X]. We say that f is d-universal for some positive integer d if for any positive integer ℓ≤(f), there exists t_0 in _q^d such that f(X)-t_0 has an irreducible factor of degree ℓ. We say that f is universal if it is d-universal for some d.In this notation, in <cit.> it is suggested that X^q+X^2 is 2-universal for any odd q (see section Finding appropriate h_0, h_1 of <cit.>).In what follows we will use notation and terminology of <cit.>. First, we need a classical result from algebraic number theory, which will be used to transfer splitting conditions of places into into group theoretical properties of a certain Galois group.Let L:K be a finite separable extension of global function fields and let M be its Galois closure with Galois group G. Let P be a place of K and 𝒬 be the set of places of L lying over P. Let R be a place of M lying over P. There is a natural bijection between 𝒬 and the set of orbits of H=_K(L,M) under the action of the decomposition group D(R|P)={g∈ G |g(R)=R}. In addition, let Q∈𝒬 and let H_Q be the orbit corresponding to Q. Then |H_Q|=e(Q|P)f(Q|P) where e(Q|P) and f(Q|P) are ramification index and relative degree respectively.A proof of Theorem <ref> can be found for example in <cit.>. For a finite Galois extension of function fields M:K with Galois group G, let P be a degree 1 place of K and R be a place of M lying over P. Let ϕ be the topological generator of (k:k) defined by y↦ y^q. Let k_R be the residue field at R and let ϕ_R be the image of ϕ in (k_R:k). If (R,M:K) is the set of elements in D(R|P) mapping to ϕ_R, we denote by (P,M:K) the set {gxg^-1: g∈ G, x∈ (R,M:K)}.We are now ready to state the other fundamental tool, which can easily be adapted from <cit.>.Let M:K be a finite Galois extension of function fields over a finite field k of cardinality q and let k̃ be the constant field of M. Let A=(M:K) and G=(M:k̃ K). Let γ∈ A such that γ acts as u↦ u^q when restricted to k̃. Let g∈ Gγ, Γ be the conjugacy class of g and let S_K be the set of places in K which are unramified in M. Then we have|{P∈ S_K| _k(P)=1, (P,M:K)=Γ}|=|Γ|/|G|q+2|Γ|/|G|𝔤_M q^1/2where𝔤_M is the genus of M.Theorem <ref> combined with Theorem <ref> is used to push group theoretical information to splitting statistics: the key fact is that the number of elements in the Galois group with a certain cycle decomposition (and in the correct coset of the geometric Galois group) determines the statistics of the unramified places that split according to the given cycle decomposition.Let us give an example that clarifies the procedure for the class of extensions we are interested in. Let f be a polynomial of degree n≥ 6 in _q[X] and consider the polynomial f-t∈_q(t)[X]. Set L=_q(x)=_q(t)[x]/(f(x)-t), K=_q(t), and M_f as in the notation section (i.e. the Galois closure of L:K). Observe first that _K(L,M_f) is in natural correspondence with the roots of f(x)-t in M_f, and in turn the action of A_f on _K(L,M_f) is equivalent to the action of A_f on the roots of f-t.Now, suppose for example we want to know an estimate for the number of t_0's in _q such that f-t_0 splits into two degree 2 irreducible factors and a degree n-4 irreducible factor in _q[x]. Let now γ∈ A be the Frobenius (i.e. x↦ x^q) for the field k_f. Take now the coset G_fγ and take the set Z of all elements in G_fγ with disjointcycle decomposition (-,-)(-,-)(-,…,-)_n-4 when you look at their action on the roots of f-t. Notice that Z⊆ G_fγ is a union of A_f-conjugacy classes, as A_f/G_f is cyclic. Applying Chebotarev for each of the conjugacy classes and adding the estimates together gives that the number of t_0∈_q such that f-t_0 has the wanted factorization pattern is then q|Z|/|G_f| +O(√(q)), where the implied constant can be chosen independent of q. In what follows we will only need the following special version of Chebotarev density theorem,which can be also derived from <cit.>. Let M:K be a finite Galois extension of function fields over a finite field k of cardinality q. Let G=(M:K) and assume that the field of constants of M is exactly k.Let Γ be a conjugacy class of G and let S_K be the set of places in K which are unramified in M. Then we have|{P∈ S_K| _k(P)=1, (P,M:K)=Γ}|=|Γ|/|G|q+O(q^1/2). The following easy lemma simplifies some of the proofs of the results in this paper.Let f be a separable polynomial, let k' be an extension of k, and k̃':=k'∩k̃. Then(k'M_f:k'(t))≅(M_f:k̃'(t)).First we observe that if F_1=M_f and F_2=k'(t), then F_1∩ F_2=k̃'(t). In addition, we know the Galois group of the compositum:(F_1F_2: F_1∩ F_2)=(k'M_f:k̃'(t))≅(M_f:k̃'(t))×(k'(t):k̃'(t))where the isomorphism is defined by the restriction map to M_f and k'(t). It follows easily that (k'M_f:k'(t))≅(M_f:k̃'(t)).§.§ Short Group Theory InterludeLet X be a finite set and G be a finite group. An action of G on X is said to be non-primitive if there exists an integer ℓ∈{2,… ,|G|-1} and a partition of X into X_1,… X_ℓ such that for any i∈{1,…, ℓ} and any g∈ G we have g(X_i)=X_i_g for some i_g∈{1,…,ℓ}. An action is said to be primitive if is not non-primitive.Roughly, the above definition states that an action of a group G on a set X is primitive if it does not preserve any non-trivial partition of X. We will also need the following group theory lemma, of which we include the proof for completeness. LetG be a subgroup of _n acting on U={1,…,n}. Suppose that G acts transitively on U and it contains a cycle of prime order r with r>n/2. Then G acts primitively on U. Let X_1⊔ X_2⊔…⊔ X_ℓ be a system of imprimitivity. This is the partition induced by a non trivial equivalence relation ∼ which is G-invariant (i.e. x∼ y implies gx∼ gy). Since G acts transitively, we recall that |X_i|=|X_1| for all i∈{1,…ℓ}. We argue by contradiction, by assuming 1<|X_1|<n. Consider now the cycle σ of order r and take X_j which intersects the support of σ (i.e. σ acts non trivially on X_j). Consider the orbit of X_j via σ:X_j,σ(X_j),…, σ^v-1(X_j),where v is the orbit of X_j via σ. We have that v necessarily divides r. Then both v=1 and v=r are impossible.§ A CHARACTERIZATION OF UNIVERSAL POLYNOMIALS We are now ready to prove the main result. Let f∈_q[X]. Suppose that n=(f)≥ 8, then f is universal if and only if A_f=G_f=𝒮_n. First, let us assume that f is d-universal for some positive integer d. Consider first A_f'=(_q^dM_f:_q^d(t))≤𝒮_n. Let x be any zero of f(X)-t over _q(t). From now on, we will look at A_f' as a subgroup of the permutation group of the roots of f(X)-t (or equivalently of the set H=__q^d(t)(_q^d(x),_q^d M_f)). Our first purpose is indeed to show that A_f'=_n.Let r be a prime in {⌊n/2⌋+1,…,n-3}. Such prime always exists by Bertrand Postulate (also known as Chebyshev's Theorem). Fix now t_0∈_q^d in such a way that f(X)-t_0 has an irreducible factor h(X) of degree r (over _q^d[X]). This impliesimmediatelythat the ramification at t_0 is one, as h(X)^e would have degree larger than n for any e>1. We claim that there exists γ∈ A_f' which is a cycle of order r.Let P be the place corresponding to t_0, Q be the place of_q^d(x) corresponding to the irreducible factor of degree rlying over P, and R be a place of _q^dM_f lying over Q. Let g∈ D(R|P) be such that its image in (_R/R:_P/P) under the natural reduction modulo R is the Frobenius automorphism. The order of g is then divisible byr, since an orbit of g acting on H=__q^d(t)(_q^d(x),_q^d M_f) has size r (by the natural correspondence given by Theorem <ref>). As r is prime, the only chance is that g has a cycle of order r in its decomposition in disjoint cycles. Now, as r>n/2, a certain power of g will be a cycle of order r: this is our element γ.Let us now summarize the properties of A_f' given by the d-universality: * It contains a cycle of order n/2<r<n-2 (by the previous argument and a direct application of Theorem <ref>).* Since f(X)-t_1 is irreducible for some t_1, we get thatA_f' contains a cycle of order n by a direct application of Theorem <ref>. * Analogously, it contains a cycle of order n-1. (1)+Lemma <ref> implies that A_f' is primitive, therefore, (1)+(2) implies that A_f' contains the alternating group thanks to a theorem of Jordan <cit.>. Then (2)+(3) implies that A_f' is not the alternating group. It follows that A_f'= 𝒮_n. Let us now show that A_f=A_f'. Recall that k̃ is the constant field of M_f. Let k'=_q^d and k̃'=k̃∩ k'. By Lemma <ref>𝒮_n=A'_f=(k'M_f:k'(t))=(M_f:k̃'(t)).Now, by observing (M_f:k̃'(t))≤(M_f:_q(t))=A_f≤𝒮_n we conclude A_f'=A_f. We have now to show that the field of constants of M_f is indeed _q. The only other possibility is that the field of constants is k̃=_q^2 as for n≥ 5,𝒮_n has no normal subgroups other than the alternating group 𝒜_n.The reader should notice that if d is even then k̃'=k̃=_q^2, therefore we are done by the fact that G_f=(M_f:k̃'(t))=(M_f:k̃(t)) =_n. Thus, we restrict to the case d odd. Let us argue by contradiction by supposing k'k̃=_q^2d. Suppose that n=(f) is odd, and let t_1∈_q^d for which f(x)-t_1 is irreducible of degree n. Let us denote by P_1 the place corresponding to t_1 in _q^d(t), Q⊂_q^2d(x) be the place over P_1 corresponding to the irreducible polynomial f(x)-t_1, and R a place of _q^dM_f lying over Q. Since Q is unique and unramified, then R is unramified. Therefore, D(R|P_1) is cyclic and it has exactly one orbit of order n corresponding to Q under the bijection given by Theorem <ref>. It follows that any generator of D(R|P_1) is a cycle of order n, so D(R|P_1) has order n. On the other hand, the order of D(R|P_1) is also f(R|P_1), which is divisible by[_q^2d:_q^d]=2 thus we have a contradiction. If n is even, then take t_2 for which f(x)-t_2 has an irreducible factor h(X) of degree n-1 (and therefore also a factor of degree 1). Let P_2 be the place corresponding to t_2 and Q_1, Q_2 be the places of _q^2d(x) corresponding respectively to h(X) and to the factor of degree one of f(x)-t_2. Let R be a place of _q^dM_f lying over P_2. Since Q_1 and Q_2 are the unique places of _q^2d(x) lying over P_2 and they are both unramified, then any place R lying above P_2 is unramified. Arguing similarly as before, we get that D(R|P_2) is cyclic and it has a cycle of order n-1, therefore f(R|P_2)=|D(R|P_2)|=n-1. On the other hand, since the size of the decomposition group is divided by [_q^2d:_q^d]=2, we get the contradiction we wanted.This shows that the constant field of _q^d M_f is _q^d. On the other hand, the field of constants of _q^dM_f is k̃_q^d: as d is odd, this forces k̃=_q (as the only other chance was k'=_q^2d). Let us prove the other implication. Suppose that G_f=A_f=_n and fix ℓ∈{1,…,n}. Let now γ be a cycle of G_f of order ℓ and let Γ be its conjugacy class. In the notation of Theorem <ref>, for any d∈ we have that|{P∈ S__q^d(t)| __q^d(P)=1, (P,M:K)=Γ}|=|Γ|/|G_f|q^d+O(q^d/2),where the implied constant is independent of d and q. This shows immediately that, when d is large enough, there is an unramified place P of degree 1 in _q^d(t) (corresponding to an element t_0∈_q^d) for which γ is the Frobenius (for some place of _q^dM_f lying over P). As γ is a cycle of order ℓ, by applying Theorem <ref> we get thatf(X)-t_0 has a factor of degree ℓ in _q^d[X].The philosophy behind the proof of the first implication of Theorem <ref> can be applied to prove similar statements, so we highlight the two most important steps here.The first step is to use the property we are interested in to obtain a bunch of group theoretical conditions via Theorem <ref> (conditions 1,2,3 in the proof of Theorem <ref>). Then, once the candidate arithmetic monodromy group is described, we have to understand how the property we requirefrom f (in this case universality), combined with the group theoretical properties we found, affect the possible field of constants of M_f (in our case we could prove that k_f is trivial). The critical advantage of the method is that one can use powerful group theoretical machinery to obtain complete characterization of monodromy groups. Moreover, via Chebotarev Density Theorem, the monodromy groups capture all the splitting statistics of the map f as long as the base field is large enough compared with the degree of f.The reader should notice that the second implication i.e. the last paragraph of the above proof can also be deduced by <cit.>.Suppose that f is d-universal for some d, then there exists d_0 for which f is d-universal for every d>d_0. Suppose that f is d-universal, then G_f=A_f=S_n. By the same argument as in the proof the second implication of Theorem <ref>, it follows that when d is large enough the number of t_0∈_q^d for which f(x) has an irreducible factor of degree ℓ can be estimated with |Γ|/|G_f|q^d for Γ the conjugacy class of an element having a cycle of order ℓ in its decomposition in disjoint cycles (one can actually directly select an element which "is" a cycle of order ℓ).A universal polynomial f of degree greater than or equal to 8 is indecomposable, i.e. it cannot be written as composition of lower degree polynomials By Theorem <ref>, it is enough to observe that 𝒮_n acts primitively on the roots of f. This forces the polynomial to be indecomposable (see for example <cit.>). § UNIVERSALITY FOR X^Q+X^2-TIn this section let us specialize to the polynomial f=X^q+X^2, as this is the one suggested for the function field sieve <cit.> and experimentally is believed to be2-universal, see last paragraph in <cit.>. For this section we will restrict to q odd. Let us recall a result due to Turnwald <cit.>.Let k be a field of characteristic different from 2 andg∈ k[X]. Suppose that the derivative g' of g has at least a simple root andfor any pair of roots α,β of g' over k we have that g(α)≠ g(β).In addition suppose that (k) ∤(g). Then the Galois Group of g-t over k(t) is 𝒮_(g). With this tool in hand, we are able to compute thearithmetic and the geometric monodromy group of X^q+X^2. Let (_q)≠ 2 and f=X^q+X^2∈_q[X]. The Galois Group A_f of f-t∈_q(t)[X] over _q(t) is𝒮_q. Moreover G_f=A_f. Clearly, Theorem <ref> does not apply to the polynomial above as its degree is divisible by the characteristic of the field. Let us consider the extension _q(x):_q(t) where x is a root of f-t, and then verifies f(x)=x^q+x^2=t. Let {y_1,…,y_q-1} be the set of roots of X^q-1+X+2x∈_q(x)[X]. They are all distinct, as the polynomial is separable. It is easy to see that x+y_i is a root of f-t for any i∈{1,… q-1}. Therefore, the splitting field M_f of f-t over _q(t) is exactly _q(x,y_1,…,y_q-1).Let us now considerB=(M_f:_q(x)) which is a subgroup of A_f=(M_f:_q(t)). The Galois Group B is the same as the Galois group of the polynomial X^q-1+X/-2-x over _q(x), for which Turnwald theorem applies with base field _q(x) since * (_q)≠ 2* The roots of X^q-2-1 are ξ^i for ξ a primitive (q-2)-root of unity and i∈{0,…,q-3}.* ξ^i(q-1)+ξ^i/-2=ξ^j(q-1)+ξ^j/-2 implies ξ^i=ξ^j but then i=j.We are now sure that the Galois Group of X^q-1+X/-2-xis B=_q-1. Observe that B≤ A_f and A_f acts transitively on the set of roots {x,x+y_1,x+y_2,…, x+y_q-1} and the stabilizer of x contains B. By the orbit-stabilizer theorem we have thatq=|A_f|/|_A_f(x)|≤|A_f|/|B|=|A_f|/(q-1)!Therefore |A_f|≥ q! but also |A_f|≤ q! as A_f is a subgroup of _q, so A_f=_q.We have now to show that G_f=A_f. Suppose that the constant field of M_f is k̃ and notice that all the arguments above apply again by replacing _q with k̃. Hence this immediately shows G_f=𝒮_q.There exists d_0∈ such that X^q+X^2 is d-universal for anyd>d_0. By individually checking the cases q<8 we can assume q≥ 8. By the previous result we have that Theorem <ref> applies, therefore it also applies Corollary <ref>, which is exactly the claim.The reader should notice now that the first occurence of d_1 for which X^q+X^2 is d_1 universal might be strictly less than d_0. What would be ideal to show, is thatd_0 is indeed “small" enough (conjecturally it is 2), on the other hand the above corollary at least shows that such d exists.§ CONSTRUCTING D-UNIVERSAL POLYNOMIALS IN ODD CHARACTERISTIC The combination of Theorem <ref> and Theorem <ref> gives a deterministic easy way to construct polynomials which are likely to build up any extension between the base field and the degree of the polynomial satisfying Conjecture <ref>. We give a class of examples in the next subsection. For the rest of this section, q will be an odd prime power.§.§ Universality for X^q+j-jX In this subsection we show a large class of polynomials which can be shown to be universal. In addition such polynomials appear to be always d-universal for a small d. Let q be an odd prime power and _p be its prime subfield. Let j∈∖{0,1,pk}_k∈. The polynomial f=X^q+j-jX∈_q[X] is universal.We would like to verify the conditions of Theorem <ref> for the geometric monodromy group of f, then it will follow that also the arithmetic monodromy group of f is the symmetric group, for which Theorem <ref> now applies, showing the universality of f.The derivative of f is f'=j X^q+j-1-j. Since j is different from 1, then f' has all single roots in _q. Now, any root of f has the form ξ^u, where ξ is a fixed primitive q+j-1 root of unity, and u is an integer in {0,…, q+j-2}. It is now enough to observe thatf(ξ^u)= ξ^u -j ξ^u=(1-j)ξ^u≠ (1-j)ξ^v=f(ξ^v) for u≠ vq+j-1.The conditions of Theorem <ref> are now verified and then Theorem <ref> applies, leading to the claim.The experiments show that this class of polynomials actually verifies a stronger property, i.e. each of them seems to be d-universal for d=j+1.In particular, for j=2, the polynomial X^q+2-2X is 3-universal for any prime q less than[The computations were performed in SAGE and the code is available upon request] 401 therefore building up suitable extensions of size up to 401^401. § ACKNOWLEDGEMENTSThe author is grateful to Michael Zieve for many interesting discussions and especially for introducing him to the version of Chebotarev Density Theorem used in this paper. The author also wants to thank Swiss National Science Foundation grant number 171248.plainnat | http://arxiv.org/abs/1706.08447v3 | {
"authors": [
"Giacomo Micheli"
],
"categories": [
"math.NT",
"cs.CR",
"11T71, 11T06, 11R09, 11R45"
],
"primary_category": "math.NT",
"published": "20170626155213",
"title": "On the selection of polynomials for the DLP quasi-polynomial time algorithm in small characteristic"
} |
III. A Saturn-size low-mass star at the hydrogen-burning [email protected] Laboratory, J J Thomson Avenue, Cambridge, CB3 0HE, UK Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK Observatoire Astronomique de l'Université de Genève, Chemin des Maillettes 51, CH-1290 Sauverny, Switzerland Astrophysics Group, Keele University, Staffordshire, ST55BG, UK Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, 8042, Graz, Austria Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, GermanySUPA, School of Physics & Astronomy, University of St Andrews, North Haugh, KY16 9SS, St Andrews, Fife, Scotland, UK Department of Physics, University of Warwick, Coventry CV4 7AL, UK Université de Liège, Allée du 6 août 17, Sart Tilman, 4000, Liège 1, Belgium Instituto de Astronomía, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México, 04510, México Hobart and William Smith Colleges, Department of Physics, Geneva, NY 14456, USA We report the discovery of an eclipsing binary system with mass-ratio q ∼ 0.07. After identifying a periodic photometric signal received by WASP, we obtained CORALIE spectroscopic radial velocities and follow-up light curves with the Euler and TRAPPIST telescopes. From a joint fit of these data we determine that EBLM J0555-57 consists of a sun-like primary star that is eclipsed by a low-mass companion, on a weakly eccentric 7.8-day orbit. Using a mass estimate for the primary star derived from stellar models, we determine a companion mass of 85 ± 4 M_ Jup (0.081M_⊙) and a radius of 0.84^+0.14_-0.04 R_ Jup (0.084 R_⊙) that is comparable to that of Saturn. EBLM J0555-57Ab has a surface gravity log g_2 = 5.50^+0.03_-0.13 and is one of the densest non-stellar-remnant objects currently known. These measurements are consistent with models of low-mass stars.short title name(s) of author(s) The EBLM projectThe data is publicly available at the CDS Strasbourg and on demand from the first author. Alexander von Boetticher1,2 Amaury H.M.J. Triaud2 Didier Queloz1,3 Sam Gill4 Monika Lendl5,6 Laetitia Delrez1, 9 David R. Anderson4 Andrew Collier Cameron7 Francesca Faedi8 Michaël Gillon9 Yilen Gómez Maqueo Chew10 Leslie Hebb11 Coel Hellier4 Emmanuël Jehin9 Pierre F.L. Maxted4 David V. Martin3 Francesco Pepe 3 Don Pollacco8 Damien Ségransan 3 Barry Smalley4 Stéphane Udry 3 Richard West812. June, 2017 ===========================================================================================================================================================================================================================================================================================================================================================================================================Eclipsing binary stars enable empirical measurements of the stellar mass-radius relation. The low-mass regime, down to the hydrogen-burning mass limit, is poorly constrained by measurements of mass and radius, but is of particular relevance to the study of exoplanets. Stars with masses below 0.25 M_⊙ are the most common stellar objects <cit.> and prove to be excellent candidates for the detection of Earth-sized planets <cit.> and their atmospheric characterization <cit.>. Determining the properties of exoplanets requires an accurate knowledge of their host star parameters, in particular the stellar mass. This motivates the study of low-mass eclipsing binaries (henceforth EBLMs) <cit.>, to empirically measure the mass-radius relation. In this context, we report our results on the eclipsing binary EBLM J0555-57. The system was detected by the Wide Angle Search for Planets <cit.>, and was identified as a non-planetary false-positive through follow-up measurements with the CORALIE spectrograph. We use radial velocities and two eclipse observations by the TRAPPIST and Euler telescopes, to determine the mass and radius of EBLM J0555-57Ab, to 85.2^+4.0_-3.0 M_ Jup (0.081M_⊙) and 0.84^+0.14_-0.04 R_ Jup (0.084R_⊙). This places EBLM J0555-57Ab at the minimum of the stellar mass-radius relation.§ OBSERVATIONS The source 1SWASPJ055532.69-571726.0 (EBLM J0555-57, J0555-57 for brevity) was observed by WASP-South between 2008-09-29 and 2012-03-22. The Hunter algorithm <cit.> detected 17 transit-like signals from 34 091 observations over four seasons, at a period of 7.7576 days. We obtained 30 spectra of EBLM J0555-57A, using the high-resolution fibre-fed CORALIE échelle-spectrograph <cit.>, mounted on the Euler telescope, between 2013-11-14 and 2017-01-21. Two eclipse observations in the near-infrared z'-band were obtained with the Euler <cit.> and TRAPPIST <cit.> telescopes, on the nights of 2014-02-24 and 2015-12-23 respectively. The observations reveal that our target was blended by a star that we label as J0555-57B. To confirm the source of the transit signal we compared observations with a large 38-pixel (px) aperture encompassing both stars, and a small 16 px aperture centred on the brighter star. A deeper transit signal was observed with the small aperture, identifying J0555-57A as the source of the eclipse signal. One spectrum of EBLM J0555-57B was obtained. The systemic radial velocities of the A and B components, γ_A = 19.537 ± 0.015 kms^-1 and γ_B = 19.968 ± 0.021 kms^-1, are nearly identical. A very similar position angle of the B component is observed on (blended) 2MASS images from 1999 and the Euler image, which shows that A and B also share the same proper motion. This confirms that EBLM J0555-57A, B, and the transiting EBLM J0555-57Ab constitute a hierarchical triple system. Focused images in the B, V, R, and z'-bands were obtained with Euler on 2014-02-23 and 2016-01-10. We measured the separation between the primary and blend star to be 2.48 ± 0.01”, with a position angle, PA = -105.57 ± 0.23^∘. The magnitude difference, Δ z' =0.753± 0.035 mag, translates into a flux-dilution of the eclipse depth by a factor 1.500 ± 0.016. The eclipse observations were reduced to obtain a photometric light-curve, as described in <cit.> and <cit.> for Euler and TRAPPIST, respectively.A significant out-of-transit observation before ingress was obtained, but few out-of-transit measurements after egress could be made. Using literature broadband optical photometry and 2MASS J, H, and K magnitudes for the A and B components combined, together with the multi-colour magnitude differences (Δ b =0.95± 0.01 mag, Δ r =0.786± 0.017 mag, Δ v = 0.832± 0.014 mag), we estimated IRFM temperatures <cit.> of 6450 ± 200 K and 5950 ± 200 K for the A and B components, respectively. A comparison to stellar model fluxes <cit.> was used, assuming a solar composition. The multi-colour observations and a GAIA DR1 parallax measurement <cit.> were used to derive individual radii, R_ A = 1.17 ± 0.10 R_⊙, and R_ B = 0.94 ± 0.08 R_⊙. The parallax and angular separation of the A and B components determine a projected outer semi-major axis a_AB,p = 479 ± 38 au.§ DATA ANALYSIS§.§ Radial velocitiesRadial velocities of EBLM J0555-57A were extracted by cross-correlating individual spectra with a numerical G2 mask <cit.>. Varying seeing conditions resulted in fluctuations in the amount of flux from J0555-57B that enters the CORALIE fibre. This contamination can be identified by the full-width at half-maximum (FWHM) of the cross-correlation function (CCF). To select the non-contaminated spectra, we assumed two populations of points, a contaminated sample and a clean sample, with distinct means and variances.Following <cit.>, a Markov chain Monte Carlo (MCMC) sampler was used to marginalise over the clean sample mean and variance, the contaminated sample mean and variance, and the prior probability that any point comes from the contaminated sample. We rejected a radial-velocity measurement and its associated spectrum when the FWHM had a posterior probability < 1% to originate from the clean distribution, as indicated in Fig. <ref>. We excluded one point with a discrepant value in the span of the bisector inverse slope. §.§ Spectral AnalysisAtmospheric parameters were obtained via a wavelet-based Monte Carlo method<cit.>. The 18 spectra identified as uncontaminated were median-combined onto an identically sampled wavelength grid. After continuum regions were determined and normalised with spline functions, the spectrum was decomposed using a discrete Daubechies (k = 4) wavelet transform. We filtered out wavelet coefficients that corresponded to high-order noise and low-order systematics, associated with poor continuum placement. A grid of models was generated with the radiative transfer code SPECTRUM <cit.>, using MARCS model atmospheres <cit.>, and version 5 of the GES atomic line list using iSpec <cit.>, with solar abundances from <cit.>. Filtered coefficients were compared to those from the grid of models using an MCMC sampler implemented in emcee <cit.>. We used four free parameters, T_ eff, [Fe/H], log g, and vsin i_⋆, in the range 4000 – 8000 K (250 K steps), l3.5 – 5 dex (logg, 0.25 dex steps) and -1 – 1 dex (Fe/H, 0.5 dex steps).The median value of the cumulative posterior probability distribution was used to estimate the atmospheric parameters for J0555-57A (Table <ref>). The precision associated with the wavelet method underestimates the uncertainty, so we adopt uncertainties by <cit.> for the synthetic spectral fitting technique of GAIA FGK benchmark stars (124 K, 0.21 dex, and 0.14 dex for T_ eff, log g, and [Fe/H], respectively). The spectroscopic temperature measurements,T_ effA = 6461 ± 124 K (18 spectra) and T_ effB = 5717 ± 124 K (1 spectrum) are consistent with the initial IRFM estimates.§ MODEL OF THE DATA The radial velocity and light curves were modeled using the ellc binary star model <cit.>[We validated ellc on two EBLM systems published in <cit.>, reaching a 1σ-agreement on the derived parameters.].An MCMC sampler <cit.>, was used to fit the transit light-curves and radial velocities in a framework similar to that described in <cit.>. We used the Bayesian information criterion (BIC) <cit.> to compare detrending baselines of varying complexity in time-, position-, FWHM-, and background-dependence. A flat baseline with a linear background subtraction is preferred for both light curves. Indeed both observations show a large fluctuation in the background flux. The parameters used in the MCMC sampling are the period P, the mid-transit time t_0, the observed transit depth D_ obs, the transit duration W, the impact parameter b, the semi-amplitude K, the parameters √(e)sinω, √(e)cosω, and the systemic velocity γ. The RV sample was separated into two parts, with distinct systemic velocities, to account for a change in the zero-point of CORALIE after a recent upgrade <cit.>. The geometric parameters of the system, R_1 / a, R_2 / a,and i were derived from the MCMC parameters using the formalism from <cit.>, and were then passed to ellc.The TRAPPIST sequence was interrupted by a meridian flip; to account for a possible systematic offset in the flux measurement, an offset-factor was included for measurements before the flip.We used a quadratic limb-darkening law in the MCMC analysis, with a Gaussian prior on coefficients that were interpolated from <cit.>, using the spectroscopic parameters T_ eff, [Fe/H] and log g. We included nuisance parameters in the MCMC sampler, that scale uncertainties in the photometry and radial velocity to account for white noise. The baseline parameters for a linear background subtraction, meridian flip, and normalization are fitted by a least-squares algorithm.Where not explicitly stated otherwise, we used unbounded, or sensibly bounded uniform priors to constrain parameters to physical intervals, for instance (0 < e < 1). The B-component dilutes the transit depth by a factor f_d = f_ A/(f_ A + f_ B), where f_A and f_B denote the flux from the A and B components respectively. We sampled a Gaussian prior on this depth dilution factor, f_d = 1.500 ± 0.016, to compute the true transit depth D_ calc at every step in the Markov chains. This calculated transit depth was used in the derivation of the physical parameters.We analyzed this first global fit for correlated noise in the photometry <cit.>. The light curve was binned in the range of 10 to 30 min and the maximum root-mean square (RMS) deviation of the residuals in this bin range was determined. The flux uncertainties were then rescaled by the ratio of the maximum binned RMS deviation to the RMS deviation of the un-binned residuals. This increased the uncertainties by factors of 2.02 and 1.37 for TRAPPIST and Euler respectively. We then performed a global MCMC fit using 100 chains of 10 000 steps each.The modes of the marginalised posterior distributions for each jump parameter are reported with upper and lower 68% confidence intervals.The physical parameters of the system were derived from the MCMC parameters, in particular the parameter log g_2, which is independent of the primary star mass <cit.>. We used the primary star density to iteratively refine the primary mass estimate M_1. An initial primary density was estimated from the transit and was used to determine a primary mass using bagemass <cit.>. bagemass uses stellar evolution models by <cit.>. The primary star mass was then used with the transit and radial velocity model, to compute an updated density, and we proceeded iteratively. The calculated density was found to be consistent from the first iteration step. § RESULTS Independently of any assumptions for the primary star, we obtain a surface gravity log g_2 =5.50^+0.03_-0.13 for EBLM J0555-57Ab, comparable to that of the recently announced brown dwarf EPIC 201702477b <cit.>.We determine a mass function f(m) = 0.0003686 ^+0.0000037_-0.0000049 M_⊙. Using the primary star mass determined with bagemass, we find a stellar companion with mass 85.2^+4.0_-3.9 M_ Jup (0.0813^+0.0038_-0.0037 M_⊙) and radius 0.84^+0.14_-0.04 R_ Jup (0.084^+0.014_-0.004 R_⊙). This implies a mass ratio q = 0.0721^+0.0019_-0.0017. A lower uncertainty in the radius measurement may be achievable by high-precision photometry <cit.>. The fit of the radial velocity results in anRMS deviation of 65 ms^-1, and our analysis reveals a low but significant orbital eccentricity, e = 0.0894^+0.0035_-0.0036. The BIC of a forced circular fit, and the Lucy-Sweeney test <cit.> validate this orbital eccentricity, since its measurement is significant at ∼25σ. The non-zero eccentricity of EBLM J0555-57Ab could indicate a previous orbital decay, for instance by Kozai-Lidov oscillations <cit.> induced by J0555-57B, or an undetected body, followed by tidal friction <cit.>. At the current semi-major axis, a = 0.0817 au, such Kozai-Lidov oscillations are likely suppressed by general-relativistic precession <cit.>. It is unlikely that a contamination of the spectra causes the measured non-zero eccentricity, but further spectroscopic observations with a fibre of smaller diameter can clarify this. We note a discrepancy between the spectroscopic logg_ 1spec = 4.18 ± 0.21 and that derived from the calculated radius and prior mass, logg_1 = 4.5^+0.03_-0.13. Spectroscopic measurements of logg are known to be poorly constrained <cit.>. We verify that adopting a prior on logg_1 for the spectroscopic analysis, using the derived value, leads to a primary and companion mass and radius that are consistent with the previous result.We conclude that EBLM J0555-57Ab is located just above the hydrogen-burning mass limit that separates stellar and sub-stellar objects (∼83 M_ Jup for objects with [M/H] = -0.5; <cit.>). In Figure <ref> we show the posterior distribution of J0555-57Ab on the mass-radius diagram for brown dwarfs and low-mass stars. Our results using bagemass indicate an age of 1.9 ± 1.2 Gy for J0555-57A. The mass and radius of J0555-57Ab are consistent with models of a metal-poor, low-mass star. J0555-57Ab does not show evidence of a radius that is inflated, for instance by magnetic fields, as hypothesized by <cit.> for low-mass stars. With its location on the lower bound of the mass-radius relation for stellar objects, J0555-57Ab is a critical object in the empirical calibration of the mass-radius relation in this regime. J0555-57Ab has a mass similar to that of TRAPPIST-1A. <cit.>. The low radius of EBLM J0555-57Ab, comparable to that of the low-mass star 2MASS J0523-1403 <cit.>, demonstrates the size dispersion for low-mass stars. It is essential that such variations are understood as we prepare for the detection of multi-planetary systems orbiting ultra-cool dwarfs by experiments such as SPECULOOS <cit.>. We thank the anonymous referee for valuable comments that improved the manuscript. The Swiss Euler Telescope is funded by the Swiss National Science Foundation. TRAPPIST-South is a project funded by the Belgian Fonds (National) de la Recherche Scientifique (F.R.S.-FNRS) under grant FRFC 2.5.594.09.F, with the participation of the Swiss National Science Foundation (FNS/SNSF). WASP-South is hosted by the South African Astronomical Observatory and we are grateful for their ongoing support and assistance. L. Delrez acknowledges support from the Gruber Foundation Fellowship. M. Gillon and E. Jehin are Belgian F.R.S.-FNRS Research Associates. This work was partially supported by a grant from the Simons Foundation (PI Queloz, grant number 327127). aa § RADIAL-VELOCITY DATA | http://arxiv.org/abs/1706.08781v2 | {
"authors": [
"Alexander von Boetticher",
"Amaury H. M. J. Triaud",
"Didier Queloz",
"Sam Gill",
"Monika Lendl",
"Laetitia Delrez",
"David R. Anderson",
"Andrew Collier Cameron",
"Francesca Faedi",
"Michaël Gillon",
"Yilen Gómez Maqueo Chew",
"Leslie Hebb",
"Coel Hellier",
"Emmanuël Jehin",
"Pierre F. L. Maxted",
"David V. Martin",
"Francesco Pepe",
"Don Pollacco",
"Damien Ségransan",
"Barry Smalley",
"Stéphane Udry",
"Richard West"
],
"categories": [
"astro-ph.SR",
"astro-ph.EP"
],
"primary_category": "astro-ph.SR",
"published": "20170627111712",
"title": "The EBLM project III. A Saturn-size low-mass star at the hydrogen-burning limit"
} |
A Fully Quaternion-Valued Capon Beamformer Based on Crossed-Dipole Arrays Xiang Lan and Wei LiuCommunications Research GroupDepartment of Electronic and Electrical EngineeringUniversity of Sheffield, UK December 30, 2023 ========================================================================================================================================== Citation metrics are analytic measures used to evaluate the usage, impact and dissemination of scientific research. Traditionally, citation metrics have been independently measured at each level of the publication pyramid, namely: at the article-level, at the author-level, and at the journal-level. The most commonly used metrics have been focused on journal-level measurements, such as the impact factor (IF) and the Eigenfactor, as well as on researcher-level metrics like the Hirsch index (h-index) and i10-index. On the other hand, reliable article-level metrics are less widespread, and are often reserved to non-standardized characteristics of individual articles, such as views, citations, downloads, and mentions in social and news media. These characteristics are known as “altmetrics”. However, when the number of views and citations are similar between two articles, no discriminating measure currently exists with which to assess and compare each article’s individual impact. Given the modern exponentially-growing scientific literature,scientists and readers of science need reliable, objective methods for managing, measuring and comparing research outputs and publications. To this end, I hereby describe and propose a new standardized article-level metric henceforth known as the “Individual Impact Index (i^3) Statistic”. The i^3 is a weighted algorithm that takes advantage of the peer-review process, and considers a number of characteristics of individual scientific publications in order to yield a standardized and readily comparable measure of impact and dissemination. The strengths, limitations and potential uses of this novel metric are also discussed.§ INTRODUCTIONIn scholarly and scientific publishing, citation metrics are used to evaluate the usage and impact of scientific research <cit.>. Traditional citation metrics have usually been focused on journal-level measurements such as the impact factor (IF) and the Eigenfactor, as well as on researcher-level metrics like the Hirsch index (h-index) <cit.>. On the other hand, standardized article-level metrics are less widespread, and are often reserved to non-standardized characteristics of individual articles, such as views, citations, downloads, and mentions in social and news media <cit.>. These are commonly referred to as “altmetrics”, and provide a record of attention, a measure of dissemination and an indicator of influence and impact of a given article <cit.>. However, there are important limitations to such article-level metrics since they do not distinguish between self-views and views by others, downloads by scientists or by laypersons, and self-disseminations through social media or press releases <cit.>.Additionally, when the number of views and citations are similar between two articles, no discriminating measure currently exists with which to assess and compare each article's individual impact. Yet, two publications with a similar number of citations can vary immensely in terms of quality and reach. Indeed, a comprehensive tool designed to appraise the individual impact of scholarly works is necessary to aid in such assessment. Given the aforementioned considerations, I hereby describe and propose a new article-level metric henceforth known as the “Individual Impact Index (i^3) Statistic”: a weighted algorithm that takes into account the scientific source and domain of the publication, the number of citations, as well as the provenance of those citations in order to yield a standardized and readily comparable measure of impact and dissemination for scholarly publications.§.§ What is understood by Impact in Science?The proper meaning of ‘impact’ in Science is not conveyed by a single definition. Rather, how one defines impact in Science may vary depending on the circumstance. A scholarly work that describes a new discovery or potential novel solution to a longstanding problem may change the way in which a scientific or technological process is undertaken. Certainly, that is impactful in the proper sense. However, in publication metrics, impact is not traditionally measured by the effect of a particular research, but rather, impact can more readily be defined as a measure of dissemination and reach of the scientific information in question <cit.>. Although works of major impact are invariably well disseminated, characterizing and standardizing that reach is a difficult process to undertake, as no reliable method or tool currently exists. The i^3 statistic seeks to fill that void. §.§ What are Citation Metrics?How do we then assess quality in scientific research and scholarly publications? The only true way to assess quality would be to subject and compare individual publications to a given standard, assigning an arbitrary measure of quality to a number of characteristics shared with the standard by the publication in question, such as presentation, methodology, and validity of conclusions <cit.>. However, imagine that as an avid reader of Science, you seek to learn something new. How can a novice assess the quality of a publication on a subject he/she knows nothing about? Even if an individual mastered the topic at hand, establishing true quality is cumbersome, as it would imply having to read, analyse, and re-test every conclusion in the publication in question. Instead, we establish subjective quality with the help of citation metrics <cit.>.Citation metrics are specific measurements of activity that quantify the usage and dissemination of scholarly works <cit.>. Greater usage implies that other scientists and individuals with expertise have accessed, shared and cited the publication in question. Because greater usage implies greater acceptance by fellow scientists, usage and dissemination are commonly used as a proxy for quality in Science.§.§ Levels of Citation MetricsCitation metrics can be studied at different levels in the publication pathway: the journal, the author and the article <cit.>. Essentially, the same question is asked at each level: is the journal/author/paper good/impactful? Journal-level metrics reveal the influence of a journal in communicating the most relevant research. Most scientists will point to Thomson Reuters's Journal of Citation Reports (JCR) Impact Factor (IF) as an external and objective measure for ranking the impact of specific journals <cit.>. The impact factor is a measure reflecting the yearly average number of citations to recent articles published in a given journal. Journals with higher impact factors are often deemed to be more important and reputable than those with lower ones. Though their use is less widespread, other journal-level metrics include the Eigenfactor and the 5-year Impact Factor. Author-level metrics are less widespread than journal-level metrics. They serve a more indirect purpose than journal- or article-level metrics. These metrics allow for comparisons between researchers on productivity and impact. While these may aid in funding of grants, distribution of resources, and hiring decisions, they are traditionally of little use to other scientists. The Hirsch Index (h-index) is the main author-level metric used today. It is a measure reflecting the maximum number of articles in an author's repertoire having at least that same number of citations. On the other hand, as mentioned previously, scientifically rigorous article-level metrics are less widespread, and are often reserved to non-standardized characteristics of individual articles. Although many attempts at developing new article-level metrics have been undertaken, the number of citations remains at the helm of the majority of these. §.§ Journal of Citation Reports (JCR)The Journal Citation Reports (JCR) is an annual publication by Clarivate Analytics (previously the Intellectual Property and Science business of Thomson Reuters), which provides systematic and objective means to evaluate the world's leading scientific and scholarly journals <cit.>. The JCR is an authoritative resource for impact factors in academic journals in the sciences (SCIE) and social sciences (SSCI) disciplines, publishing annual information on impact factors and other journal-level metrics, such as the immediacy index <cit.>. Other information provided by the JCR includes basic bibliographic information of the journals such as the publisher, title abbreviation, language, and ISSN identification. Furthermore, JCR categorizes journals into 171 categories in the sciences and 54 in the social sciences. Simply put, these subject categories refer to the different disciplines into which academic journals are classified <cit.>. The impact factor information allows therefore for each journal to be ranked within its own subject category.§ THE I^3 STATISTIC[THE I^3 STATISTIC IS PROVISIONALLY PATENT PROTECTED THROUGH THE UNITED STATES PATENT AND TRADEMARK OFFICE UNDER THE SERIAL: USPTO 62/506,119]The i^3 statistic is hereby proposed as the weighted measure of the impact of an individual article. Currently, none of the established article-level metrics provide a distinction between the number of citations and the impact of an article. Instead they are often used as a proxy for one another. The i^3 statistic is a measurement that for any given publication, considers: the JCR category of the journal where the article is published, the impact factor of the publishing journal, as well as the number of citations and the provenance of those citations, assigning higher value or “impact” to citations in journals with higher impact factors than lower ones. If we use the number of citations as an objective measure of impact then one could argue that a paper published in a remote, unknown journal with x citations has more impact than a paper in a prestigious journal with x-b citations. Intuitively, we know this to not systematically be the case. When considering the impact of a scholarly work, citation number textitand provenance should be equally important to rank publications. The statistic's algorithm accounting for the aforementioned considerations is the following:i^3 = G(f(x)) = 1-e^-βf(x)wheref(x) = ψ_a + ∑_i=0^x η_xψ_x = ψ_a + η_1ψ_1 + η_2ψ_2 +... + η_xψ_xand β = 1/3πϕ Equation (1) depicts the main algorithm to calculate the i^3 statistic index. e refers to Euler's number, an irrational and transcendental constant playing a crucial role in mathematics and number theory, which can de defined as follows: e = ∑_n=0^∞1n! = 10! + 11! + 12! +...+ 1n!+ 1n+1! + ... β refers to the Balayla coefficient, which is described below. Equation (2) is a linear equation, which takes into account the impact factor of the publishing journal (ψ_a) as well as the sum product of citation number and provenance, counting the individual number of citations η and the impact factor of the journals where they are found (ψ_x). As stated before, Equation (3) depicts the Balayla coefficient, a JCR category-specific, unitless coefficient, which is inversely proportional to ϕ, the number of journal titles in a given JCR category. §.§ Properties of the i^3 Statistic The equation for the i^3 statistic is complement to the exponential decay function, with an asymptote at y = 1, the maximum theoretical i^3 value a scientific paper can have. Given the above equation (1), it follows that:lim_f(x)→ 0 G(f(x)) = 0andlim_f(x)→∞ G(f(x)) = 1 As such, the i^3 yields a value between 0 and 1. The i^3 = 0 when no citations have taken place, and the i^3 = 1 when, theoretically, a single article has infinite citations. Articles with an i^3 value closer to 1 are deemed to have more impact than those closer to 0. By standardizing the citation information the i^3 can be used as a ranking algorithm and as a tool to optimize literature searches. Following is the graphic representation of the i^3 function, for an average β coefficient of 0.00115, which corresponds to a JCR category with ϕ = 92 titles. §.§ Graphic representation of the i^3 Statistic[assumes an average β coefficient of 0.0011.] [ axis lines = center, xlabel = f(x), ylabel = i^3,ymin=0, ymax=1.8, legend pos = north east,ymajorgrids=false, grid style=dashed, width=13cm, height=6cm,] [ domain= 0:5000, color= blue, ] 1-2.1778^(-0.00115*x); + [ dashed, domain= 0:5000, color = black, mark size = 0pt] 1; i^3 y=1 Note the slow rise of the i^3 statistic as a function of f(x) in blue, and the asymptote at y=1 in the black dashed line. §.§ The Balayla (β) Coefficient The Balayla coefficient, an eponym named after the author, constitutes a positive rational number, which assigns a value as an inverse function of the number of journal titles ϕ in a given JCR category. By definition, ϕ is a natural integer greater or equal to 1, as the journal where the article in question is found yields at a minimum, a value of ϕ = 1. The purpose of the β coefficient is to compensate i^3 scores in JCR categories where the number of titles is lower, thereby lowering opportunity to publish in domain-specific titles. Though articles can be published in general journals that fit domains other than their own, preliminary analysis shows that the vast majority of research articles are published in journals within their own domain categories. Given the above equation (3), it follows that: lim_ϕ→ 1β≃ 0.1 and lim_ϕ→∞β = 0 The β coefficient therefore takes on values between 0 and ≈0.1 §.§ How was the Balayla Coefficient determined? The Balayla coefficient was determined empirically using real data from JCR's 2015 report. A preliminary search determined that in the citation distribution curve of all publications, 1000 citations falls on average at the 90th-95th percentile. A corresponding i^3 value of 0.90 should evoke a similar percentile in the i^3 score distribution. I have calculated an mean value of ψ = 2.00 for the impact factor of a average journal, leading to f(x) ≈ 2,000. Similar calculations reveals the average number of journals perJCR category to be ϕ = 79.4. Isolating β and inputting the above values we obtain: i^3 = G(f(x)) = 1-e^-βf(x) 0.90 = G(f(x)) = 1-e^-2000β 0.1 = e^-2000β ln(0.1) = ln(e^-2000β) β≈ 0.00115 Let λ be the coefficient relating β to ϕ. It therefore follows that: λ = βϕ↔β=λ/ϕ The inverse relationship between an average β coefficient = 0.00115 and an average number of titles per category in the JCR ϕ = 79.4 leads to the following determination: 0.00115 = 179.4x⇒λ≈13π⇔β=1/3πϕ §.§ Density distribution of the Balayla Coefficient [ axis lines = box, xlabel = ϕ, ylabel = ,ymin=0, ymax=0.005,domain= 0:500,ymajorgrids=true,grid style=dashed,mark size = 1.1 pt,width=10cm, height=4.5cm, legend pos = north east] +[scatter,only marks, samples=70,scatter src=x] 1/(3*pi*x); B scaled y ticks=false §.§ Dynamics of the i^3 StatisticThe i^3 serves multiple purposes. One of these, is allowing the tracking of an individual paper's impact over time through fluctuations in the i^3 score. We define the following notations in this regard: i^3_x → i^3_1, i^3_5, i^3_10where, i^3_1 = An article's i^3 score one year after its original date of publication. i^3_5 = An article's i^3 score five years after its original date of publication. i^3_10 = An article's i^3 score ten years after its original date of publication.The i^3_CR(t) stands for the i^3 citation ratio at time t. We can use the i^3_t formulation above, where t = year since original publication off and divide it by the total i^3 score amassed by the publication throughout its history up until the point of interest. Of importance, this ratio uses information about the number and provenance of citations and journal impact factors both at time t for the numerator and at the present time for the denominator. We can use this information to create distribution curves of citations, and thus determine whether the impact of a scholarly work is evenly distributed throughout its history or whether its concentrated at any one individual or multiple points. i^3_CR(t) = i^3_ti^3 However, because the i^3 is not a linear function a more precise measure of the ratio i^3_CR(t) and citation distribution is obtained by integrating each of the i^3 functions, namelyi^3_t and i^3, and dividing them to obtain the i^3_CR(t).i^3_CR(t) = ∫ _ 0 ^ ti _t^ 3 dt∫ _ 0 ^ f(x)i ^ 3df(x) i^3_CR(t)=1/βe^-βt + t + C/1/βe^-βf(x) +f(x) + C The algorithm for the i^3 remains the same. The f(x) equation is replaced by the variable t in the numerator to denote the period of interest up to which the i^3_CR(t) is being calculated. t = ψ_a + ∑_i=0^t η_tψ_t = ψ_a + η_1ψ_1 + η_2ψ_2 +... + η_tψ_tBy definition, regarding equations (2) and (7):∑_i=0^tη_tψ_t ≤∑_i=0^xη_xψ_x Considering the solved integral in equation (6) we deduce that as the number of citations increases to infinity, the i^3_CR(t) approaches the simple ratio of the sum-product of the number and provenance of citations, as stipulated in equation (2):lim _ x,t→∞∫ _ 0 ^ tI _ t ^ 3dt /∫ _ 0 ^ f(x)I ^ 3df(x)= t / f(x)= f(x) _ t / f(x)§.§ Circling around the asymptote As shown in the graphic representation of the i^3 statistic (Section 2.2), the curve reaches an asymptote at y=1 when f(x) approaches ≈ 3,500. The latter is of course dependent on the β coefficient, and therefore, of the subject category of the article in question. Given the asymptote of the i^3 function, the derivative approaches 0 as f(x) goes to infinity.G'f(x) = i^3ddx = 1 - e^-βf(x)ddxG'f(x) = βe^-βf(x)lim_f(x)→∞ G'(f(x)) = 0 In other words, the f(x) of a publication increases at a slower rate beyond a certain point. Though in practical terms this means the impact of the publication in question is very large, it makes comparison with other like-publications more cumbersome. To get around this limitation, we can integrate the functions again to determine whether the small difference in i^3 values corresponds to actual large discrepancies in impact. [ axis lines = center, xlabel = f(x), ylabel = i^3,ymin=0, ymax=1.8, legend pos = north east,ymajorgrids=false, grid style=dashed, width=13cm, height=6cm, fill = blue,] [ domain= 0:5000, color= blue, name path = A ] 1-2.1778^(-0.00115*x); + [ dashed, domain= 0:5000, color = red, mark size = 0pt, name path = B] 1-2.1778^(-0.00140*x); +[mark=none, color=blue] coordinates (3500, 0) (3500, 0.9687);+[mark=none, color=red, dashed] coordinates (4000, 0) (4000, 0.9687); [name path=xaxis] (/pgfplots/xmin, 0) – (/pgfplots/xmax,0);[red, pattern=north west lines] fill between[of=B and xaxis, soft clip=domain=3530:3970];[blue, pattern=north west lines] fill between[of=A and xaxis, soft clip=domain=3530:3970];Paper A Paper BNote how for an f(x) = 3500 (Paper A) and an f(x) = 4000 (Paper B), thei^3 values are similar. Integrating the i^3 functions and calculating the area under the curve (AUC) facilitates ranking and comparison. The dashed area in grey demonstrates excess area of paper B, indicating its impact is greater.As noted, the area under the curve (AUC) measurements at the extreme of the i^3 function can assist in the comparison of highly cited works. This calculation serves as an adjunct measure of the dynamic history of the publication, in that the AUC increases with increasing f(x), independently of the function's (i^3) asymptote. §.§ Rationale behind the i^3 Statistic The main driving force behind the development of the i^3 algorithm was the need for a optimized method to navigate the exponentially-growing scientific literature. Simply put, faced with several articles on a similar topic, a simple measure such as the i^3, a index number between 0 and 1, can allow for the ranking of said publications based on their perceived impact, indicating which is likely to be of higher quality. Because the i^3 is based on the number and provenance of citations in other scientific sources that have likely undergone peer review, it can effectively amass the validation and approval of a whole community of experts who have read the publication in question and considered it worth of a citation in their own work. Indeed, relative to non-scientific publications, which are not regulated and where no standardized evaluation of content takes place, the i^3 statistic takes advantage of peer-review and peer-expertise to evaluate and validate research findings, which are then disseminated through citations. If journals with a higher impact factor are deemed more reputable than those with lower impact factors, then it can be argued that the articles cited in those journals ought to be considered more reputable than those which aren't cited. Similarly, if journals with a higher impact factor are deemed more reputable, then the readership of those journals ought to be greater, and so is the exposure of articles cited in those journals. Finally, if journals with a higher impact factor are presumably more difficult to publish in, then the articles cited in those journals are likely subject to a higher standard, thus better reflecting their true worth. In one sentence, the premise of the i^3 is that an individual article's impact is a composite measure of its own original publication and the characteristics of its citation history.§ USES, STRENGTHS AND LIMITATIONS OF THE I^3 The potential uses of the i^3 statistic are numerous. First, it can stratify scientific work by a rigorous measure of dissemination and perceived quality. Secondly, it can be used as a screening litmus test to assess a scientist's work and as a tool to rank a scientist's portfolio. It is conceivable that by allowing for the ranking of individual publications, the i^3 may aid in optimizing literature searches and reading tasks, and more generally, establishing a hierarchy to attribute awards, funding and grant money when funds are limited. Similarly, since the i^3 will be category-specific, it can potentially be used to develop reference values of impact percentiles by discipline, allowing scientists to set reasonable objectives for the expected impact of their work. Furthermore, journals can use the i^3 to estimate a distribution of the citation pattern of the articles they publish. Finally, the i^3 may be used to counter the "Matthew Effect", a primarily sociological phenomenon whereby in certain scenarios in Society "the rich get richer, and the poor get poorer" <cit.>. The analogy in scientific publishing is the following: In most databases where research is accessed, the papers that have the most citations appear first, and are therefore more likely to be accessed and cited themselves, perpetuating the self-serving cycle. Inevitably, this also leads to papers with lower citation counts being left behind. By taking into account both the number and the provenance of citations, the i^3 can limit the occurrence of this phenomenon. Indeed, some publications with lower citation counts may still fare up at the top if the citations are found in high-impact journals or in JCR categories that have a low number of journal titles. The strengths of the i^3 are also multiple. First, the i^3 is the only standardized article-level metric that uses and distinguishes citation provenance. Secondly, unlike altmetrics, the i^3 focuses solely on scientific references, giving credence to the notion that the impact it seeks to evoke comes from rigorous sources. Thirdly, the i^3 is author-blind and independent, and compared to altmetrics, where authors can tweet and access their own work, the i^3 is less amenable to self-intervention by the authors. But perhaps the biggest strength comes from the simplicity of its nature: a simple number between 0 and 1 that is easily used to compare individual publications. On the other hand, the i^3 does have a number of limitations. First, citations can come from journals in different disciplines with highly variable impact factors, which may introduce bias. Similarly, not all press is good press: some citations are actually critiques and may actually imply that the article in question evokes the opposite of 'quality'. Furthermore, while the i^3 is based on the assumption that citations in prestigious journals with greater readership implies higher potential for impact, there is no guarantee that readers will actually engage with bibliographic material. In addition, the i^3 does not distinguish between years of publication, self-citations or order of authorship. And finally, the i^3 may not be a reflection of an article's worth, but rather of the topic at hand, the accessibility of the publishing journal, or the popularity of the journal where it is published.§.§ How to use the i^3Imagine that, as a reader of Science, you are interested in reading about a particular subject, or answering a specific question on a given topic. How do you go about finding the right literature? Most Scientists will access official, well-known databases, where the subject's literature is likely to be found. Once in the database, one may limit the vast number of results by performing an advanced search, limiting findings by language, year of publication, or authorship, amongst others. Given the vast literature that is available, dozens, perhaps hundreds or thousands of results will be available. The task of going through all of that literature is daunting and certainly time-consuming. Having an i^3 value associated with each publication would provide a rapid and simple way to compare the impact of individual publications, serving as an additional way to restrict findings and increase the likelihood of finding the right literature to answer their query. By simply comparing two or more i^3 values, simple numbers between 0 and 1, and ranking publications from the highest i^3 to the lowest, the reader can guide his search, by counting on the validation of their peers who, through citations and perpetuation of findings, have endorsed the scholarly work in question. § CONCLUSION The i^3 is a novel article-level metric that can be used to assess and compare a standardized measure of the impact of individual scholarly publications. In addition to its main objective, the adoption of this new metric may have implications in hiring practices, distribution of funds and grants, and ranking scientists' portfolios at large.§ ADDENDUM §.§ Balayla (β) Coefficients in sample JCR categories§.§ Plot of varied β coefficients in independent i^3 functions [ axis lines = center, xlabel = f(x), ylabel = i^3,ymin=0, ymax=1.8, legend pos = north east,ymajorgrids=false, grid style=dashed, width=13cm, height=6cm,] [ domain= 0:5000, color= blue, ] 1-2.1778^(-0.00115*x); + [ domain= 0:5000, color = black, mark size = 0pt] 1-2.1778^(-0.00085*x); + [ domain= 0:5000, color = red, mark size = 0pt] 1-2.1778^(-0.00067643*x); + [ domain= 0:5000, color = green, mark size = 0pt] 1-2.1778^(-0.0024*x); + [ domain= 0:5000, color = orange, mark size = 0pt] 1-2.1778^(-0.00171543*x); + [ domain= 0:5000, dashed, color = black, mark size = 0pt] 1; i^3 y=1 Each i^3 curve rises at different velocities. This is the nature of the Balayla coefficient. Nonetheless, note how independent of the β coefficient, all i^3 formulations have an asymptote at y=1.apalike | http://arxiv.org/abs/1706.08806v2 | {
"authors": [
"Jacques Balayla"
],
"categories": [
"cs.DL"
],
"primary_category": "cs.DL",
"published": "20170626121409",
"title": "The Individual Impact Index ($i^3$) Statistic: A Novel Article-Level Citation Metric"
} |
Peter Vereš [email protected] Present address: Minor Planet Center, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109We have conducted a detailed simulation of LSST's ability to link near-Earth and main belt asteroid detections into orbits. The key elements of the study were a high-fidelity detection model and the presence of false detections in the form of both statistical noise and difference image artifacts. We employed the Moving Object Processing System (MOPS) to generate tracklets, tracks and orbits with a realistic detection density for one month of the LSST survey. The main goals of the study were to understand whether a) the linking of Near-Earth Objects (NEOs) into orbits can succeed in a realistic survey, b) the number of false tracks and orbits will be manageable, and c) the accuracy of linked orbits would be sufficient for automated processing of discoveries and attributions. We found that the overall density of asteroids was more than 5000 per LSST field near opposition on the ecliptic, plus up to 3000 false detections per field in good seeing. We achieved 93.6% NEO linking efficiency for H<22 on tracks composed of tracklets from at least three distinct nights within a 12-day interval. The derived NEO catalog was comprised of 96% correct linkages. Less than 0.1% of orbits included false detections, and the remainder of false linkages stemmed from main belt confusion, which was an artifact of the short time span of the simulation. The MOPS linking efficiency can be improved by refined attribution of detections to known objects and by improved tuning of the internal kd-tree linking algorithms.§ INTRODUCTIONThe origin of asteroid orbit determination is closely tied to the discovery of Ceres by Giuseppe Piazzi on January 1, 1801. The new object, believed to be a comet, was followed briefly and lost after going through solar conjunction. To resolve the problem, Carl Friedrich Gauss developed a new method of orbit determination later the same year. With his prediction, the orbit of Ceres was determined and it was recovered at the end of 1801. Nowadays, more than 700,000 asteroid orbits are known and managed by The International Astronomical Union's Minor Planet Center (MPC). With the current CCD surveys, dedicated search telescopes, sub-arcsecond astrometry, and improved orbit determination techniques, the derivation of an initial orbit for a single object observed multiple times within a few days is relatively simple, with some known caveats. Subsequent follow-up and archival matching usually extends the observed arc to months or even years. However, new telescopes and deeper search, mostly for Near-Earth Objects (NEOs), make contributions more challenging for the follow-up community. NEOs have been a focus of attention for at least two decades, mostly motivated by the goal of reducing the Earth impact hazard by cataloging the population, but lately also for sample-return missions, proposed asteroid mining and crewed missions. Therefore, the MPC's NEO Confirmation Page is being saturated, often leaving objects either unconfirmed or with very short arcs, necessitating archival linking or searches. In 2022, the Large Synoptic Survey Telescope <cit.> will start to operate and provide unprecedented numbers of asteroid and comet detections at magnitudes beyond the reach of the current follow-up telescopes. LSST will operate in almost real-time in an automated mode, including identification of known moving objects and linking detections of newly discovered asteroids. The complexity of the problem lies in the fact that instead of careful treatment of a single object, LSST will have to treat millions of detections, including the spurious and false ones, and successfully link them into orbits, while rejecting false linkages. The latter part has yet to be demonstrated on a real asteroid survey comparable to LSST. The idea of automated asteroid detection, linkage and identification was implemented for Pan-STARRS <cit.> in its Moving Object Processing System <cit.>. MOPS was developed as an end-to-end collection of algorithms and programs, able to process individual detections, link detections into single-night tracklets, combine tracklets into tracks and then derive multi-night orbits. Its features and capabilities include propagation and simulation of synthetic orbits, efficiency studies, identification of detections with a catalog of known or derived orbits, providing alerts and an interactive web-interface. Its track-finding algorithm is based on scalable and variable kd-tree algorithms <cit.>. The MOPS linking performance was tested in simulations, including all types of Solar System objects (150,000 orbits by <cit.>) and even with expected false detection and asteroid density rates (15 million orbits by <cit.>). The resulting linking efficiency was close to 100% and the high accuracy suggested that MOPS will work for an advanced asteroid survey. However, the Pan-STARRS project was never completed into its original design of 4-telescopes, its 3-year mission with a single telescope was not Solar System optimized and was different from the proposed and tested survey cadence, its limiting magnitude was below the predicted value, and the rates of spurious detections were orders of magnitudes larger than predicted. It was the false detection rates in particular, that did not allow MOPS to derive orbits due to the dramatic increase in the number of false tracks that overwhelmingly outnumbered the real ones. Notably, this experience might be a source of skepticism regarding LSST 's ability to manage the large load of real and false detections into a working linking algorithm, which has not been demonstrated yet. Still, some components of MOPS are being successfully used by many (Pan-STARRS1, Pan-STARRS2, NEOWISE, DECAM) and MOPS is planned to be used with its full capabilities, including linking, for LSST. However, the effectiveness of MOPS is crucial. Without successful linking, the expected numbers of LSST discovered Solar System objects could be significantly decreased.Our goal was to test MOPS for LSST with a realistic density of moving objects and false detections and to understand whether MOPS can handle the expected large number of false tracks. We emphasized the realistic approach by employing the baseline survey cadence, the exact shape of the field of view, several observational constraints and parameters, as well as the most recent population models of the Solar System objects. § LARGE SYNOPTIC SURVEY TELESCOPE LSST is a next generation all-sky survey telescope currently being constructed atop CerroPachón in Chile. Its first light is scheduled for 2020 and its 10-year nominal survey will start 2 years later. This 8.4-meter, wide-field telescope, with a 3.2 Gigapixel camera and real-time image processing, data acquisition and transport, is mainly motivated by the study of Dark Matter and Dark Energy. However, its nightly coverage of 6,000 square degrees, mostly done in two visits to a given field in the same night, provides an excellent opportunity for a deep survey of the small bodies of the Solar System. Because of its limiting magnitude, reaching to 24.5 in r-band in 30-seconds, and large load of detected asteroids and comets, the discovery and characterization of Solar System objects must be done automatically, by identifying with known objects and correct linking of new objects.For our simulations, we selected one month of the 10-year baseline survey (see <cit.> for a description of ) created by the Operations Simulator <cit.>. OpSim provides a list of fields with information on their positions, epochs, limiting magnitudes, filters, seeing, etc. Fields also avoid the Moon and filters are sensitive to the phase of the Moon and its presence above the horizon. The selected dates covered the 28th observing cycle (OC 28) of . An observing cycle is a MOPS-defined interval of time, from a full moon to a full moon. OC 28 spannedthrough the months of May when the ecliptic has the largest altitude above the horizon around midnight and also the nights are the longest in the summer at the LSST site (Figure <ref>). Thus, the density of NEOs and Main Belt Asteroids (MBAs) is at its greatest. Some nights were removed by OpSim to simulate weather, resulting in 27 clear nights.A small fraction of fields were observed only once per night; these singletons were removed from our simulation. The mean and maximum limiting magnitude of the selected observing cycle as well as the time spent in individual filters are denoted in Table <ref>. The survey spends most its time in the r, i and z-band, and only 3% of time in the u-band. The LSST camera consists of 21 platforms called rafts, each consisting of a 3×3 array of 9 CCD chips, yielding a total of 189 CCDs. Each chip comprises 4096×4096 10-micron pixels, and so the total number of active pixels is 3,170,893,824. Because there are gaps between chips within the 3×3 rafts and also between the rafts, some fraction of the focal plane is not useable. The total active area is equal to 9.50 deg^2, whereas the total raft area yields to 10.45 deg^2, resulting in a fill factor of 0.9089. Gaps can be simulated by an exact mask or by a statistical treatment of detections. The pixel mask approach is computationally more expensive, because it requires building and matching up the fields with the 3.2 billion pixels. This work used the probabilistic approach, where fill factor represents the probability of a potential detection to be found in a single frame. To simulate the field, we employed a square layout of 25 rafts with the area of 12.445deg^2 and then applied a mask for the four corner rafts to obtain the above mentioned 10.45 deg^2. Finally, 90.89% of detections were randomly selected to form the detection list.LSST utilizes an altitude-azimuthal mount and the camera is able to rotate, and thus the fields are not generally aligned with the local RA-DEC frame. In fact, due to desired dithering, each exposure is observed in a randomized field orientation. The field rotation affects the probability of the detection to be visible in multiple visits, because some of the detection can hit the masked area in the second visit. This aspect of the survey is fully modeled in our simulations. § FIELD DENSITY§.§ Asteroid detections We generated synthetic detections for NEO and MBA population modelsby propagation of the orbits to the epochs of the OpSim fields. The propagation used JPL's small body codes with the DE405 planetary ephemerides, where all planets, plus Pluto and the Moon were perturbing bodies. We did not use any asteroids as perturbers. Only detections inside of the rotated field were analyzed and filtered based on the field limiting magnitude and other selected parameters of the detection model. Some details of the detection model are described in <cit.>. We utilized a <cit.> NEO population containing 801,959 Keplerian orbits with absolute magnitude down to H<25. The distribution of its orbital elements is roughly similar to earlier work by <cit.>, however, the <cit.> population is size-dependent and its size-frequency distribution covers the H>22 space better than the previous work which underestimated the count. (See Figure <ref>.) The orbital and size-frequency distribution properties of “Granvik's" NEO population were derived from analysis of NEO observations by the Catalina Sky Survey and Mt. Lemmon telescopes. Our NEO population is artificially deficient of large NEO with H<17; however, these are believed to be essentially all discovered and there are only about 500 of them, thus they are a negligible fraction of other detections in an LSST field of view. Initially, we were also using the earlier NEO model by <cit.>, which we denote as “Bottke's”, that also contains objects down to H<25; however, its total number is only 268,896 and is thus deficient in small objects, particularly for H>22. MBAs will dominate the number density of moving objects in the LSST field of view, and they represent a source of background noise and possible confusion for NEO identification. In our LSST simulations, we used the <cit.> model of the main-belt population (see Figure <ref>). This population contains 13,883,361 orbits and is the most robust population model available to date.In the Grav MBA model, the cumulative distribution slope is equal to α=0.28±0.01 for H between 16 and 20. However, the population was created for a Pan-STARRS4-like survey with a limiting magnitude of m_V=24.5, and so it is truncated to remove MBAs that are fainter than m_V=24.5 when at perihelion and at opposition. This truncation results in an artificial break, seen in Figure <ref>, in the Grav population size-frequency distribution at H≃21. To investigate how this break affects the areal density of MBAs in the LSST survey simulation, we compared the simulated MBA density in LSST fields to the predicted number density by <cit.> who had observed MBAs with the 3.8-meter Mayall telescope at Kitt Peak National Observatory in 2001 within the so-called SKADS survey. SKADS detected asteroids in a fixed 8.4 deg^2 patch of sky around the opposition point in the Johnson-Cousins R-band down to limiting magnitude of 23.0–23.5 on six nights spanning an 11-night baseline. Based on <cit.>, the debiased cumulativenumber of MBAs follows the equation N(>H)∝10^α Hwhere α=0.30±0.02. This slope was derived for H in the range 15–18, with assumed validity to at least H=20. <cit.> derived the areal density of MBAs asN(<m_R)=210*10^0.27*(m_R-23)where N(<m_R) is the cumulative number of asteroids brighter than m_R per square degree. The derived detection efficiency was 98% at m_R=17. To compare with our modeled number density of MBAs, we selected LSST fields withsolar elongation greater than 178 and within one degree from the ecliptic from OC 28, yielding 27 fields. This simulation was run with fill factor of ϵ_0=90.89%, fading and color transformation assuming all asteroids are of a spectroscopic S-type (see <cit.>). There was a slight difference in the definition of detection efficiency.Our modeled detections are subject to so-called fading that reduces detection efficiency asϵ(m) = ϵ_0/1+e^m-m_5/wwhere ϵ(m) is the detection efficiency, m the apparent magnitude, m_5 the limiting magnitude defined for SNR=5 and w=0.1 the width of the fading function. SKADS defined its detection efficiency by a similar relationϵ(m) = η_0-c(m-17)^2/1+e^m-m_5/wwhere, based on observations, η≈0.98 and c≈0.005. Here c measures the strength of the quadratic drop and the remaining parameters are the same as in the previous equation.Additionally, there are a number of sources of uncertainties that must to be considered in the estimate of the MBA density:* A different slope of the population. α=0.28 and 0.30 for Grav and Gladman, respectively.* The transformation from the LSST bands and the SKADS R-band to V-band. The term V-R in SKADS was 0.37±0.15 mag, leading to relative uncertainty of about 9% in areal density when transforming to V-band.* The scaling of the detection efficiency. This work used a different model than SKADS for fading. Figure <ref> shows the number density of MBAs near opposition as a function of limiting magnitude of the field in V-band based on the SKADS survey and the simulated LSST survey with the synthetic Grav MBA population. Note that at m_5>24.5 the simulated MBA density drops because of the artificially truncated Grav's population. In , 14% of the fields have a limiting magnitude fainter than 24.5 in V-band. Depending on the limiting magnitude and the elongation from ecliptic and opposition, the MBA density in our simulation was underestimated by up to 12% in those fields. However, few of the 14% fields fainter than 24.5 mag were taken at opposition near the ecliptic, and so the effect of the truncation in Grav's MBA population is presumed negligible. The density of MBAs decreases significantly as a function of ecliptic latitude (Figure <ref>).§.§ Measurement errors Each ephemeris-based position in the field was altered by adding realistic astrometric and photometric errors based on the computed signal-to-noise ratio (SNR). The limiting magnitude of the field m_5 is defined for SNR=5. The SNR of a detection <cit.> is computed from the difference between the computed magnitude m and m_5 asSNR = 1/√((0.04-γ).χ+γχ^2)where γ=0.038 and χ=10^0.5(m-m_5). Then, photometric uncertainty is derived asσ_m=2.5log_10(1+1/SNR) .and the computed m is combined with an error drawn from a normal distribution with a mean of zero and variance σ_m^2.We have assumed that LSST astrometry is measured relative to a post-Gaia star catalog and so absolute systematic errors are negligible while relative errors are expected at a floor level of 10 mas. The astrometric error σ_astr for any detection is therefore computed as quadrature combination of 10 mas and the ratio of the seeing Θ and SNRσ_astr^2=(10 mas)^2+(Θ/SNR)^2.Asteroids are moving targets and so, depending on the rate of motion, their shape deviates from a stellar PSF and is in fact a convolution of the motion and the PSF. The faster the object moves, the larger the astrometric error. Therefore, if the trail length L>Θ, the seeing term Θ in Eq. <ref> is replaced by the geometric mean of seeing and trail length as Θ'=√(Θ L).To obtain realistic astrometry, we combine the computed position with an astrometric error term drawn from a normal distribution with a zero mean and variance of σ_astr^2. Figure <ref> shows histograms of astrometric uncertainties, in both linear and log-scale. The latter shows that there are two populations of NEA detections, those with high SNR and therefore low uncertainty, around 10 mas, and another centered around 100 mas from low SNR detections, which presumably also includes most of the objects with relatively fast rates of motion. The median astrometric error obtained for NEOs is 47 mas.To simulate observational constraints and limitations of the LSST processing pipeline and CCD effects, we employed a set of filters that determined whether a detection that fulfilled the limiting magnitude was still visible. We included vignetting, which reduces sensitivity to detections that are far from the optical axis of the field. The LSST optical design minimizes vignetting, with only 7% of the collecting area having a penalty above 0.1 mag. In CCD surveys the limiting magnitude does not behave like a step function that strictly determines the visibility. In fact, the detection limit follows a fading function, e.g., Eq. (<ref>) that defines the limiting magnitude as a 50% probability of detection. In our work, this value is taken at SNR=5 and denoted as m_5. The fading function is multiplied by a fill factor, simulating the focal plane gaps. Because of the sidereal tracking rate, all asteroids will move, and particularly fast moving NEOs will look trailed. The detected trails are described by a convolution of a point-spread-function with the motion vector. The longer the trail, the fainter the peak signal and the SNR decreases. This loss effectively decreases the magnitude of asteroids as a function of their on-sky rate of motion. We assumed that all NEOs and MBAs are of S-types for the purpose of the ephemeris computed V-band magnitude transformed to the LSST filter system. The details of this detection model are discussed by <cit.>.§.§ False detections The LSST transient detection data stream will include many detections that are not associated with solar system objects, and the objective of linking only real LSST detections of moving objects to form tracks and orbits represents a significant challenge. There are three broad categories of non-solar system transients that are expected from LSST. The first category of LSST transient detections arise from real astrophysical phenomena (e.g., variable stars, supernovae, etc.) that appear in the same location in multiple instances. Such astrophysical transients will be filtered out of the MOPS input stream by virtue of their stationary appearance and thus will not affect the asteroid linking problem. The remaining two categories of non-solar system transients consist of spurious detections arising from either random noise or image differencing artifacts, both of which will enter the MOPS input stream. The first source of false detections, from random fluctuations in the sky background and from detector noise, is driven by gaussian statistics at the individual pixel level. The number N_>η of these random sources above a given signal-to-noise threshold η in the CCD image where Gaussian noise is convolved with a Gaussian PSF follows the formula by <cit.>N_>η=S/2^5/2π^3/2σ_g^2η e^-η^2/2,where S is the total number of pixels in the focal plane array, σ_g≃Θ/2.35, and Θ is the FWHM seeing measured in pixels. The number of random false positives depends strongly on the seeing (Figure <ref>), with the better the seeing the larger the number of random false positives. The average seeing of 0.80 arcsecond leads to 650 random false positives with SNR>5 in one LSST image. We generated random false positives following Equation <ref> in random x-y positions in the field. The number of random false positives for a given field was selected from a normal distribution with a mean and variance of N from equation <ref>. Then, magnitudes were assigned to the generated random noise as follows: We generated a random number p from a uniform distribution [0,1]. This number corresponds to the normalized cumulative distribution N(>η)/N_TOTAL. Then η=√(η_0^2-2log(1-p)) which can be directly transformed to a magnitude as V=V_LIM-2.5log(η/η_0) where V_LIM is the m_5 limiting magnitude at η_0=5. The number density of random false positives has a strong dependence on SNR; therefore, most of the random noise sources will be near the the limiting magnitude (Figure <ref>).The second source of false detections comes from difference image artifacts, which arise from differencing a field image with a fiducial image of the static sky that has been derived from a stack of several (or a great many) images of the same field over some time period. This differencing removes stationary objects so that only transient sky phenomena, including moving objects, appear as detections in the difference image. However, registration errors across the field can leave dipole-shaped artifacts in the difference image at the location of a static source. Artifacts may also originate from a poor convolution kernel, variable seeing across the field, stray light in the optical system or reflections in the lenses. Artifacts are often concentrated around bright sources due to saturation or diffraction spikes, and masking around these sources can be an efficient means of substantially reducing the rate of artifacts. Although an improved optical configuration and machine learning can remove many of these false artifacts, some fraction will always remain in the data stream. For this work we assumed the estimated density of differencing artifacts derived by <cit.>, who used actual imagery obtained by the Dark Energy Camera (DECAM) on Cerro Tololo <cit.> and processed them with a nascent version of the LSST image processing pipeline. <cit.> report that the primary result of their study is that “the LSST pipeline is capable of producing a clean sample of difference image detections, at roughly the 200–400 per square degree level.” This is their final result, but our work used a preliminary estimate as the point of departure for our linking simulations. This earlier estimate allowed for roughly 90–380 artifacts per square degree, and we took the geometric mean of this range as the starting point, which leads to 185/deg^2 or 1777 artifacts per LSST field. <cit.> did find far higher concentrations of artifacts near bright stationary sources, which they eliminated by masking the area around them, thus allowing the reported low artifact density. Following their result, we modeled bright source masking by reducing the effective fill factor by 1%. To seed the detection list with artifacts, we selected the number of artifacts in each field according to a gaussian distribution with mean and variance 1777 and distributed them randomly across the field. Thus our artifact rate was roughly 3× the rate from random noise in typical seeing (Figure <ref>), and about half of the upper bound derived by <cit.> from processing actual DECam data. Our model for difference artifacts is independent of observing conditions such as seeing and field density. However, we note that the most dense regions of the galactic plane are relegated to the Galactic Plane proposal observations in , which happens to be mostly covered by a single-visit-per-night cadence, and is anyway only a few percent of observing time. If we remove all Galactic Plane proposal fields from there is a negligible effect on NEO completeness. Thus our linking and completeness results do not require or assume operation in star fields with extreme density.Based on the <cit.> report, we model that the SNR distribution of differencing artifacts follows ∝SNR^-2.5. The algorithm computes the SNR η from η=η_0(1-p)^-2/3 where p is a randomly generated number from a uniform distribution [0,1]. (See Figure <ref>.) The magnitude of a simulated artifact is then derived according to V=V_LIM-2.5log(η/η_0) where V_LIM is the m_5 limiting magnitude at η_0=5. Artifacts have much shallower dependence on η, and therefore tend to be farbrighter than random noise sources. Roughly half of modeled artifacts have SNR>10, while virtually none of the random false detections had SNR>7. The brightness distribution of artifacts suggests that at least some potential false tracklets that include artifacts can be immediately eliminated by enforcing consistency in the photometry. However, according to Figure <ref>, about 90% of artifacts have SNR<20, and if a bright artifact with SNR=20 is paired with a faint asteroid detection having SNR=5 the magnitude difference will be Δ m = 2.5log_1020/5≃ 1.5 mag. As it happens, MOPS limits the photometric variation among tracklet components to Δ m < 1.5 mag by default, which suggests that few false tracklets in our simulation have been eliminated in this way. This criteria could be made more strict, which would reduce the false tracklet rate at the risk of removing real objects that are actually more interesting by virtue of a large light-curve amplitude. Thus, as a rule, the photometric consistency requirement should be as relaxed as much as feasible in order to avoid eliminating real tracklets. We suspect that this requirement can be dropped altogether without significantly impacting linking performance.We note that our work neglects the possibility that artifacts are spatially correlated in RA-DEC, which could introduce difficulties in the linking process whereby artifacts could reappear near the same RA-DEC location and mimic the motion of asteroids. RA-DEC correlation among artifacts could arise from two causes, either camera defects or stationary sources. For LSST, the rotational dithering of the camera serves to break the correlation from any defects in the instrument, most of which would already be masked in processing, and the masking of bright stationary sources serves to remove them as a source of artifacts. <cit.> found that the rate of correlated detections in the DECam data stream was low enough to be negligible for our purposes, only ∼2/deg^2. This no-correlation assumption is at variance with the Pan-STARRS1 experience, but appears to be well justified for LSST. § MOVING OBJECT PROCESSING SYSTEM A central question for this work is whether the linking of tracklets into tracks and orbits will prove successful with real LSST data. LSST MOPS will receive full-density lists of detections of moving and transient targets, including NEOs, MBAs and false detections. From these inputs MOPS must create tracklets, tracks and orbits, despite the fact that the data stream is contaminated by potentially large numbers of false detections, which leads to high rates of false tracklets. Our simulation synthesized detections in the LSST fields from a full-density NEO model (∼850,000 orbits), an MBA model (∼11 million orbits) and false detections (both random noise and differencing artifacts). The final detection lists were submitted to the MOPS routine, and tracklets were created. Finally, tracklets were submitted to the linking stage, the most challenging step. §.§ Tracklets The list of detections for a given field that has been multiple times in a night is submitted to the part of MOPS. A tracklet is created for a detection in the first image if there is a second detection in its vicinity in the second image. The radius of the search circle is defined by the lower and upper velocity thresholds of , which were set to 0.05/day and 2.0/day, respectively, in this study. If there are more possible connections in the circle, in addition to the “CLEAN" tracklet, consisting of detections of one object, then a “MIXED" tracklet consisting of detections of two objects or a “BAD" tracklet that includes a false detection is created as well. Increasing the upper velocity limit increases the search area and thus the number of false tracklets. In some simulations, for velocities of 1.2–2.0/day, we used the information on the trail length to limit the search area for companion detections. At 1.2/day, a detection will have a non-PSF shape and its length will be 1.8 times the PSF for the average 0.86 arcsec seeing, and so its length and orientation can be determined. Thus, instead of a large circular search area around trailed detections, smaller regions consistent with the anticipated velocity and direction of the trails are searched, and any matching detections must have a compatible trail length and direction. See Figure <ref> for a graphical depiction.The number of tracklets depends on the density of detections, which can be large (Figure <ref>). To understand the feasibility of the simulation we gradually increased the number of detections in OC 28. The following steps are also summarized as Cases A-E in Table <ref>. * Initially, we only used NEO orbits from Bottke's model (Case A, Table <ref>). Switching to Granvik's NEO model increased the number of detections by 35% and tracklets by 55% (Case B). Because Granvik's NEO model is more current and has many more objects we used that population in the simulations. At this stage, with only NEO orbits, nearly all tracklets were CLEAN, with only 4 MIXED tracklets (99.97% tracklet purity).* Adding the MBA population to Granvik's NEOs (Case C) increased the number of detections in one month to 15 million and the number of tracklets to 6 million. Most of the tracklets were for MBAs; however, about 17% of tracklets were MIXED, i.e., derived from different objects. The large number of MIXED tracklets was substantially reduced by taking advantage of trail-related velocity information when in the velocity range 1.2–2.0/day (Case D). In this dual velocity mode of , 1.2/day is the upper threshold for creating a tracklet by searching in a circle. If the detection is trailed and the trail length implies a velocity >1.2/day, then its matching pair in the second image must be in a predicted location, based on the time between exposures, and the position and velocity of the first detection (Figure <ref>). Thus, the number of randomly linked detections in a large circle decreased dramatically. This increased the number of good NEO tracklets by 20% and decreased the number of MIXED tracklets by a factor of 5.* The next step added false detections from random noise to the full-density NEO and MBA detection list (Case E). This doubled the number of detections to 30 million, and so the synthetic to false detection ratio was about 1:1. However, the number of tracklets only increased from 6 million in case C to 7.5 million in case E. In this scenario tracklets were created up to the 2/day limit without the use of velocity information. In addition to 1 million MIXED tracklets, the simulation generated about 700,000 BAD tracklets (i.e., those with both synthetic and false detections) and 600,000 NONSYNTH tracklets consisting solely of false detections.* The final, full-density simulation was achieved by also injecting differencing artifacts, which more than doubled again the total number of detections, to 66 million (Case F). Now, over 77% of detections were false, and so the ratio between synthetic and false detections was about 1:3.5. NEOs represent only 0.07% of the detection list. The full-density simulation was challenging for the tracklet stage. Therefore, we used trail-derived velocity information for tracklets created in the velocity range of 1.2–2.0/day. Still, the total number of tracklets was very large, ∼11.9 million. Out of this sample, about 57% of tracklets were somehow erroneous, either including at least one false detection or detections of different objects. This simulation revealed that artifacts related to false positives create the majority of the linking challenge. Though we did not directly test it, the use of trail-related velocity information presumably leads to a dramatic reduction in the false tracklet rate for the full-density simulation. [ht!]Number of detections and tracklets for OC 28 in various simulations. MIXED tracklets include detections from at least two distinct objects, BAD tracklets include detections from both false detections and moving objects, and NONSYNTH tracklets consist entirely of false detections.4c Detections 6c TrackletsCase NEO MBA False Det Total%NEO %MBA %False Total%NEO %MBA %MIXED %BAD %NONSYNTH A BottkeNo None 36k 1000 0 11k 100 0 0 00B Granvik No None 49k 1000 0 17k 100 0 0 00C Granvik YesNone 15M 0.399.70 6.2M0.2382.816.900D [Tracklet generation used rate information from 1.2–2.0/day. Otherwise rate information was ignored over entire range 0.05–2.0/day.] Granvik YesNone 15M 0.399.70 5.4M0.3194.84.9 00E Granvik YesRandom only30M 0.250.649.27.5M0.1968.2149.77.9F 1sttablefootGranvikYesRandom + artifacts66M 0.122.677.312M 0.1442.72.2 6.148.8 §.§.§ The Linking ProcessAutomated linking of tracklets is a crucial element of LSST's NEO discovery pipeline. Without an automated linking stage, the NEO discovery rate would suffer and would rely heavily on follow-up observers, which will be impractical given the faint limit and volume of the LSST detections. The MOPS linking algorithm connects tracklets from three distinct nights into candidate tracks that are subsequently tested through orbit determination. The process consists of the following four distinct steps:* Assemble tracklet list. The first step collects, for a given field, all of the tracklets from the last N nights for which the earlier position and velocity project into the destination field. The forward mapping of tracklets is based on linear motion, and acceleration that leads to nonlinear motion is not accounted for. Thus some NEO tracklets may be neglected, especially those very near the Earth with a rapid change in geometry and observable acceleration.The combinatorics of linking strongly favor small N, but the objective of NEO completeness favors large N, which allows more missed detection opportunities. For LSST, N usually ranges from 12–20, though 30 has been contemplated as a stretch goal. This work used N=12 days for linking tests, consistent with our objective of understanding whether linkage could be at all successful in the presence of large numbers of false detections. NEO linkage of nearby objects is not likely to succeed for large N unless MOPS is extended so that some plane-of-sky acceleration is allowed when assembling the field-by-field tracklet lists. This would likely lead to a modest increase in the NEO discovery rate at the expense of many more false tracklets and increased linking overhead.* Assemble candidate track list. The second step in linkage generates a list of candidate tracks based on the input tracklets. Generally, there are hundreds of available fields per night, each being processed in parallel. The algorithm is based on a kd-tree search <cit.> that reduces the number of potential tracks to be tested from n^2 to nlog n, where n is the number of tracklets available for linking on the given field. This saves a significant amount of computational resources, but the problem remains challenging.has multiple tunable parameters, such as the minimum number of nights, the minimum number of detections, the minimum and maximum velocities, and some kd-tree linking parameters (, , ). The “vtrees" finds 2 compatible tracklets from which to estimate the endpoints of the track. The initial search pruning is done with respect to a maximum error denoted as . The track is confirmed when additional “support tracklets" are found. is a threshold for the goodness of fit for the support tracklets to the model estimated from the 2 initial tracklets. in days flattens the tracklets epoch to the same time, if they fall within this margin. Different parameter values led to vastly different CPU and memory requirements, and markedly different numbers of candidate tracks. However, optimization of this stage is complex. The ideal parameter settings depended on the number of detections and varied from field to field. For instance, experiments with only synthetic NEO orbits led to 99% linking efficiency. Adding noise and MBAs and running tests for selected target fields and tracks and varying parameters led to inconclusive results because the correct parameters depend on the field, and optimizing on a full lunation was infeasible. We explored the optimization of the kd-tree parameters on a single, dense field in the middle of OC28. The total number of candidate tracks increased as a function of and , and there was only a weak dependence on , at least for <0.01 (Figure <ref>). However, the most correct NEO tracks were derived for =0.003 and=0.003 (Figure <ref>). Pushing the kd-parameters to obtain as many NEOs as possible led to an extreme increase in the number of false candidate tracks (Figures <ref>–<ref>). Also, the memory and CPU load increased dramatically (Figures <ref>–<ref>).This work was conducted with a single 8-core Linux workstation with 96 GB of memory (upgraded from 32 GB during the course of the work), and a crucial part of the challenge of linking was avoiding out-of-memory crashes. The final values utilized for the main linking simulation in this work were therefore a combination of feasibility and available computational resources: (, , )= (0.001, 0.001, 0.003). This corresponds to the lower left corner of the upper right plot in Figures <ref>–<ref>. Better performance could have been obtained for, say, (, , )= (0.003, 0.003, 0.003), but this would require use of a large cluster with more memory per core, something that will be readily available to LSST. * Derive preliminary orbit. The third step took the candidate tracks derived by and submitted them for Initial Orbit Determination (IOD). MOPS uses Gauss' method to generate potential initial orbits from the astrometry, and for each track the best fitting IOD is selected. Most false tracks were eliminated at this stage with no valid IOD. * Perform differential corrections. The fourth stage was Orbit Determination (OD), which used JPL OD routines to obtain converged orbits. This includes sophisticated fall-back logic to try to obtain 4- or 5-parameter fits if the 6-parameter orbit fit diverged. MOPS filtered out some false tracks at this stage based on rudimentary screening on post-fit residual statistics. As discussed below, MOPS's built-in orbit quality filtering is not strict and is agnostic regarding the expected errors in the astrometry, and thus relatively few false orbits were rejected at this stage. All orbits that passed the MOPS default quality screening were added to the MOPS derived object table, which was the basis for understanding the overall linking performance. §.§.§ Linking Performance Linking tests were conducted on observing cycle 28 of the baseline survey, with Granvik's NEO model, MBAs and the full false detection lists (Case F, Table <ref>). The NEO linking efficiency is defined as the number of unique NEOs present in the post-linking, derived-object catalog divided by the number of unique NEOs with possible 12-day tracks in the detection list. The linking efficiency was 93.6% for H<22 NEOs and 84.0% for all NEOs (i.e., H<25). These numbers were lower than the case without the false detections, where we achieved >99% linking efficiency, similar to previous work <cit.>. The lower efficiency for all NEOs arises from the fact that the vast majority of NEOs were of the smallest diameters, e.g., 23<H<25. Also, smaller objects tend to have faster rates and greater acceleration because they are seen at closer geocentric distances, and they tend to have shorter observability windows. Note that the derived linking efficiency was for a single set of selected kd-tree parameters with a single 8-core workstation. With more powerful computational facilities and a more optimized kd-tree search (possibly on a per-field basis), there is excellent reason to believe that the linking efficiency can be significantly improved.Many derived NEO orbits stemmed from objects in the MBA input catalog. Table <ref> shows the makeup of the 5348 NEO orbits (defined by q<1.3) derived from OC 28 alone. Among these orbits, 2222 originated from CLEAN linkages of actual NEOs, 1896 were CLEAN orbits associated with MBAs and 1230 were erroneous (“Not CLEAN") linkages. Nearly all of the erroneous linkages combined detections of different MBAs to form an NEO orbit; few were contaminated by false detections. At first blush this implies a purity of 77.0% in the NEO catalog, but we describe below why this apparently low accuracy is mostly a manifestation of an ineffective orbit quality screening applied by MOPS. Correct interpretation of the orbits and improved screening increases the accuracy to 96%. In contrast to the NEO orbits, Table <ref> reveals that the MBA catalog has 99.8% purity already at this stage, without more refined filtering on orbit quality. Only 6 NEOs appear in the non-NEO orbit catalog, and most of these are borderline cases where q≃1.3.§.§.§ Orbit Quality Filtering The large fraction of erroneous linkages that appear in the NEO orbit catalog stem from a weak orbit quality filter implemented by MOPS, which requires the post-fit RMS of astrometric residuals to be less than 0.4 arcsec, a criterion that is too readily met for astrometry with a median error less than 0.05 arcsec. Moreover, because the RMS is not normalized by the reported astrometric uncertainty, it fails to take into account the varying quality of astrometry within and between tracklets in a candidate track. The upshot of this approach is that most such erroneous linkages show residuals clearly inconsistent with the astrometric uncertainty, and yet they pass the MOPS quality control test. Rather than modifying MOPS and re-running the simulation, we post-processed the post-fit astrometric residuals, with their associated uncertainties, to derive the sum of squares of the normalized residuals for each orbit in the NEO catalog. This provided the so-called χ^2 of residuals, from which it is straightforward from classical statistics to calculate the probability p_val that the fit is valid, which is to say, the likelihood of of getting a higher value of χ^2 by chance. A higher post-fit χ^2 naturally leads to a lower p_val because the increased residuals reflect a poorer fit that has a lower probability.Figure <ref> depicts the distribution of p_val among the 5348 cataloged NEO orbits. The histogram reveals that few erroneous linkages appear for p_val>0.25 and that few NEOs appear for p_val<0.25, thus we selected 25% as the p_val cutoff for acceptable orbits. This criterion led to rejection of 7% of clean and 87% of not clean orbits. Most of the clean orbits that were filtered out were MBAs mis-classified as NEOs, 14% of which were filtered out. Only 2% of clean NEO orbits were removed by this filter. As tabulated in Table <ref>, more aggressive p_val filtering—at the 50% or 90% level—is less effective at removing erroneous linkages, even as the loss of clean NEOs becomes unacceptable. Thus a modest modification of MOPS is necessary to allow a more statistically rigorous orbit quality filtering, but the rudimentary approach described here leads to a 96% purity (3816/3979, see Table <ref>) in the NEO catalog. In the context of accuracy, the clean MBAs that appear in the NEO orbit catalog are accounted as correctly linked, which is, in fact, the case. The rate of contamination of NEO orbits by false positives is extremely low, despite the large numbers of false positives injected into the detection stream. As shown in Table <ref>, after filtering at p_val>25%, only 5 false detections appear in the NEO catalog. This can be compared to the total of over 29,000 detections that form the NEO catalog and the 51M false detections polluting the data stream. This result demonstrates that NEOs can be successfully linked with high efficiency and high accuracy when surveying with the baseline LSST cadence, even in the presence of significant numbers of false detections. §.§.§ Confusion from MBAs To better understand the issue of the large fraction of NEO orbits stemming from correctly linked non-NEO objects, we used systematic ranging to explore the full orbit determination problem for these cases.Systematic ranging is an orbit estimation technique designed to analyze poorly constrained orbits, typically with only one or a few nights of astrometry, for which the conventional least squares orbit determination can fail due to nonlinearity <cit.>. We tested hundreds of cases and found that nearly all showed a characteristic “V”-shaped orbital uncertainty pattern in e vs. q that allowed both NEO and MBA orbits (left panel, Figure <ref>). In some cases the “V” shape was broken at the vertex so that there were two distinct orbital solutions (center panel, Figure <ref>). The systematic ranging technique affords a statistically rigorous estimate of the probability that the track represents an NEO orbit, and for these correctly-linked MBAs that appear with NEO orbits, few have high NEO probabilities, reflective of the fact that the data are compatible with the non-NEO (truth) orbits (Figure <ref>). It is also important to note that most of these MBAs that appear as NEOs are detected far from opposition. Figure <ref> shows that only ∼10% of these cases are found within 60 from opposition, and that about half are detected at 80 or farther from opposition. This result is merely reflecting the classical result that orbital ambiguities result from three-night orbits of objects far from opposition. It is an unavoidable feature of observing at low solar elongations, and is generally corrected after a fourth night of data is obtained. However, as described below, the current MOPS configuration does not efficiently attribute a fourth night of data to the already cataloged orbit, and so the ambiguity is often not resolved in our simulations. We note also that this confusion is an artifact of simulating only a single observing cycle. In actual operations, MBAs seen at low solar elongation would eventually move into the opposition region and appear even brighter there. These MBAs would be readily cataloged with their correct orbits because there is little ambiguity in the opposition region, at which point it becomes straightforward to link to the ambiguous orbits arising from near-sun detections. We also conducted systematic ranging analyses on some of the erroneous linkages leading to NEO orbits, almost all of which were erroneous MBA-MBA linkages, and these revealed a very different characteristic pattern in the e vs. q uncertainty space (right panel, Figure <ref>). The uncertainty region was typically very small, leading to a high computed probability that the orbit is of an NEO (“Not Clean” in Figure <ref>). In these cases, the uncertainty regions were also elongated and with one side having a sharp cutoff. In many such cases the heliocentric orbits were hyperbolic. This points to a likelihood that more effective screening tests can be developed to eliminate these false MBA-MBA linkages, despite the fact that some pass even strict orbit quality tests. For example, Table <ref> shows that even for p_val>90% a few dozen erroneous linkages remain in the NEO catalog. However, most of these erroneous MBA-MBA linkages are readily repaired when the individual MBAs are eventually re-observed at other epochs and correctly linked through other tracklets.§.§.§ Duplicate Orbits Table <ref> indicates that there were 4118 clean linkages in the NEO catalog, but not all of these are unique. Table <ref> shows that 8.7% of these are actually duplicate entries of the same object. In Figure <ref> we see that the duplicate NEO entries are of almost identical orbits, with 95% of duplicates matching in both eccentricity and perihelion distance (in au) to within 0.02. The non-NEO catalog has an even greater rate of duplication (17.3%). Virtually all of these duplicates are readily linked with standard orbit-to-orbit identifications techniques <cit.>, which are already part of MOPS. Most duplicates can be avoided altogether with a more efficient application of the MOPS attribution algorithm <cit.>. Within the linking process, a tracklet is first checked to see if it is can be attributed to an object already in the catalog. If so then it is linked to that object and removed from the tracklet list so that it is not passed along to kd-tree linking. The fact that so many objects in our simulation are linked into multiple independent tracks in a single observing cycle implies, first, that there are at least six tracklets in the lunation, indicating a very solid discovery, and second, that the attribution algorithm can easily be tuned to attribute these extra tracklets before they are even linked into tracks. Not only would such a re-tuning keep the orbit catalog cleaner, it would also cut down on the computational expense of kd-tree searches by removing tracklets from the search that are associated with already discovered objects. The problem of duplicate orbits is likely to be easily resolved through testing and tuning of existing MOPS functionality.§ DISCUSSION We performed a high-fidelity simulation of linking NEO and MBA detections into orbits in a realistic density scenario with false detections and constraints of the LSST survey in one observing cycle. Tracklet generation created false tracklets at a rate of 57% being false. This rate can be larger if one neglects the information on trail length and orientation when creating tracklets. We used this velocity information for the velocity range of 1.2–2.0 deg/day. Optimization of kd-tree parameters to provide maximum number of clean tracks is correlated with large number of false tracks and varies from field to field. It is also CPU and memory intensive, though it can be managed by distributed and multi-core or cloud computing.On a single-lunation, full-density simulation, with NEOs, MBAs and false detections, we obtained a linking efficiency of 93.6% for H<22 NEOs with 12-day tracks. Linking efficiency on the full population down to H<25 was lower. We believe that, with modest revision and tuning of the MOPS linking algorithms andan appropriate allocation of computational resources that this number can be significantly increased, probably to99% or more. On the same simulation, the derived NEO catalog was comprised of 96% correct linkages. The remaining 4% of linkages were almost exclusively incorrect MBA-MBA links, most of which should be eliminated over a longer duration simulation. Less than 0.1% of orbits in the derived NEO catalog included false detections.Some enhancements to MOPS are needed in the linking stage to eliminate duplicate and false orbits. This includes improving the orbit quality filter and tuning of the attribution, precovery[Here “precovery” refers to a search of the MOPS database for tracklets observed previously that did not form a derived object because not enough tracklets were observed at the time. It is similar to attribution of new detections, but operates on past observations.] and orbit-orbit identification modules. Together with optimization of the kd-tree track search, this would increase the linking efficiency and thus increase the number of cataloged NEOs. The linking efficiency directly affects the discovery completeness as discussed in <cit.>.Acknowledgments The Moving Object Processing System was crucial to the successful completion of this study. This tool is the product of a massive development effort involving many contributors, whom we do not list here but may be found in the author list of <cit.>. This report identifies a few deficiencies in MOPS, but our remarks should not be viewed in a pejorative sense. The software has so far never been fielded for its designed purpose, and we expect that minor improvements and tuning can resolve the issues that we have mentioned. We thank Larry Denneau (IfA, Univ. Hawaii) for his tremendous support in installing and running the MOPS software.This study benefited from extensive interactions with Zeljko Ivezic, Lynne Jones and Mario Juric, all from the University of Washington. As members of the LSST project, they provided vital guidance in understanding the performance and operation of LSST. They also provided important insight into the expected interpretation and reliability of LSST data. And they reviewed with us their early results on DECam image processing, which allowed us to include credible image differencing artifacts in the simulated LSST detection stream.Davide Farnocchia (JPL) supported the systematic ranging analyses of linking products described in this report.Mikael Granvik (Univ. Helsinki) kindly provided an early version of the <cit.> NEO population model, which was used extensively in this study.ThisresearchwasconductedattheJetPropulsionLaboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.Copyright 2017 California Institute of Technology. Government sponsorship acknowledged. [Bottke et al.(2002)]2002Icar..156..399B Bottke, W. F., Morbidelli, A., Jedicke, R., et al. 2002, , 156, 399 [Delgado et al.(2014)]2014SPIE.9150E..15D Delgado, F., Saha, A., Chandrasekharan, S., et al. 2014, , 9150, 915015 [Denneau et al.(2013)]2013PASP..125..357D Denneau, L., Jedicke, R., Grav, T., et al. 2013, , 125, 357 [Farnocchia et al.(2015)]2015Icar..258...18F Farnocchia, D., Chesley, S. R., & Micheli, M. 2015, , 258, 18 [Flaugher et al.(2015)]2015AJ....150..150F Flaugher, B., Diehl, H. T., Honscheid, K., et al. 2015, , 150, 150 [Gladman et al.(2009)]2009Icar..202..104G Gladman, B. J., Davis, D. R., Neese, C., et al. 2009, , 202, 104 [Granvik et al.(2016)]2016Natur.530..303G Granvik, M., Morbidelli, A., Jedicke, R., et al. 2016, , 530, 303 [Grav et al.(2011)]2011PASP..123..423G Grav, T., Jedicke, R., Denneau, L., et al. 2011, , 123, 423 [Hodapp et al.(2004)]2004SPIE.5489..667H Hodapp, K. W., Siegmund, W. A., Kaiser, N., et al. 2004, , 5489, 667 [Ivezić et al.(2014)]Ivezic2014 Ivezić, Ž., Tyson, J. A., Abel, B. et al. 2014, arXiv:0805.2366[Ivezic et al.(2009)]2009AAS...21346003I Ivezic, Z., Tyson, J. A., Axelrod, T., et al. 2009, Bulletin of the American Astronomical Society, 41, 460.03 [Jones et al.(2017)]jones2017 Jones, R. L., Slater, C. Moeyens, J., Allen, L., Jurić, M., & Ivezić, Ž. 2017, “The Large Synoptic Survey Telescope as a Near-Earth Object Discovery Machine,” submitted to Icarus.[Kaiser et al.(2002)]2002SPIE.4836..154K Kaiser, N., Aussel, H., Burke, B. E., et al. 2002, , 4836, 154 [Kaiser(2004)]Kaiser04 Kaiser, N., The Likelihood of Point Sources in Pixellated Images, Pan-STARRS Project Management System, PSDS-002-010-00.[Kubica et al.(2007)]2007ASPC..376..395K Kubica, J., Denneau, L., Jr., Moore, A., Jedicke, R., & Connolly, A. 2007, Astronomical Data Analysis Software and Systems XVI, 376, 395 [LSST Science Collaboration et al.(2009)]2009arXiv0912.0201L LSST Science Collaboration, Abell, P. A., Allison, J., et al. 2009, arXiv:0912.0201 [Milani et al.(2000)]2000Icar..144...39M Milani, A., Spina, A. L., Sansaturio, M. E., & Chesley, S. R. 2000, , 144, 39 [Milani et al.(2001)]2001Icar..151..150M Milani, A., Sansaturio, M. E., & Chesley, S. R. 2001, , 151, 150 [Slater et al.(2016)]Slater Slater, C., Jurić, M.,Ivezić, Ž., Jones, L., 2016.<http://dmtn-006.lsst.io>[Vereš & Chesley(2017)]2017Veres_1 Vereš, P. &Chesley, S.R.,2017, , submitted | http://arxiv.org/abs/1706.09397v1 | {
"authors": [
"Peter Vereš",
"Steven R. Chesley"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170627185441",
"title": "Near-Earth Object Orbit Linking with the Large Synoptic Survey Telescope"
} |
APS/123-QED School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Material Science and Engineering, Nanchang University, 999 Xuefu Road, Nanchang, Jiangxi, China, 330031 School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandSchool of Physics, Trinity College Dublin, Dublin 2, IrelandSchool of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandCentre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland [email protected] School of Physics, Trinity College Dublin, Dublin 2, Ireland Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, IrelandAdvanced Materials and BioEngineering Research Centre (AMBER), Trinity College Dublin, Dublin 2, Ireland Precise tunability of electronic properties of 2D nanomaterials is a key goal of current research in this field of materials science. Chemical modification of layered transition metal dichalcogenides leads to the creation of heterostructures of low-dimensional variants of these materials. In particular, the effect of oxygen-containing plasma treatment on molybdenum disulfide (MoS_2) has long been thought to be detrimental to the electrical performance of the material. Here we show that the mobility and conductivity of MoS_2 can be precisely controlled and improved by systematic exposure to oxygen:argon plasma, and characterise the material utilising advanced spectroscopy and microscopy. Through complementary theoretical modelling which confirms conductivity enhancement, we uncover the role of a two-dimensional phase of molybdenum trioxide (2D-MoO_3) in improving the electronic behaviour of the material. Deduction of the beneficial role of MoO_3 will serve to open the field to new approaches with regard to the tunability of 2D semiconductors by their low-dimensional oxides in nano-modified heterostructures. Oxide-mediated self-limiting recovery of field effect mobility in plasma-treated MoS_2 Hongzhou Zhang December 30, 2023 ======================================================================================[lines=3]The recent decade has produced intense research into layered two-dimensional nanomaterials, with transition metal dichalcogenides (TMDs) such as MoS_2 being the prime focus in the area of novel nanoelectronics <cit.>. Progress demands that a nanofabrication methodology is developed to control the structure and properties of semiconducting layered crystals so that desired functionalities are obtained for these materials in the future. These may include phase transitions <cit.> or conductivity modulation for next-generation data storage <cit.>. In particular, the interaction of low energy RF-generated plasma ions with MoS_2 has already led to the creation of a multitude of devices, including rectifying diodes, photovoltaics and non-volatile memories <cit.>. Plasma power and exposure time have emerged as key variables to delineate between chemical etching and physical sputtering regimes <cit.>. Treatment with oxygen-containing plasma leads to the formation of molybdenum trioxide (MoO_3) centres which have been reported to increase the resistivity of the material and inhibit carrier transport, while retaining relative structural integrity of the now oxide-containing MoS_2 heterostructure <cit.>.Here we demonstrate the tuning of electrical resistivity of few-layer MoS_2 by treatment with O_2:Ar (1:3) plasma. The field effect mobility, μ_FE, of the MoS_2 channel is seen to deteriorate initially but recovers to above-original levels after 6 seconds of exposure to the plasma. The associated electrical conductivity of the devices is noted to increase by an order of magnitude at this stage. Upon further treatment, the conductivity and mobility drop again and no subsequent recovery is seen. In the limited literature regarding this phenomenon, the reason for the apparent recovery remains under debate <cit.>. In this article, we propose a mechanism of impurity-mediated electrical tuning facilitated by a two-dimensional phase of MoO_3, with advanced spectroscopic and microscopic studies to support electrical characterisation. We infer the presence of this 2D molybdenum trioxide phase, which serves to provide additional charge carriers to the MoS_2 channel at 6 seconds of plasma treatment which increases its conductivity. Complementary mathematical modelling of conductive networks reveals the beneficial effect of the freshly-incorporated oxide in the MoS_2 matrix. Recent theoretical work has predicted the two-dimensional phase of MoO_3 to be a material with a distinctly high acoustic phonon-limited carrier mobility (> 3000 cm^2 V^-1 s^-1) <cit.>, while experimental 2D FETs made of sub-stoichiometric exfoliated MoO_3 have reported mobilities far exceeding that of MoS_2 ( > 1100cm^2 V^-1 s^-1) <cit.>. The advantageous effect of the 2D phase of MoO_3 on the electrical properties of MoS_2 may play a key role in the applications of planar heterostructures of layered TMDs in novel electronic devices. Future research into this area must consider the benefits of defect-mediated transport in 2D nanoelectronics. Recovery of field effect mobility in plasma-treated MoS_2 For the initial plasma exposures, the level of drain-source current for a 4 layer (4L) device varies slightly until 6 seconds, when a significant rise in output current is noted indicating an increase in the conductivity of thechannel ([fig:electrical]Fig.1a). Subsequent exposures cause a continuing drop in current level until the noise floor of the instrument (10^-11 A) is reached past 12 seconds of plasma treatment. The gate characteristics ([fig:electrical]Fig.1b) of the n-type MoS_2 change significantly after 6 seconds. The level of output current at negative gate values increases by several orders of magnitude at 6 s, implying a drastic shift in the threshold voltage (V_TH) to negative gate biases. Correlated with this is the change in the sensitivity of the output current to the applied gate-source bias (V_GS), with a much more gradual increase in output current throughout the sweep. [fig:electrical]Figures 1c,d track the evolution of the threshold gate voltage (V_TH) and the subthreshold swing (S_sub) over plasma exposure time. The threshold voltage is seen to shift from ∼ -21 V for the untreated device to ∼ -37 V at 6 s of exposure and subsequently to large positive gate biases of ∼ 22 V at 10 s and ∼ 60 V at > 10 s. The shift towards negative threshold voltages at 6 s implies increased depletion mode functionality for n-type devices, while the upshift of V_TH after further exposure denotes an increase of p-type doping. S_sub, in turn, initially shows little change until it increases 5-fold at 6 s and up to 8-fold at 10 s relative to the values before treatment. Upon further exposure, it drops again to ∼ 25 V dec^-1 as V_TH is shifted to large positive gate biases. At 6 seconds, all samples show a marked increase in S_sub, indicating that they are less sensitive to variations in gate field around the region where the FET conductive channel is formed. The field effect mobility, μ_FE, of the device is plotted as a function of gate bias in [fig:electrical]Fig.1e. The peak value of mobility is seen to drop with exposure time, and the gate bias necessary to reach saturation shifts towards larger V_GS. Importantly, for V_GS in the region from -60 V to 5 V (highlighted green area in the plot), the mobility at 6 s can now be tuned to much higher values than for the untreated device, with corresponding conductance of the device increasing by over one order of magnitude in this region. Subsequent treatments to 8 s and 10 s decrease μ_FE markedly and no recovery is seen beyond this point.In addition, the change in μ_FE over exposure time at each applied gate bias between -60 V and 60 V is charted in [fig:electrical]Fig.1f in 1 V steps. The extracted curves are color-mapped to the palette seen on the right, scaling from extremely low gate biases (-60 V) to extremely high (60 V). Inside the decay envelope of the peak mobility evident from the edge contour of this graph, we observe a series of recovery peaks around the region corresponding to a treatment time of 6 seconds. This recovery is pronounced in the linear regime near V_TH, i.e. where μ_FE rises above initial values extracted for the untreated device. This corresponds directly to the green region in [fig:electrical]Fig.1e where the red curve (6 s) attains higher values than the other curves, i.e. across V_GS∈ [-60 V, 5 V]. The subsequent drop in μ_FE and conductance is a direct consequence of material etching and introduction of scattering centres that happens after 10 seconds. The notable increase in current density seen in [fig:electrical]Fig.1a, the close-to-linear response to the variation in gate bias at 6, 8 and 10 s (red, purple and orange curves in [fig:electrical]Fig.1b), and the increase in S_sub all hint at the presence of a highly conductive phase in the material at 6 seconds of exposure, which is responsible for the recovery. In the following sections, we show this phase is a two-dimensional form of MoO_3 produced by a chemical reaction with the plasma.MoS_2 surface modification by oxygen insertion We use atomic force microscopy (AFM) to track the thickness variation and surface roughness of the plasma-treated flake. Phase maps of the same region on a 4L flake are shown in [fig:structural]Figs.2a-c, with notable change in contrast indicating material difference over time. [fig:structural]Fig.2d charts the change in the edge heights evaluated from line profiles across the edges of 4L and 5L regions (see Supplementary Fig. 9). The initial edge height on the 4L portion increases by ∼ 30% from 0 s to 6 s, and on the 5L area by ∼ 10%. This is followed by a subsequent drop in height for longer exposure times. The surface roughness ([fig:structural]Fig.2e) stays constant within instrument precision, and does not vary more than the thickness of one layer of MoS_2 or MoO_3 in the first 8 seconds of exposure. The peak in edge height at 6 s is a critical point at which the etching mechanism shifts from one largely dominated by chemical oxygen insertion into the lattice, to one where argon-dominated sputtering of material and removal of species from the surface takes over. With increasing dose of the plasma, the integration of oxygen into the mechanically-exfoliated 2H-MoS_2 structure will introduce considerable change; including rearrangement of electronic density and effective lattice deformation which increases the interlayer distance and raises the thickness of the thin MoS_2-xO_x film <cit.> while forming oxide-containing patches on the surface. These fine oxide patches, spectroscopically determined to exist by Ko et al. <cit.>, can be seen in the scanning electron micrograph in [fig:structural]Fig.2g. The contrast is due to the higher work function of MoO_3 (6.6 eV) <cit.> compared with that of MoS_2 (∼4.04-4.47 eV) <cit.>. The structural modification undergone by the MoS_2 in the plasma chamber can also be linked to the change in its optical contrast over time (see Supplementary Fig.10). In this sputtering-dominated regime, the surface roughness is seen to increase by over 1 nm at 28 seconds of exposure. However, the unchanging surface roughness up until 8 seconds indicates initial direct conversion of MoS_2 into a planar oxide. Most importantly, the edge height trend correlates with the electrical recovery discussed in the previous section, with a peak at 6 seconds. Spectroscopic analyses of the surface-bound oxide We investigate the change to the chemical content of our MoS_2 devices by employing Raman, photoluminescence (PL), energy dispersive X-ray (EDX) and X-ray photoelectron (XPS) spectroscopies. The Raman spectrum ([fig:spectroscopies]Fig.3a) shows notable shifts in the characteristic peaks corresponding to the A^'_1 mode at ∼404 cm^-1 and the E^' mode at ∼386.0 cm^-1 once the sample is exposed to the plasma. Accompanied by a 6-fold decrease in amplitude, the E^' peak downshifts to ∼384 cm^-1 while the intensity of the A^'_1 peak decreases 5-fold and its position upshifts to ∼410 cm^-1, increasing the fitted peak separation from Δω = 18 cm^-1 (characteristic for monolayer <cit.>) to Δω = 26 cm^-1 (see inset and fits in Supplementary Fig.12). The insertion of oxygen into the MoS_2 crystal lattice by the plasma can account for the change in dielectric screening environment and the restoring forces between adjacent MoS_2 molecular layers, thereby affecting the frequencies of both characteristic modes. An increase in Δω occurs when MoO_3 replaces MoS_2 on the surface of the material <cit.>. Conversely, the peak separation remains constant or is reduced when no oxides are detected after plasma treatment <cit.>. In addition, the asymmetric peak broadening of both peaks over time seen in [fig:spectroscopies]Fig.3a has been associated with the presence of additional defect-induced phonons originating from oxide centres in plasma-treated MoS_2 <cit.>. The PL intensity is reduced considerably, with a notable shift in peak positions, even just after 2 seconds of plasma exposure ([fig:spectroscopies]Fig.3b). With further exposure, the emission is nearly fully quenched after 8 seconds. The bidirectional shifts of the A peak over time serve to illustrate band structure distortion induced by the plasma treatment. The photoluminescence quenching is due to the defect-induced midgap states that inhibit direct excitonic recombination <cit.>. The associated quenching rate increases greatly with defect density, inhibiting the radiative recombination completely after 10 seconds of exposure (see Supplementary Figs.13,14). Areal EDX mapping of the MoS_2 flakes [fig:spectroscopies](Fig.3c) unveils the insertion of oxygen into the MoS_2 structure in a patch-like pattern, as has previously been suggested <cit.>. We note that plasma exposure time correlates with the gain in the oxygen K_α line relative to the sulfur K_α peak. We plot the relationship of the oxygen and sulfur peak intensities to the plasma exposure time ([fig:spectroscopies]Fig.3d). Oxygen content has increased 3-fold in the first 10 seconds of exposure to the plasma. Importantly, areas high in O signal also show a reduced S signal, suggesting that the oxygen has replaced the sulfur in the MoS_2 lattice through an oxide-forming chemical reaction.[fig:spectroscopies]Figure 3e shows XPS spectra of the Mo 3d region, indicating the increased presence of oxide species over exposure time. For the pristine sample, the peaks around 229 eV and 232 eV correspond to, respectively, the Mo^4+ 3d_5/2 and Mo^4+ 3d_3/2 spin-orbit split components. The 6 s spectrum shows a characteristic Mo^6+ 3d doublet attributed to MoO_3 <cit.>. After 10 s of exposure, the intensity of the trioxide-associated doublet increases further, with a significant ratio of the surface now containing MoO_3 (estimated at 30-40% from areas of each fitted component). In addition, a thickogram calculation <cit.> reveals that the intensity attenuation of the 10 s spectrum is consistent, within known parameters, with the presence of a bilayer of 2D-MoO_3, i.e. a bulk unit cell of α-MoO_3 on the surface at this exposure time, and 61% of the bilayer of 2D-MoO_3 at 6 s (see details in Supplementary Section 3). [fig:spectroscopies]Figure 3f demonstrates the S 2p region. Peak broadening is evident with increased plasma exposure time, indicating a change in chemical order of the surface. Sub-stoichiometric MoS_2-x has also been reported to cause the characteristic broadening of the S 2p doublet <cit.>, consistent with the picture of sulfur atoms being removed from the surface of the MoS_2 flakes.Most interestingly, both the Mo 3d and S 2p signals are upshifted after 6 s and downshifted after 10 s. It is widely accepted that MoO_3 can induce hole doping and concomitant downshifting of the MoS_2 peaks due to Fermi level realignment <cit.>. This is in agreement with our transfer curves, with significant threshold voltage shift to positive gate biases at higher plasma exposure times ([fig:electrical]Fig.1c). The upshift at 6 seconds may correlate with the n-type doping observed in the transfer curves in [fig:electrical]Fig.1b. All the results demonstrate that the plasma-treated MoS_2 undergoes a continuing oxygen insertion and crystal structure distortion. However, plasma-generated MoO_3 is an insulator <cit.>. The electrical recovery at the 6 s mark indicates that an intermediary phase must exist between the pristine MoS_2 semiconductor and the MoO_3-rich insulator. This indicates that the 2D-MoO_3 phase at 6 s is markedly different from its bulk counterpart <cit.>.Nanoscale effects of plasma etching at recovery time Many etching mechanisms, some contradictory, have been proposed for the surface reaction of oxygen-containing plasma with MoS_2 <cit.>. To uncover the nature of the elusive two-dimensional oxide phase, we go on to study the effects of the plasma etching on the nanoscale by aberration corrected scanning transmission electron microscopy (AC-STEM). [fig:microscopy]Figure 4a shows a region of a bilayer MoS_2 flake after plasma treatment of 6 seconds. Notable damage occurs to the MoS_2 lattice at this exposure time. Regions of MoS_2 material are completely removed, in nanometre-sized regions. These pits deepen with increasing plasma dose and eventually become perforations. This etching phenomenon is seen to nucleate from individual defective sites, spanning only a few nanometres across. Some of these voids are missing a part of the top molecular layer of MoS_2 after 6 s, leaving behind a bare monolayer region underneath (as confirmed by simulation in Supplementary Fig 28). Large-scale AC-STEM micrographs are presented in Supplementary Fig.25. These images were used to obtain statistics on the dimensions of voids formed by the plasma in the MoS_2 at the recovery time. [fig:microscopy]Figure 4b demonstrates the distributions of the extracted widths and lengths of imaged voids on the bilayer flake. Yellow (green) histograms show the width (length) distributions. Length is here defined as the largest void dimension, while width is the dimension perpendicular to it. A positive correlation between the lateral dimensions of the etched voids is extracted from data fitting (see residuals in Supplementary Fig.29), showing the close-to-isotropic growth of the voids. The average area of a pore at 6 s is 12.5 ± 0.1 nm^2. At this time, the relative total percentage area covered by the voids from images sampled in the AC-STEM is ∼ 3.6%.Markedly, the nanoscale EDX and EELS mapping performed with the 60 pm electron beam probe return spectra suggesting very little oxygen presence (see Supplementary Fig. 33). It has been demonstrated that oxygen plasma interaction with molybdenum metal leads to the creation of volatile Mo oxides <cit.>. We find that the plasma-formed oxide phase studied presently is volatile under ultra high vacuum. When the sample is left overnight in the in-situ testing system (∼10^-9 mbar), the n-type depletion mode functionality is reversed by a drastic shift of V_TH towards positive gate biases (see Supplementary Fig. 6). Similarly, when inserted into the vacuum chamber of the AC-STEM overnight (∼10^-9 mbar), the free-standing flakes lose their weakly-bound surface oxides. The inability to detect oxygen in these atomically-resolved voids leads us to infer that the oxide which was present initially and is responsible for the electrical recovery was sublimated at UHV conditions, leaving behind the underlying MoS_2 structure. Resistive network modelling of conductivity over time Modelling of classical conductive networks has proven to be an excellent avenue to describe the global conductive properties of nanoscale devices based on local properties of a network <cit.>. An appropriate conductive network model was thus applied to approximate the effect of plasma-etching on the sheet conductance of MoS_2. [fig:simulation]Figure 5a shows a plot of the relative conductance of a simulated MoS_2 network, whose nodes undergo conversion to highly conductive 2D-MoO_3 and insulating MoO_3 phases over time (in arbitrary units). The increase in sheet conductance yielded at the start of the simulation qualitatively mirrors the recovery peaks in [fig:electrical]Fig. 1f. This can only be observed if the conductance of the 2D phase is much higher than that of MoS_2 and the relative transition rates between phases follow a relationship such that already defective sites are more likely to convert (see Methods). The relative concentration of conductors that exist in each phase is sampled throughout the simulation and visualised through the color-mapped lattices in i-iv. The sheet conductance rises initially as the concentration of 2D-MoO_3 increases. The conductance (and associated field effect mobility) reaches a peak, just as observed in experimental results in [fig:electrical]Fig.1f, and proceedsto drop off as conductors begin to transition to the insulating MoO_3.The differing transition rates affect not only the size of the conductive peak but phase concentrations that it occurs at (see Supplementary Section 5). The network map sampled at the conductance peak (ii) shows that the spike in sheet conductance occurs with a relative network coverage of 2D-MoO_3 at approximately 15%. This differs from the experimentally determined areal void coverage of ∼ 4% at 6 s, but remains well under the site percolation threshold for a square lattice <cit.>. The difference may originate in the underestimation of the area of oxide-born defective sites from AC-STEM images of bilayer flakes. Some voids can become filled with adventitious hydrocarbons over time and/or be obscured by a top molecular layer depending on sample orientation when transferring to TEM grid. This resistive network model demonstrates qualitatively that a highly conductive intermediate oxide phase will serve to facilitate a recovery in the conductance and associated μ_FE of a MoS_2 sheet over plasma exposure time. Outlook and conclusion We have demonstrated a simple and reliable method to tune the electronic properties of few-layer MoS_2 FETs by using O_2:Ar (1:3) plasma. The apparent recovery of electrical conductivity is attributed to the temporary presence of a volatile two-dimensional phase of MoO_3, whose effect on the performance of the FET is self-limiting as further exposure results in physical etching and removal of the oxide. We have also inferred the existence of this 2D phase of MoO_3 from evidence collected from advanced spectroscopic and microscopic studies. Additionally, we have demonstrated with a robust simulation that the presence of a conductive phase on the surface of MoS_2 will induce a dose-dependent recovery in the conductance of the material network. Our results are of great importance to groups studying novel 2D TMDs and their low-dimensional oxides for future heterostructure van der Waals devices.Methods MoS_2 exfoliation, identification and transfer: MoS_2 flakes were mechanically exfoliated from commercially available bulk molybdenite crystals (SPI Supplies) using the adhesive tape method and deposited on a pre-cleaned Si substrate capped by 285 nm of SiO_2. Samples of up to 10 layers in thickness were identified through optical contrast measurements and Raman spectroscopy. Electron beam lithography was employed to define contacts, followed by deposition of metal film (5 nm Ti/35 nm Au) and lift off in acetone. Suspended MoS_2 samples were prepared utilising the stamp-transfer methodology <cit.> to move flakes from substrates onto TEM grids by etching away the SiO_2 surface underneath a polymer-embedded MoS_2 flake.Plasma treatment: The on-chip MoS_2 FET devices were modified in a Fischione Instruments 1020 plasma cleaner, producing a 13.52 MHz field to ionise a 1:3 mixture of O_2:Ar gas at a constant chamber pressure of ∼5 mbar. The samples were always exposed to the plasma for 2 seconds at a time, at the same position in the chamber (to within ± 1 mm), to control the accuracy of the experiment. After each exposure, the samples were removed and characterised electrically. Electrical measurements: The devices were globally back-gated through the highly doped Si substrate and measured in a two-probe configuration at a pressure of 10^-4 mbar in the vacuum chamber of a scanning electron microscope. The source and drain terminals were provided by tungsten nanomanipulator tips (Imina miBot) connecting the deposited contacts to an Agilent B2912A dual channel sourcemeter.Microscopy: Transmission electron microscopy was carried out in an FEI Titan 80-300 operated at 300 keV, at a chamber pressure of 4 · 10^-7 mbar. Atomic force microscopy was performed at ambient pressure in an Oxford Asylum system using cantilevers calibrated at 140 kHz. Aberration corrected scanning transmission electron microscopy was carried out in a NION UltraSTEM 200 system operated at 60 keV, at a chamber pressure of ∼ 10^-9 mbar.Spectroscopy: As sampling efficiency from mechanically exfoliated flakes is extremely low, XPS was performed on larger flakes whose surface (∼2 mm^2) was plasma-treated in the same way as all FET samples after deposition on Si/SiO_2 substrates. The system utilised a monochromated Al K_α X-ray source with an Omicron EA 125 hemispherical analyser set to a pass energy of 19 eV, giving a combined instrumental and source resolution of 0.50 eV. The spectra for these samples were fitted with 2H polytype peaks, as is usual for mechanically exfoliated MoS_2 flakes. PL spectroscopy was performed on substrate-supported flakes using an excitation wavelength of 405 nm. Raman spectroscopy was carried out at atmospheric pressure with a Horiba Jobin-Yvon 488 nm laser equipped with 1200 grooves/mm and a CCD camera. Acquisition time was fixed at 10 acquisitions per second. A 100 × objective lens was used. The laser spot size was ∼1 μm, while the power of the laser was kept below 1 mW. EDX mapping was done on suspended samples using a Bruker Nano XFlash 5030 detector in a Zeiss Supra SEM at 5 keV, with a step size of 0.7nm/px. Simulation of conductive networks: The computational model begins with a resistive network of identical conductors of magnitude g_S_2 = 1. During each iteration of the simulation, one random conductor transitions from its current phase to the next phase with a certain probability, unless that conductor is already in the final MoO_3 phase. If it does not transition then one of the adjacent sites is chosen and the transition check process is repeated. The probabilities represent the differing transition rates that occur between phases. While the transition rates are experimentally unknown, the assumption is made that they progress such that MoS_2 p_1→ 2D-MoO_3 p_2→ MoO_3, where p_1, p_2 indicate relative conversion probabilities for each process and p_2 > p_1. This relationship stems from the fact that the MoS_2 basal plane is chemically unreactive, but any defective nucleation sites will be more likely to facilitate chemical reactions once they are formed <cit.>. The sheet conductance is then calculated using Kirchhoff's and Ohm's laws (see Supplementary Section 5). The iterations are continued until all conductors are in the insulating MoO_3 phase. 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Bie, author Y.-B. Zhou, author Z.-M. Liao, author K. Yan, author S. Liu, author Q. Zhao, author S. Kumar, author H.-C. Wu, author G. S. Duesberg, author G. L. Cross,et al.,@noopjournal journal Advanced Materials volume 23, pages 3938 (year 2011)NoStop [KC et al.(2015)KC, Longo, Wallace, and Cho]kc2015surface author author S. KC, author R. C. Longo, author R. M. Wallace,andauthor K. Cho, @noopjournal journal Journal of Applied Physicsvolume 117, pages 135301 (year 2015)NoStop§ ACKNOWLEDGEMENTS We are grateful to members of staff at the Advanced Microscopy Laboratory, CRANN, Trinity College Dublin for their continued technical support. We thank C. P. Cullen and S. Callaghan for fruitful discussions regarding XPS. The work at the School of Physics and the Centre for Research on Adaptive Nanostructures and Nanodevices at Trinity College Dublin is supported by Science Foundation Ireland [grant No: 11/PI/1105, 12/TIDA/I2433 07/SK/I1220a and 08/CE/I1432] and the Irish Research Council [grant No: GOIPG/2014/972 and EPSPG/2011/239]. § AUTHOR CONTRIBUTIONS J.J analysed the data, created figures and wrote the manuscript with input from C.O'C and H.Z. I. O'R. discovered the phenomenon. J.J and Y.Z. conducted subsequent plasma exposures and electrical tests. C.O'C. and E.W. carried out the resistor network modelling. Y.Z. performed Raman experiments. P.M. and J.J. carried out SEM imaging and EDX measurements. D.F. and J.J. performed HRTEM and STEM. C.D. acquired AC-STEM images and associated EELS and EDX maps. A.S analysed the AC-STEM results and performed QSTEM simulations. J.J., Y.Z. and J.G. carried out the PL measurements. C. McG. and J.J. performed XPS experiments and analysis. D.K. and J.J. carried out AFM experiments. M.S.F., A.L.B., J.J.B. and V.N. oversaw the experimental work. H.Z. conceived the study and supervised the project. All authors have given approval for the final version of the manuscript.§ ADDITIONAL INFORMATION Supplementary information containing additional experimental data for each of the sections discussed in this manuscript is available online. Data availability: Correspondence and requests for materials should be addressed to H.Z. Raw data (code and source data for graphs) generated and/or analysed during the current study are available in the Zenodo repository at: https://www.zenodo.org/record/809442DOI:10.5281/zenodo.809442§ COMPETING FINANCIAL INTERESTSThe authors declare no competing financial interests. | http://arxiv.org/abs/1706.08573v1 | {
"authors": [
"Jakub Jadwiszczak",
"Colin O'Callaghan",
"Yangbo Zhou",
"Daniel S. Fox",
"Eamonn Weitz",
"Darragh Keane",
"Ian O'Reilly",
"Clive Downing",
"Aleksey Shmeliov",
"Pierce Maguire",
"John J. Gough",
"Cormac McGuinness",
"Mauro S. Ferreira",
"A. Louise Bradley",
"John J. Boland",
"Valeria Nicolosi",
"Hongzhou Zhang"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170626193158",
"title": "Oxide-mediated self-limiting recovery of field effect mobility in plasma-treated MoS$_2$"
} |
2.05 2020 Equations de conservation et lois de comportement d'un polymre lectroactif1420 M. TIXIERa, J. POUGETb1220 a. Dpartement de Physique, Universit de Versailles Saint Quentin, 45, avenue des Etats-Unis, F-78035 Versailles, France; [email protected]. Institut Jean le Rond d'Alembert, UMR 7190, Universit Pierre et Marie Curie, CNRS, F-75005 Paris, France; [email protected] Résumé :Un polymère électro-actif ionique (le Nafion par exemple) peut ê tre utilisé comme capteur ou comme actionneur. Pour ce faire, on place une fine couche de ce matériau saturé d'eau entre deux é lectrodes. La saturation en eau entraîne une dissociation quasi compl ète du polymère et la libération de cations de petite taille. L'application d'un champ électrique perpendiculairement à la lame provoque la flexion de celle-ci. Inversement, le fléchissement de la lame fait apparaître une différence de potentiel entre les é lectrodes. Ce phénomène fait intervenir des couplages multiphysiques de type électro-mécano-chimiques. Nous avons modélisé ce système par un milieu poreux dé formable dans lequel s'écoule une solution ionique et nous avons utiliséune approche "milieu continu". Les équations de Maxwell et de conservation de lamasse, de la quantité de mouvement et de l'énergie sont écrites d'abord àl'échelle microscopique pour chaque phase et pour les interfaces, puis pour le matériau complet grâce à une technique de moyenne. La thermodynamique des processus irréversibles linéaires nous permet d'en déduire les lois de comportement : une loi rhéologique de type Kelvin-Voigt, des lois de Fourier et de Darcy généralisées et une équation de type Nernst-Planck. 1620 Abstract :Ionic electro-active polymer (Nafion for example) can be used as sensor or actuator. To this end, a thin film of the water-saturated material is sandwiched between two electrodes. Water saturation causes a quasi-complete dissociation of the polymer and the release of small cations. The application of an electric field across the thickness results in the bending of the strip. Conversely, a voltage can be detected between the two electrodes when the strip is bent. This phenomenon involves multiphysics couplings of electro-mechanical-chemical type.The system is modeled by a deformable porous medium in which flows an ionic solution and we use a continuous medium approach. Maxwell's equations and conservation laws of mass, linear momentum and energy are first written at the microscopic scale for each phase and interfaces, then for the complete material using an average technique. Thermodynamics of linear irreversible processes provides the constitutive equations : a Kelvin-Voigt stress-strain relation, generalized Fourier's and Darcy's laws and a Nernst-Planck equation. 1420 Mots clefs :Electro-active polymers - Multiphysics coupling - Deformable porous media - Balance laws - Constitutive relations - Polymer mechanics - Nafion§ INTRODUCTIONLes polymères électro-actifs sont des matériaux innovants trè s intéressants, notamment dans les domaines de la bio-inspiration (conception d'ailes battantes inspirées du vol des insectes), de la biom écanique (muscles artificiels) et de la récupération d'é nergie. Nous nous sommes plus particulièrement intéressés à une lame de polyélectrolyte de type Nafion recouverte sur ses deux faces d'une fine couche de métal servant d'électrodes (I.P.M.C.). Un tel système présente des déformations de grande amplitude lorsqu'il est soumis à des différences de potentiel de quelques volts; inversement, la flexion de la lame fait apparaître une différence de potentiel entre les électrodes. Il peut donc être utilisé comme capteur ou actionneur.Lorsque le polyélectrolyte est saturé d'eau, il se dissocie quasi complètement, libérant des cations de petite taille (H^+, Li^+ ou Na^+); les anions restent fixés sur le squelette du polym ère. Lorsqu'un champ électrique est appliqué perpendiculairement aux électrodes, les cations se déplacent vers l'électrode né gative (cathode), entraînant avec eux le solvant par un phénomè ne d'osmose. Ceci provoque un gonflement du polymère au voisinage de la cathode et une diminution de volume du côté opposé. Il en ré sulte un fléchissement de la lame vers l'anode. Une lame de 200 μ m d'épaisseur de quelques centimètres de long fléchit ainsi de quelques millimètres en une seconde sous l'action d'une différence de potentiel de quelques volts <cit.>. La modélisation de ce matériau doit donc prendre en compte les couplages entre phénomè nes électriques, chimiques et mécaniques.§ MODLISATIONNotre modélisation est basée sur la thermomécanique des milieux continus. Les chaînes polymères chargées négativement sont assimilées à un milieu poreux déformable, homogène et isotrope. Ce milieu poreux est saturé d'une solution ionique constitu ée par l'eau et les cations. Le matériau apparaît donc comme la superposition de trois systèmes mobiles les uns par rapport aux autres : les cations, le solvant et le solide poreux. Les grandeurs physiques relatives à ces trois systèmes sont identifiées respectivement par les indices 1, 2 et 3, l'indice 4 étant relatif à la phase liquide (eau + cations) et l'absence d'indice au matériau complet. Les phases solide et liquide sont séparées par une interface d'é paisseur négligeable (indice i). Les constituants 2, 3 et 4sont assimilés à des milieux continus, de même que le matériau complet. Nous supposerons en outre que la gravité et le champ et l'induction magné tiques sont négligeables. §.§ Processus de moyenneNous avons utilisé un modèle à gros grains développé pour les mélanges à deux constituants <cit.>.Les différentes grandeurs physiques sont tout d'abord définies à l'échelle microscopique. Cette échelle doit être suffisamment petite pour que le volume correspondant ne contienne qu'une seule phase, mais suffisamment grande pour légitimer l'hypothèse de milieu continu. A l'échelle macroscopique, on définit un volume élémentaire repré sentatif (V.E.R.) contenant les deux phases; ce volume doit être suffisamment grand pour que les grandeurs relatives au matériau complet aient un sens, mais suffisamment petit pour que l'on puisse les considé rer comme locales. Dans le cas du Nafion, l'échelle microscopique est de l'ordre d'une centaine d'Angströms et l'échelle macroscopique de l'ordre du micron <cit.>. Pour chacune des phases 3 et 4, on dé finit une fonction de présence χ _k( r ,t) de type Heaviside :χ _k=1 si la phase k occupe le point rau temps t,χ _k=0sinonLes grandeurs physiques microscopiques sont repérées par un exposant^0, les grandeurs macroscopiques sont sans exposant. Une grandeur physique macroscopique g_k est obtenue en calculant la moyenne statistique ⟨⟩ _k sur le V.E.R. d'une grandeur microscopique g_k^0 relative à la phase k. Nous supposons que cette moyenne est équivalente à une moyenne en volume (hypothèse d'ergodicité) et qu'elle commute avec les dérivées spatiales et temporelles <cit.> : g_k=⟨χ _kg_k^0⟩ =ϕ _k⟨ g_k^0⟩ _koù ϕ _k=⟨χ _k⟩ désigne la fraction volumique de la phase k. On remarquera que les grandeurs macroscopiques sont définies sur tout le matériau, alors que les grandeurs microscopiques ne sont définies que sur une phase. Une grandeur macroscopique g relative au matériau complet est obtenue par sommation des grandeurs macroscopiques relatives aux différentes phases et interfaces :g=∑_k=3,4,ig_k§.§ Modlisation de l'interfaceDans la réalité, la zone de contact entre les deux phases a une certaine épaisseur et les grandeurs physiques extensives varient continû ment d'une phase à l'autre. On peut remplacer cette réalit é complexe par deux phases volumiques dont les paramètres microscopiques peuvent être considérés comme localement constants séparées par une surface de discontinuité Σ de position arbitraire. Soit Ω, un cylindre de bases parallèlesà Σ à cheval sur l'interface. Ω est donc divisé en deux volumes Ω _3 et Ω _4 appartenant respectivement aux phases 3 et 4.Les grandeurs continues décrivant la zone de contact seront repér ées par un exposant ^0 et pas d'indice. Une grandeur surfacique microscopique g_i^0 attachée l'interface peut être définie par :g_i^0=Σ⟶ 0lim1/Σ{∫_Ωg^0dv-∫_Ω _3g_3^0dv-∫_Ω _4g_4^0dv}La grandeur macroscopique correspondante est donnée par :g_i=⟨ g_i^0χ _i⟩où χ _i=-gradχ _k.n _k désignela fonction de présence de l'interface et n _k la normalesortante de la phase k. La position de Σ est fixée de telle sorte que l'interface n'ait pas de masse volumique (ρ _i^0=0). On suppose par ailleurs qu'il n'y a pas de flux de matière d'une phase dans l'autre, d'où :V_1^0=V_2^0= V_3^0=V_4^0=V_i^0où V_k^0 désigne la vitesse microscopique de la phase k. On négligera en outre les fluctuations de vitesses de tous les constituants et de l'interface à l'échelle du V.E.R. §.§ Drives particulaires et matriellesPour écrire les équations de bilan, il est nécessaire de calculer les variations d'une grandeur extensive g_k en suivant le mouvementde la phase correspondante. C'est ce que nous appellerons une dérivée particulaire d_k/dt :d_kg_k/dt=∂ g_k/∂ t+grad g_k.V_kCette dérivée peut être définie à l'échelle microscopique ou macroscopique.Les différentes phases ne se déplaçent pas à la même vitesse. Pour calculer la variation d'une grandeur extensive g relative au matériau complet, nous définissons une dérivée en suivant le mouvement des différents constituants appelée dérivée mat érielle <cit.> :ρD/Dt( g/ρ) =ρ∑_k ρ _k/ρd_k/dt( g_k/ρ _k) =∑_3,4,i∂ g_k/∂ t+div( g_k V_k) ρD/Dt( g/ρ) =∑_3,4,i∂g_k/∂ t +div( g_k⊗V_k)où ρ _k désigne la masse volumique de la phase k rapport ée au volume du matériau complet. Cette dérivée n'a bien s ûr de sens qu'à l'échelle macroscopique. Elle ne doit pas ê tre confondue avec la dérivée particulaire d/dt en suivant la vitesse barycentrique V=∑_k=3,4ρ _k/ρV_k du matériau. §.§ Equations de bilanL'équation de bilan d'une grandeur extensive microscopique relative à la phase k de densité volumique g_k^0( x ,t) est de la forme :∂ g_k^0/∂ t+div( g_k^0 V_k^0) =-divA_k^0+B_k^0où A_k^0 désigne le flux de g_k^0 li é à des phénomènes autres que la convection et B_k^0 la production volumique de g_k^0 (terme source). A l'échelle macroscopique, cette équation devient :∂ g_k/∂ t+div( g_kV_k ) =-divA_k+B_k-⟨ A_k^0.n_kχ _i⟩où :A_k=⟨χ _kA_k^0 ⟩ B_k=⟨χ _kB_k^0⟩ Pour une interface, l'équation de bilan s'écrit <cit.> :∂ g_i^0/∂ t+div_s( g_i^0 V_i^0) =∑_3,4[ g_k^0(V_k-V_i^0) .n_k+ A_k^0.n_k] -div_s A_i^0+B_i^0où g_i^0 est une grandeur surfacique. div_s désigne la divergence le long de l'interface, A_i^0 est le flux de g_i^0 le long de l'interface lié à des phénomènes autres que la convection et B_i^0 la production surfacique de g_i^0. A l'échelle macroscopique : ∂ g_i/∂ t+div( g_iV_i ) =∑_3,4⟨χ _iA_k^0. n_k⟩ -divA_i+B_iOn notera que g_i, A_i et B_i sont des grandeurs volumiques.Pour le matériau complet, on obtient par sommation :ρD/Dt( g/ρ) =-divA+Boù :A=∑_3,4,iA_k B=∑_3,4,iB_k § EQUATIONS DE CONSERVATION§.§ Conservation de la masseL'obtention de ces équations est détaillée dans <cit.>.La conservation de la masse s'écrit pour chacune des phases et pour le matériau complet :∂ρ _k/∂ t+div( ρ _k V_k) =0 (k=2,3) ∂ρ/∂ t+div( ρV)=0avec :ρ _1=ϕ _4CM_1ρ _k=ϕ _kρ _k^0 (k=2,3)ρ _4=ρ _2+ϕ _4CM_1M_k désigne la masse molaire et C la concentration en cations relative au volume de la solution. On suppose les fluctuations de C négligeables sur le V.E.R.. §.§ Equations lectriquesL'équation de conservation de la charge électrique s'écrit :divI+∂ρ Z/∂ t=0où :ρ Z=∑_3,4ρ _kZ_k+Z_i I_3=ρ _3Z_3V_3 I_4=ρ _1Z_1V_1Z_k désigne la charge électrique massique et I_k la densité volumique de courant de la phase k; Z_i est la densité surfacique de charges de l'interface. Le champ électrique E_k^0 et l'induction électrique D_k^0 vérifient les équations de Maxwell et leurs conditions aux limites. On admet que les fluctuations du champ électrique à l'échelle du V.E.R. sont négligeables et qu'il a même valeur danstoutes les phases. On en dduit que le polymère se comportecomme un milieu diélectrique linéaire, homogène et isotrope :rotE=0 divD=ρ Z D =εEo la permittivité diélectrique ε de la phase k s'crit :ε =∑_3,4ϕ _k⟨ε _k^0⟩ _k§.§ Bilan de la quantit de mouvementLa seule force volumique appliquée est la force électrique. Pour le matériau complet :ρDV/Dt=divσ +ρ ZEOn vérifie que σ, tenseur des contraintes du maté riau global, est symétrique et qu'en l'absence de forces extérieures,la quantité de mouvement se conserve. §.§ Equations de bilan d'nergieLes deux phases peuvent être assimilées à des milieux liné aires isotropes non dissipatifs. Le théorème de Poynting prend alors la forme intégrale suivante si aucune charge ne sort du volume Ω de frontière ∂Ω <cit.> :d/dt∫_Ω1/2( E·D+B·H) dv=-∮_∂Ω( E∧H) ·nds-∫_Ω E·IdvLe membre de gauche représente la variation de l'énergie potentielle du domaine Ω en suivant le mouvement des charges. Pour le maté riau complet l'énergie potentielle E_p=1/2D ·E vérifie :ρD/Dt( E_p/ρ) =-E·I Les vitesses relatives des différentes phases V_k- V sont petites par rapport aux vitesses V_k mesurées dans le référentiel du laboratoire; dans l'exp érience évoquée précédemment pour le Nafion, on obtient ainsi des vitesses relatives de l'ordre de 2 10 ^-4 m s^-1 et des vitesses absolues voisines de 410^-3 m s^-1. En première approximation, on peut donc identifier l'énergie cinétique du matériau complet E_c=1/2ρ V^2 et la somme des énergies cinétiques des constituants ∑_3,41/2ρ _kV_k^2. L'équation de bilan de l'énergie cinétique se déduit de l'équation de bilan de la quantité de mouvement et s'écrit, pour le matériau complet :ρD/Dt( E_c/ρ) =∑_3,4[ div( σ_k·V_k) - σ_k:gradV_k] +(I-i) ·Eoù :i=I-∑_k=3,4( ρ _kZ_kV_k) -Z_iV_iest le courant de diffusion des cations dans la solution.L'équation de conservation de l'énergie totale E s'écrit :ρD/Dt( E/ρ) =div( ∑_k=3,4 σ_k·V_k) -divQoù Q désigne le flux de chaleur par conduction.L'énergie interne U s'obtient par différence entre les é nergies totale, potentielle et cinétique :ρD/Dt( U/ρ) =∑_3,4(σ_k:gradV_k) + i·E-divQavec U=E-E_c-E_p. On peut également écrire cette équation en utilisant la dérivée en suivant le mouvement du barycentre des constituants du système :ρd/dt( U/ρ) =σ: gradV+i^'·E-divQ^'où :i^'=I-ρ ZV ≃∑_k=1,3ρ _kZ_k( V_k- V) Q^'=Q-∑_k=3,4U_k ( V-V_k) -∑_k=3,4 σ _k·( V_k-V ) On peut résumer les échanges d'énergies grâce au tableau suivant :fluxE_c⟷ E_p U⟷ E_p E_c⟷ U E_p-( I-i) ·E -i·EE_c ∑_3,4σ_k·V_k +( I- i) ·E-∑_3,4 σ_k:gradV_k U -Q+i·E +∑_3,4( σ_k: gradV_k) E ∑_3,4σ_k· V_k-QLe flux d'énergie cinétique est égal au travail des forces de contact, le flux d'énergie interne est le flux de chaleur et le flux d' énergie totale est la somme des deux; on vérifie qu'il n'y a pas de terme source dans cette dernière équation. ( I- i) ·E est le travail des forces électriques et correspond à une conversion d'énergie potentielle en énergie cinétique. i·E représente l'énergie potentielle convertie en chaleur par effet Joule. ∑_3,4( σ_k: gradV_k) traduit la dissipation visqueuse, c'est à dire la conversion d'énergie cinétique en chaleur.§.§ Equation de bilan d'entropieL'équation de bilan de l'entropie volumique S s'écrit :ρD/Dt( S/ρ) =s-divΣoù Σ et s désignent respectivement le flux et la création d'entropie. Dans le référentiel barycentrique, cette équation devient :ρd/dt( S/ρ) =s-divΣ ^'où :Σ ^'=Σ -∑_k=3,4S_k( V-V_k ) § FONCTION DE DISSIPATION§.§ Relations thermodynamiquesLa relation de Gibbs s'écrit pour les phases solide et liquide <cit.> :ρ _3^0d_3^0/dt( U_3^0/ρ _3^0 ) =p_3^01/ρ _3^0d_3^0ρ _3^0/dt+ σ _3^0e^s:d_3^0ε _3^0 ^s/dt+ρ _3^0T_3^0d_3^0/dt( S_3^0/ ρ _3^0) d_4^0/dt( U_4^0/ρ _4^0) =T_4^0 d_4^0/dt( S_4^0/ρ _4^0) -p_4^0 d_4^0/dt( 1/ρ _4^0) +∑_k=1,2μ _k^0d_4^0/dt( ρ _k^'/ρ _4^0)où T_k^0 désigne la température absolue, ε _3^0 et σ _3^0e les tenseursdes déformations et des contraintes d'équilibre du solide.ε _3^0s et σ _3^0essont les parties symétriques de traces nulles de ces mêmes tenseurs.La pression p_3^0 du solide est d éfinie à partir du tenseur des contraintes et vérifie la relation d'Euler, de même que celle du liquide :p_3^0=-1/3tr( σ_3^0e) =T_3^0S_3^0-U_3^0+μ _3^0ρ _3^0 U_4^0-T_4^0S_4^0+p_4^0=∑_k=1,2μ _k^0ρ _k^'où μ _k^0 est le potentiel chimique massique du constituant k.ρ _k^' désigne la masse volumique des constituants rapport ée au volume de la solution :ρ _k^'/ρ _4^0=ρ _k/ρ _4 ρ _1^'=CM_1 ρ _2^'=ρ _2^0ϕ _2/ϕ _4 On peut raisonnablement supposer que les fluctuations des grandeurs intensives (pressions, températures, potentiels chimiques), des tenseursdes dformations ε _3^0 et des contraintes d' équilibre σ _3^0e sont négligeables à l'é chelle du V.E.R.. Si l'on fait de plus l'hypothèse de l'équilibre thermodynamique local, il vient :p=p_3=p_4=p_3^0=p_4^0 T=T_3=T_4=T_i=T_3^0=T_4^0Cette hypothèse suppose entre autres que l'équilibre thermique s' établit suffisamment rapidement. Si les déformations du solide sont petites, on obtient les relations de Gibbs, d'Euler et de Gibbs-Duhem du mat ériau global :TD/Dt( S/ρ) =D/Dt( U/ ρ) +pD/Dt( 1/ρ) -1/ρ σ _3^e^s:d_3ε _3^s/dt -∑_1,2μ _kρ _4/ρd_4/dt(ρ _k/ρ _4) p=TS-U+∑_k=1,2,3μ _kρ _k dp/dt=SdT/dt+∑_k=1,2,3ρ _kdμ _k/ dt-σ ^e^s:gradVLa relation de Gibbs peut également s'écrire dans le réfé rentiel barycentrique :Td/dt( S/ρ) =d/dt( U/ρ) +pd/dt( 1/ρ) -∑_k=1,2,3μ _kd/dt( ρ _k/ρ ) -1/ρσ ^e^s:grad V§.§ Forces et flux gnralissLe tenseur des contraintes est l'addition du tenseur des contraintes d'é quilibre σ ^e et du tenseur des contraintes visqueuses σ ^v, cette seconde partie étant nulle à l'é quilibre : σ=σ ^e+σ ^v=-p1+ σ ^e^s+σ ^vEn combinant les bilans d'énergie interne (<ref>) et d'entropie (<ref>) avec la relation de Gibbs (<ref>), on peut identifier la cré ation et le flux d'entropie : s=σ ^v:gradV/T+ i^'·E/T- Q^'·gradT/T^2+∑ _k=1,2,3ρ _k( V-V_k ) ·gradμ _k/T Σ ^'=Q^'/T +∑_k=1,2,3μ _kρ _k/T( V- V_k)Introduisons dans s les flux de diffusion de masse des cations dans la solution J_1 et de la solution dans le solide J_4 :J_1=ρ _1( V_1- V_2) J_4 =ρ _4( V_4-V_3)Ces deux flux sont linéairement indépendants. On identifieun flux scalaire, trois flux vectoriels et un flux tensoriel ainsi que les forces généralisées associées :Flux Forces1/3trσ ^v 1/TdivV Q^' grad1/T J_1 ρ _2/ρ _4[ 1/T Z_1E-gradμ _1/T+ gradμ _2/T] J_4 ρ _3/ρ[ 1/T(ρ _1/ρ _4Z_1-Z_3) E-ρ _1/ρ _4gradμ _1/T-ρ _2/ ρ _4gradμ _2/T+grad μ _3/T] σ ^v^s 1/TgradV^s § EQUATIONS CONSTITUTIVES§.§ Loi rhologiqueLe milieu est isotrope. D'après le principe de Curie, il ne peut donc pas y avoir de couplage entre des forces et des flux d'ordres tensoriels diff érents <cit.>. Compte tenu de la symétrie du tenseur σ^v^s, les flux scalaire et tensoriel d'ordre 2 s'é crivent donc :1/3tr( σ ^v) =L_1/Tdiv V σ ^v^s=L_2/TgradV^soù L_1 et L_2 sont deux coefficients phénoménologiques scalaires. gradV^s désigne la partie sym étrique sans trace du tenseur gradV. En supposant que le tenseur des contraintes d'équilibre vérifie la loi de Hooke, on obtient :σ=λ( trε) 1+2G ε+L_1^'/T( tr∙ ε) 1+L_2/T∙ εoù L_1^'=L_1-L_2/3; λ est le premier coefficient de Lamé, G le module de cisaillement et ε le tenseur des déformations du matériau complet. Si la phase liquide est un fluide newtonien et stokésien, la pression v érifie :p=-1/3tr( σ ^e) =( λ +2/ 3G) trεLa loi rhéologique obtenue s'identifie avec un modèle de Kelvin -Voigt de coefficients viscoélastiques λ _v et μ _v :λ _v≡L_1^'/T 2μ _v≡L_2/T Les différents coefficients peuvent être estimés en se basant sur le cas du Nafion saturé en eau, bien documenté dans la litté rature. On obtient les ordres de grandeur suivants <cit.> :G∼ 4.5 10^7 Paλ∼ 3 10^8 Pa E∼ 1.3 10^8 Paν∼ 0.435où E désigne le module d'Young et ν le coefficient de Poisson. Le temps de relaxation en traction est de l'ordre de 15𝑠 <cit.> et il est proche de celui en relaxation en cisaillement <cit.>. Les coefficients viscoélastiques peuvent être déduits d'essais de traction <cit.> et des temps de relaxation :λ _v∼ 7 10^8 Pa sμ _v∼ 10^8 Pa s On en déduit :L_1^^'∼ 2.1 10^11 Pa s K L_2∼ 6 10^10 Pa s K Ces coefficients dépendent fortement de la température, qui est ici assez proche de la température de transition vitreuse. §.§ Loi de Fourier gnraliseLa premire équation constitutive vectorielle s'écrit :Q^'=L_3grad1/T+L_4 ρ _2/ρ _4[ 1/TZ_1E- gradμ _1/T+gradμ _2/ T] +L_5ρ _3/ρ[ 1/T( ρ _1 /ρ _4Z_1-Z_3) E-ρ _1/ρ _4 gradμ _1/T-ρ _2/ρ _4 gradμ _2/T+gradμ _3/ T] La phase liquide est une solution diluée d'électrolyte fort. D'aprs <cit.> :μ _1( T,p,x) ≃μ _1^0( T,p) +RT/ M_1ln( CM_2/ρ _2^0) μ _2( T,p,x) ≃μ _2^0( T,p) -RT/ ρ _2^0C μ _3( T,p,x) =μ _3^0( T)où μ _2^0 et μ _3^0 désignent les potentiels chimiques du solide et du solvant pur et où μ _1^0 dépend du solvant et du soluté. Ces expressions permettent de calculer gradμ _k compte tenu des relations de Gibbs-Duhem des phases solide et liquide.§ DISCUSSION§.§ Proprits physiques du NafionNous allons approximer les deux autres relations vectorielles en nous limitantau cas isotherme et en nous basant sur le cas du Nafion<cit.>. On obtient les valeurs suivantes :Cations Solvant SolideM_k ( kg mol^-1) 10^-2 18 10^-3 10^2 - 10^3 ρ _k^0 ( kg m^-3)10^3 2.1 10^3 v_k ( m^3 mol^-1) M_1/ρ _4^0 ∼10^-5 18 10^-6ρ _k ( kg m^-3) 14 0.35 10 ^3 1.4 10^3 Z_k ( C kg^-1) 10^7 0 9 10^4où v_k est le volume molaire partiel du constituant k. C∼ 4 10^3 mol m^-3 et la viscosit dynamique de l'eau estη _2=10^-3 Pa s. La masse équivalente du polymère ou masse de polymère par mole de sites ioniques vaut M_eq ∼1.1 kg eq^-1. On prendra en outre ϕ _4∼35%, T=300 K et ‖E‖∼ 10^4 V m^-1. On en déduit ρ∼ 1.8 10^3 kg m^-3. En première approximation, on peut donc considérer que :Z_1>>Z_3ρ∼ρ _2∼ρ _3>>ρ _1ρ _1Z_1∼ρ _4Z_3§.§ Equation de Nernst-PlanckLes coefficients phénoménologiques diagonaux sont géné ralement grands devant les coefficients non diagonaux :L_6≳ L_7On en déduit :J_1≃L_6Z_1/TE+1/ T[ v_2/M_2( L_6-ρ _3L_7/ρ) -v_1L_6/M_1] gradp-RL_6/M_1C gradCexpression que l'on peut identifier avec la relation de Nernst-Planck <cit.> :V_1=-D/C[ gradC- Z_1M_1C/RTE+Cv_1/RT( 1-M_1/ M_2v_2/v_1) gradp] + V_2R=8,314 J K^-1 est la constante universelle des gaz parfaits.D=RL_6/M_1ρ _1∼2 10^-9 m^2 s^-1 est le coefficient de diffusion de masse des cations <cit.>. On en déduit :L_6∼ 3,5 10^-11 kg K s m^-3 L_7<<L_6On peut estimer les différents termes de l'équation. L'ordre de grandeur du gradient de concentration des cations peut être d éduit de <cit.> et <cit.>. Le gradient de pression estde l'ordre de 10^9 Pa m^-1. D'où :‖gradC‖ ≲ 10^8 mol m^-4 M_1C/RTZ_1‖E‖ ∼ 1.6 10^9 mol m^-4 Cv_1/RT( 1-M_1/M_2v_2/v_1) ‖gradp‖ ∼ 1.1 10^3 mol m^-4Le champ électrique et la diffusion de masse sont donc les principaux responsables du déplacement des cations; le gradient de pression a un effet négligeable. §.§ Loi de Darcy gnraliseCompte tenu des hypothèses faites, le flux J_4 peut s'écrire : J_4≃1/T[ L_7Z_1+L_8ρ _3/ρ( ρ _1/ρ _4Z_1-Z_3) ]E-R/M_1CL_7gradC +1/T[ L_7( v_2/M_2-v_1/M_1 ) -ρ _3ϕ _4/ρρ _4L_8]gradpEn identifiant le terme de pression à la loi de Darcy, on obtient :L_8∼KT/η _2ϕ _4^2ρ _2^2ρ/ρ _3∼ 3.8 10^-5 kg s K m^-3>>L_7où K désigne la perméabilité intrinsèque de la phase solide et η _2 la viscosité dynamique du solvant; compte tenu de la taille des pores (100 Å; <cit.>), K∼10 ^-16 m^2. Les ordres de grandeur des différents termes de l'équation sont les suivants :K/η _2ϕ _4‖gradp‖ ∼ 2.8 10^-4 m s^-1 Kρ _2^0/η _2ϕ _4( ρ _1/ρ _4 Z_1-Z_3) ‖E‖ ∼ 1.1 m s^-1 R/M_1Cρ _4L^8‖gradC‖ <<2 10^-6 m s^-1La relation phénoménologique obtenue s'identifie avec une loi de Darcy généralisée :V_4-V_3≃ -K/η _2ϕ _4[ gradp-ρ _4^0( Z_4-Z_3)E]Le second terme de cette expression traduit le mouvement de la solution sous l'action du champ électrique; il s'agit donc d'un terme d'osmose. La distribution des cations est très hétérogène <cit.> : ils s'accumulent près de l'électrode négative où l'on peutécrire Z_4>>Z_3. L'équation constitutive obtenue coïncide alors avec celle de M.A. Biot <cit.>. Au voisinage de l'é lectrode négative, Z_4<<Z_3 et l'on retrouve le résultat de Grimshaw et al <cit.>. Au centre de la lame, les deux termes ont le même ordre de grandeur.§ CONCLUSIONNous avons modélisé le comportement d'un polymère é lectroactif ionique saturé d'eau de type Nafion. La présence d'eau provoque une dissociation quasi totale du polymère et la libération de cations de petite taille. Nous avons représenté ce milieu par la superposition de trois systèmes ayant des champs de vitesses diffé rents : les cations, le solvant et le solide assimilé à un milieu poreux déformable. Nous avons écrit les équations de bilan de la masse, de la quantité de mouvement, de l'énergie et de l'entropie et les équations de Maxwell pour chaque phase (solide et liquide) à l' échelle microscopique. Un processus de moyenne nous a permis d'en dé duire les équations relatives au milieu complet à l'échelle macroscopique. L'écriture des relations thermodynamiques nous permet d'obtenir la fonction de dissipation du matériau.Nous en avons déduit ses lois de comportement : une loi rhéologique de type Kelvin-Voigt, une loi de Fourier et une loi de Darcy géné ralisées et une équation de Nernst-Planck. Nous avons fait une é valuation des coefficients phénoménologiques et des différents termes de ces relations en fonction des paramètres physiques des constituants.Nous envisageons maintenant de comparer les résultats fournis par ce mod èle aux données expérimentales publiées dans la litté rature. Une piste d'amélioration pourrait être de remplacer la loi rh éologique par un modèle de Zener, mieux adapté au comportement viscoélastique des polymères.§ NOTATIONSLes indices k=1,2,3,4,i désignent respectivement les cations, le solvant, le solide, la solution et l'interface. Les quantités non indic ées sont relatives au matériau complet. L'exposant ^0 indique une quantité à l'échelle microscopique, l'absence d'exposant correspond à l'échelle macroscopique. Les quantités microscopiques sont rapportées au volume de la phase correspondante, les quantités macroscopique au volume du matériau complet. L'exposant ^s désigne la partie symétrique sans trace d'un tenseur du second ordre, et ^T sa transposée.C : concentration molaire en cations (relative à la phase liquide);D : coefficient de diffusion de masse des cations dans la phase liquide;D (D_k^0) : induction électrique;E (E_k^0) : champ électrique;E (E_c, E_p, U, U_k^0) : énergie volumique totale (cinétique, potentielle, interne);G, λ, E, ν : coefficients élastiques;I(I_k) : densité volumique de courant;i(i^') : courant électrique de diffusion;J_k : flux de diffusion massique;K : perméabilité intrinsèque de la phase solide;L_i,L_i^' : coefficients phénoménologiques;M_k : masse molaire du constituant k;n_k : normale sortante de la phase k;p (p_k, p_k^0) : pression;Q (Q^') : flux de chaleur;s : production volumique d'entropie;S (S_k) : entropie volumique;T (T_k, T_k^0) : température absolue;v_k : volume molaire partiel du constituant k (relatif à la phase liquide);V (V_k, V_k^0) : vitesse;Z (Z_k) : charge électrique massique;ε (ε _k^0) : permittivité diélectrique;ε (ε _k, ε _k^0) : tenseur des déformations;η _2 : viscosité dynamique de l'eau;λ _v, μ _v : coefficients viscoelastiques;μ _k (μ _k^0) : potentiel chimique massique;ρ (ρ _k, ρ _k^', ρ _k^0) : masse volumique;σ (σ ^v, σ ^e, σ_k, σ_k^0e) : tenseur des contraintestotales (dynamiques, d'équilibre);Σ (Σ ^') : flux d'entropie par conduction;ϕ _k : fraction volumique de la phase k;χ _k : fonction de présence de la phase k ;9 nemat2000 Nemat-Nasser S., Li J., Electromechanical response of ionic polymers metal composites, Journal of Applied Physics, 87 (2000) 3321–3331Nigmatulin79 Nigmatulin R.I., Spatial averaging in the mechanics of heterogeneous and dispersed systems, Int. 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Phys., 43 (2005) 786–795Silberstein2008 Silberstein N.N., Mechanics of Proton Exchange Membranes : Time, Temperature and Hydration Dependence of the Stress-Strain Behavior of Persulfonated Polytetrafluoroethylene, Thesis, Massachusetts Institut of Technology, Cambridge, MA, 2008Silberstein2011 Silberstein M. N., Pillai P. V., Boyce M. C., Biaxial elastic-viscoplastic behavior of Nafion membranes. Polymer, 52 (2011) 529–539Combette Combette P., Ernoult I., Pysique des polymères, Hermann, Paris, 2006Diu Diu B., Guthmann C., Lederer D., Roulet B., Thermodynamique, Hermann, Paris, 2007Heitner-Wirguin Heitner-Wirguin C., Recent advances in perfluorinated ionomer membranes: structure, properties and applications. J. Membrane Sci., 120 (1996) 1–33Gebel Gebel G., Structural evolution of water swollen perfluorosulfonated ionomers from dry membrane to solution. Polymer, 41 (2000) 5829–5838Cappadonia Cappadonia M., Erning J., Stimming U., Proton conduction of Nafion@ 117 membrane between 140 K and room temperature. J. Electroanal. Chem., 376 (1994) 189–193Choi Choi P., Jalani N.H., Datta R., Thermodynamics and proton transport in Nafion I. Membrane swelling, sorption and ion-exchange equilibrium. J. Electrochem. Soc. 152 (2005) 84–89Lakshmi Lakshminarayanaiah N., Transport phenomena in membranes. Academic Press, New-York, 1969Zawodsinski Zawodsinski T.A., Neeman M., Sillerud L.O. and Gottesfeld S., Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes. J. Phys. Chem.-US 95 (1991) 6040–6044nemat2002 Nemat-Nasser S., Micro-mechanics of actuator of ionic polymer-metal composites Journal of Applied Physics, 92 (2002) 2899–2915Biot Biot M. A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys., 26 2 (1955) 182–185 Grimshaw Grimshaw P.E., Nussbaum J.H., Grodzinsky A.J., Yarmush M.L.: Kinetics of electrically and chemically induced swelling in polyelectrolyte gels. J. Chem. Phys., 93 6 (1990) 4462–4472 | http://arxiv.org/abs/1708.05056v1 | {
"authors": [
"Mireille Tixier",
"Joël Pouget"
],
"categories": [
"physics.app-ph",
"physics.chem-ph"
],
"primary_category": "physics.app-ph",
"published": "20170626143219",
"title": "Conservation laws and constitutive equations for an electro-active polymer"
} |
The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin Texas 478712–1229 [email protected] 0002-1736-4009Department of Computer Science University of Dayton 300 College Park Dayton Ohio [email protected] We present a full implementation of the parareal algorithm—an integration technique to solve differential equations in parallel—in the Julia programming language for a fully general, first-order, initial-value problem.We provide a brief overview of Julia—a concurrent programming language for scientific computing. Our implementation of the parareal algorithm accepts both coarse and fine integrators as functional arguments.We use Euler's method and another Runge-Kutta integration technique as the integrators in our experiments.We also present a simulation of the algorithm for purposes of pedagogy and as a tool for investigating the performance of the algorithm. Parareal Algorithm Implementation and Simulation in Julia Saverio Perugini December 30, 2023 =========================================================§ INTRODUCTION The parareal algorithm was first proposed in 2001 by Lions, Maday, and Turinici <cit.> as an integration technique to solve differential equations in parallel.We present a full implementation of the parareal algorithm in the Julia programming language (<https://julialang.org>) <cit.> for a fully general, first-order, initial-value problem.Furthermore, we present a simulation of the algorithm for purposes of pedagogy and as a tool for investigating the performance of the algorithm.Our implementation accepts both coarse and fine integrators as functional arguments.We use Euler's method and another Runge-Kutta integration technique as the integrators in our experiments.We start with a brief introduction to the Julia programming language.§ AN INTRODUCTION TO JULIA: DYNAMIC, YET EFFICIENT, SCIENTIFIC/NUMERICAL PROGRAMMING Julia is a multi-paradigm language designed for scientific computing; it supports multidimensional arrays, concurrency, and metaprogramming.Due to both Julia's LLVM-based Just-In-Time compiler and the language design, Julia programs run computationally efficient—approaching and sometimes matching the speed of languages like C.See <cit.> for a graph depicting the relative performance of Julia compared to other common languages for scientific computing on a set of micro-benchmarks. §.§ Coroutines and Channels in Julia Coroutines are typically referred to as tasks in Julia, and are not scheduled to run on separate CPU cores.Channels in Julia can be either synchronous or asynchronous, and can be typed.However, if no type is specified in the definition of a channel, then values of any type can be written to that channel, much like unix pipes. Messages are passed between coroutines through channels with theandfunctions. To add tasks to be automatically scheduled, use thefunction, or theandmacros. Of course, coroutines have little overhead, but will always run on the same cpu. The current version of Julia multiplexes all tasks onto a single os thread. Thus, while tasks involving i/o operations benefit from parallel execution, compute bound tasks are effectively executed sequentially on a single os thread. Future versions of Julia may support scheduling of tasks on multiple threads, in which case compute bound tasks will see benefits of parallel execution too<cit.>.§.§ Parallel Computing In addition to tasks, Julia supports parallel computing—functionsrunning on multiple cpus or distributed computers. New processes are spawned with <>, where <> is the number of processes desired. The functionreturns the pids of the created processes. The functionreturns a list of the processes. Alternatively, the Julia interpreter can be started with the<> option, where <> is the number of processes desired. For instance:[language=Matlab] julia julia> addprocs(3) 3-element ArrayInt64,1:234julia> workers() 3-element ArrayInt64,1:234^D julia -p 3 julia> workers() 3-element ArrayInt64,1:234^D Note that the process ids start at 2 because the Julia REPL shell is process 1.Processes in Julia, which are either locally running or remotely distributed, communicate with each other through message passing.The function <> <> <> executes <> on worker <> and returns a value of thetype, which contains a reference to a location from which the return value can be retrieved, once <> has completed its execution.Thevalue can be extracted with the function , which blocks until the result is available. Thus, the functionis used to send a message while the functionis used to receive a message. For instance:[language=Matlab] julia> addprocs(2) julia> future = remotecall(sqrt, 2, 4) Future(2,1,3,NullableAny()) julia> fetch(future) 2.0 After the functionis run, the worker process simply waits for the next call to .[language=Matlab] julia> counter1 = new_counter(3) (::#1) (generic function with 1 method) julia> future = remotecall(counter1, 2) Future(2,1,23,NullableAny()) julia> fetch(future) 4 The Julia macrosimplifies this message-passing protocol for the programmer and obviates the need for explicit use of the low-levelfunction.Similarly, the macrocan be used to run each iteration of a (for) loop in its own process.[language=Matlab] julia> future = @spawn sqrt(4) julia> fetch(future) 2.0 julia> addprocs(2) 2-element ArrayInt64,1:34 julia> @everywhere function fib(n) if (n < 2) return n elsereturn fib(n-1) + fib(n-2) end end julia> @everywhere function fib_parallel(n) if (n < 35) return fib(n) else x = @spawn fib_parallel(n-1) y = fib_parallel(n-2) return fetch(x) + y end end julia> @time fib(42) 2.271563 seconds (793 allocations: 40.718 KB) 267914296 julia> @time fib_parallel(42) 3.483601 seconds (344.48 k allocations: 15.344 MB, 0.25 There are also remote channels which are writable for more control over synchronizing processes. §.§ Multidimensional Arrays Julia supports multidimensional arrays, an important data structure in scientific computing applications, with a simple syntax and their efficient creation and interpretation over many dimensions <cit.>.The function call <> creates an array, where the n^th argument in <> specifies the size of the n^th dimension of the array.Similarly, the programmer manipulates these arrays using function calls that support infinite-dimensional arrays given only limitations on computational time.In summary, Julia incorporates concepts and mechanisms—particularly concurrency and multidimensional arrays—which support efficient scientific computing.§ THE PARAREAL ALGORITHM The parareal algorithm is designed to perform parallel-in-time integration for a first-order initial-value problem.The algorithm involves two integration techniques, often known as the `coarse' integrator and the `fine' integrator. For the algorithm to be effective, the coarse integrator must be of substantially lower computational cost than the fine integrator. The reason will become apparent later in this section. Consider the differential equation (<ref>) given by y'(t) = f(t,y(t))t ∈ [a,b] with its associated initial-value problem (<ref>)y(t^*) = y^* t^*∈ [a,b]. For simplicity, let us assume t^* = a, so that the solution only extends rightward.To obtain an approximate solution to equation (<ref>) satisfying the initial condition (<ref>), we partition our domain into [t_0=a,...,t_N=b] with uniform step size Δ. We now precisely define an `integrator' as a function from (0,∞) ×ℝ^2×ℛ to ℝ where ℛ is the set of all Riemann integrable functions. For example, the integrator I given by I(δ,x_0,y_0,g) = y_0 + g(x_0,y_0)δ is the integrator corresponding to Euler's method with step size δ. Let 𝒞 and ℱ be the coarse and fine integrators, respectively. Definey_0,1 = y(t_0) = y^*y_n+1,1 = y(t_n+1) = 𝒞(Δ,t_n,y_n,1,f). Since y_n+1,1 depends on y_n,1, this algorithm is inherently sequential. Partition [t_n,t_n+1] into {t_n^0=t_n,...,t_n^m,...t_n^M=t_n+1} with uniform step size δ < Δ. Define z_n,1^0 = y(t_n^0) = y_n,1z_n,1^m+1 = y(t_n^m+1) = ℱ(δ,t_n^m,z_n,1^m,f). This yields an approximate solution {z_n,1^0,...,z_n,1^M} to (<ref>) over [t_n,t_n+1] with initial conditionsy(t_n) = y_n,1. Since z_n_1,1^m_1 does not depend on z_n_2,1^m_2 for n_1 ≠ n_2, we can compute these approximations in parallel. After the last subproblem is solved, we simply combine the solutions on each subdomain to obtain a solution over the whole interval. However, our values { y_1,1,...,y_n,1} are relatively inaccurate. The vertical spikes in the orange graph separating the coarse and fine predictions in Figure <ref> illustrate this error.However, z_n-1,1^M is a better approximation for ϕ(t_n)where ϕ is the exact solution to the differential equation. We use this to obtain abetter set of points {y_n,2} for the coarse approximation. We do this by first defining w_n,1 = y_n,1 and then definingw_1,2 = y_1,1 = y_1,2 = y^*w_n,2 = 𝒞(Δ,t_n-1,y_n-1,2,f)y_n,2 = (w_n,2-w_n,1) + z_n-1,1^M.Thus, w_n+1,2 serves as a new prediction given a more accurate previous prediction from y_n,2 since z_n-1,1^M has now been taken into account in calculating y_n,2. In general, we continue evaluating so that for k > 1, we havew_1,k = y_1,k = y^*w_n,k = 𝒞(Δ,t_n-1,y_n-1,k-1,f)y_n,k = (w_n,k-w_n,k-1) + z_n-1,k-1^M. Note that since y_n,k is dependent on w_n,k, this step must be done sequentially. As k increases, w_n,k - w_n,k-1→ 0, which means that y_n,k converges to the value that the fine integrator would predict if fine integration were simply done sequentially. Thus, each k denotes fine integration over the whole interval. This means that the totalcomputation performed is much greater than if fine integration were performed sequentially.However, the time efficiency of each iteration has the potential to be improved through concurrency. Since fine integration is more computationally intensive, this improvement in the run-time efficiency may compensate for the cumulative computation performed.Let K be the total number of iterations necessary to achieve a desired accuracy of solution and P be the number of subintervals into which we divideaccording to the coarse integrator. If K=1, then we achieve perfect parallel efficiency. If K = P, then we likely slowed the computation down. The parareal algorithm is guaranteed to converge to the solution given by the sequential fine integrator within P iterations. For a more complete treatment of this convergence analysis, we refer the reader to <cit.>. For fully general pseudocode, we refer the reader to <cit.>. § PARAREAL ALGORITHM IMPLEMENTATION IN JULIA [frame=tblr,basicstyle=,float=*, caption=Implementation of the parareal algorithm in Julia., label=lst:euler] @everywhere function parareal(a,b,nC,nF,K,y0,f,coarseIntegrator,fineIntegrator) #initialize coarse information xC = linspace(a,b,nC+1); yC = zeros(size(xC,1),K); deltaC = (b-a) / (nC + 1); yC[1,:] = y0;#"coarse integrator partially evaluated" ciPEvaled = ((x1,y1) -> coarseIntegrator(deltaC,x1,y1,f));#get initial coarse integration solution for i=2:(nC+1)yC[i,1] = ciPEvaled(xC[i-1],yC[i-1,1]); end correctC = copy(yC);#initialize fine information xF = zeros(nC,nF+1); for i=1:nCxF[i,:] = linspace(xC[i],xC[i+1],nF+1); end sub = zeros(nC,nF+1,K); deltaF = xF[1,2] - xF[1,1];#"fine integrator partially evaluated" fiPEvaled = ((x1,y1) -> fineIntegrator(deltaF,x1,y1,f));for k=2:K#run fine integration on each subdomaintic();@sync for i=1:nC sub[i,1,k] = correctC[i,k-1]; @async for j=2:(nF+1)sub[i,j,k] = fiPEvaled(xF[i,j-1],sub[i,j-1,k]); endendtoc();#predict and correctfor i=1:nC yC[i+1,k] = ciPEvaled(xC[i],correctC[i,k]); correctC[i+1,k] = yC[i+1,k] - yC[i+1,k-1] + sub[i,nF+1,k];end endyF = zeros(nC*(nF+1),K-1); for k=2:KyF[:,k-1] = reshape(sub[:,:,k]',nC*(nF+1)); endreturn reshape(xF',nC*(nF+1)),reshape(sub[:,:,K]',nC*(nF+1)),yF,sub,xC,correctC,yC; endListing <ref> presents a sequential implementation of the parareal algorithm (from the prior section) in Julia. Macros available in Julia can be used to parallelize this program.Use of these macros in the following modification to the first inner loop within the main loop make the implementation multi-threaded.[basicstyle=,numbers=none] @sync for i=1:N@async xF[i,:],sub[i,:,k] = eulerMethod(M,xC[i],xC[i+1], correctC[i,k-1],f); endListing <ref> presents an implementation of the parareal algorithm (from the prior section) in Julia. Themacro within the loop causes the program to evaluate the first expression to its right as a concurrent task (i.e., theloop assigning values to ). Themacro causes the main program thread to wait until all tasks (spawned in the the first expression to its right with anormacro) complete.Once all concurrent tasks are complete, execution of the program proceeds sequentially. Given the semantics of these macros, the program in Listing <ref> correctly perform concurrent integration. The sequential and parallel versions of this implementation have no significant differences in run-time efficiency.However, if astatement is placed in the argument of , the parallel version runs much faster.This demonstrates that use of those two macros does lead to concurrent program execution.§ GRAPHICAL ALGORITHM SIMULATION [frame=tblr,basicstyle=,float=*, caption=Implementation of a graphical simulator of the parareal algorithm in Julia., label=lst:simulate] @everywhere function fullMethod(n,a,b,y0,f,integrator)#setup domain and range space x = linspace(a,b,n+1);deltaX = x[2] - x[1]; y = ones(n+1,1);#initialize left endpoint y[1] = y0;#integrate each point for i=1:n y[i+1] = integrator(deltaX,x[i],y[i],f); endreturn x,y; endfunction simulate(a,b,N,M,K,y0,f,coarseInt,fineInt,showPrev)x1,y1 = fullMethod(N*(M+1),a,b,y0,f,fineInt);x,y,yF,sub,xC,yC,iC = parareal(a,b,N,M,K,y0,f,coarseInt,fineInt);xF = (reshape(x,M+1,N))';fine = M+1;for k=2:K display(plot(x1,y1)); if(showPrevk > 2 )display(scatter!(xC,yC[:,k-2],color="red",legend=false)); end display(scatter!(xC,yC[:,k-1],color="green",legend=false)); done = zeros(Int64,N,1); workingSubdomains = 1:N; while(done != (M+1) * ones(N,1) )index = Int64(ceil(size(workingSubdomains,1)*rand()));currThread = workingSubdomains[index];while( done[currThread] == M+1 ) currThread = Int64(ceil(N * rand()));endcurrThreadPlot = Int64(ceil(fine*rand()));totalAdvance = done[currThread] + currThreadPlot;if(totalAdvance > fine) totalAdvance = fine; endnewP = (done[currThread]+1):totalAdvance;display(plot!(xF[currThread,newP],sub[currThread,newP,k],color="black"));done[currThread] = totalAdvance;workingSubdomains = find( ((x)->x != M+1), done );print(join(["Working on subdomain #", currThread, "...", "Pending Subdomains: ", workingSubdomains', ""])); end display(plot!(x,yF[:,k-1],color="orange")); sleep(5);end end# Implementation schemes. function euler(delta,x0,y0,f)return y0 + delta * f(x0,y0); endfunction rungeKutta(delta,x0,y0,f)k1 = f(x0,y0);k2 = f(x0+delta/2,y0 + (delta/2)*k1);k3 = f(x0+delta/2,y0 + (delta/2)*k2);k4 = f(x0+delta,y0+delta*k3);return y0 + (delta/6)*(k1+2*k2+2*k3+k4); endThe functionin Listing <ref> creates a graphical simulator of the parareal algorithm.This function can be used to introduce the parareal algorithm to students in a numerical analysis course. The first line gets the sequential solution from the fine integrator (the `ideal' solution) and the second line gets the history of the computations that took place during the parareal execution. The main loop over the variable k then displays the inner workings of the algorithm. The ideal solution is plotted, with a scatter plot of the points obtained from the coarse integrator. To simulate the parallel nature of the algorithm, random progress is made on randomly selected subdomains. Thus, the plot dynamically makes partial progress on different subdomains until all subdomains are finished with the fine integration. After this, the plots are connected into the current iteration's approximation. During the next iteration, the previous guesses from the coarse integrator are displayed in red and the new guesses from the coarse integrator are displayed in green. As k increases, these guesses converge to the ideal solution.In addition to the use of this function for pedagogical purposes, it can be used to investigate the types of curves for which the parareal algorithm might be practical.For instance, consider the differential equationy'(x) = sin(xy),x ∈ [-20,20] with y(-20) = 10, Δ = 4 (10 points), and δ = 0.008 (500 points).Figure <ref> shows the first and ninth iterations.The ninth iteration's large error on the right end of the interval shows that this is an example where parareal convergence is slow. This is as inefficient as possible, needing as many iterations as subdomains in order for the solution to converge.However, the simulation also shows that if f(x,y) = sin(x)e^x, then the solution converges after just one iteration.These two examples show that the algorithm's efficiency can be highly dependent on the integrand. Below the simulation function are Euler's method and another Runge-Kutta integration technique that can be used as examples to be passed as first-class functions as coarse or fine integration techniques to the `parareal' or `simulate' functions.A Git repository of both the implementation and graphical simulation is available in BitBucket at <https://bitbucket.org/sperugin/parareal-implementation-and-simulation-in-julia>. All of the graphical plots are generated with the Julia Plots package available at <https://juliaplots.github.io/>.A video describing this application of Julia is available on YouTube at <https://www.youtube.com/watch?v=MtgbeLO6ZM4>.00 #1 #1DOI:0#1#1 #1 #1 #1 #1#1#1#1 #1#1 [??web] website titleJulia Micro-Benchmarks. howpublishedAvailable: <https://julialang.org/benchmarks/> [Last accessed: 14 November 2018]. (year????).[??Jul] JuliaParallelComputingMan titleJulia v1.0 Documentation: Parallel Computing. howpublishedAvailable: <https://docs.julialang.org/en/v1/manual/parallel-computing/> [Last accessed: 14 November 2018]. (year????).[AubanelAubanel2011] Aubanel authorpersonE. Aubanel. year2011. Scheduling of tasks in the parareal algorithm. journalParallel Comput. volume37, number3 (year2011), pages172–182.[Bezanson, Chen, Karpinski, Shah, and EdelmanBezanson et al2014] Operators:Julia authorpersonJ. Bezanson, personJ. Chen, personS. Karpinski, personV. Shah, and personA. Edelman. year2014. Array Operators Using Multiple Dispatch: A Design Methodology for Array Implementations in Dynamic Languages. In booktitleProceedings of ACM SIGPLAN International Workshop on Libraries, Languages, and Compilers for Array Programming. publisherACM Press, addressNew York, NY, pages56–61.[Gander and VandewalleGander and Vandewalle2007] Gander authorpersonM.J. Gander and personS. Vandewalle. year2007. Analysis of the parareal time-parallel time-integration method. journalSIAM Journal on Scientific Computing volume29, number2 (year2007), pages556–578.[Lions, Maday, and TuriniciLions et al2001] LionsEtAl authorpersonJ.-L. Lions, personY. Maday, and personG. Turinici. year2001. A “parareal” in time discretization of PDE's. journalComptes Rendus de l'Académie des Sciences - Series I - Mathematicsvolume332 (year2001), pages661–668.[Moffit and TateMoffit and Tate2014] Julia authorpersonJ. Moffit and personB.A. Tate. year2014. Julia. In booktitleSeven more languages in seven weeks: Languages that are shaping the future, editorpersonB.A. Tate, personF. Daoud, personI. Dees, and personJ. Moffit (Eds.). publisherPragmatic Bookshelf, addressDallas, TX, Chapter 5, pages171–207.[NielsenNielsen2012] Nielsen authorpersonA.S. Nielsen. year2012. titleFeasibility study of the parareal algorithm. thesistypeMaster's thesis. schoolTechnical University of Denmark. | http://arxiv.org/abs/1706.08569v2 | {
"authors": [
"Tyler M. Masthay",
"Saverio Perugini"
],
"categories": [
"cs.MS",
"cs.DC",
"cs.NA"
],
"primary_category": "cs.MS",
"published": "20170626192758",
"title": "Parareal Algorithm Implementation and Simulation in Julia"
} |
The IoT energy challenge: A software perspective Kyriakos Georgiou1, Samuel Xavier-de-Souza2, Kerstin Eder1 December 30, 2023 ============================================================== We discuss the volume of Voronoï cells defined by two marked vertices picked randomly at a fixed given mutual distance 2s in random planar quadrangulations.We consider the regime where the mutual distance 2s is kept finite while the total volume of the quadrangulation tends to infinity. In this regime, exactly one of the Voronoï cells keeps a finite volume, which scales as s^4 for large s.We analyze the universal probability distributionof this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.§ INTRODUCTION In a recent paper <cit.>, we analyzed the volume distribution of Voronoï cells for some families of random bi-pointed planar maps. Recall that a planar map is a connected graph embedded in the sphere: it is bi-pointed if it has two marked distinct vertices. These marked vertices allow us to partition the map into two Voronoï cells, where each cell corresponds, so to say, to the part of the map closer to one marked vertex than to the other. The volume of, say the second Voronoï cell (that centered around the second marked vertex) is then a finite fraction ϕ of the total volume of the map, with 0≤ϕ≤ 1, while the first cellclearly spans the complementary fraction 1-ϕ. The main result proven in <cit.> is that, for several families of random bi-pointed maps with a fixed total volume, and in the limit where this volume becomes infinitely large, the law for the fraction ϕ of the total volume spanned by the second Voronoï cell is uniform in the interval [0,1], a property conjectured by Chapuy in <cit.> among other more general conjectures. Here it is important to stress that the above result holds when the two marked vertices are chosen uniformly at random in the map. In particular,their mutual distance is left arbitrary[For one family on maps considered in <cit.>, it was assumed for convenience thatthe mutual distance be even, but lifting this constraint has no influence on the obtained result.].This paper deals on the contrary with Voronoï cells within random bi-pointed maps where the two marked vertices are picked randomlyat a fixed given mutual distance. Considering again the limit of maps with an infinitely large volume and keeping the(fixed) mutual distance between the marked vertices finite, we find that only one of the Voronoï cells becomes infinitely large while the volume of the other remains finite. In particular, the fraction of the total volume spanned bythis latter cell tends to 0 while that of the infinite cell tends to 1. In other words, having imposed a fixed finite mutual distance between the marked vertices drastically modifies the law forthe fraction ϕ which is now concentrated at ϕ=0 if it is precisely the second Voronoï cell which remains finite orat ϕ=1 if this second cell becomes infinite.In this regime of fixed mutual distance, a good measure of theVoronoï cell extent is now provided bythe volume of that of the two Voronoï cells which remains finite. The main goal of this paper is to compute the law for this finite volume, in particular in a universal regime where the mutual distance, although kept finite, is large.The paper is organized as follows: we first introduce in Section 2 the family of bi-pointed maps that will shall study (i.e. bi-pointed quadrangulations), define the volumes of the associated Voronoï cells and introduce some generating function with some control on these volumes (Section 2.1). We then discuss the scaling function which captures the properties of this generating function in some particular scaling regime (Section 2.2), and whose knowledge is the key of the subsequent calculations.Section 3 is devoted to our analysis ofVoronoï cell volumes in the regime of interest in this paper, namely when the maps become infinitely large and the mutual distance between the marked vertices remains finite. We first analyze (Section 3.1) the law for the fraction ϕ of the total volume of the map spanned by the second Voronoï cell and show, as announced above, that it is evenly concentrated atϕ=0 or ϕ=1. We then analyze (Section 3.2) map configurations for which the volume of the second Voronoï cell remains finite and show how to obtain, fromthe simple knowledge of the scaling function introduced above, the law for this (properly rescaled) volume when the mutual distance becomes large. This leads to an explicit universal expression (Section 3.3) for theprobability distribution of the finite Voronoï cell volume (in practice for its Laplace transform), whose properties are discussed in details. Section 4 proposesan instructive comparison of our result with that, much simpler, obtained for Voronoï cells within bi-pointed random trees. Section 5 discusses the case of asymmetric Voronoï cells where some explicit bias in the evaluation of distances is introduced. Our conclusions are gathered in Section 6. A few technical details, as well as explicit but heavy intermediate expressions, are given in various appendices. § VORONOÏ CELLS IN BI-POINTED MAPS§.§ A generating function for bi-pointed maps with a control on their Voronoï cell volumes The objects under study in this paper are bi-pointed planar quadrangulations, namely planar maps whose all faces have degree 4, and with two marked distinct vertices. We moreover demand that these vertices, distinguished as v_1 and v_2, be at some even graph distance d(v_1,v_2), namelyd(v_1,v_2)=2sfor some fixed given integer s≥ 1.Given v_1 and v_2, the corresponding two Voronoï cells are obtained via some splitting of the map into two domains which, so to say, regroup vertices which are closer to one marked vertex than to the other.As discussed in details in <cit.>, a canonical way to perform this splitting consists in applying the well-know Miermont bijection <cit.> whichtransforms a bi-pointed planar quadrangulation into a so-called planar iso-labelled two-face map (i-l.2.f.m), namely a planar map with exactly two faces, distinguished asf_1 and f_2 and with vertices labelled by positive integers satisfying: * labels on adjacent vertices differ by 0 or ±1;* the minimum label for the set of vertices incident to f_1 is 1;* the minimum label for the set of vertices incident to f_2 is 1.As recalled in <cit.>, theMiermont bijection provides a one-to-one correspondence between bi-pointed planar quadrangulations and planar i-l.2.f.m, the labels of the vertices corresponding precisely to their distance to the closest marked vertex in the quadrangulation. More interestingly, by drawing the original quadrangulation on top of its image, the two faces f_1 and f_2 define de facto two domains in the quadrangulation which are perfect realizations of the desired two Voronoï cells as, by construction, each of these domains regroups vertices closer to one marked vertex.Since faces of the quadrangulation are, under the Miermont bijection, in correspondence with edges of the i-l.2.f.m,the volume (= number of faces) of a given cell in the quadrangulation is measured by half the number of edge sides incident to the corresponding face in the i-l.2.f.m.Note that this volume is ingeneral some half-integer since a number of faces of the quadrangulation may be shared by the two cells (see <cit.> for details).To be precise, an i-l.2.f.m is made of a simple closed loop ℒ separating its two faces f_1 and f_2[This loop is simply formed by the cyclic sequence of edges incident to both faces.] together with a number of subtrees attached to vertices along ℒ, possibly on each side of the loop. If we call e_1 and e_2 the total number of edges for subtrees in the face f_1 and f_2 respectively, and e the length (= number of edges) of the loop ℒ, the volumes n_1 and n_2 of the Voronoï cells are respectivelyn_i=e_i+e/2,i=1,2,for a total volumeN= n_1+n_2= e_1+e_2+e .Finally, the requirement that d(v_1,v_2)=2s translates into the following fourth label constraint: * the minimum label for the set of vertices incident to ℒ is s. Having defined Voronoï cells, we may control their volume by considering the generating function F(s,g,h) of bi-pointed planar quadrangulations where d(v_1,v_2)=2s, with a weight g^n_1h^n_2 .From the Miermont bijection and the associated canonical construction of Voronoï cells, F(s,g,h) is also the generating function of i-l.2.f.m satisfying the extra requirement (L_4) with a weight g^e_1h^e_2 (√(gh))^e .As such, F(s,g,h) may, via some appropriate decomposition of the i.l.2.f.m, be written as (see <cit.>)F(s,g,h)= Δ_sΔ_t log(X_s,t(g,h))|_t=s =log(X_s,s(g,h)X_s-1,s-1(g,h)/X_s-1,s(g,h)X_s,s-1(g,h))(here Δ_s is the finite difference operator Δ_s f(s) ≡ f(s)-f(s-1)),where X_s,t(g,h) is some generating function for appropriate chains of labelled trees (which correspond to appropriate open sequences of edges with subtrees attached on either side of the incident vertices). Without entering into details, it is enough for the scope of this paper to know that the generating function X_s,t(g,h)is entirely determined[This relation fully determines X_s,t(g,h) for all s,t≥ 0 order by order in g and h, i.e.X_s,t(ρ g,ρ h) is fully determined order by order in ρ.] by the relation (obtained by a simple splitting of the chains)X_s,t(g,h)=1+√(gh)R_s(g)R_t(h)X_s,t(g,h)(1+√(gh)R_s+1(g)R_t+1(h)X_s+1,t+1(g,h))for s,t≥ 0, wherethe quantity R_s(g) (as well as its analog R_t(h)) is a well known generating function for appropriatelabelled trees. It is given explicitly by R_s(g)=1+4x+x^2/1+x+x^2(1-x^s)(1-x^s+3)/(1-x^s+1)(1-x^s+2)for g= x1+x+x^2/(1+4x+x^2)^2 ,where x is taken in the range 0≤ x≤ 1 and parametrizes g (in the range 0≤ g≤ 1/12 for a proper convergence of the generating function). For h=g, the solution of (<ref>) can be made explicit and readsX_s,t(g,g)=(1-x^3)(1-x^s+1)(1-x^t+1)(1-x^s+t+3)/(1-x)(1-x^s+3)(1-x^t+3)(1-x^s+t+1) .Unfortunately, no such explicit expression is known for X_s,t(g,h) when h≠ g and the relation (<ref>) might thus appear of no practical use at a first glance. As discussed in <cit.>, this is not quite true as we may recourse to appropriate scaling limits of all the above generating functions to extract explicit statistics on Voronoï cell volumes in a limit where the maps become (infinitely) large. Let us now discuss this point.§.§ The associated scaling functionThe limit of large quadrangulations (i.e. with a large number N of faces) is captured by the singularity of F(s,g,h) whenever g or h tends toward its critical value 1/12. As we shall see, in all cases of interest, this singularity may be analyzed by settingg=G(a,ϵ) , h=G(b,ϵ) , where G(c,ϵ)≡1/12(1-c^4/36ϵ^4) ,and letting ϵ tend to 0. In this limit, we have for instance the following expansion for the quantity x parametrizing g in (<ref>):x=1-a ϵ +a^2 ϵ ^2/2-5a^3 ϵ ^3/24+a^4 ϵ ^4/12-13a^5 ϵ ^5/384+a^6 ϵ ^6/72-157a^7 ϵ ^7/27648+a^8 ϵ^8/432+O(ϵ ^9) ,so that, for h=g (i.e. b=a), we easily get from the exact expression (<ref>) of X_s,t(g,g) the expansionF(s,g,g)=log(s^2 (2 s+3)/(s+1)^2 (2 s-1))-(2 s+1)a^4ϵ ^4/60 3.cm +(2 s+1) (10 s^2+10 s+1) a^6ϵ ^6/1890+O(ϵ ^8) .Since a^4ϵ^4=36 (1-12g) is regular when g→ 1/12, the most singular part of this generating function is given by(2 s+1) (10 s^2+10 s+1) a^6ϵ ^6/1890=4 (2 s+1) (10 s^2+10 s+1)/35(1-12g)^3/2 and we thus deduce that the number F_N(s) of bi-pointed planar quadrangulations with N faces and with their two marked vertices at distance 2s behaves at large N as-10pt F_N(s)≡[g^N]F(s,g,g)N →∞∼3/412^N/√(π) N^5/2𝔣_3(s) ,𝔣_3(s)=4 (2 s+1) (10 s^2+10 s+1)/35 . When s itself becomes large, this number behaves as 3/4 12^N/√(π) N^5/2×16/7s^3 . Note that this later estimate assumes that N becomes first arbitrarily large with a value of s remaining finite, and only then is s set to be large. This order of limitscorresponds to what is usually called the local limit. In particular, N and s do not scale with each other..3cm Now it is interesting to note that getting this last result (<ref>) does not require the full knowledge of F(s,g,g) and may be obtained upon using insteadsome simpler scaling function which captures the behavior of F(s,g,g)in a particular scaling regime. Consider indeed the generating function X_s,t(g,g) in a regime where g→ 1/12 as above by letting ϵ→ 0 in (<ref>), but where we let simultaneously s and t become large upon settings=S/ϵ ,t=T/ϵ ,with S and T kept finite. In this scaling regime, we have the expansionX_⌊ S/ϵ⌋,⌊ T/ϵ⌋(g,g)=3+x(S,T,a) ϵ+O(ϵ^2) ,where the function x(S,T,a) is given explicitly from (<ref>) byx(S,T,a) =-3a-6 a (e^-a S+e^-a T-3 e^-a (S+T)+e^-2a (S+T))/(1-e^-a S) (1-e^-a T) (1-e^-a (S+T)) .This in turn implies the expansionΔ_sΔ_t log(X_⌊ S/ϵ⌋,⌊ T/ϵ⌋(g,g) ) |_T=S =∂_S∂_T log(3+x (S,T,a) ϵ)|_T=S×ϵ^2+O(ϵ^4)=1/3 ∂_S∂_T x(S,T,a)|_T=S×ϵ^3 +O(ϵ^4)which yieldsF(⌊ S/ϵ⌋,g,g )= ℱ(S,a) ϵ^3+O(ϵ^4) ,where the scaling function ℱ(S,a) associated with F[s,g,g] reads explicitlyℱ(S,a) =1/3 ∂_S∂_T x(S,T,a)|_T=S = 2 a^3e^-2 a S(1+e^-2 a S)/(1-e^-2 a S)^3= (1/2S^3-a^4 S/30+2 a^6 S^3/189+O(S^5)) .A crucial remark is that we recognize in this latter small S expansion of ℱ(S,a) the large s leading behavior[In particular, we have the large s expansion: log(s^2 (2 s+3)/(s+1)^2 (2 s-1))= 1/2s^3+O(1/s^4) .] of the coefficients in the expansion (<ref>) for F[s,g,g] in the local limit. For instance, the large sbehavior of the singular term (proportional to ϵ^6) in (<ref>) is given by (2 s+1) (10 s^2+10 s+1) a^6/1890s →∞∼2/189 s^3a^6 = s^3 × [S^3] ℱ(S,a) .For a=√(6) (in which case we have the direct identification ϵ^6=(1-12g)^3/2),the left hand side is precisely the coefficient 𝔣_3(s) in (<ref>), so that the result (<ref>) may thus be read off directly on the expression of the scaling function ℱ(S,a) via𝔣_3(s)s →∞∼s^3 × [S^3] ℱ(S,√(6))=s^3 ×2/189 (√(6))^6 = 16/7s^3 ,without recourse to the explicit knowledge of the full generating function F[s,g,g].The origin of this “scaling correspondence", which connects the local limit at large s to the scaling limit at small S is explained in details in the next section. This correspondence is in fact a general property and can be applied in the situation where h≠ g. It therefore allows us to access the large s limit of the large N asymptotics of [g^N]F(s,g,h) (again sending N→∞ first) from the simple knowledge of the scaling function associatedwith F(s,g,h). .3cm As of now, let us already fix our notations for scaling functions when g and h are arbitrary: parametrizing g and h as in (<ref>) above, we have when ϵ→ 0 the expansionX_⌊ S/ϵ⌋,⌊ T/ϵ⌋(g,h)=3+x(S,T,a,b) ϵ+O(ϵ^2)with a scaling function x(S,T,a,b) which, from (<ref>) expanded at lowest non-trivial order in ϵ, is solution of the non-linear partial differential equation 2 (x(S,T,a,b))^2+6 (∂_Sx(S,T,a,b)+∂_Tx(S,T,a,b))+27 (r(S,a)+r(T,b))=0 .Here r(S,a) is the first non-trivial term in the small ϵ expansion of R_⌊ S/ϵ⌋(g), namely, from its explicit expression (<ref>),R_⌊ S/ϵ⌋(g) =2+r(S,a) ϵ^2+O(ϵ^3) ,r(S,a)=-a^2 (1+10 e^-a S+e^-2 a S)/3 (1-e^-a S)^2 .As for F(s,g,h), we may now use (<ref>) to relate the associated scaling function ℱ(S,a,b) to x(S,T,a,b), namelyF(⌊ S/ϵ⌋,g,h)= ℱ(S,a,b) ϵ^3+O(ϵ^4) where ℱ(S,a,b) = 1/3 ∂_S∂_T x(S,T,a,b)|_T=S .Scaling functions are in general much simpler than the associated full generating functions. In particular, although we have no formula for X_s,t(g,h) for arbitrary g and h, an explicit expression for x(S,T,a,b) is known for arbitrary a and b,as first obtained in <cit.> upon solving (<ref>) with appropriate boundary conditions. We may thus recourse to this result to get an explicit expression for the scaling function ℱ(S,a,b) itself via (<ref>). The corresponding formula is quite heavy and its form is not quite illuminating. Still, we display it in Appendix A for completeness (the reader may refer to this expression to check the various limits and expansions of ℱ(S,a,b)displayed hereafter in the paper).As opposed to F(s,g,h), the scaling function ℱ(S,a,b) is thus known exactly and we will now show in details how to use the scaling correspondence to deduce from its small S expansion the large s limit of the large N asymptotics of [g^N]F(s,g,h) and control the volume of, say, the second Voronoï cell in large quadrangulations, by some appropriate choice of h.§ INFINITELY LARGE MAPS WITH TWO VERTICES AT FINITE DISTANCE This section is devoted to estimating the law for the volumes spanned by the Voronoï cells in bi-pointed quadrangulations whose total volume N (= number of faces) tends to infinity. Calling n_1 and n_2 the two Voronoï cell volumes, we have n_1+n_2=N so it is enough to control one of two volumes, say n_2. Here the distance 2s between the two marked vertices is kept finite (possibly large) when N→∞. §.§ The law for the proportion of the total volume spanned by one Voronoï cellFor large N and finite s, the first natural way to measure n_2 is to express it in units of N, i.e. consider the proportionϕ≡n_2/Nof the total volume spanned by the second Voronoï cell. We have of course 0≤ϕ≤ 1 and the large N asymptotic probability law 𝒫_s(ϕ) for ϕ may be obtained from F(s,g,h) via∫_0^1dϕ 𝒫_s(ϕ)e^μ ϕ= lim_N→∞[g^N]F(s,g,ge^μ/N)/[g^N]F(s,g,g)since g^n_1(ge^μ/N)^n_2=g^Ne^μ ϕ. From the scaling correspondence, the large N asymptoticsof [g^N]F(s,g,ge^μ/N) is, at large s, encoded in the small S expansion of the scaling function ℱ(S,a,b) for some appropriate b≡ b(a,μ). The right hand side of the above equality may thus be computed explicitly at large s from the knowledge of ℱ(S,a,b).This computation, together with the precise correspondence between the large s local limit and the small S scaling limit, is discussed in details in Appendix B. We decided however not to develop the calculation here since the resulting law is in fact trivial. As might have been guessed by the reader, we indeed find∫_0^1dϕ 𝒫_s(ϕ)e^μ ϕs →∞∼1/2 (1+e^μ)or equivalently𝒫_s(ϕ) s →∞∼1/2 (δ(ϕ)+δ(ϕ-1)) .This result simply states that, for N→∞and s finite large, only one of the Voronoï cells has a volume of order Nwith, by symmetry,{n_1=N-o(N) ,n_2=o(N) with probability 1/2 , n_1=o(N) ,n_2=N-o(N) with probability 1/2 . .The main purpose of this paper, discussed in the following sections,is precisely to characterize the volume of the Voronoï cell which is an o(N).As we shall see, the volume of this Voronoï cell remains actually finite and scales as s^4 when s becomes large. §.§ Infinitely large maps with a finite Voronoï cellThis section and the next one present our main result, namely the law for the (properly rescaled) volume of the Voronoï cell which is not of order N when N→∞. More precisely, we will concentrate here on map configurations for which the total volume N=n_1+n_2 tends to infinity butthe volume n_2 is kept finite. We will then verify a posteriori that the number of these configurations represents 1/2 of the total number of bi-pointed maps whenever s is large. This will de facto prove that the configurations for which n_2=o(N) in (<ref>) are in fact, with probability 1, configurations for which n_2 is finite.Let us denote byF_n_1,n_2(s)≡ [g^n_1h^n_2]F(s,g,h)the number of planar bi-pointed quadrangulations with fixed given values of n_1 and n_2.In the limit N=n_1+n_2→∞ with afixed finite n_2, this number may be estimated from the leading singularity of F(s,g,h) wheng→ 1/12 for a fixed value of h< 1/12 (see <cit.> for a detailed argument of a fully similar estimate in the context of hull volumes). We have indeedF_N-n_2,n_2(s)N →∞∼3/412^N/√(π) N^5/2× 12^-n_2[h^n_2] 𝔣_3(s,h) ,where 𝔣_3(s,h) is the coefficient of the leading singularity of F(s,g,h) when g→ 1/12 at fixed h,hence is obtained from the expansion[The precise form of this expansion is dictated by the similar explicit expansion (<ref>) for F(s,g,g).In particular, the absence of singular term ∝ (1-12g)^1/2 is imposed by the fact that such a term, if present, would imply thatF_N-n_2,n_2(s)be of order const.× 12^N/ N^3/2at large N while this quantity is clearly bounded by [g^N]F[s,g,g] which, as we have seen, is of order const'.× 12^N/ N^5/2 only. ]F(s,g,h)=𝔣_0(s,h)+𝔣_2(s,h)(1-12g)+𝔣_3(s,h)(1-12g)^3/2+O((1-12g)^2) .Upon normalizing by the total number of bi-pointed maps F_N(s) with fixed N and s, whose asymptotic behavior is given by (<ref>),we deduce the N→∞ limiting probability 𝔭_s(n_2) that the second Voronoï cell has volume n_2:𝔭_s(n_2)=lim_N→∞F_N-n_2,n_2(s)/F_N(s) = 1/𝔣_3(s)12^-n_2[h^n_2] 𝔣_3(s,h) .This probability for arbitrary finite n_2 may be encoded in the generating function∑_n_2𝔭_s(n_2) ρ^n_2=𝔣_3(s,ρ/12)/𝔣_3(s) ,where ρ∈ ]0,1] is a weight per unit volume. Recall that n_2 may take half integer values so that the sum on the left hand side above actually runs over all (positive) half-integers.Let us now discuss the scaling correspondence in details[We discuss here the general case where h≠ g is fixed while g→ 1/12. Our arguments could be repeated verbatim to the case h=g→ 1/12 to explain the scaling correspondence in this case, as observeddirectly from the explicit expressions of F(s,g,g) and ℱ(S,a).]. Its origin is best understood by considering the all order expansion of F(s,g,h) for g→ 1/12, namelyF(s,g,h)=∑_i≥ 0𝔣_i(s,h)(1-12g)^i/2(with 𝔣_1(s,h)=0 as discussed in the footnote <ref>). We may indeed, via the identification (1-12g)^1/2=(a/√(6))^2 ϵ^2 for g=G(a,ϵ),relate the scaling function ℱ(S,a,b) to this expansion upon writing ℱ(S,a,b) =lim_ϵ→ 0 1/ϵ^3F(⌊ S/ϵ⌋,G(a,ϵ),G(b,ϵ))=lim_ϵ→ 0 ∑_i≥ 0(a/√(6))^2i ϵ^2i-3 𝔣_i(⌊ S/ϵ⌋,G(b,ϵ)) .Since G(b,ϵ) depends only on the product b ϵ, the quantity 𝔣_i(⌊ S/ϵ⌋,G(b,ϵ)), which depends a priori on S, b and ϵ, is actually a functionof the two variables S/ϵ and b ϵ only,or equivalently of the two variablesS/ϵ and S/ϵ× b ϵ = bS. We deduce from the very existence of the scaling function above that[The fact that all the φ_i, i≠ 1, are not zero is verified a posteriori by the fact that ℱ(S,a,τ/S) has asmall S expansion involving non vanishing S^2i-3 coefficients for all i≥ 0, i≠ 1.]𝔣_i(⌊ S/ϵ⌋,G(b,ϵ)) ϵ→ 0∼(S/ϵ)^2i-3φ_i(bS)for i≠ 1 (while 𝔣_1=0) with, moreover, the direct identificationℱ(S,a,b)=∑_i≥ 0(a/√(6))^2iS^2i-3 φ_i(bS)with φ_1=0.This latter identity allows us in turn to identify the functions φ_i(τ) viaφ_i(τ)=(a/√(6))^-2i [S^2i-3] ℱ(S,a,τ/S) =[S^2i-3]ℱ(S,√(6),τ/S) ,where the last term was obtained by setting a=√(6) in the middle term since, φ_i(τ) being independent of a, the middle term should be too for consistency. .3cm We may now come back to our estimate of (<ref>) when s is large. To obtain a non trivial law at large s, we must measure n_2 in units of s^4, i.e. consider the probability distribution for the rescaled volume V defined byV≡n_2/s^4 .This law is indeed captured by choosing ρ=e^-σ/s^4, in which case the second argument, ρ/12, of the numerator in (<ref>) behaves as1/12e^-σ/s^4s →∞∼1/12(1-σ/S^4(S/s)^4) = G(√(6) σ^1/4/S,S/s) .Taking ϵ=S/s and b=√(6) σ^1/4/S in the above estimate (<ref>) and using the identification (<ref>), we may now write𝔣_i(s,1/12e^-σ/s^4) s →∞∼ s^2i-3 φ_i(√(6) σ^1/4)= s^2i-3 [S^2i-3]ℱ(S,√(6),√(6) σ^1/4/S) ,leading eventually, using (<ref>) and (<ref>), to-10pt ∑_n_2𝔭_s(n_2)e^-σVs →∞∼[S^3]ℱ(S,√(6),√(6)σ^1/4/S)/[S^3]ℱ(S,√(6))=7/16 [S^3]ℱ(S,√(6),√(6)σ^1/4/S) .Since we have at our disposal an explicit expression for ℱ(S,a,b), this equation will give us a direct access to the desired law for V.§.§ Explicit expressions and plots As a first, rather trivial, check of our expression (<ref>), let us estimate the probability that n_2 remains finite in infinitely large planar bi-pointed quadrangulations. This probability is obtained by summing 𝔭_s(n_2) over all allowed finite values of n_2, i.e. by setting ρ=1 in (<ref>), i.e. σ=0 in (<ref>). It therefore takes the large s value∑_n_2𝔭_s(n_2) s →∞∼ [S^3]ℱ(S,√(6),0)/[S^3]ℱ(S,√(6))=7/16[S^3]ℱ(S,√(6),0) .For b=0, the explicit expression for ℱ(S,a,b) simplifies intoℱ(S,a,0)= -36√(2)a^3 ∑_m=1^5 p_m(aS)e^-maS/(577+408√(2))(∑_m=0^2 q_m(aS)e^-maS)^3 ,where the p_m(r) (1≤ m≤ 5) and q_m(r) (0≤ m≤ 2) are polynomials of degree 3 and 2 respectively in the variable r, given by-30ptp_1(r)=-6 (816+577 √(2))-6 (915+647 √(2)) r-3 (618+437 √(2)) r^2-2 (99+70 √(2)) r^3 -30pt p_2(r)=-24 (222+157 √(2))-12(126+89 √(2)) r+12 (27+19 √(2)) r^2+4(24+17 √(2)) r^3 -30ptp_3(r)=-108 (4+3 √(2))-180 (3+2 √(2)) r-54 (4+3 √(2)) r^2-12(3+2√(2)) r^3 -30pt p_4(r)=-24 (-6+5 √(2))+12 (-6+√(2)) r-12 (3+√(2)) r^2-4 √(2) r^3 -30pt p_5(r)=-6(-24+17 √(2))+6(-27+19 √(2)) r-3 (-18+13 √(2)) r^2+2(-3+2 √(2)) r^3 -30pt q_0(r)=6+3 √(2) r+r^2 -30pt q_1(r)=-24(-4+3 √(2))-12 (-3+2 √(2)) r+2(-4+3 √(2)) r^2 -30pt q_2(r)=6(-17+12 √(2))-3(-24+17 √(2)) r+(-17+12 √(2)) r^2 . We have in particular the small S expansion ℱ(S,a,0)=1/2 S^3-a^4 S/60+a^6 S^3/189+O(S^4) which leads to a probability that n_2 be finite equal to ∑_n_2𝔭_s(n_2) s →∞∼7/16 (√(6))^6/189=1/2 .We thus see that configurations for which the second Voronoï cell remains finite whenever N→∞ represent at large s precisely 1/2 of all the configurations. This is fully consistent with our result (<ref>) provided that the configurations for which we found n_2=o(N) are actually configurations for which n_2 remains finite. Otherwise stated, configurations for which both n_1 and n_2 would diverge at large N are negligible at (large) finite s. .3cmBeyond this first result at σ=0, we can consider, for any σ≥ 0,the expectation value of e^-σV for bi-pointed quadrangulations with N→∞ and finite s, conditioned to have their second Voronoï cell finite[Alternatively, we may lift this conditioning and interpretE_s[e^-σV] as the expectation value of e^-σV where V is the rescaled volume of the smallest Voronoï cell. ]. It is given by E_s[e^-σV]= ∑_n_2𝔭_s(n_2)e^-σV/∑_n_2𝔭_s(n_2) ,and has a large s limiting valueE_s[e^-σV] s →∞∼ E[e^-σV] =[S^3]ℱ(S,√(6),√(6)σ^1/4/S)/[S^3]ℱ(S,√(6),0) = 7/8[S^3]ℱ(S,√(6),√(6)σ^1/4/S) .From the explicit expression of ℱ(S,a,b) displayed in Appendix A, we deduce after some quite heavy computation the following expression for E[e^-σV]:-10pt E[e^-σV]=3/2P(σ^1/4)+ ∑_m=1^3( P_m(σ^1/4,√(2))e^m√(6) σ^1/4 + P_m(σ^1/4,-√(2)) e^-m√(6) σ^1/4)/(Q(σ^1/4)(4+(4+3√(2))e^√(6)σ^1/4+(4-3√(2)) e^-√(6)σ^1/4)-12)^4where P(r), Q(r) and P_m(r,γ) (m=1,2,3) are polynomials in r of degree at most 8 (with coefficients linear in γ for the last threepolynomials), given explicitly by -32ptP(r)=96 (-252-399 √(3) r-756 r^2-161 √(3) r^3+170 r^4+153 √(3) r^5+144 r^6+22 √(3) r^7+4 r^8)-30ptP_1(r,γ) =126 (168+85 γ )+63 √(3)r(867+596γ )+1323r^2(132+95 γ )+28 √(3)r^3(3153+2300 γ ) +24r^4(2463+1843 γ ) +√(3)r^5(588+905 γ )-36r^6(177+124 γ )-6 √(3)r^7(174+127 γ ) -36r^8 (4+3 γ )-30pt P_2(r,γ) =-8 (63 (24+17 γ )+63 √(3)r(105+74 γ )+378r^2 (78+55 γ )+14√(3)r^3(1569+1108 γ )..+12 r^4 (2337+1652 γ )+ √(3)r^5(6954+4919 γ )+18r^6 (154+109 γ )+6 √(3)r^7 (24+17 γ ))-30pt P_3(r,γ) =126 (24+17 γ )+63 √(3)r(277+196 γ )+189r^2(516+365 γ )+28 √(3)r^3(3399+2404 γ )+24r^4(7193+5087 γ )+√(3) r^5 (68436+48397 γ )+36r^6(1465+1036 γ )+6 √(3)r^7(1342+949 γ ) +12 r^8 (140+99 γ ) -30ptQ(r)= (1+√(3) r+r^2) . The function E[e^-σV] is plotted in Figure <ref> for illustration. .3cm The actual probability distribution 𝔓_s(V) for the rescaled volume V is the inverse Laplace transform of E_s[e^-σV], hence its large s limit 𝔓(V) is given by the inverse Laplace transform of E[e^-σV]. From the quite involved form(<ref>) above, there is no real hope to get an explicit expression for 𝔓(V) but it may still be plotted thanks to appropriate numerical tools <cit.>. The resulting shape is displayed in Figure <ref>. .3cm A few analytic properties of 𝔓(V) may be obtained from its explicit Laplace transform (<ref>) above: in particular, we may easily find large and small Vasymptotic equivalents of 𝔓(V), as discussed now. .3cm ∙ The large volume limit. For small σ, we have the expansionE[e^-σV] = 1-665 √(3)/1024σ^1/4+49/768√(3)σ^3/4+63/80σ+O(σ^5/4) so that E[e^-σV] is not analytic at σ=0, with all its derivative infinite at this point. We first deduce that all the (positive) moments of 𝔓(V) are infinite. By a standard argument using the famous Karamata's tauberian theorem, the large V tail of 𝔓(V) is estimated from the leading (∝σ^1/4) small σ singularity above as𝔓(V)V →∞∼665 √(3)/4096Γ(3/4) 1/V^5/4 . A comparison between 𝔓(V) (as obtained numerically) and its large V equivalent is displayed in Figure <ref>. .3cm∙ The mall volume limit. For large σ, we have the asymptotic equivalenceE[e^-σV] σ→∞∼9/2(3√(2)-4)e^-√(6) σ^1/4 . By a simple saddle point calculation (see Appendix C), we deduce the small V estimate𝔓(V)V → 0∼3^11/6 (3 -2 √(2))/2 √(π) 1/V^7/6e^-3^5/3/41/V^1/3which is a flat function at V=0. A comparison between 𝔓(V) (as obtained numerically) and its small V equivalent is displayed in Figure <ref>. § A COMPARISON WITH VORONOÏ CELLS IN INFINITELY LARGE BI-POINTED TREESAs an exercise, it is interesting to compare our result for 𝔓(V) to that, much simpler, obtained for another family of maps, namely bi-pointed plane trees, which are planar maps with a single face and with two marked vertices v_1 and v_2, taken again at some fixed even distance d(v_1,v_2)=2s along the tree. Any such map is made of a simple path 𝒫, formed by the edges joining v_1 to v_2, completed by trees attached to the internal vertices of 𝒫 on both side of the path and at its extremities v_1 and v_2. The two Voronoï cells are now trivially defined by splitting the tree at the “central vertex" in 𝒫, which is the vertex along 𝒫 lying at distance s from both v_1 and v_2(there are in general two subtrees attached to this vertex and we may decide to split the tree so as to assign one of these subtrees to the first Voronoï cell and the other subtree to the second cell). The volumes n_1 and n_2 of the two Voronoï cells are now measured by their number of edges and the generating function F_ tree(s,g,h) enumerating these maps with a weight g^n_1h^n_2 reads simplyF_ tree(s,g,h)=(g (R_ tree(g))^2)^s (h (R_ tree(h))^2)^swhere R_ tree(g) is the generating function for planted trees with a weight g per edge, namely[It is solution of R_ tree(g)=1+g(R_ tree(g))^2.]R_ tree(g)=1-√(1-4g)/2g .The scaling function associated with F_ tree(s,g,h) is obtained by settingg=G_ tree(a,ϵ) , h=G_ tree(b,ϵ) , where G_ tree(c,ϵ)≡1/4(1-c^2/4ϵ^2) ,and reads simplyℱ_ tree(S,a,b)=lim_ϵ→ 0 F_ tree(⌊ S/ϵ⌋,G_ tree(a,ϵ),G_ tree(b,ϵ)) =e^-(a+b)S .By repeating and adapting the arguments of previous sections, here with leading singularities of type (1-4g)^1/2, we can find the large N (= total number of edges) asymptotic law𝔓_ tree(V_ tree) for the rescaled volumeV_ tree≡n_2/s^2among bi-pointed trees with d(v_1,v_2)=2s, conditioned to have their second Voronoï cell finite (which again represent 1/2 of all bi-pointed trees with fixed s). For large s, we find (with obvious notations) the expectation valueE_ tree[e^-σV_ tree]=[S]ℱ_ tree(S,2,2σ^1/2/S)/[S]ℱ_ tree(S,2,0)=e^-2σ^1/2 . We may now deduce by inverse Laplace transform the exact law for V_ tree 𝔓_ tree(V_ tree)=1/√(π) V_ tree^3/2e^-1/V_ tree , which is nothing but a simple Lévy distribution. In particular, all its (positive) moments are infinite. This distribution is a particular member of the more general family of one-sided Lévy distributions with parameter α, namely distributions whose Laplace transform is e^- const.p^α. For 0<α<1, such distributions are flat at small volume V and vanish asconst./V^(2-α)/(2(1-α))exp(-const.'/V^α/(1-α)) .For large volume, they present a fat tail with an algebraic decay of the form 1/V^1+α.The simple law 𝔓_ tree(V) for trees corresponds precisely to the situation where α=1/2. As for the distribution 𝔓(V) of quadrangulations, it is obviously not a Lévy distribution butits small and large V behaviors are nevertheless similar to those obtained for a Lévy distribution with α=1/4. Clearly,the value of α appearing in both the small and large V asymptotics is related to the fractal dimension D of the mapsat hand (D=2 for trees and D=4 for quadrangulations) viaα=1/D .It is tempting to conjecture that the above forms for small and large V asymptotics should be generic and hold for other families of maps, possibly within more involved universality classes with more general fractal dimensions, hence more general values of α. § ASYMMETRIC VORONOÏ CELLS In Section <ref>, we estimated the large N asymptotic proportion of bi-pointed planar quadrangulations for which the second Voronoï cell is finite. The obtained value 1/2 is trivial by symmetry if we assume that configurations for which both Voronoï cells become infinite are negligible (this latter property being de facto proven by the result itself). Note that, in this respect, our computation was performed here in the “worst" situation where the value of the distance 2s between the two markedvertices is large. We may now explicitly break the symmetry and define asymmetric Voronoï cells upon introducing some biasin the measurement of distances. The bijection between bi-pointed planar quadrangulations and planar i-l.2.f.m is indeed only one particular instance of the Miermont bijection. The Miermont bijection allows us to introduce more generally what are called delays, which are integers associated with the marked vertices and allow for some asymmetry in the evaluation of distances <cit.>. In the case of two marked vertices, two delays may in principle be introduced but, in practice, only their difference (called θ below) does matter. In the presence of delays, the resulting image of the bi-pointed quadrangulation is again a two-face map, but now with a more general labelling of its vertices by integers. If we insist on keeping a distance d(v_1,v_2)=2s between the marked vertices in the original quadrangulation, the labelling of the two-face map, which now involves some additional integer parameter θ, is characterized by the followingfour properties: * labels on adjacent vertices differ by 0 or ±1;* the minimum label for the set of vertices incident to f_1 is 1-θ;* the minimum label for the set of vertices incident to f_2 is 1+θ;* the minimum label for the set of vertices incident to ℒ is s;if f_1 and f_2 denote the two faces of the map and ℒ the loop made of edges incident to both faces.The Miermont bijection is a one-to-one correspondence between bi-pointed planar quadrangulations with d(v_1,v_2)=2sand planar two-face maps with a labelling satisfying (L_1)-(L_4) above for any fixed θin the range <cit.>-s<θ<s .All the vertices v of the original quadrangulation but v_1 and v_2 are recovered in the two-face map, andtheir label is related to the distance in the quadrangulation viaℓ(v)=min(d(v,v_1)-θ,d(v,v_2)+θ) .Again the two domains of the original quadrangulation covered by f_1 and f_2 respectively (upon drawing the quadrangulation and its image via the bijection on top of each other)naturally define two cells in the map. For some generic θ, those are howeverasymmetric Voronoï cells with the following properties: the first cell now contains all the verticesvsuch that d(v_,v_1) < d(v,v_2)+ 2θ (cell 1)(this includes the vertex v_1), as well as a number of vertices satisfying d(v_,v_1)=d(v,v_2)+2θ. The second cell contains all the vertices vsuch thatd(v_,v_1) > d(v,v_2)+ 2θ (cell 2)(including v_2) as well as a number of vertices satisfying d(v_,v_1)=d(v,v_2)+2θ. In particular, the loop ℒ, whose vertices belong to both f_1 and f_2, contains only vertices satisfyingd(v_,v_1)=d(v,v_2)+2θ.Taking θ> 0 therefore “favors" cell 1 whose volume is, on average, larger than that of cell 2. A control on these volumes is again obtained directly via the bijection by assigning a weight g per edge in f_1, h per edge in f_2 and √(g h) per edge along ℒ. The corresponding generating function reads then -1.cm F(s,θ,g,h)= Δ_uΔ_v log(X_u,v(g,h))|_u=s+θ v=s-θ =log(X_s+θ,s-θ(g,h)X_s+θ-1,s-θ-1(g,h)/X_s+θ-1,s-θ(g,h)X_s+θ,s-θ-1(g,h)) ,giving rise to a scaling function ℱ(S,Θ,a,b) via-1.cm F(⌊ S/ϵ⌋,⌊Θ/ϵ⌋,G(a,ϵ),G(b,ϵ))= ϵ^3 ℱ(S,Θ,a,b) +O(ϵ^4) where ℱ(S,Θ,a,b) = 1/3 ∂_U∂_V x(U,V,a,b)|_U=S+Θ V=S-Θ .Defining the asymmetry factor ω by ω≡θ/s ,-1≤ω≤ 1 ,the local limit of configurations with fixed s and ω is, at large s, encoded in the small S expansion ofℱ(S,ωS, a,b). In particular,we easily get the b=0 small S expansion generalizing (<ref>) ℱ(S,ωS, a,0) =1/2 S^3-a^4S/480 (1+ω )^3 (8-9 ω +3 ω ^2)+a^6S^3/6048(1+ω )^3 (32-33 ω +3 ω ^2+9 ω ^3-3 ω ^4)+O[S]^4 .By the same argument as in Section <ref>, we directly read from the S^3 coefficient of this expansion the large s probability Π(ω) that, for N→∞, the volume of the second (now asymmetric with a fixed value of the asymmetry factor ω) Voronoï cell remains finite:Π(ω)=1/64(1+ω )^3 (32-33 ω +3 ω ^2+9 ω ^3-3 ω ^4) .This probability is displayed in Figure <ref>. It satisfies of course Π(ω)+Π(-ω)=1 as expected by symmetry, and in particular Π(0)=1/2 for the symmetric case. For, say ω→ 1, the second Voronoï cell, which is maximally unfavored by the asymmetry is finite with probability Π(1)=1.For a better understanding of the meaning of Π(ω), we may quote Miermont in <cit.> and “let water flow at unit speed from the sources" v_1 and v_2 at given mutual distance 2s “in such a way that the water starts diffusing from" v_1at time -ωs, from v_2 at time +ωs, “and takes unit time to go through an edge. When water currents emanating from different edges meet at a vertex (whenever the water initially comes from the same source of from different sources), they can go on flowing into unvisited edges only[Miermont's bijection is so designed that meeting currents “can go on flowing into unvisited edges only respecting the rules of a roundabout, i.e. edges that can be attained by turning around the vertex counterclockwise and not crossing any other current."] (...). The process ends when the water cannot flow any more (...)."In this language, Voronoï cells correspond to domains covered by a given current. When the map volume tends to infinity, only one of the currents flows all the way to infinity, the other current remaining trapped within a finite region. We may then view Π(ω) as the probability for the second current (emanating from the source v_2) to remain trapped within a finite domain or, equivalently, as the probability for the first current (emanating from the source v_1) to escape to infinity. § CONCLUSIONIn this paper, we computed explicitly the value of the expectation value E[e^-σV] for infinitely large bi-pointed planar maps, where V is the rescaled volume of that of the two Voronoï cells which remains finite. This law describes Voronoï cells constructed from randomly picked vertices at a prescribed finite mutual distance, and in the limit where this distance is large. Although it may look quite involved, the expression (<ref>) is nevertheless expected to be universal since its derivation entirely relies on properties of scaling functions, which are in fact characteristic of the Brownian map rather than the specific realization at hand (here quadrangulations).In other words, we expect that the same expression (<ref>), up to some possible non-universal normalization for the parameter σ, would be obtained, in the same regime,for all bi-pointed planar map families in the universality class ofso-called pure gravity[This includes maps with bounded face degrees with possible degree dependent weights, as well asmaps with unbounded face degrees with degree dependent weights which restrain the proliferation of large faces.].We then deduced from this result a number of features of the associated, universal, volume probability distribution𝔓(V), such as its large and small V behaviors (<ref>) and (<ref>).Although we have not been able to give a tractable explicit formula for this law for arbitrary V, we thereby showed that its nature is comparable to that of a simple one-sided Lévy distribution with parameter α=1/4. .3cm Let us conclude by briefly discussing Voronoï cells, now in the so called scaling regime: here we continue to fix the mutual distance 2s between the marked vertices but we now let s and N tend simultaneously to infinity with the ratio S=s/N^1/4 kept finite. In this regime, the fraction ϕ=n_2/N of the total volume spanned by the second Voronoï cell is again a good measure of the cell volume distribution and the asymptotic law[We use a slightly different notation 𝒫_{S}(ϕ) with curly bracketsto distinguish this law from the local limit law 𝒫_s(ϕ) at fixed s.] 𝒫_{S}(ϕ) for ϕ at fixed S may be obtained as in Section 3.1 via∫_0^1dϕ 𝒫_{S }(ϕ)e^μ ϕ= lim_N→∞[g^N]F(⌊ S N^1/4⌋,g,ge^μ/N)/[g^N]F(⌊ S N^1/4⌋,g,g) .As explained in Appendix B in the context of the local limit law 𝒫_s(ϕ), the g^N coefficient of the numerator above may be extracted via some simple contour integral which, upon taking g=G(a,1/N^1/4), involves at large N the scaling function ℱ(S,a,b) atb=(a^4-36μ)^1/4. This leads to ∫_0^1dϕ 𝒫_{S }(ϕ)e^μ ϕ=∫_𝒞_μ da a^3 e^a^4/36ℱ(S,a,(a^4-36 μ)^1/4) /∫_𝒞_0 da a^3 e^a^4/36ℱ(S,a,a) ,where the integral over a is over some appropriate contour 𝒞_μ depending on μ (see Appendix B).Due to the involved expression of ℱ(S,a,b), we were not able to perform the above contour integralsfor general S. Still, for S→ 0, we recover precisely via a small S expansion the result of Section 3.2 (as expected from thescaling correspondence), now in the form lim_S→ 0∫_0^1dϕ 𝒫_{S }(ϕ)e^μ ϕ= 1/2 (1+e^μ) ⇔lim_S→ 0𝒫_{S }(ϕ)= 1/2 (δ(ϕ)+δ(ϕ-1)) .As for the S→∞ limit, it may be obtained as follows (here we simply give a sketch of the calculation and leave to the reader the task of filling the gaps): from its explicit expression, we haveℱ(S,a,b)S →∞∼ℨ(a,b) e^-(a+b)S ,where the value of the coefficient ℨ(a,b) may easily be obtained but is unimportant for our calculation (apart from the, easily verified property that it has a non-zero limit ℨ(a,a) when b→ a). For large S, the above contour integrals may be evaluated by a saddle-point method.For the denominator (with b=a), we writed/da(a^4/36-2aS)|_a=a^*=0⇔ (a^*)^4=(18 S)^4/3and deform the contour 𝒞_0 so as to pass via the positive real saddle point at a^*=(18 S)^1/3. This gives a denominator[Since the denominator is directly proportional to the S-dependent two-point function (i.e. the distance profile) in large maps, we recognize here the well-known Fisher's law which states that this function decays at large Slike e^-const. S^δ with the exponent δ=1/(1-1/D)=4/3 at D=4.] proportional at large S toℨ(a^*,a^*)e^-(18 S)^4/3/12. As for the numerator, it is dominated by the same saddle point but the replacementb→((a^*)^4-36μ)^1/4= (18 S)^1/3-μ/2S+O(1/S^2)creates a μ-dependent correction. At large S, this leads to a numerator now proportional toℨ(a^*,a^*)e^-(18 S)^4/3/12+μ/2 with the sameproportionality constant as for the denominator, giving eventually lim_S→∞∫_0^1dϕ 𝒫_{S }(ϕ)e^μ ϕ=e^μ/2⇔lim_S→∞𝒫_{S }(ϕ)= δ(ϕ-1/2) .This result is quite natural since, heuristically, the limit S→∞ describes maps with an elongated shape, with the two marked vertices sitting at its extremities. The frontier between the Voronoï cells for such an elongated map is typically a small cycle sitting halfway along the elongated direction, hence splitting the map into two domains of the same volume, equal to half the total volume. It would be interesting to visualize and follow the continuous passage, for increasing S, of the distribution 𝒫_{S }(ϕ) from its S→ 0 to its S→∞ limit above and to better understand how its average over arbitrary S (properly weighted by the S-dependent two-point function, i.e. the distance profile of the Brownian map) creates the uniform distribution for ϕ∈[0,1].§ ACKNOWLEDGEMENTS The author acknowledges the support of the grant ANR-14-CE25-0014 (ANR GRAAL).§ EXPRESSION FOR THE SCALING FUNCTION ℱ(S,A,B) The scaling function x(S,T,a,b) was computed in <cit.> as the appropriate solution of (<ref>). Its explicit expression is quite heavy and is not reproduced here. From this expression, we may obtain ℱ(S,a,b) directly via (<ref>). It takes the following form:ℱ(S,a,b)=-e^-(a+b)S/6 𝔗(e^-aS,e^-bS,a,b)/𝔇(a,b) ( 𝔈(a,b)/𝔘(e^-aS,e^-bS,a,b) )^3 ,where, introducing the notationc ≡√(a^2+b^2/2) ,we have explicitly-1.2cm 𝔈(a,b) = 6ab (a-b)^2(a+b) (2 a^2+b^2)(a^2+2 b^2) ,-1.2cm 𝔇(a,b) =(a+2 c) (b+2 c) (5 a^3+7 a^2 c+ 4 a b^2 +2 b^2 c) (4a^2 b+2a^2 c+5 b^3 +7 b^2 c)×(17 a^2 (a^2 + b^2)+12 a (2 a^2+b^2) c +2 b^4)(2 a^4+12 b (a^2+2 b^2) c+ 17 b^2 (a^2 + b^2)) ,while𝔗(σ,τ,a,b) and 𝔘(σ,τ,a,b) are polynomials of respective degree 4 and 2 in both σ and τ, namely𝔗(σ,τ,a,b)=∑_i=0^4∑_j=0^4 t_i,j σ^iτ^jand𝔘(σ,τ,a,b) =∑_i=0^2∑_j=0^2 u_i,j σ^iτ^j.The coefficients t_i,j≡ t_i,j(a,b) and u_i,j≡ u_i,j(a,b) may be written for convenience as sums of two contributions:t_i,j=t_i,j^(0)+ct_i,j^(1) ,u_i,j=u_i,j^(0)+cu_i,j^(1) ,where we have the explicit expressions-1.2cm t_0,0^(0)=(a+b)^3 (396 a^10+1448 a^9 b+3672 a^8 b^2+6520 a^7 b^3+9135 a^6 b^4+10146 a^5 b^5+9135 a^4 b^6+6520 a^3 b^7+3672 a^2 b^8+1448 a b^9+396 b^10) -1.2cm t_0,1^(0)=-4 (a^2-b^2)^2 (198 a^9+502 a^8 b+1099 a^7 b^2+1551 a^6 b^3+1806 a^5 b^4+1596 a^4 b^5+1128 a^3 b^6+596 a^2 b^7+224 a b^8+48 b^9) -1.2cm t_0,2^(0)=-6 b (a^2-b^2) (2 a^2+b^2) (198 a^8+280 a^7 b+611 a^6 b^2+584 a^5 b^3+599 a^4 b^4+368 a^3 b^5+200 a^2 b^6+64 a b^7+12 b^8) -1.2cm t_0,3^(0)=4 a (a^2-b^2)^2(2 a^2+b^2) (99 a^6+29 a^5 b+186 a^4 b^2+40 a^3 b^3+104 a^2 b^4+12 a b^5+16 b^6) -1.2cm t_0,4^(0)=-(a-b)^3 (2 a^2+b^2)^2 (99 a^6-82 a^5 b+191 a^4 b^2-120 a^3 b^3+104 a^2 b^4-40 a b^5+12 b^6) -1.2cm t_1,0^(0)= -4 (a^2-b^2)^2(48 a^9+224 a^8 b+596 a^7 b^2+1128 a^6 b^3+1596 a^5 b^4+1806 a^4 b^5+1551 a^3 b^6+1099 a^2 b^7+502 a b^8+198 b^9) -1.2cm t_1,1^(0)=8 (a+b)^3 (a^4+7 a^2 b^2+b^4) (48 a^6+74 a^5 b+168 a^4 b^2+149 a^3 b^3+168 a^2 b^4+74 a b^5+48 b^6)-1.2cm t_1,2^(0)=-24 b (a^2-b^2)^2 (2 a^2+b^2) (24 a^6+34 a^5 b+62 a^4 b^2+54 a^3 b^3+43 a^2 b^4+20 a b^5+6 b^6) -1.2cm t_1,3^(0)=-8 a (a-b)^2 (2 a^2+b^2) (a^4+7 a^2 b^2+b^4) (24 a^4+7 a^3 b+33 a^2 b^2+6 a b^3+10 b^4)-1.2cm t_1,4^(0)= 4 (a^2-b^2)^2 (2 a^2+b^2)^2 (12 a^5-22 a^4 b+27 a^3 b^2-27 a^2 b^3+14 a b^4-6 b^5) -1.2cm t_2,0^(0)=6 a (a^2-b^2) (a^2+2 b^2) (12 a^8+64 a^7 b+200 a^6 b^2+368 a^5 b^3+599 a^4 b^4+584 a^3 b^5+611 a^2 b^6+280 a b^7+198 b^8)-1.2cm t_2,1^(0)= -24 a (a^2-b^2)^2(a^2+2 b^2) (6 a^6+20 a^5 b+43 a^4 b^2+54 a^3 b^3+62 a^2 b^4+34 a b^5+24 b^6) -1.2cm t_2,2^(0)= 0 -1.2cm t_2,3^(0)=24 a^2 (a^2-b^2)^2 (2 a^2+b^2) (a^2+2 b^2) (3 a^3-2 a^2 b+2 a b^2-2 b^3) -1.2cm t_2,4^(0)=-6 a (a^2-b^2)(2 a^2+b^2)^2 (a^2+2 b^2) (3 a^4-8 a^3 b+11 a^2 b^2-8 a b^3+6 b^4) -1.2cm t_3,0^(0)=4 b (a^2-b^2)^2(a^2+2 b^2) (16 a^6+12 a^5 b+104 a^4 b^2+40 a^3 b^3+186 a^2 b^4+29 a b^5+99 b^6) -1.2cm t_3,1^(0)= -8 b (a-b)^2 (a^2+2 b^2) (a^4+7 a^2 b^2+b^4) (10 a^4+6 a^3 b+33 a^2 b^2+7 a b^3+24 b^4) -1.2cm t_3,2^(0)=-24 b^2 (a^2-b^2)^2(2 a^2+b^2) (a^2+2 b^2) (2 a^3-2 a^2 b+2 a b^2-3 b^3) -1.2cm t_3,3^(0)= -8 a b (a+b)^3 (2 a^2+b^2) (a^2+2 b^2) (a^4+7 a^2 b^2+b^4) -1.2cm t_3,4^(0)= 4 b (a^2-b^2)^2(2 a^2+b^2)^2 (a^2+2 b^2) (2 a^2-a b+3 b^2) -1.2cm t_4,0^(0)=(a-b)^3 (a^2+2 b^2)^2 (12 a^6-40 a^5 b+104 a^4 b^2-120 a^3 b^3+191 a^2 b^4-82 a b^5+99 b^6) -1.2cm t_4,1^(0)=-4 (a^2-b^2)^2(a^2+2 b^2)^2 (6 a^5-14 a^4 b+27 a^3 b^2-27 a^2 b^3+22 a b^4-12 b^5) -1.2cm t_4,2^(0)= 6 b(a^2-b^2)(2 a^2+b^2) (a^2+2 b^2)^2 (6 a^4-8 a^3 b+11 a^2 b^2-8 a b^3+3 b^4) -1.2cm t_4,3^(0)= 4 a (a^2-b^2)^2 (2 a^2+b^2) (a^2+2 b^2)^2 (3 a^2-a b+2 b^2) -1.2cm t_4,4^(0)= -(a+b)^3 (2 a^2+b^2)^2 (a^2+2 b^2)^2 (3 a^2+2 a b+3 b^2)and-1.2cm t_0,0^(1)=4 (a+b)^4 (10 a^4+18 a^3 b+25 a^2 b^2+18 a b^3+10 b^4) (14 a^4+12 a^3 b+29 a^2 b^2+12 a b^3+14 b^4) -1.2cm t_0,1^(1)=-4 (a^2-b^2)^2(280 a^8+710 a^7 b+1414 a^6 b^2+1839 a^5 b^3+1881 a^4 b^4+1428 a^3 b^5+812 a^2 b^6+316 a b^7+68 b^8) -1.2cm t_0,2^(1)=-24 b (a^2-b^2)(2 a^2+b^2) (10 a^3+7 a^2 b+8 a b^2+2 b^3) (7 a^4+5 a^3 b+9 a^2 b^2+4 a b^3+2 b^4) -1.2cm t_0,3^(1)=4 (a^2-b^2)^2(2 a^2+b^2) (140 a^6+41 a^5 b+193 a^4 b^2+36 a^3 b^3+68 a^2 b^4+4 a b^5+4 b^6) -1.2cm t_0,4^(1)=-4 (a-b)^3 (2 a^2+b^2)^2 (5 a^2-2 a b+2 b^2) (7 a^3-3 a^2 b+6 a b^2-2 b^3) -1.2cm t_1,0^(1)= -4 (a^2-b^2)^2 (68 a^8+316 a^7 b+812 a^6 b^2+1428 a^5 b^3+1881 a^4 b^4+1839 a^3 b^5+1414 a^2 b^6+710 a b^7+280 b^8) -1.2cm t_1,1^(1)= 16 (a+b)^2 (a^4+7 a^2 b^2+b^4) (34 a^6+86 a^5 b+155 a^4 b^2+179 a^3 b^3+155 a^2 b^4+86 a b^5+34 b^6) -1.2cm t_1,2^(1)=-24 b(a^2-b^2)^2(2 a^2+b^2) (34 a^5+48 a^4 b+71 a^3 b^2+52 a^2 b^3+30 a b^4+8 b^5) -1.2cm t_1,3^(1)=-16 (a-b)^2 (2 a^2+b^2) (a^4+7 a^2 b^2+b^4) (17 a^4+5 a^3 b+15 a^2 b^2+2 a b^3+2 b^4)-1.2cm t_1,4^(1)= 4 (a^2-b^2)^2(2 a^2+b^2)^2 (17 a^4-31 a^3 b+30 a^2 b^2-22 a b^3+8 b^4) -1.2cm t_2,0^(1)= 24 a (a^2-b^2)(a^2+2 b^2) (2 a^3+8 a^2 b+7 a b^2+10 b^3) (2 a^4+4 a^3 b+9 a^2 b^2+5 a b^3+7 b^4) -1.2cm t_2,1^(1)= -24 a (a^2-b^2)^2(a^2+2 b^2) (8 a^5+30 a^4 b+52 a^3 b^2+71 a^2 b^3+48 a b^4+34 b^5) -1.2cm t_2,2^(1)= 0 -1.2cm t_2,3^(1)=24 a (a^2-b^2)^2 (2 a^2+b^2) (a^2+2 b^2) (4 a^3-3 a^2 b-2 b^3) -1.2cm t_2,4^(1)=-24 a (a-2 b) (a^2-b^2)(2 a^2+b^2)^2 (a^2+2 b^2)(a^2-a b+b^2) -1.2cm t_3,0^(1)=4 (a^2-b^2)^2 (a^2+2 b^2) (4 a^6+4 a^5 b+68 a^4 b^2+36 a^3 b^3+193 a^2 b^4+41 a b^5+140 b^6) -1.2cm t_3,1^(1)= -16 (a-b)^2 (a^2+2 b^2) (a^4+7 a^2 b^2+b^4) (2 a^4+2 a^3 b+15 a^2 b^2+5 a b^3+17 b^4) -1.2cm t_3,2^(1)=-24 b (a^2-b^2)^2 (2 a^2+b^2) (a^2+2 b^2) (2 a^3+3 a b^2-4 b^3)-1.2cm t_3,3^(1)= 16 (a+b)^2 (2 a^2+b^2) (a^2+2 b^2) (a^2-a b+b^2) (a^4+7 a^2 b^2+b^4) -1.2cm t_3,4^(1)= -4 (a^2-b^2)^2(2 a^2+b^2)^2 (a^2+2 b^2) (a^2-a b+4 b^2) -1.2cm t_4,0^(1)=-4 (a-b)^3 (a^2+2 b^2)^2 (2 a^2-2 a b+5 b^2) (2 a^3-6 a^2 b+3 a b^2-7 b^3) -1.2cm t_4,1^(1)= 4 (a^2-b^2)^2 (a^2+2 b^2)^2 (8 a^4-22 a^3 b+30 a^2 b^2-31 a b^3+17 b^4) -1.2cm t_4,2^(1)=-24 b(2 a-b)(a^2-b^2) (2 a^2+b^2) (a^2+2 b^2)^2 (a^2-a b+b^2) -1.2cm t_4,3^(1)= -4 (a^2-b^2)^2(2 a^2+b^2) (a^2+2 b^2)^2(4 a^2-a b+b^2) -1.2cm t_4,4^(1)=4 (a+b)^4 (2 a^2+b^2)^2 (a^2+2 b^2)^2,together with -1.2cm u_0,0^(0)=-(a-b)^2 (a+b) (2 a^2+b^2) (a^2+2 b^2) -1.2cm u_0,1^(0)=4 (a-b)^2 (a+b) (a^2+2 b^2)^2 -1.2cm u_0,2^(0)=-(a-b) (a^2+2 b^2) (2 a^4+17 a^2 b^2+17 b^4) -1.2cm u_1,0^(0)=4 (a-b)^2 (a+b) (2 a^2+b^2)^2 -1.2cm u_1,1^(0)=-8 (a+b) (4 a^2+a b+4 b^2) (a^4+7 a^2 b^2+b^4) -1.2cm u_1,2^(0)=4 (a^2-b^2) (4 a^5+14 a^4 b+22 a^3 b^2+32 a^2 b^3+19 a b^4+17 b^5) -1.2cm u_2,0^(0)= (a-b) (2 a^2+b^2) (17 a^4+17 a^2 b^2+2 b^4) -1.2cm u_2,1^(0)=-4 (a^2-b^2) (17 a^5+19 a^4 b+32 a^3 b^2+22 a^2 b^3+14 a b^4+4 b^5) -1.2cm u_2,2^(0)=(a+b) (34 a^6+76 a^5 b+137 a^4 b^2+154 a^3 b^3+137 a^2 b^4+76 a b^5+34 b^6)and -1.2cm u_0,0^(1)=0 -1.2cm u_0,1^(1)=-12 b (a-b)^2 (a+b) (a^2+2 b^2) -1.2cm u_0,2^(1)=12 b (a-b)(a^2+2 b^2)^2-1.2cm u_1,0^(1)=-12 a (a-b)^2 (a+b) (2 a^2+b^2) -1.2cm u_1,1^(1)=48 (a^2+a b+b^2) (a^4+7 a^2 b^2+b^4) -1.2cm u_1,2^(1)=-12 (a^2-b^2)(2 a^4+6 a^3 b+11 a^2 b^2+9 a b^3+8 b^4) -1.2cm u_2,0^(1)= -12 a (a-b) (2 a^2+b^2)^2 -1.2cm u_2,1^(1)=12 (a^2-b^2) (8 a^4+9 a^3 b+11 a^2 b^2+6 a b^3+2 b^4) -1.2cm u_2,2^(1)= -12 (a+b)^2 (a^2+a b+b^2) (4 a^2+a b+4 b^2) . § CALCULATION OF THE LAW FOR Φ=N_2/N IN THE LOCAL LIMIT Our starting point is the expansion of X_s,t(G(a,ϵ),G(b,ϵ)) when ϵ→ 0, with G(c,ϵ) as in (<ref>). Expanding the equation (<ref>) to increasing orders in ϵ, we deduce the expansionX_s,t(G(a,ϵ),G(b,ϵ))=3 (s+1) (t+1) (s+t+3)/(s+3) (t+3) (s+t+1)+∑_i≥ 1𝔛_i(s,t,a,b) ϵ^2i , where the first term is easily deduced from ϵ the exact expression (<ref>) with g=h=1/12 and where the 𝔛_i's are obtained recursively, order by order in ϵ^2.Expanding (<ref>) at order ϵ^2 shows that 𝔛_1(s,t,a,b)=0 and,at order ϵ^4, that 𝔛_2 may be written (by linearity) as 𝔛_2(s,t,a,b)=a^4 ξ(s,t)+ b^4 ξ(t,s) ,where ξ(s,t) is the solution of some appropriate partial differential equation. We thus have the following form for the first terms in the expansion:X_s,t(G(a,ϵ),G(b,ϵ))=3 (s+1) (t+1) (s+t+3)/(s+3) (t+3) (s+t+1)+(a^4 ξ(s,t)+ b^4 ξ(t,s)) ϵ^4 + 𝔛_3(s,t,a,b) ϵ^6 +O(ϵ^8) , from which we deduce via (<ref>) the expansionF(s,G(a,ϵ),G(b,ϵ))=log(s^2 (2 s+3)/(s+1)^2 (2 s-1))-(a^4+b^4) ψ(s)ϵ ^4+𝔉_3(s,a,b) ϵ^6 +O(ϵ^8) ,where ψ(s) is directly related to ξ(s,t) and 𝔉_3(s,a,b) to 𝔛_3(s,t,a,b). From the very existence of the scaling function, we may writeℱ(S,a,b) =lim_ϵ→ 0 1/ϵ^3F(⌊ S/ϵ⌋,G(a,ϵ),G(b,ϵ))=1/2S^3+(a^4+b^4) lim_ϵ→ 0ϵ ψ(⌊ S/ϵ⌋)+lim_ϵ→ 0(ϵ^3𝔉_3(⌊ S/ϵ⌋,a,b) +O(ϵ^5)) .This is to be compared with the small S expansion of ℱ(S,a,b), as obtained from its exact expression of Appendix A, namelyℱ(S,a,b)=1/2S^3-(a^4+b^4) S/60+(a^6+b^6) S^3/189+O(S^5) .We readily deduce that ψ(s)∼ s/60 when s→∞ and 𝔉_3(s,a,b) s →∞∼ (a^6+b^6) s^3/189(note that the O(ϵ^5) term in (<ref>) necessarily leads to an O(S^5) term in ℱ(S,a,b)), which is precisely the announced scaling correspondence. We may now estimate [g^N]F(s,g,ge^μ/N).This quantity is obtained by a contour integral around g=0, namely1/2iπ∮dg/g^N+1 F(s,g,ge^μ/N)and, at large N, we may change variable from g to a by taking g=G(a,ϵ) with5.cm ϵ=1/N^1/4 .Setting h=ge^μ/N =g(1+ μϵ^4) then amounts to choosing-b^4/36=-a^4/36+μ+O(1/N) ⇔ b^4=a^4-36μ +O(1/N) .Using dg=-(1/12)a^3/(9 N) and g^N+1∼ (1/12)^N+1e^-a^4/36, we eventually arrive at[g^N]F(s,g,ge^μ/N) =1/2iπ12^N/N∫_𝒞_μ da -a^3/9e^a^4/36{log(s^2 (2 s+3)/(s+1)^2 (2 s-1)) -1/N(2a^4-36μ ) ψ(s)+1/N^3/2𝔉_3(s,a,(a^4-36 μ)^1/4) +O(1/N^2) }with some appropriate integration contour 𝒞_μ in the complex plane. As explained in <cit.> and illustrated in Figure <ref>, this contour is made of a first part 𝒞_μ,1 consisting of two half straight lines at ± 45^∘ meeting at the origin, and a part 𝒞_μ,2which makes a back and forth excursion from 0 to (36μ)^1/4.Both the constant (i.e. independent of a) terms and the a^4 term in-between the curly brackets lead to integrals along this contour which vanish identically by symmetry[This vanishing holds for any finite s, i.e. even before taking the s→∞ limit.],so that[g^N]F(s,g,ge^μ/N) N →∞∼1/2iπ12^N/N^5/2∫_𝒞_μ da -a^3/9e^a^4/36𝔉_3(s,a,(a^4-36 μ)^1/4)with a right hand side which behaves at large s as-1.2cm1/2iπ∫_𝒞_μ da -a^3/9e^a^4/36𝔉_3(s,a,(b^4-36 μ)^1/4)s →∞∼s^3/1891/2iπ∫_𝒞_μ da -a^3/9e^a^4/36 (a^6+(a^4-36 μ)^3/2) .The contribution of the two terms in this latter integral were computed in <cit.>, namely1/2iπ∫_𝒞_μ da -a^3/9e^a^4/36 a^6 = 432/π ∫_0^∞ dtt^4e^-t^2=162/√(π) , 1/2iπ∫_𝒞_μ da -a^3/9e^a^4/36 (a^4-36 μ)^3/2= 432e^μ/π ∫_0^∞ dtt^4e^-t^2=162/√(π)e^μ .This yields eventually[g^N]F(s,g,ge^μ/N) N →∞∼12^N/√(π)N^5/2(6/7 (1+e^μ)s^3 + O(s^2))at large s. At μ=0, we recover the estimate (<ref>). Taking the appropriate ratio, we arrive immediately at the desired result (<ref>).§ ESTIMATE OF 𝔓(V) AT SMALL V Since the quantity 𝔓̂(σ)≡ E[e^-σV]is the Laplace transform of the probability distribution 𝔓(V), with V taking its values in [0,∞),it has no singularity for real non-negative σ. Its inverse Laplace transform, the probability distribution 𝔓(V) itself, may thus be obtained via𝔓(V)= 1/2iπ∫_γ-i∞^γ+i∞dσe^σV 𝔓̂(σ)for any real non-negative γ.At small V, this integral may be evaluated via a saddle point approximation as follows. For large σ, we have the asymptotic equivalence𝔓̂(σ) σ→∞∼9/2(3√(2)-4)e^-√(6) σ^1/4and the integral is dominated by its saddle point σ^* given byd /dσ( σV -√(6) σ^1/4)|_σ=σ^*=0⇔σ^*= 3^2/3/4 V^4/3 .The use of the asymptotic equivalent above for 𝔓̂(σ) is fully consistent if σ^* becomes large, i.e. when V itself becomes small. Settingσ= σ^*+i ηin the integral, with η real (i.e. choosing implicitly γ=σ^*), we may use the expansionσV -√(6) σ^1/4= -3^5/3/4V^1/3-3^1/3/2 V^7/3 η ^2+O(η ^3)to write𝔓(V)V → 0∼9/2(3√(2)-4) e^-3^5/3/4V^1/31/2 π∫_-∞^∞dηe^-3^1/3/2 V^7/3 η ^2 = 3^11/6 (3 -2 √(2))/2 √(π) 1/V^7/6e^-3^5/3/41/V^1/3 .plain | http://arxiv.org/abs/1706.08809v1 | {
"authors": [
"Emmanuel Guitter"
],
"categories": [
"math.CO",
"math-ph",
"math.MP"
],
"primary_category": "math.CO",
"published": "20170627121108",
"title": "A universal law for Voronoi cell volumes in infinitely large maps"
} |
=1mm | http://arxiv.org/abs/1706.08792v1 | {
"authors": [
"Felix Lochner",
"Felix Ahn",
"Tilmann Hickel",
"Ilya Eremin"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.str-el"
],
"primary_category": "cond-mat.supr-con",
"published": "20170627114144",
"title": "Electronic properties, low-energy Hamiltonian and superconducting instabilities in CaKFe$_4$As$_4$"
} |
firstpage–lastpage 000 Top Down Electroweak Dipole Operators Michael Ramsey-Musolf^1,2 December 30, 2023 =====================================A newly formed magnetar has been proposed as the central engine of short GRBs to explain on-going energy injection giving observed plateau phases in the X-ray lightcurves. These rapidly spinning magnetars may be capable of emitting pulsed emission comparable to known pulsars and magnetars. In this paper we show that, if present, a periodic signal would be detectable during the plateau phases observed using the Swift X-Ray Telescope recording data in Window Timing mode. We conduct a targeted deceleration search for a periodic signal from a newly formed magnetar in 2 Swift short GRBs and rule out any periodic signals in the frequency band 10–285 Hz to ≈15–30% rms. These results demonstrate that we would able to detect pulsations from the magnetar central engine of short GRBs if they contribute to 15-30% of the total emission. We consider these constraints in the context of the potential emission mechanisms. The non-detection is consistent with the emission being reprocessed in the surrounding environment or with the rotation axis being highly aligned with the observing angle. As the emission may be reprocessed, the expected periodic emission may only constitute a few percent of the total emission and be undetectable in our observations. Applying this strategy to future observations of the plateau phases with more sensitive X-ray telescopes may lead to the detection of the periodic signal. Gamma-Ray Bursts, Magnetars § INTRODUCTION The launch of the Swift satellite <cit.> has led to a reformation in our understanding of early afterglow emission from gamma-ray bursts (GRBs). Particularly, Swift highlighted that central engine activity is often long lived, powering flares and plateaus <cit.>. Prolonged central engine activity is often explained as ongoing accretion onto the newly formed black hole (BH) following the collapse of a massive star <cit.>.For Short GRBs (SGRBs), typically with durations of T_90≤ 2 s[Though we cannot rely upon prompt emission alone to unambiguously identify SGRBs <cit.>]<cit.>, prolonged accretion is not expected within the standard progenitor model. They are thought to originate from the merger of a compact binary system constituting of two neutron stars (NSs) or a NS and a BH <cit.>. In this model the accretion is expected to end within ∼ 2 s <cit.> powering the prompt gamma-ray emission. Possible late time accretion of material on highly eccentric orbits could lead to flares in the X-ray lightcurve but cannot power prolonged plateau phase <cit.>. However, studies of SGRB X-ray lightcurves has shown that there is evidence of plateau phases signifying prolonged energy injection that cannot be explained by this theory <cit.>. An alternative model is that the central engine of GRBs is a newly formed millisecond pulsar with a high magnetic field and sufficient rotational energy to prevent gravitational collapse <cit.>. The magnetar can be formed in a variety of ways; during the collapse of a massive star <cit.>, via accretion induced collapse of a NS or a white dwarf <cit.> or the merger of two NSs <cit.>. This model predicts a plateau phase in the X-ray lightcurves originating from dipole emission from the rapidly spinning down magnetar <cit.>. As the magnetar spins down, the plateau slowly turns over to a powerlaw decline. If the newly formed magentar is unstable (i.e. the mass supported by its rapid rotation is greater than the maximum allowed mass of a NS), then it will reach a critical point at which it is unable to support itself and will instead collapse to form a BH. At that point, the energy injection is rapidly turned off leading to a steep decay phase in the X-ray lightcurve rather than a shallow decay phase <cit.>.This model has been fitted to large samples of Long GRBs <cit.>, all Swift SGRBs with sufficient X-ray observations <cit.> and has been proposed to explain energy injection in the class of SGRBs with extended emission <cit.>. The fitted magnetar parameters for all of these candidates are consistent with the expected values for newly formed magnetars, although there is no conclusive proof to date that magnetars are the central engines. <cit.> suggested that the next generation gravitational wave detectors may be able to provide this proof, as a newly formed magnetar has been predicted to produce an additional gravitational wave signal following the initial inspiral signal <cit.>, however the expected detection rates are very low.Alternatively, if a magnetar is the central engine powering GRBs, we might expect to see periodic features in the emission. Known magnetars have clear periodic signals in their emission caused by their rotation periods <cit.>. The X-ray pulsations typically contribute to 30% of the signal, with a range of 10–80% <cit.>. There is an energy dependence on the pulsed fraction of the signal, where low energies tend to have smaller pulsed frations <cit.>. Detection of a periodic signal during the plateau phase in the X-ray lightcurve would provide excellent supporting evidence for the magnetar central engine model. There have been searches for a periodic signal in the prompt emission of GRBs with a number of instruments with no success, for example: BATSE GRBs <cit.>, INTEGRAL GRBs <cit.>, GRB 051103 <cit.> and BAT GRBs <cit.>. <cit.> searched the prompt emission of a number of short GRBs for evidence of a precessing jet <cit.>. However, these searches typically target the prompt emission and have not probed the regime where we might expect periodic signals from a magnetar central engine (i.e. during the plateau phase). Only two GRBs have been searched for periodic emission during the X-ray observations when the magnetar central engine may dominate the emission, GRB 060218 <cit.> and GRB 090709A <cit.>. The prompt emission of GRB 090709A possibly showed evidence of a periodic signal <cit.>, however this was ruled out with more careful analysis of the prompt data from BAT, XRT (X-ray Telescope) and XMM (X-ray Multi-mirror Mission) observations of the X-ray afterglow <cit.>. However, in the majority of these studies, the authors have targeted a constant spin period whereas a magnetar central engine is expected to have a rapidly decelerating spin period which would be very difficult to detect in standard searches for periodic signals. <cit.> did conduct a deceleration search, however they were targeting signals in the prompt emission where we do not expect the signal from a spinning down magnetar. In this paper, we present the first targeted deceleration search for a periodic signal associated with a spinning down magnetar central engine. For a successful periodicity search we require: * A GRB which is not in a high density environment or have a progenitor which may have blown off a large amount of material, as this could lead to reprocessing of the emission which may dilute the periodic signal.* A plateau phase showing evidence of energy injection within the X-ray observations. The magnetar component should dominate the lightcurve in order to get the largest periodic signal, so we need GRBs which have a minimal standard afterglow component. * Window Timing <cit.> mode observations covering part of the plateau phase. WT mode provides the timing resolution required for a millisecond periodicity search. * A good redshift constraint.We propose that SGRBs provide the ideal dataset for this analysis as they are expected to occur in low density environments and typically have a faint afterglow. From the analysis in <cit.>, we identified two, unambiguously short, SGRBs which satisfied these criteria: GRB 090510 and GRB 090515 <cit.>. Section 2 describes the periodic signals predicted from the magnetar central engines that are consistent with the X-ray lightcurves of these SGRBs. In Section 3 we describe the periodicity search conducted and provide the results, while Section 4 discusses the theoretical implications of our observations and the likelihood of the production of a detectable signal. § PERIODIC SIGNAL PREDICTIONS The magnetar spin period and spin-down rate are analytically predictable using the dipole radiation model which is fitted to the X-ray lightcurves of the SGRBs. The initial magnetic field strength and spin period are given by <cit.>:B^2_p,15=4.2025 I_45^2R^-6_6L_0,49^-1T_em,3^-2(ϵ/1-cosθ), P^2_0,-3=2.05 I_45L_0,49^-1T_em,3^-1(ϵ/1-cosθ). Where B_15 is the magnetic field strength of the newly born magnetar in 10^15 G, P_0,-3 is the initial spin period of the magnetar in ms, I_45∼ M_1.4R_6^2 is the moment of inertia of a NS where I = 10^45 g cm^2 I_45, R_6 is the radius of the magnetar in 10^6 cm, M_1.4 is the mass of the magnetar in 1.4 M_⊙, L_0,49 is the plateau luminosity in 10^45 erg s^-1 and T_em,3 is the plateau duration in 10^3 s. In both equations, we also include the dependence on the beaming angle (θ) and the efficiency in conversion of the rotational energy into the observed X-ray emission (ϵ). Additionally, we do not know the mass or radius of the newly formed magnetar. The mass of the newly formed magnetar is expected to be 1 ≤ M_1.4≤ 1.5 therefore, as P_0,-3∝ M_1.4^0.5, the spin period is only expected to vary by ∼20% which is not significant in comparison to the other uncertainties caused by efficiencies and beaming. Magnetars may be formed with radii up to ∼ 30 km <cit.>, however it is expected that they will stabilise at a typical NS radius of R_6∼ 1 within the first few seconds <cit.>. Therefore, in this paper we assume M_1.4=1.5, as the newly formed magnetar is most likely to be a massive neutron star, and R_6=1.The magnetar emission is assumed to be isotropic and 100% efficient for fitting purposes, however it is important to note that this is an idealised situation and changes to this assumption can cause significant differences in the output values for B_15 and P_0,-3 <cit.>. We account for these uncertainties later in this section. Using the method described in <cit.>, the observed 0.3–10 keV lightcurves of the SGRBs were converted into restframe 1–10000 keV lightcurves and fitted using the magnetar central engine plateau model, given by equations <ref> and <ref>, and the parameters for each of these GRBs are provided in Table <ref>. These fits determine the initial spin period of the magnetar, however the magnetar is rapidly spinning down so we also need to predict the spin down evolution. To do this we can use, from <cit.>,dΩ/dt = N_ dip/IN_ dip=-1.5×10^45μ_33^2P^-3_-3assuming there is no ongoing accretion. Where μ is the dipole magnetic moment, μ_33=B_15R_6^3=10^33 G cm^3 μ, Ω=2π/P_-3 is the angular velocity and N_ dip is the torque from the dipole emission <cit.>. We note <cit.> derive a more complex torque from dipole emission, taking into account open magnetic flux tubes in an accreting magnetar system, however the accretion is expected to have ended prior to the emission we observe and this additional complexity not required. By substitution of Equations <ref> and <ref> into Equations <ref> and <ref> followed by integration, we can predict that the spin period evolution with time can be described by: ν=(5×10^-7xt + 10^-6P_0,-3^2)^-1/2 s,where ν≡ 1/P andx=B_15^2R_6^4/2π M_1.4.By differentiation we can determine the spin down rate to be given by:ν̇=-5×10^-7x/2ν^3 Hz s^-1Additionally, we assume that the magnetar is spinning down purely via dipole radiation so the relationship between the spin and its spin down properties are well defined using the breaking index:n = νν̈/ν̇^2 (= 3 for dipole spindown).This assumption is intrinsic to the magnetar model typically fitted to the X-ray plateaus <cit.>, however known young pulsars are known to be spinning down differently to this, with braking indices n<3, <cit.>. Recently, <cit.> extended the magnetar model to use the late time decay slope to constrain the spin down of a magnetar central engine in two SGRBs. One of their sample is fitted with n=2.6 ± 0.1 (GRB 140903A), consistent with the observed n<3 braking indices in millisecond magnetars. The other, GRB 130603B, has n=3.0 ± 0.1 as expected for dipole radiation. Therefore, the assumption of pure dipole radiation is likely to be consistent for at least some of the magnetar engines fitted in the SGRB sample but likely not all. Unfortunately, neither GRB in <cit.> have sufficient WT mode data to be included in our sample. However, it is promising that in the future we may be able to directly measure the braking index for SGRBs and, combined with the required WT mode data, obtain a much deeper constraint on periodic emission. In this paper, we consider the impact of different braking indices and this issue will be discussed further in Section 3. Using Equations <ref>, <ref> and <ref> alongside the magnetic field strengths and restframe spin periods obtained, we can describe how the spin frequency of the newly formed magnetar evolves with time. The restframe spin frequencies are then converted into observed frame spin frequencies that we might expect to detect from the timing analysis conducted in Section 3. In Figure <ref> we show how the spin frequency evolves during the WT mode observation for each of the SGRBs. However, from these plots it is clear that the spin frequency can decrease significantly from the start of the WT observation to the end of the WT observation so any periodicity searches will need to account for this rapid spin down. As previously stated, the efficiency in converting the rotational energy into the observed plateau and the beaming angle of the emission have a significant impact on the spin periods predicted. However, both of these are currently unknown; here efficiencies are assumed to lie within the range 1–100% while jet opening angles for SGRBs are thought to range from 1–20 degrees or more <cit.>. Figure <ref> shows the periodicity at the start of the WT mode observation for each of the GRBs as a function of both the efficiency and beaming of the observation. The region above the blue dash-dot line illustrates the region that cannot be probed using the observations due to the timing resolution of the Swift WT mode observations. We also show the spin break up frequency of a 1.4 M_⊙ NS in the observer frame for each of the GRBs (red dotted line) above which no NSs can exist. <cit.> showed that the observed correlation between the plateau luminosity and duration for GRBs <cit.> can be used to tightly constrain the efficiency and beaming angle of the emission from the magnetar central engine. A probability contour plot was produced by this analysis, providing the probability that the magnetar model is consistent with the observed dataset as a function of different beaming angles and efficiencies. We use the 50% probability contours from the analysis of <cit.> to reject regions of the beaming and efficiency parameter space, thus more tightly constraining the properties of the magnetar. The upper and lower 50% contours are well fitted with simple exponential equations and we use these fits to incorporate the allowed region of the parameter space into the modelling of the periodic signal. All combinations of beaming angles and efficiencies that do not lie within these contours are excluded from the modelling. After applying these constraints, we note that all of the expected spin periods for the SGRBs lie within the detectable range for the WT mode data. Using the allowed spin periods, we extract the range of values for ν, ν̇ and ν̈ that we want to probe for each GRB, provided in Table <ref>. Although these numbers are strongly related (see equations <ref>–<ref>), we search the entire region of this parameter space for simplicity.This analysis has shown that a periodic signal resulting from a magnetar spinning down via dipole radiation would be detectable by Swift XRT in WT mode observations for these SGRBs and reasonable combinations of the efficiency and beaming angle. Note this analysis assumes that the plateau emission contains a highly pulsed component which is detectable above the continuum emission, in Section 4 we discuss the likelihood of this.§ TIMING ANALYSIS METHOD AND RESULTS§.§ X-Ray Data In view of our preceding discussion, we analyzed two targeted Swift/XRT observations on the two GRBs previously discussed. We used only data recorded with a time resolution of 1.766 ms (WT-mode) and we extracted the source counts from a circular region of radius 30 arcsec centered on the brightest pixel and in the energy range 0.5–10 keV. The background was calculated from a similar extraction region placed as far as possible from the source location.A summary of the total length of the observations, the total number of photons, the background count-rate and the observation ID are summarized in Table <ref>. §.§ Simple Periodicity Search The first type of pulse search we adopted is the simplest one and is based on a by-eye inspection of power spectra of different length. We calculated Fourier transforms with length between 4 s and 128 s with no background subtraction and/or dead time correction applied prior to the calculation. Then we averaged each power spectrum by Leahy-normalizing them by subtracting a Poissonian (counting) noise level incorporating dead-time effects as explained in <cit.>.We first looked for candidate pulsations with a power exceeding a threshold power of 30, which would correspond to a 3-σ detection (single trial). We then produced dynamical power spectra of 4-s length each and looked for patterns in the peak powers. In neither case we had a candidate to follow up.If we assume that the pulse power remains in one Fourier frequency bin during the entire observation, then we can place upper limits on the root-mean-square (rms) pulse amplitude under the assumption that the power spectrum contains only white (counting) noise <cit.>: rms = [2(S/N)·(S+B)]^1/2 S^-1T_ obs^-1/2where S and B are the signal and background count rates, respectively, S/N is the target signal-to-noise of the pulsations (i.e., the single trial significance) and T_ obs is the length of the power spectrum (in seconds).Upper limits for a S/N≈ 5 are of the order of 10% rms for both observations. When looking at the 4-s long dynamical power spectra, we would have detected a signal with a S/N≈3 if the rms amplitude of the pulsations had been in excess of approximately 50%. We caution that since we expect a very rapid drift of the pulse frequency over time the power will spread across multiple bins. Therefore our assumption of having the power in one Fourier frequency bin breaks down and the aforementioned 10% upper limits become unrealistic. The amount of bins over which the power spreads depends on the deceleration of the pulsar.§.§ Deceleration Search As a first approximation we can consider a neutron star decelerating at a constant rate. The maximum number of bins z_ max over which the spin frequency power will spread is thus <cit.>:z_ max = a_ max T_ obs^2 N_ harm ν/cwhere a_ max is the maximum allowed (radial) deceleration, N_ harm is the harmonic number and c is the speed of light.The acceleration can be calculated from our estimated ν̇ in the preceding sections. Since the maximum ν̇ is of the order of -0.01 Hz s^-1, our maximum acceleration wouldgive a drift of the order of 50,000 m s^-2 and a maximum number of bins of the order of a few hundreds. To begin with, we performed a deceleration search with the software PRESTO (v.17Mar15) on our Swift/XRT time series <cit.>. The search uses matched filtering techniques to add power of a drifting spin frequency under the assumption that the drift is approximately constant in time (i.e., there is a constant deceleration).The search was carried for frequencies in the range 10 to 283 Hz (i.e., our Nyquist frequency) and for z_ max=800. We searched pulsations under the assumptions that no harmonic content was present in the data, which is a good assumption if the expected pulse emission patters is nearly sinusoidal (as is the case for a Lambertian emitter like a hot spot). The significance is calculated by looking at the power returned by the matched filtering technique and then it is transformed into a false alarm probability from a chi-square distribution. No candidate above 3σ was found in any of the GRB used.§.§ Sensitivity and Upper Limits To determine the sensitivity of our search we performed a set of Monte-Carlo simulations where we generated simulated time series having the same sampling time, number of photons (following a Poisson distribution) and duration of the original Swift/XRT time series. The simulated time series contain an injected sinusoidal signal whose phase evolves in time. The time evolution is described in terms of a frequency, frequency derivative and braking-index, whose values cover a 3D grid (see Table <ref>). The deceleration search is then applied to the time series. The procedure is then repeated by increasing the amplitude of the signal from a minimum of 10% rms in steps of 2% up to 50% rms.Since we are working under the assumption that our deceleration is constant, we also investigated the effect of the braking index n, by setting it to zero while exploring the other grid parameters. The simulations show that the effect of a varying braking index on the detection sensitivity is small. This is indeed expected, since the total length of the observations is short and thus the variation of ν̇ is not dominant. For both GRB090515 and GRB090510, the deceleration search shows a robust detection (>3σ) when the the rms amplitude of the pulsations is larger than about 30% rms in almost all grid points. The minimum rms amplitude for which we have a detection is about 15% rms. This means that if a signal of 30% rms or more had been present in one of the two GRBs analyzed, we would have certainly detected a signal at any of the frequencies accessible. We summarize the results in Figure <ref>. § DISCUSSION OF POSSIBLE ORIGINS OF PULSATIONS In the previous sections, we assumed that there would be a periodic signal associated with the spin frequency of a rapidly spinning down magnetar in the lightcurve. We showed that this periodic signal would be detectable in the WT mode observations when using a deceleration search and found no periodic signal in excess of ≈15–30% rms of the total flux. In this section, we discuss the potential sources of periodic emission and the likelihood that they would be detectable.Our first consideration is the environment of the magnetar. If it is surrounded by an optically thick cloud of material, the fractional amplitude of the pulsed emission drops exponentially with optical depth and, hence, very difficult to detect. During the merger process, a very dense ejecta is expected and this has been modeled in simulations. The ejecta is not isotropic and with a preferred direction along the equatorial plane <cit.>, leaving the region along the rotation axis reasonably clean. As we are observing emission from the relativistic jet, we know the viewing angle is close to the rotational axis, while the relativistic jet itself is optically thin <cit.>. So our viewing angle is most favourable for the periodic emission to escape. §.§ Quasi-periodic emission from disk processionThe accretion disk around the central object (black hole or magnetar) may become warped via differential precession and the amplitude peaks when the spin axis of the central object is highly misaligned relative to the accretion disk <cit.>. The initial quasi-periodic signals have spin periods of the order of 50 ms <cit.>. However, for SGRBs the accretion disk is expected to be gone within a few seconds <cit.> and hence this signal would only be expected during the prompt emission and not during our observations so we rule out this mechanism for our analysis. This signal has been sought in periodicity searches of a set of BAT, GBM and BATSE SGRBs by <cit.> but remains undetected to date. §.§ Pulsar emissionThe magnetar central engine is a highly magnetised, millisecond pulsar so we might expect it to emit pulses similar to those observed from known pulsars and magnetars, assuming that the observed emission originates directly from the magnetar. We observe periodic emission from pulsars due to a misalignment between the magnetic axis and the rotation axis; a hot spot at the magnetic poles sweeps in and out of view as the neutron star rotates giving a characteristic pulse. The maximal signal occurs when the magnetic axis is orthogonal to the rotation axis and the viewer is also orthogonal to the rotation axis. However, as we have observed a SGRB, we know that the viewing angle is along the initial jet and, hence, close to the rotation axis so very little pulsed emission is expected. There is a chance that the observing axis is off the rotation axis as the jet has a particular opening angle <cit.>, so there may still be a periodic component to the emission.However, the magnetic fields and rotation axis are also expected to be highly aligned due to the dynamo mechanism that produces the high magnetic fields <cit.>. <cit.> show that the rotation axis and dipole field can become orthogonal on a given timescale, the dissipation timescale, if this is less than the electromagnetic spindown timescale (i.e. <10^3T_em,3 s). The dissipation timescale is defined by <cit.> as:1/τ_ DIS = 3×10^-8 s^-1(10^4/n) (ν/300 Hz) (ϵ_B/10^-7),where n is a factor related to the spin down mechanism and ϵ_B is the quadrupolar distortion of the neutron star due to the magnetic field. This is ∼ 10^7 s with typical parameters and hence is orders of magnitude longer than the electromagnetic spindown timescales of the magnetars considered in this paper. <cit.> extend this analysis to consider the special case of new born magnetars and show the condition for the two axes to become orthogonal is given by:E_B/10^50 erg≲ 2.1 M_1.4/P_-3^2(3+lnP_-3/10B_15+lnM_1.4/0.48 R_6^4)where E_B is the internal magnetic energy. Using typical parameters alongside the predicted magnetic fields and spin periods for the magnetars considered in this analysis, we find the magnetars considered in this paper are typically rotating too slowly for their axes to become completely misaligned on the spindown timescale. Therefore, we expect the magnetic axis and the rotational axis to be close to aligned throughout the spindown timescale <cit.>. Observations of known magnetars suggest that the magnetic field and spin axes are typically slightly misaligned <cit.>.Therefore, very little periodic emission from a pulsar component may be expected due to two reasons: * The viewing angle is very close to the rotation axis, so only a small proportion of the emission is expected to be pulsed even if the magnetic and rotation axes are completely orthogonal.* The magnetic and rotation axes are likely to be highly aligned at birth and are very unlikely to become orthogonal on the spindown timescales of the magnetars studied in this paper.In their studies of PSRJ0821-4300, <cit.> calculated the pulsed fraction of the X-ray emission as a function of the viewing angle and hot-spot angle from the rotation axis. They show that once these angles are greater than≃5 degrees, the pulsed X-ray fraction exceeds ∼20% (note there is also an energy band dependence). Therefore, assuming we are directly observing hot-spot emission (similar to that in standard pulsars), our upper limits on the pulsed fraction show that the observing angle and magnetic field axis need to be ≲5 degrees from the rotation axis.Considering pulsed emission from magnetars, our limits of 15–30% are probing many of the typical pulsed fractions observed in known magnetars <cit.>. We have used the full energy band of Swift to obtain sufficient photons, 0.3–10 keV, where the pulsed fraction may be lower <cit.>. However, we note that these photons we observe are redshifted and hence we are probing higher energy emission where the pulsed component is expected to be larger. §.§ Pulsations from time-dependent scattering in the magnetosphereThe detection of X-ray pulsations during the radio quiet mode of PSR B0943+10 <cit.> presents an alternative to the standard pulsar model described in Section 4.2. During the radio quiet mode, the X-ray data has a 100% pulsed thermal component in addition to a non-thermal component, consisting of ∼ 50% of the total X-ray emission. PSR B0943+10 has a rotation axis which is thought to be only 9 degrees away from the observer angle and has a nearly aligned magnetic axis, similar to the expected configuration for the magnetar central engine model. <cit.> suggest that the X-ray pulsations originate from a scattered component from within closed magnetic field lines. This model could also be applicable to the magnetars considered in this analysis and, assuming we are able to directly observe the pulsar magnetosphere, may lead to a detectable pulsation signal during the X-ray plateau.As we rule out a pulsed fraction of ∼15–30%,we are not directly observing this emission. §.§ Electron acceleration along field lines<cit.> detected pulsations in the hard X-ray component of the emission from the magnetised white dwarf AE Aquarii. The authors propose that the rotating magnetic white dwarf is accelerating electrons along its magnetic field lines, assuming the surrounding medium is a relatively low density plasma <cit.>. Although the magnetic fields of the magnetars in this paper are orders of magnitude larger than the white dwarf, a similar mechanism could potentially work in the plasma surrounding the newly born magnetar. However, it is not clear if this mechanism would still produce a periodic component if the rotation and magnetic axes are aligned. §.§ Reprocessing of emission The observed lightcurves are consistent with the energy originating from the spin-down luminosity of a magnetar <cit.> however it is not clear where or how the observed X-ray photons are emitted. The mechanisms discussed in Sections 4.2–4.4 assumed we were directly observing the magnetar or its immediate surroundings, however this is unlikely and the emission is most likely to be reprocessed. The magnetar central engine is expected to emit a strong wind that interacts with itself and the local environment <cit.>. This magnetar wind could produce the observed emission via magnetic reconnection or shocks and we consider the likelihood that a periodic signal, from one of the mechanisms outlined earlier, could be retained after reprocessing via these mechanisms:* Direct energy injection via forced reconnection: This theory was originally proposed to explain emission observed in the Crab nebula <cit.>. The neutron star emits a magnetised wind which interacts with a surrounding nebula giving a shock at ∼ 10^17 cm. As the neutron star rotates the magnetic field within the wind alternates so, when it reaches the shock front, magnetic reconnection occurs. However, the alternating magnetic fields may not be present due to the alignment of the magnetic field and rotation axis (as discussed in Section 4.2). This model is comparable to models proposed for the prompt emission of GRBs via turbulent magnetic reconnection when there is a collision between two shells with differing magnetic fields <cit.> and would occur at ∼ 10^15–10^16 cm. This model is consistent with the steep decay phase observed in some lightcurves, as the magnetar wind will stop rapidly when the magnetar collapses to form a black hole. The model proposed by <cit.> suggests there will be two variability timescales, one from the central engine and the second from random relativistic turbulence within the emitting regions. In this model there would be some imprint of the millisecond periodic signal from the central engine assuming that the magnetic and rotation axes are misaligned. However, this is likely to be on similar timescales to the relativistic turbulence and hence only constitute a small percentage of the observed signal, making it undetectable in our observations. Even if it is present, a signal of this size would be extremely difficult to detect with current X-ray facilities.* Direct energy injection via up-scattering of photons in the forward shock: There is a continued outflow from the central engine, e.g. a magnetar wind, which up-scatters the synchrotron photons left behind the forward shock <cit.>. If there is a pulsed component in the magnetar wind, due to misaligned magnetic and rotation axes, this could potentially cause a periodic up-scattering of the photons but is likely to be cancelled out due to the variability timescales in the forward shock only being weakly dependent on the input signal timescale <cit.>. This emission is predicted to occur at ∼ 10^16–10^17 cm and, if the incoming electrons are hot, this can lead to a scattered signal which is significantly higher luminosity than the standard forward shock emission. The model can explain rapid steep decay phases after the plateau if the scattering outflow suddenly decreases significantly, consistent with the magnetar wind rapidly switching off as the source collapses to a black hole. However, this signal is expected to be brighter for a wind environment, which is not expected for SGRBs. Another disadvantage of this theory for SGRBs is the expectation that SGRBs occur in a very low density environment, hence the forward shock component is expected to be faint - i.e. few photons are available to be up-scattered.* Indirect energy injection via a refreshed forward shock: In this scenario, the energy from the magnetar wind is injected directly into the forward shock and hence contributes to the standard forward shock emission <cit.>. Therefore, as with the up-scattering mechanism, it is unlikely to retain the periodic component <cit.>. However, this model cannot explain the steep decay phases sometimes observed when the central engine rapidly stops injecting energy into the system. Additionally, this mechanism requires a standard forward shock, which is expected to be weak for SGRBs occurring in low density environments.* Indirect energy injection via a reverse shock: Alternatively energy injection, such as from a magnetar wind, is expected to boost the reverse shock <cit.>. This model is compatible with the low density environments expected with short GRBs and is capable of explaining the steep decay following the plateau phase <cit.>. This is a very promising mechanism as it is consistent with the observed emission properties and the magnetar central engine model. Unfortunately, as with the forward shock, this mechanism of reprocessing the emission would most likely obliterate any periodic component in the energy injection. All these are viable emission mechanisms within the magnetar central engine model, but only the forced reconnection model holds the potential of retaining some of the underlying temporal structure from the central engine, however the periodicities we are searching for are comparable to the random reconnection timescales. Therefore, with the sensitivity of current X-ray facilities, if the magnetar emission has been reprocessed we are very unlikely to be able to extract a periodic signal from the random noise component. § CONCLUSIONS Plateaus in the X-ray lightcurves of short GRBs are signatures of energy injection that are thought to originate from a newly formed magnetars rapidly spinning down due to the emission of dipole radiation. Using the magnetar central engine model, we are able to predict the spin-down frequency and rate that may result in an evolving periodic component in the observed X-ray emission (similar to that observed in pulsars and Galactic magnetars). In this paper, we show that the frequency of the periodic component is detectable within the capabilities of the WT mode of the XRT onboard the Swift Satellite and calculate the optimal parameter space to search for 2 SGRBs.We have conducted a deceleration search for a periodic signal during X-ray plateaus following these SGRBs and, taking into account rapid spin-down via dipole radiation, do not detect any periodic component to a limit of ≈15–30% rms. The rotation and magnetic axes of the magnetar are likely to be close to alignment, unfavourable for the production of a significant periodic component. We show that this signal is still potentially attainable if we are directly observing emission from the magnetar central engine. However, the emission is likely to be reprocessed by the magnetar wind interacting with the forward or reverse shocks and the reprocessing mechanisms are likely to reduce any periodic component to a few percent of the total emission. With future, more sensitive instrumentation <cit.> and more complex search models it will be possible to place much more stringent limits on the presence of a periodic component or lead to a detection that would confirm the magnetar central engine model. § ACKNOWLEDGEMENTSWe are grateful to Phil Uttley for his suggestion to look into this and to Yuri Cavecchi, Daniela Huppenkothen, Anna Watts and Ralph Wijers for very useful discussions. AP acknowledges support from a Netherlands Organization for Scientific Research (NWO) Vidi Fellowship. PTO would like to acknowledge funding from STFC. This work makes use of data supplied by the UK Swift Science Data Centre at the University of Leicester and the Swift satellite. Swift, launched in November 2004, is a NASA mission in partnership with the Italian Space Agency and the UK Space Agency. Swift is managed by NASA Goddard. Penn State University controls science and flight operations from the Mission Operations Center in University Park, Pennsylvania. Los Alamos National Laboratory provides gamma-ray imaging analysis.999[Aloy, Janka, & Müller2005]aloy2005 Aloy M. A., Janka H.-T., Müller E., 2005, A&A, 436, 273[Barthelmy et al.2009]barthelmy2009 Barthelmy S. D., et al., 2009, GCN Circ., 9364, 1 [Berger2010]berger2010 Berger, E., 2010, ApJ, 722, 1946[Bernardini et al.2012]bernardini2012 Bernardini M. G., Margutti R., Mao J., Zaninoni E., Chincarini G., 2012, A&A, 539, A3 [Blackman, Yi, & Field1996]blackman1996 Blackman E. G., Yi I., Field G. B., 1996, ApJ, 473, L79[Bromberg et al.2013]bromberg2013 Bromberg O., Nakar E., Piran T., Sari R., 2013, ApJ, 764, 179[Bucciantini et al.2012]bucciantini2012 Bucciantini N., Metzger B. D., Thompson T. A., Quataert E., 2012, MNRAS, 419, 1537[Bucciantini et al.2006]bucciantini2006 Bucciantini N., Thompson T. 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"authors": [
"A. Rowlinson",
"A. Patruno",
"P. T. O'Brien"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170626180008",
"title": "A deceleration search for magnetar pulsations in the X-ray plateaus of Short GRBs"
} |
Berkeley Center for Theoretical Physics, Department of Physics, University of California, Berkeley, CA 94720, USA Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan We point out that a simple inflationary model in which the axionic inflatoncouples to a pure Yang-Mills theory may give the scalar spectral index(n_s) and tensor-to-scalar ratio (r) in complete agreement with thecurrent observational data. Pure Natural Inflation Masahito Yamazaki December 30, 2023 ====================== Cosmic inflation plays an important role in explaining observable featuresof our universe, including its extreme flatness, as well as the originof primordial curvature perturbations.The detailed predictions of inflation,however, depend on the potential V(ϕ) of the inflaton field ϕ.An important issue, therefore, is to understand what is the correct modelof inflation and how it emerges from the underlying physics.Recent observations by Planck <cit.> and BICEP2/KeckArray <cit.> have started constraining simple modelsof inflation.In particular, arguably the simplest model of inflationV(ϕ) = m^2 ϕ^2/2 <cit.>—which gives the correctvalue for the scalar spectral index n_s ≃ 0.96—is now excludedat about the 3σ level because of the non-observation of tensormodes.This raises the following questions.Does the model of inflationneed to be significantly complicated?Is the agreement of n_s of thequadratic potential with the data purely accidental?In this letter, we argue that the answers to these questions may both beno.In particular, we argue that a simple inflationary model in which theinflaton ϕ couples to the gauge field of a pure Yang-Mills theory ℒ = 1/32π^2ϕ/f ϵ^μνρσ TrF_μν F_ρσ, may give the values of n_s and the tensor-to-scalar ratio, r, inperfect agreement with the current observational data.Here, ϕ isa pseudo-Nambu-Goldstone boson—axion—of a shift symmetry ϕ→ϕ + const., and f is the axion decay constant.For nowwe assume that the gauge group of the Yang-Mills theory is SU(N),but the model also works for other gauge groups; see later.Conventionally, the potential of the axion field as in Eq. (<ref>)is assumed to take the form generated by non-perturbative instantons V(ϕ) = Λ^4 [1 -cos(ϕ/f) ], where Λ is the dynamical scale of the Yang-Mills theory.The resultinginflation model is called natural inflation <cit.>,which has been extensively studied in the literature.The potential ofEq. (<ref>), however, is not favored by the current data, andit would soon be excluded at a higher confidence level if the bound on rimproves with n_s staying at the current value; see Fig. <ref>. It is known since long ago, however, that the cosine potentialin Eq. (<ref>) is not correct in general, as arguedby Witten <cit.> in the large Nlimit <cit.> with the 't Hooft coupling λ≡ g^2 N held fixed. [See, e.g., Refs. <cit.> for related discussion in the context of inflation.] In particular, while the physics is periodic in ϕ with the periodof 2π f (because θ≡ϕ/f is the θ angle ofthe Yang-Mills theory), the multi-valued nature of the potential allowsfor the potential of ϕ in a single branch V(ϕ) = N^2 Λ^4 𝒱(λϕ/8π^2 N f), not to respect the periodicity under ϕ→ϕ + 2π f.Here, the combination x ≡λϕ/8π^2 N f, appearing in the argument of 𝒱(x) is determined by analyzingthe large N limit.This allows for building axionic models of inflationin which the range of the field excursion exceeds the decay constantf <cit.>.The potential of Eq. (<ref>) has an expansion of the form V(ϕ) = ∑_n=1^∞ b_2n(ϕ/F)^2n, where F ∝ f.The values of the coefficients b_2n—more preciselytheir signs and double ratios—are important for how the predictionsfor n_s and r change as F is varied.If the cosine potential inEq. (<ref>) were valid, then we would obtain sgn(b_2n) = (-1)^n-1, and b_6/b_4/b_4/b_2 = 2/5, b_8/b_6/b_6/b_4 = 15/28, ⋯, which lead to the curves labeled as “cosine” in Fig. <ref>.The correct values of the double ratios, however, are expected to bedifferent from these values.In fact, b_2n's obtained by latticegauge theory disfavor the cosine form of Eq. (<ref>)and are rather consistent with those expected from large Nexpansion <cit.>.While b_2n's may in principle be determined by lattice calculations,their errors are still large.Instead, we may infer the form of thepotential by the following arguments.First, invariance under the CPtransformation ϕ→ -ϕ implies that 𝒱(x)is a function of x^2, where we have absorbed a possible bare θparameter in the definition of ϕ.Second, 𝒱(x) is expectedto flatten as the potential energy approaches the point of the deconfiningphase transition with increasing |ϕ| (since the dynamics generatingthe potential will become weaker).Assuming that the potential is givenby a simple power law, we thus expect 𝒱(x) ∼ 1/(x^2)^p(p > 0).This potential is singular at x → 0, and a simpleway to regulate it is to replace x^2 with x^2 + const.Aftersetting the minimum of the potential to be zero, these considerations give V(x) = M^4 [ 1 - 1/( 1 + c x^2 )^p] (p > 0), where M ∼√(N)Λ, and c > 0 is a parameter of order unity.Here, we have used the well-established fact that the coefficient of x^2is positive when 𝒱(x) is expanded around x=0.We call themodel of inflation in which the axionic inflaton potential is generatedby a pure Yang-Mills theory (whose potential we expect to take the formof Eq. (<ref>)) pure natural inflation.As in the cosine potential, the potential of Eq. (<ref>) givessgn(b_2n) = (-1)^n-1.It, however, gives different valuesof the double ratios b_6/b_4/b_4/b_2 = 2(p+2)/3(p+1), ⋯, b_2n+4/b_2n+2/b_2n+2/b_2n= (n+1)(p+n+1)/(n+2)(p+n), ⋯. Therefore, predictions of this model are different from those of conventionalnatural inflation.(For example, by equating (b_6/b_4)/(b_4/b_2) we obtainp = -7/2 < 0.)Here, we have assumed that the effect of a transitionbetween different branches can be neglected, which we will argue to bethe case.The potential of Eq. (<ref>) can be obtained by a holographiccalculation <cit.>, which is applicable inthe limit of large N and 't Hooft coupling.In this calculation, ND4-branes in type IIA string theory are considered, with the D4-braneswrapping a circle.Below the Kaluza-Klein scale M_ KK for thecircle, the theory reduces to a 4d (non-supersymmetric) pure SU(N)Yang-Mills theory, with the dynamical scale Λ = M_ KKe^-24 π^2/11 λ, where λ is the 't Hooft coupling at M_ KK.Consideringthe backreaction to the geometry of the constant Wilson line of theRamond-Ramond one-form, which represents the θ angle of the gaugetheory, the potential of the form of Eq. (<ref>) is obtainedwith c=1 and p=3.Specifically, the potential of ϕ for a singlebranch is given by V(ϕ) = M^4 [ 1- 1/( 1 + (ϕ/F)^2 )^3], where [The 't Hooft coupling (and the gauge coupling squared) defined in Ref. <cit.> is a factor of 2 smaller than λ (g^2) here.] M^4 = λ N^2/3^7 π^2 M_ KK^4, F = 8π^2 N/λ f. The potentials for the other branches are obtained by replacing ϕ withϕ + 2π k f (k ∈ℤ); see Fig. <ref>. To illustrate the parameter region we consider, let us choose λ≃ 8π^2. Strictly speaking, the holographic calculation is not quite valid with thisvalue of the 't Hooft coupling—it requires a larger value of λ.However, we may expect, e.g. based on the success of the AdS/QCDprogram <cit.>, that this reasonablyapproximates the true dynamics of the 4d Yang-Mills theory.With thischoice of λ, we find M ≈√(N)Λ,F ≈ N f, as one naively expects from dimensional and N-scaling considerations.From the analysis of Ref. <cit.>, we expect that forsufficiently large N the effect of a transition between differentbranches is not important, unless ϕ becomes much larger than F(which does not occur in our analysis below).To reproduce the observedamplitude of the scalar perturbation, we will need to take M ∼ 10^16 GeV. The precise value depends on other parameters, e.g. F.The second expression in Eq. (<ref>) implies that for N ≫ 1the characteristic scale for the field excursion, F, can be much largerthan the axion decay constant f.This, however, does not mean that Fcan be much larger than the Planck scale M_ Pl≃ 1.22 ×10^19 GeV.In general, the decay constant f is expected to besmaller than the field theoretic cutoff (string) scale M_* f ≲ M_*. On the other hand, the Planck scale is related with M_* as M_ Pl^2∼ N^2 M_*^2 (see, e.g., Ref. <cit.>), so that N drops fromthe relation between F and M_ Pl: F ≲ O(M_ Pl). We argue that this is a desired feature.If F ≫ M_ Pl, theinflaton potential would be well approximated by the first, quadraticterm in Eq. (<ref>), which is excluded by the data.Becauseof Eq. (<ref>), however, we expect that higher terms inEq. (<ref>) are important.This makes the predictions ofthe model deviate from those of the quadratic potential V(ϕ) =m^2 ϕ^2/2.In Fig. <ref>, we plot the values of n_s and r predictedby the potential of Eq. (<ref>).For F ≫ M_ Pl,the predictions approach those of the quadratic potential, denoted by theblack dots for e-folding N_e = 50 (left) and 60 (right).As Fdecreases, however, they deviate from these values.In particular, thetensor-to-scalar ratio r decreases while n_s being (almost) kept, asrepresented by the lines indicated as “pure natural.”This is becausehigher terms in Eq. (<ref>) start contributing. [A special case of V(ϕ) ≈ A - B/ϕ^6 leading to n_s ≃ 0.965 and r ≃ 8 × 10^-4 was discussed in Ref. <cit.>.] In the figure, we have varied F from 100 M_ Pl to 0.1 M_ Pl,with the predictions for F/M_ Pl = 10, 5, 1 indicated by the reddots (from top to bottom).We find that the model gives the values ofn_s and r consistent with the data at the 95% (68%) CL forF/M_ Pl≲ 3.3 (0.7) for N_e = 50 and F/M_ Pl≲ 6.8 (4.4) for N_e = 60.Given Eq. (<ref>), thisis quite satisfactory. In Fig. <ref>, we plot the predictions arising from the potentialin which p in Eq. (<ref>) takes more general values p = 1, 2, 3,4, 10.We have parameterized the potentials as V(ϕ) = M^4 [ 1- 1/( 1 + (ϕ/F)^2 )^p], and, as in Fig. <ref>, varied F/M_ Pl in the rangebetween 100 and 0.1, with the dots representing F/M_ Pl =10, 5, 1 (from top to bottom).We find that the success of the model isrobust for a wide range of p ≈ 1 O(10): the predictedvalues of n_s and r agree well with the current data for F/M_ Pl≲ O(1).We thus conclude that pure natural inflation is consistentwith the data even if the true potential does not take exactly the form ofEq. (<ref>) as suggested by the holographic analysis.So far, we have focused on the case that the gauge group of the Yang-Millstheory is SU(N).However, our basic arguments, e.g. those aroundEq. (<ref>), do not depend on this specific choice.We thusexpect that similar predictions also result for other gauge groups,with N replaced by the dual Coxeter number of the group.Finally, we mention that a large value of N—despite the fact that itdoes not help to make F larger than M_ Pl—can make the decayconstant f smaller than F; see Eq. (<ref>).This allowsfor enhancing couplings of the inflaton ϕ to the standard modelgauge fields for fixed F, i.e. for a fixed inflation potential.Thisin turn allows for raising reheating temperature T_R.The reheatingtemperature is given by T_R ≃ 0.2 √(Γ_ϕ M_ Pl),where Γ_ϕ≈n m_ϕ^3/4096 π^5 f^2, is the inflaton decay width.Here,m_ϕ = √(2p)M^2/F≃10^13 GeV/F/M_ Pl, is the inflaton mass, and n is the final state multiplicity.(n = 12if ϕ decays only to the standard model gauge fields.)This yields T_R ∼ 10^9 GeV( N/10)( 0.5/F/M_ Pl)^5/2, The model can, therefore, be made consistent with thermal leptogenesis, whichrequires T_R ≳ 2 × 10^9 GeV <cit.>,even for F/M_ Pl≈ O(1).This work was supported in part by the WPI Research Center Initiative (MEXT,Japan).The work of Y.N. was also supported in part by the National ScienceFoundation under grants PHY-1521446, by MEXT KAKENHI Grant Number 15H05895,and by the Department of Energy (DOE), Office of Science, Office of High EnergyPhysics, under contract No. DE-AC02-05CH11231.The work of M.Y. was supportedby JSPS KAKENHI (15K17634) and JSPS-NRF Joint Research Project.99Ade:2015lrj P. A. R. Ade et al. [Planck Collaboration], “Planck 2015 results. XX. Constraints on inflation,” Astron. Astrophys.594, A20 (2016)[arXiv:1502.02114 [astro-ph.CO]]. Array:2015xqhP. A. R. Ade et al. [BICEP2 and Keck Array Collaborations], “Improved constraints on cosmology and foregrounds from BICEP2 and Keck Array cosmic microwave background data with inclusion of 95 GHz band,” Phys. Rev. Lett.116, 031302 (2016)[arXiv:1510.09217 [astro-ph.CO]]. Linde:1983gd A. D. Linde, “Chaotic inflation,” Phys. Lett.129B, 177 (1983).Freese:1990rb K. Freese, J. A. Frieman and A. V. Olinto, “Natural inflation with pseudo Nambu-Goldstone bosons,” Phys. Rev. Lett.65, 3233 (1990).Adams:1992bn F. C. Adams, J. R. Bond, K. Freese, J. A. Frieman and A. V. Olinto, “Natural inflation: particle physics models, power law spectra for large scale structure, and constraints from COBE,” Phys. Rev. D 47, 426 (1993)[hep-ph/9207245]. Witten:1979vv E. Witten, “Current algebra theorems for the U(1) Goldstone boson,” Nucl. Phys. B 156, 269 (1979).Witten:1980sp E. Witten, “Large N chiral dynamics,” Annals Phys. 128, 363 (1980).tHooft:1973alw G. 't Hooft, “A planar diagram theory for strong interactions,” Nucl. Phys. B 72, 461 (1974).Kaloper:2011jz N. Kaloper, A. Lawrence and L. Sorbo, “An ignoble approach to large field inflation,” JCAP 03, 023 (2011)[arXiv:1101.0026 [hep-th]]. Dubovsky:2011tu S. Dubovsky, A. Lawrence and M. M. Roberts, “Axion monodromy in a model of holographic gluodynamics,” JHEP 02, 053 (2012)[arXiv:1105.3740 [hep-th]]. Dine:2014hwa M. Dine, P. Draper and A. Monteux, “Monodromy inflation in SUSY QCD,” JHEP 07, 146 (2014)[arXiv:1405.0068 [hep-th]]. Yonekura:2014oja K. Yonekura, “Notes on natural inflation,” JCAP 10, 054 (2014)[arXiv:1405.0734 [hep-th]]. Kaloper:2016fbr N. Kaloper and A. Lawrence, “London equation for monodromy inflation,” Phys. Rev. D 95, 063526 (2017)[arXiv:1607.06105 [hep-th]]. Silverstein:2008sg E. Silverstein and A. Westphal, “Monodromy in the CMB: gravity waves and string inflation,” Phys. Rev. D 78, 106003 (2008)[arXiv:0803.3085 [hep-th]]. McAllister:2008hb L. McAllister, E. Silverstein and A. Westphal, “Gravity waves and linear inflation from axion monodromy,” Phys. Rev. D 82, 046003 (2010)[arXiv:0808.0706 [hep-th]]. Kaloper:2008fb N. Kaloper and L. Sorbo, “A natural framework for chaotic inflation,” Phys. Rev. Lett. 102, 121301 (2009)[arXiv:0811.1989 [hep-th]]. Giusti:2007tu L. Giusti, S. Petrarca and B. Taglienti, “Theta dependence of the vacuum energy in the SU(3) gauge theory from the lattice,” Phys. Rev. D 76, 094510 (2007)[arXiv:0705.2352 [hep-th]]. Witten:1998uka E. Witten, “Theta dependence in the large N limit of four-dimensional gauge theories,” Phys. Rev. Lett. 81, 2862 (1998)[hep-th/9807109]. Erlich:2005qh J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, “QCD and a holographic model of hadrons,” Phys. Rev. Lett. 95, 261602 (2005)[hep-ph/0501128]. DaRold:2005mxj L. Da Rold and A. Pomarol, “Chiral symmetry breaking from five dimensional spaces,” Nucl. Phys. B 721, 79 (2005)[hep-ph/0501218]. Dvali:2007hz G. Dvali, “Black holes and large N species solution to the hierarchy problem,” Fortsch. Phys. 58, 528 (2010)[arXiv:0706.2050 [hep-th]]. Buchmuller:2005eh W. Buchmüller, R. D. Peccei and T. Yanagida, “Leptogenesis as the origin of matter,” Ann. Rev. Nucl. Part. Sci.55, 311 (2005)[hep-ph/0502169]. | http://arxiv.org/abs/1706.08522v2 | {
"authors": [
"Yasunori Nomura",
"Taizan Watari",
"Masahito Yamazaki"
],
"categories": [
"hep-ph",
"astro-ph.CO",
"hep-th"
],
"primary_category": "hep-ph",
"published": "20170626180000",
"title": "Pure Natural Inflation"
} |
http://arxiv.org/abs/1706.08934v3 | {
"authors": [
"Carlo Ciliberto",
"Dimitris Stamos",
"Massimiliano Pontil"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170627164331",
"title": "Reexamining Low Rank Matrix Factorization for Trace Norm Regularization"
} |
|
Materials Science Factory, Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Cantoblanco, E-28049 Madrid (Spain). School of Physics and Centre for Quantum Computation and Communication Technology, The University of New South Wales, Sydney 2052, AustraliaSchool of Physics and Centre for Quantum Computation and Communication Technology, The University of New South Wales, Sydney 2052, AustraliaSchool of Physics and Centre for Quantum Computation and Communication Technology, The University of New South Wales, Sydney 2052, Australia Department of Physics, University at Buffalo, SUNY, Buffalo, New York 14260-1500, USASchool of Physics and Centre for Quantum Computation and Communication Technology, The University of New South Wales, Sydney 2052, AustraliaMaterials Science Factory, Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Cantoblanco, E-28049 Madrid (Spain).School of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies, UNSW Node, The University of New South Wales, Sydney 2052, Australia ABSTRACT: Full electrical control of quantum bits could enable fast, low-power, scalable quantum computation. Although electric dipoles are highly attractive to couple spin qubits electrically over long distances, mechanisms identified to control two-qubit couplings do not permit single-qubit operations while two-qubit couplings are off. Here we identify a mechanism to modulate electrical coupling of spin qubits that overcomes this drawback for hole spin qubits in acceptors,that is based on the electrical tuning of the direction of the spin-dependent electric dipole by a gate. In this way, inter-qubit coupling can be turned off electrically by tuning to a “magic angle” of vanishing electric dipole-dipole interactions, while retaining the ability to manipulate the individual qubits. This effect stems from the interplay of the T_d symmetry of the acceptor state in the Si lattice with the magnetic field orientation, and the spin-3/2 characteristic of hole systems. Magnetic field direction also allows to greatly suppress spin relaxation by phonons that limit single qubit performance, while retaining sweet spots where the qubits are insensitive to charge noise. Our findings can be directly applied to state-of-the-art acceptor based architectures, for which we propose suitable protocols to practically achieve full electrical tunability of entanglement and the realization of a decoherence-free subspace. KEYWORDS: Qubit, holes, electrical tuning, entanglement, sweet spot, decoherence free subspaceEntanglement control and magic angles for acceptor qubits in Si Dimitrie Culcer December 30, 2023 ===============================================================A scalable quantum computer architecture requires long qubit coherence times <cit.> and fast high-fidelity control of single-qubit and two-qubit operations. For solid-state spin qubits <cit.>, silicon offers improved spin lifetimes <cit.>, elimination of nuclear spin induced decoherence by isotope purification <cit.>, absence of spin relaxation by piezoelectric phonons <cit.>, and compatibility with Si microtechnology. <cit.> Coupling of spin qubits to electric fields has recently attracted much attention to improve qubit manipulation rates <cit.>, while electric fields are significantly easier to apply and localize than magnetic fields, and use much less power. <cit.> Coupling to electric fields also open new possibilities for two qubit gates mediated by electric fields<cit.> or microwave photons.<cit.>Electrical spin manipulation can be achieved with holes in Si <cit.> thanks to the intrinsically large spin orbit interaction (SOI) in the Si valence band. <cit.> Recently, an acceptor-based quantum information processing platform was introduced <cit.>, in which inversion symmetry breaking by the interface (Fig. <ref>(a)) gives rise to a Rashba interaction that couples the spin to in-plane electric fields, enabling fast electrical spin manipulation via electric dipole spin resonance (EDSR). Importantly, sensitivity to charge noise which produces dephasing is suppressed to first order. Yet, two limitations of the conventional electrically driven spin qubits remain: (i) T_1 controlled by phonons cannot be significantly enhanced without sacrificing qubit manipulation rates, and (ii) inter-qubit couplings based on electric dipole-dipole interactions can only be turned off by deactivating one of the qubits.Here we show that these limitations can be lifted by exploiting an unconventional interaction between the acceptor bound hole and the in-plane magnetic field direction that derives from the T_d symmetry of the acceptor in the Si lattice. The T_d term's strength and dependence on the top gate electric field allows the qubit's spin polarization to be controlled by the gate rather than the magnetic field, so that magnetic field orientation allows an unusually large tuning of qubit couplings to the environment. A decoherence free subspace (DFS) is possible where the qubit is essentially decoupled from phonons, while further improving previously identified insensitivity to charge noise.<cit.>Moreover, the top gate can be used to turn on and off the electric dipole-dipole coupling while allowing both qubits to be manipulated independently when the coupling is off.The underlying mechanism is a rapid electrical control of the charge dipole orientation of the individual qubits.We analyze this new two-qubit coupling mechanism and propose two protocols that allow fast, independent, fully electrical single-qubit and long-distance two-qubit manipulations that are compatible with long coherence times. The new two-qubit coupling mechanism and enhanced spin lifetimes predicted here greatly improve the prospects for a practical implementation of quantum information using extensively studied hole spin systems. <cit.> For hole spins bound to acceptor atoms many milestones have already been achieved experimentally, from the placement of acceptors near an interface <cit.>, to the measurement of single-acceptor states <cit.>, and coupling between two acceptors. <cit.> The results reported here also highlight the advantages of acceptors versus quantum dots as a suitable platform for hole spin qubits.Figure <ref>(a) shows the layered geometry in the vertical direction of the heterostructure for a single acceptor as a qubit, while Fig. <ref>(b) shows schematically the coupling between neighboring qubits in a 2D array under one of our proposed protocols.The theoretical description of a single acceptor relies on the Hamiltonian (see Supp. Mat. <cit.>)H=H_KL+H_BP+H_c+H_inter+H_F+H_B+H_ T_dwhich contains the details of the Si valence band via the Kohn-Luttinger Hamiltonian <cit.> H_KL including cubic symmetry terms, the strain Bir-Pikus term <cit.> H_BP, the acceptor Coulomb potential H_c, the (001) Si/SiO_2 interface H_inter, the interaction with electric field H_F, and magnetic field H_B = g_1 μ_B 𝐁·𝐉 + g_2 μ_B 𝐁·𝐉^3, where 𝐉 represent the spin-3/2 matrices. The projection of the spin-3/2 onto the axis perpendicular to the interface, ẑ, is m_J=± 3/2 for the heavy holes (HH) and m_J=± 1/2 for the light holes (LH). Tensile strain gives the qubit a LH character <cit.>, ensuring a strong Zeeman interaction with an in-plane magnetic field. <cit.>The tetrahedral symmetry of the acceptors gives rise to a linear coupling to the electric field of the form H_ T_d=p/√(3)({J_y,J_z}F_x+{J_x,J_z}F_y+{J_x,J_y}F_z), where the quadrupolar terms involving products of two spin matrices have no counterpart in spin-1/2 electron systems (in which they are either the identity or zero). Here p is an effective dipole moment that can be calculated <cit.> by p=e∫_0^a f^*(r)r f(r) with a the lattice constant of the host material, and f(r) the radial bound hole envelope function. For a boron acceptor in Si p=0.26 Debye.This value is larger in deep acceptors, with a smaller Bohr radius. <cit.> H_ T_d is also linearly proportional to the strength of the electric field.We neglect other allowed T_d symmetric terms as their coupling constants are much smaller. <cit.>We use the effective mass approach described in Ref. AbadilloUrielNanotech2016, however in this work we consider a magnetic field 𝐁 with an arbitrary in-plane orientation characterised by an angle ϕ. In the basis {3/2,1/2,-1/2,-3/2}, the effective Hamiltonian is <cit.>H_ eff=[0 √(3)/2ε_Ze^-iϕ -ipF_z0;√(3)/2ε_Ze^iϕ Δ_HL ε_Ze^-iϕ -ipF_z;ipF_zε_Ze^iϕ Δ_HL √(3)/2ε_Ze^-iϕ;0ipF_z√(3)/2ε_Ze^iϕ0 ],where ϕ=0 for 𝐁 along the x-direction, given by one of the main crystal axes. The cubic g-factor g_2 (ref KopfPRL1992) is not explicitly shown in Eq. <ref>, but is included in the numerical calculations. The Zeeman term is ε_Z=g_1 μ_B B. The HH energy is set to zero for F_z=0, and Δ_HL is the HH-LH energy difference. The qubit is defined by the two levels making up the spin-split ground state. At zero fields, the strain conditions are such that the lower (qubit) branch is of LH character (Δ_HL<0) while the upper branch is of HH character. These LH and HH branches interact and they anticross at a particular value of the vertical field F_z (ref SalfiPRL2016).As Fig. <ref>(a) shows, the local tetrahedral symmetry of the acceptor makes a clear distinction between the main crystal axes and any other direction. It is represented by the term H_ T_d in the Hamiltonian, which governs the qubit interaction with electric fields, and becomes more pronounced as the top gate voltage is increased, generating a mixing between HH and LH in the two branches. The interplay between the terms with tetrahedral symmetry H_ T_d and the usual Zeeman interaction H_B gives rise to a new and counterintuitive magnetic field orientation dependence of the qubit properties with no analog for spin-1/2 electrons. Fig. <ref> shows the strong dependence of the qubit frequency on the magnetic field orientation.Adjusting the in-plane magnetic field direction influences the properties of previously identified sweet spots where the derivative of the qubit energy vanishes as a function of electric field. At ϕ=0 one sweet spot is located at F_z=0 while the other resides at a finite value of F_z (18.1 MV/m in Fig. <ref>(a) for d=4.6 nm), which depends on the acceptor depth. The sweet spot at F_z=0 moves in F_z as a function of ϕ and is hence called the anisotropic sweet spot. On the other hand, the large field sweet spot, where LH and HH levels in the qubit mix with probability amplitudes a_L=1/2 and a_H=-√(3)/2 respectively, remains at the same value of F_z=18.1 MV/m, and is thus called the isotropic sweet spot. We find that for the particular values ϕ=1/2arcsin(1/3)+nπ, and ϕ=1/2(π-arcsin(1/3))+nπ, the isotropic and anisotropic sweet spots fuse into a single sweet spot that makes the qubit insensitive to charge noise up to second order, see Fig <ref>(a). In this case the energy dispersion is flat at the sweet spot within a 2-3 MV/m window.The spin lifetime enhancement and charge dipole orientation mechanism for controlling two-qubit coupling rely on gate voltage control of the qubit's spin polarization and appropriate choice of the magnetic field relative to the polarization.Since spin polarization of a conventional spin 1/2 qubit only depends on the magnetic field, we first explain the mechanism to electrically vary spin polarization, and how the orientation of the effective spin polarization<cit.> relative to magnetic field influences single qubit properties. We note that in our acceptor qubit, the HH and LH levels have different projections of total angular momenta.Consequently, adjusting the LH-HH hybridization using the gate electric field F_z influences the spin polarization.The orientation of the magnetic field with respect to this spin polarization influences qubit dynamics including coupling to e.g. phonons by changing the magnetic coupling ε_Zo to the excited branch (see Fig.<ref>(a)).For example, when the magnetic field is colinear to the spin polarization, the latter becomes a constant of the motion and the following holds [H_ inter+H_ T_d,H_B] = 0.Here, the qubit completely decouples from the upper branch, and environmental effects on qubit dynamics are suppressed in a decoherence free subspace (DFS). This DFS differs from conventional ones, which arise from symmetries of the encoded states.<cit.> Conversely, when the magnetic field points perpendicularly to the effective spin polarization of a particular branch the hole can no longer sense the magnetic field and the effective g-factor becomes zero, as shown in Fig. <ref>(b).Given that magnetic field orientation can suppress magnetic coupling ε_Zo to the upper branch, and that acoustic phonons mix the qubit and upper branches, magnetic field orientation suppresses the spin relaxation due to acoustic phonons, see Fig.<ref>(b). The induced relaxation is proportional to the coupling ε_Zo and inversely proportional to the qubit-upper branch energy difference Δ (ref SalfiPRL2016): 1/T_1∝ E_ Larmor(ϕ)^3[ε_Zo(ϕ)/Δ]^2. For an acceptor at 4.6 nm from the interface at the isotropic sweet spot, with B=0.5 T pointing along one of the crystal axes, T_1 ≈ 20 μs. While T_1 can be enhanced trivially by suppressing the qubit frequency ħω, it is more interesting to enhance T_1 by suppressing the coupling ε_Zo, which can be quite effective since the 4×4 manifold is very well isolated from higher excited states (by ∼ 20 meV).Since the qubit coupling to in-plane electric fields providing single qubit gates by EDSR is D∝ (ε_Zo(ϕ)/2Δ), the qubit operations are slowed down. However, D decreases with ε_Zo while T_1 is enhanced much faster with 1/ε_Zo^2 (see suppl-material), so the number of single-qubit operations per qubit lifetime r is enhanced even if the operations are slowed down.<cit.> For instance, when ϕ=0, r=10^5, and it can be enhanced to 10^6 for ϕ=30^∘ or to 10^7 for ϕ=40^∘.For two-qubit operations, we consider electric dipole-dipole interaction V_dd = (𝐯_1 ·𝐯_2R^2 - 3(𝐯_1 ·𝐑)(𝐯_2 ·𝐑))/4πϵ R^5, where 𝐯_i is the spin-dependent charge dipole of qubit i (refs FlindtPRL2006, GolovachPRB2006). We note that the orientation of this spin-dependent charge dipole is affected by the top gate since, as discussed before, the effective spin-polarization is modified by the vertical electric field, see Fig. <ref>(b). For two acceptor qubits a and b separated in the acceptor z-plane by a distance R and relative orientation (cosθ_E,sinθ_E), this electric dipole-dipole interaction works in the two-qubit subspace as H_dd=J_dd(σ^a_+ +σ^a_-)(σ^b_+ + σ^b_-). The dipole-dipole coupling J_dd is inversely proportional to the qubit-upper branch energy separation Δ, and it is directly proportional to the qubit-upper branch couplings ε_Zo of each qubit, and their Rashba couplings α, such thatJ_dd=α^aα^bε_Zo^aε_Zo^b/8πϵ R^3 Δ ^a Δ ^bG(F_z^a,F_z^b,ϕ,θ_E),with G(F_z^a,F_z^b,ϕ,θ_E) a modulating function, related to the spin-dependent charge dipoles, that depends on the operating point of each qubit, the magnetic field orientation ϕ, and θ_E. As discussed previously <cit.>, adiabatically tuning the gate to turn on/off an electric dipole 𝐯_i allows to modulate the two-qubit coupling<cit.>, but this requires the deactiviation of one of the qubits. Introducing the new degree of freedom ϕ, it is possible to have J_dd vanish for particular parameter choices, independently of the modulus of 𝐯_i.Since J_dd decays with ε_Zo^2, same as 1/T_1, the number of two-qubit operations per qubit lifetime is constant for any ϕ, provided that phonons limit the qubit coherence as expected at the sweet spot. Hence, the coupling J_dd can be suppressed by choosing parameters such that G(F_z^a,F_z^b,ϕ,θ_E)=0 (see purple areas in Fig. <ref>), without deactivating single qubit operations. This condition is the cornerstone of our protocols and generalizes the concept of magic angles<cit.> where dipole-dipole interactions vanish.A conventional magic angle is the angle subtending the magnetic field and relative qubit position, when magnetic dipole-dipole interactions vanish.Its main drawback is that magnetic field cannot be rapidly swept.In our case, the magic angle describes the direction of the qubit's spin-dependent electric dipole relative to the qubit positions where spin-dependent electric dipole-dipole interactions vanish. Our protocols are defined at fixed values of ϕ. We consider three different situations: (i) both qubits at their isotropic sweet spots; (ii) one qubit is at the isotropic sweet spot and the other at the anisotropic one; (iii) both qubits at their anisotropic sweet spots. Note that gate voltages alone can switch between these cases.In case (i) the dipolar coupling is perfectly isotropic G(F_z^a,F_z^b,ϕ,θ_E)=1, hence neighboring qubits at isotropic sweet spots will always be coupled. In case (ii), see Fig. <ref>(a), we distinguish two interesting orientations in the first quadrant: ϕ=15^∘ and 75^∘. These two cases have opposite behavior in the x and y directions: with ϕ=15^∘ the couplings in the x (y)direction is minimized (maximized), whileϕ=75^∘ works in the opposite way. In case (iii), see Fig. <ref>(b), a similar behavior appears for ϕ=12^∘, 78^∘ and also for ϕ=40^∘ and 50^∘.We propose two possible protocols for fully tuning two-qubit coupling electrically in a rectangular array of acceptors arranged in the x and y directions of the Si lattice, see Figures <ref>(c) and (d). Due to the opposite behavior of the coupling in Eq. <ref> for these directions, both protocols require a much larger spacing between acceptors in the direction where J_dd is not suppressed, where qubit coupling would be performed by cQED <cit.>, while faster dipolar interactions are performed in the perpendicular direction, where they can be switched on and off. Whenever dipolar coupling is switched off, single qubit operations are performed. This general idea can be achieved by fixing the in-plane magnetic field orientation to a particular magic angle. Protocol 1: choose ϕ=12^∘ or ϕ=40^∘ (ϕ=78^∘ or ϕ=50^∘) and locate acceptors in a close range (≈ 20 nm apart for 10^4 two-qubit operations per qubit lifetime) in the x (y) direction and a longer separation in the y (x) direction. Sweeping the local gates such that two neighboring qubits in the x (y) direction are taken to their anisotropic sweet spots (case (iii)), the coupling J_dd is suppressed in that particular direction and each qubit can be addressed individually. Then, taking adiabatically both qubits to the isotropic sweet spot, the coupling is reactivated. In the perpendicular direction, J_dd cannot be turned off at this particular ϕ so acceptors need to be more separated and entanglement between any pair of qubits is performed via cQED. <cit.> Protocol 2: in a similar 2D array, choose ϕ=15^∘(ϕ=75^∘).Sweeping the local gates such that two neighboring qubits in the x (y) direction are taken to the isotropic-anisotropic sweet spot combination (case (ii)), the coupling J_dd is suppressed in that particular direction and each qubit can be addressed individually. The qubit in the anisotropic sweet spot is adiabatically swept to the isotropic sweet spot, reactivating J_dd. cQED is performed in the y (x) direction.Both protocols require adiabatically sweeping the vertical electric field to move between the two sweet spots. During the adiabatic sweep the qubit frequency changes making qubits momentarily susceptible to charge noise. The potential for decoherence allows to differentiate the two protocols.Angles ϕ=40^∘ and ϕ=50^∘ in Protocol 1 imply a magnetic field orientation very close to the DFS of the anisotropic sweet spot, which means single-qubit operations per qubit lifetime are extremely enhanced. However, the exposure to charge noise in the adiabatic sweeping is higher than in protocol 2 due to the difference in Larmor frequency between sweet spots. Angles ϕ=12^∘ and ϕ=78^∘ in Protocol 1 and the angles used in Protocol 2 are not close to the DFS, hence the single-qubit operations per time are not particularly enhanced, though T_1 is still enhanced with respect to the ϕ=0 case. Exposure to decoherence during the adiabatic sweep is strongly minimized since the value of ϕ is close to the one that merges the isotropic and anisotropic sweet spots (they are barely separated by ≈ 1.5 MV/m).Conventionally, there are two categories of approaches to turn on and off a two-qubit gate: One type achieves tuning of qubit interaction by the reduction of single-qubit dipoles <cit.>, which means single-qubit operations would also slow down.Alternatively, two-qubit gates can be effectively halted by detuning the frequencies of the two qubits <cit.>, however in this approach two-qubit dynamics is only suppressed, not turned off.Since in both our protocols two-qubit couplings can be totally suppressed, our proposal here is clearly a more efficient alternative in precisely controlling two-qubit operations.We also note that both protocols are robust against acceptor placement imprecision. As an example, for an accuracy of± 5 nm in-plane position, and inter-acceptor distance of 20 nm, the angle variation would be of ± 15^∘. Under these circumstances, J_dd, although non-zero, would be significantly suppressed (see Fig. <ref>). In summary, we have found an unexpected magnetic field orientation dependence of all the parameters involved in one and two-qubit operations with acceptors in Si. This dependence can be used to develop tailored protocols at fixed in-plane magnetic field orientations (magic angles) that allow to perform one and two-qubit operations independently, by purely electrical means, while maintaining each qubit in sweet spots where charge noise effects are suppressed. The proposed protocols are well within reach using state-of-the-art technology, paving the way for a full electrically controlled Si-based quantum computer implementation. ACKNOWLEDGMENTSJCAU and MJC acknowledge funding from Ministerio de Economía, Industria y Competitividad (Spain) via Grants No FIS2012-33521 and FIS2015-64654-P and from CSIC (Spain) via grant No 201660I031. JCAU thanks the support from grants BES-2013-065888 and EEBB-I-16-11046. DC and SR are supported by the Australian Research Council Centres of Excellence FLEET (CE170100039) and CQC2T (CE110001027), respectively.XH thanks support by US ARO through grant W911NF1210609, and Gordon Godfrey Fellowship from UNSW School of Physics. JS acknowledges financial support from an ARC DECRA fellowship (DE160101490).SUPPLEMENTARY INFORMATION§ KOHN-LUTTINGER AND BIR-PIKUS HAMILTONIANSThe Kohn-Luttinger Hamiltonian for the valence band of semiconductors <cit.>, including the Coulomb impurity, is H_ KL= [ P+Q L M 0 i/√(2)L -i√(2)M; L^* P-Q 0 M -i√(2)Qi√(3/2)L; M^* 0 P-Q-L -i√(3/2)L^* -i√(2)Q; 0 M^*-L^* P+Q -i√(2)M^*-i/√(2)L^*; -i√(2)L^*i√(2)Qi√(3/2)Li√(2)MP+Δ_SO 0;i√(2)M^* -i√(3/2)L^*i√(2)Qi√(2)L 0P+Δ_SO ]We can define the effective Rydberg unit as Ry^*=e^4m_0/2ħ^2ϵ_s^2γ_1 and the effective Bohr radius as a^*=ħ^2ϵ_sγ_1/e^2m_0. In these units the differential operators in Eq. (<ref>) areP = -k^2+2/rQ = -γ_2/γ_1(k_x^2+k_y^2-2k_z^2) L = i2√(3)γ_3/γ_1(k_x-ik_y)k_z M = -√(3)γ_2/γ_1(k_x^2-k_y^2)+i2√(3)γ_3/γ_1k_x k_y, with m_0 the free electron mass, ϵ_s the semiconductor static dielectric constant, andγ_1, γ_2 and γ_3 material dependent Luttinger parameters. The interaction with strain is given by the Bir-Pikus Hamiltonian <cit.>:H_ BP = aϵ1+ b((J_x^2-5/41)ϵ_xx+(J_y^2-5/41)ϵ_yy+ (J_z^2-5/41)ϵ_zz) + d/√(3)({J_x,J_y}ϵ_xy +{J_y,J_z}ϵ_yz+{J_x,J_z}ϵ_xz)The parameters a,b and d are the deformation potentials of the host material, and ϵ_ij are the deformation tensor components.§ EFFECTIVE 4× 4 HAMILTONIANThe results of the diagonalization of the total Hamiltonian are mapped onto an effective Hamiltonian for the four lowest states. For any magnetic field orientation ϕ the Hamiltonian in the |m_J⟩ basis is <cit.>:H_ eff=[0 √(3)/2ε_Ze^-iϕ -ipF_z0;√(3)/2ε_Ze^iϕ Δ_HL ε_Ze^-iϕ -ipF_z;ipF_zε_Ze^iϕ Δ_HL √(3)/2ε_Ze^-iϕ;0ipF_z√(3)/2ε_Ze^iϕ0 ] .For the qubit Hamiltonian, we define E_l=1/2(Δ_HL-√(Δ_HL^2+4p^2F_z^2)), E_u=1/2(Δ_HL+√(Δ_HL^2+4p^2F_z^2)), a_L=E_l/√(E_l^2+p^2F_z^2) and a_H=pF_z/√(E_l^2+p^2F_z^2). After the transformations shown in Ref. <cit.> the qubit Hamiltonian is: H_ qubit=[ E_l-1/2ε_Zl 0 Z_1 Z_2; 0 E_l+1/2ε_Zl Z_2 Z_1; Z_1-Z_2 E_u-1/2ε_Zu 0;-Z_2 Z_1 0 E_u-1/2ε_Zu ]being Z_1 = 1/2ε_Zocos(θ_l/2-θ_u/2-θ_o) Z_2 = i/2ε_Zosin(θ_l/2-θ_u/2-θ_o).Where the first two states correspond to the qubit branch and the last two to the upper branch. The different relevant couplings and the associated phases are defined as followsθ_l = arctan(a_L^2cos(ϕ)+√(3)a_La_Hsin(ϕ),-a_L^2sin(ϕ)-√(3)a_La_Hcos(ϕ))θ_u = arctan(-a_H^2cos(ϕ)+√(3)a_La_Hsin(ϕ),-a_H^2sin(ϕ)+√(3)a_La_Hcos(ϕ))θ_o = arctan(-a_La_Hsin(ϕ)+√(3)/2(a_L^2-a_H^2)cos(ϕ),-a_La_Hcos(ϕ)+√(3)/2(a_L^2-a_H^2)sin(ϕ))ε_Zl = 2ε_Z√(3a_L^2a_H^2+a_L^4+2√(3)a_L^3a_Hsin(2ϕ)) ε_Zu = 2ε_Z√(3a_L^2a_H^2+a_H^4-2√(3)a_H^3a_Lsin(2ϕ)) ε_Zo = 2ε_Z√(3(a_L^2-a_H^2)/4+a_H^2a_L^2+√(3)a_Ha_L(a_H^2-a_L^2)sin(2ϕ))Physically, ε_Zl and ε_Zu are the qubit and upper Zeeman splittings while ε_Zo is the qubit-upper branch coupling. The phases associated to these couplings θ_l, θ_u, θ_o can be related to an effective spin polarization of each Kramer doublet and the interacting term respectively.§ SWEET SPOTSThe value of the qubit Larmor frequency is given, to first order, by ε_Zl. Its value depends explicitly on both the electric field and magnetic field magnitudes, butalso depends on the magnetic field orientation. We can look for sweet spots by simply finding the solutions to dε_Zl/dF=0:dε_Zl/dF_z=ε_Z(-3+4a_L^2)(a_H+√(3)a_Lsin(2ϕ))a'_L(F_z)/a_H√(3-2a_L^2+2√(3)a_La_Hsin(2ϕ))=0.One of the solutions corresponds to the fixed sweet spot a_L=-√(3)/2 (a_H=1/2). Considering that a'_L∝ a_H, the other solution is equivalent to solve a_H+√(3)a_Lsin(2ϕ)=0. Considering positive electric fields a_H≥ 0 and a_L≤ 0, meaning that for 0≤ϕ≤π/2 and π≤ϕ≤ 3π/2 there is a ϕ dependent sweet spot solution.Particularly, at ϕ=π/4+nπ this corresponds to a_H=√(3)/2 and a_L=-1/2 while for a magnetic field aligned with the main axes of the crystal, this sweet spot corresponds to the value F_z=0.Also, when ϕ=π/4+nπ the value of the Larmor frequency at the isotropic sweet spot is zero. At this point there is an inversion of the effective spin polarization of the lower branch, implying that an effective g-factor flip occurs in this particular case.§ DECOHERENCE FREE SUBSPACE (DFS)As the qubit interacts with the upper branch states through the Zeeman interaction, we can see this subspace as a leakage submanifold. The interaction terms between the qubit states and the leakage states are then given by the off-diagonal Zeeman terms in the qubit Hamiltonian. A decoherence and relaxation free subspace would be a subspace in which the Zeeman interaction is purely diagonal in the qubit basis, or equivalently [H_T_d+H_interface,H_B]=0. In this subspace, all the interactions would be suppressed to first order. To find this subspace we use the following elements that form part of a bigger basis of the space of spin 3/2:e_1 = 1/2e_2 = 1/6(2J_z^2-J_x^2-J_y^2) e_3 = 1/√(5)J_x e_4 = 1/√(5)J_y e_5 = 1/√(12){J_x,J_y}e_6 = 1/√(12){J_y,J_z}e_7 = 1/√(12){J_z,J_x}In this basis the elements of the effective Hamiltonian areH_inter = Δ_HL(e_1-e_2) H_T_d = 2pF_ze_5 H_B = √(5)(B_xe_3+B_ye_4)The commutators are then[H_inter,H_B] =i2√(3)Δ_HL(B_ye_7-B_xe_6)[H_T_d,H_B] = -4ipF_z(B_xe_7-B_ye_6)The total commutator is[H_inter+H_T_d,H_B] = e_7(-4ipF_zB_x+i2√(3)Δ_HLB_y) + e_6 (4ipF_zB_y-i2√(3)Δ_HLB_x)As e_7 and e_6 are different elements of the basis, [H_inter+H_T_d,H_B]= 0 is equivalent to solve the following systemΔ_HLB_y-2pF_z/√(3)B_x=0 Δ_HLB_x-2pF_z/√(3)B_y=0The system has non-trivial solutions if and only if B_x=± B_y which corresponds to the orientations ±π/4+nπ. These are the most symmetric directions as the main axes of the crystal remain indistinguishable.Case B_x=-B_y, the solution requires Δ_HL=-2pF_z/√(3) which corresponds to the isotropic sweet spot. The case B_x=B_y requires Δ_HL=2pF_z/√(3) corresponding to the anisotropic sweet spot. In these two cases the Zeeman interaction can be diagonalized simultaneously with the interface and T_d symmetry terms, so the qubit becomes isolated from the upper branch.It is also interesting to express the different contributions in terms of the spherical tensors J_+ and J_-H_T_d+H_inter = -3/41-i/8(Δ_HL+2pF_z/√(3))(J_++iJ_-)^2 + i/8(Δ_HL-2pF_z/√(3))(J_+-iJ_-)^2 H_B = 1+i/4(B_x-B_y)(J_+-iJ_-)+ 1-i/4(B_x+B_y)(J_++iJ_-)From here, it can be seen how for the sweet spots, and for particular values of the magnetic fields, the non-magnetic and the magnetic terms share eigenvectors.§ EFFECTIVE SPIN POLARIZATION INDUCED BY THE SPIN-ORBIT INTERACTIONThe qubit states in the |m_J⟩ basis are:|l±⟩ =a_L|± 1/2⟩∓ ia_H|∓ 3/2⟩Transforming the m_J basis into the |l_z,s_z⟩ basis using the Clebsch-Gordan coefficients|l±⟩ =a_L/√(3)(|± 1, ∓ 1/2⟩+√(2)|0, ± 1/2⟩) ∓ ia_H|∓ 1, ∓ 1/2⟩To get the effective spin polarization of this branch we can compute the matrix of expected values for the in-plane spin operators ⟨ l'|σ_i|l⟩:⟨ l'|σ_x|l⟩= a_L/3[0 a_L+i√(3)a_H; a_L-i√(3)a_H0 ] ⟨ l'|σ_y|l⟩= a_L/3[ 0ia_L+√(3)a_H; -ia_L+√(3)a_H 0 ]At the isotropic sweet spot a_L=-√(3)/2 and a_H=1/2, making ⟨ l'|σ_x|l⟩=-⟨ l'|σ_y|l⟩, and at the same time the expected value at the direction ϕ=-π/4 becomes saturated to the maximum possible value of spin projection1/2 which is equivalent to an effective spin polarization in the ϕ=-π/4±π direction. The change of effective spin-polarization in the qubit branch by rotating the magnetic field can be seen in the supplemental file gs_spin_ polarization.mov. The other interesting possibility is a_L=-1/2 and a_H=√(3)/2 which corresponds to the anisotropic sweet spot in the particular case ϕ=π/4. In this case the upper states have the composition a_L=-√(3)/2and a_H=1/2 so ⟨ u'|σ_x|u⟩=⟨ u'|σ_y|u⟩ indicating an effective spin polarization of the upper branch for ϕ=π/4±π.§ PHONON-INDUCED SPIN RELAXATIONAt low temperatures the expression for the relaxation times of the acceptor qubit is1/T_1=(ħω)^3/20ħ^4 πρ[ ∑_i |⟨ -|D_ii|+⟩ |^2(2/v_l^5+4/3v_t^5)+ ∑_i≠ j |⟨ -|D_ij|+ ⟩ |^2(2/3v_l^5+1/v_t^5)]Going to second order in the Schrieffer-Wolff transformation <cit.> we get⟨ -|D_ij|+ ⟩ = 1/E_l-E_u(Ĥ'_-,u-Ĥ'_u-,++Ĥ'_-,u+Ĥ'_u+,+)Where Ĥ'=Ĥ'_Z+Ĥ'_ph. The elements of the Hamiltonian Ĥ'_ph areD_ii = b'(J_i^2-5/4) D_ij = 2d'/√(3){J_i,J_j}i≠ jThe values of |⟨ -|D_ij|+ ⟩|^2 at the isotropic sweet spot are independent of the angle ϕ except for the intrinsic dependence of ε_Zo. These values are |⟨ -|D_xx|+ ⟩|^2 = |⟨ -|D_yy|+ ⟩|^2=3b^2ε_Zo^2/64Δ^2|⟨ -|D_zz|+ ⟩|^2 = 3b^2ε_Zo^2/16Δ^2 |⟨ -|D_xy|+ ⟩|^2 = d^2ε_Zo^2/16Δ^2 |⟨ -|D_xz|+ ⟩|^2 = |⟨ -|D_yz|+ ⟩|^2=d^2ε_Zo^2/8Δ^2 In the case of the anisotropic sweet spot the values of |⟨ -|D_ij|+ ⟩|^2 are |⟨ -|D_xx|+ ⟩|^2 = |⟨ -|D_yy|+ ⟩|^2=3b^2ε_Zo^2/64Δ^2|⟨ -|D_zz|+ ⟩|^2 = 3b^2ε_Zo^2/16Δ^2|⟨ -|D_xy|+ ⟩|^2 = d^2ε_Zo^2/16Δ^2 |⟨ -|D_xz|+ ⟩|^2 = d^2ε_Zo^2cos^2θ_o/4Δ^2|⟨ -|D_yz|+ ⟩|^2 = d^2ε_Zo^2sin^2θ_o/4Δ^2Collecting terms, the new dependence on the phase θ_o cancels out so we arrive to the same formula except for different definitions of ε_Zo and the energy difference Δ=E_l-E_u.The general formula for phonon-induced relaxation at the sweet spots is then1/T_1=(ħω(ϕ))^3/20ħ^4 πρ(ε_Zo(ϕ)/E_l-E_u)^2 [ 3b'/32(2/v_l^5+4/3v_t^5)+5d'/48(2/3v_l^5+1/v_t^5)] § SINGLE QUBIT OPERATIONSThe lack of inversion symmetry allows the action of an in-plane linear Stark effect H_E=e(E_xx+E_yy)=eE_∥(cosθ_∥+sinθ_∥), see Ref. <cit.>. The effect of this interaction of the acceptor hole is calculated considering several excited states and mapped onto the effective 4× 4 Hamiltonian. In the qubit basis it becomesĤ_E=[ 0 0E_1RE_2R; 0 0E_2RE_1R; -E_1RE_2R 0 0;E_2R -E_1R 0 0 ]beingE_1R = iα E_∥sin(θ_∥+θ) E_2R = -α E_∥cos(θ_∥+θ)Where θ=θ_u/2-θ_l/2. Applying a second order SW transformation <cit.> we get that the qubit EDSR term isD=αε_Zo(ϕ)/E_l-E_ucos(θ_o-θ_∥).With this coupling and the limiting factor T_1 we can calculate the number of π rotations per qubit lifetime, see Fig. <ref>. The number of qubit operations diverges when T_1 tends to infinity, since T_1∝ 1/ε_Zo^2 while D∝ε_Zo.§ TWO QUBIT COUPLINGConsider two acceptors separated by a distance 𝐑. Due to the spin-orbit interaction and the electric dipole moment of each acceptor, a spin-dependent dipolar interaction is expected between the two acceptors. Let the subspace of the two qubits be {|l-^a,l-^b⟩, |l-^a,l+^b⟩, |l+^a,l-^b⟩, |l+^a,l+^b⟩}. The total Hamiltonian is H^Σ=H_op^a+H_op^2+V^12, where H_op are the single acceptor Hamiltonians and V^12 is the Hamiltonian of the electrostatic interaction given by V^12(𝐫_1-𝐫_2)=e^2/4πϵ|𝐫_1-𝐫_2|.Here we assume that each qubit may have different energies and applied fields. The single qubit Hamiltonians are⟨ m|H_op^i|m'⟩ = ( [ -ε_l^i/20Z_1^ii Z_2^i;0ε_l^i/2i Z_2^iZ_1^i;Z_1^i -i Z_2^i Δ ^i-ε_u^i/20; -i Z_2^iZ_1^i0 ε_u^i/2+Δ ^i;])where the superindex i indicates acceptor a or b.When the two acceptors are far enough we can use the multi-pole expansion for the Coulomb interaction:⟨ mn|V^12|m'n'⟩ = R^2⟨ m|e𝐫'_1|m'⟩·⟨ n|e𝐫'_2|n'⟩-3(⟨ m|e𝐫'_1|m'⟩·𝐑)(⟨ n|e𝐫'_2|n'⟩·𝐑)4πϵ R^5Being 𝐫'_i=𝐫_i-𝐑_i the hole coordinate relative to the ion, and assuming an arbitrary relative position in the xy plane 𝐑=Rcos(θ_E)x̂+Rsin(θ_E)ŷ, so⟨ mn|V^12|m'n'⟩ = (1-3cos^2θ_E)⟨ m|e x'_1|m'⟩·⟨ n|e x'_2|n'⟩+(1-3sin^2θ_E)⟨ m|e y'_1|m'⟩·⟨ n|e y'_2|n'⟩+⟨ m|e z'_1|m'⟩·⟨ n|e z'_2|n'⟩4πϵ R^3The dipole matrix elements relevant for the Coulomb interaction are⟨ m|e(x',y')^i|m'⟩ = ( [00 i q_1x,y^i q_2x,y^i;00 q_2x,y^iiq_1x,y^i; -iq_1x,y^i q_2x,y^i00; q_2x,y^i -iq_1x,y^i00;])where q_1x^i=α^isinθ^i, q_2x^i=-α^icosθ^i, q_1y^i=α^icosθ^i, and q_2y^i=α^isinθ^i (assuming the approximation α≫ p valid for both qubits).We can project the interactions into the 4× 4 subspace using a SW transformation. Working out the second order correction H^(2) we get a spin-independent shiftH^(2)=-I[((q_1^a)^2+(q_2^a)^2)((q_1^b)^2+(q_2^b)^2)/Δ ^a+Δ^b+(Z_1^a)^2+(Z_2^b)^2/Δ^a+(Z_1^b)^2+(Z_2^b)^2/Δ ^b]The third-order correction is H^(3)_mm'=-1/2∑_l,m”[H'_mlH'_lm”H'_m”m/(E_m'-E_l)(E_m”-E_l)+H'_mm”H'_m”lH'_lm'/(E_m-E_l)(E_m”-E_l)] +1/2∑_l,l'H'_ml'H'_ll'H'_l'm'[1/(E_m-E_l)(E_m-E_l')+1/(E_m'-E_l)(E_m'-E_l)].The result of the third order correction to zeroth order in ε_i isH^(3)=J_dd[ 0 0 0 1; 0 0 1 0; 0 1 0 0; 1 0 0 0 ]whereJ_dd=4 (q_2^a Z_1^a+q_1^a Z_2^a)(q_2^b Z_1^b+q_1^b Z_2^b)/Δ ^a Δ ^bwhich is an Ising type spin-spin interaction H^(3)=J_dd(σ_a++σ_a-)(σ_b++σ_b-).After adding all the contributions and substituting, the value of J_dd is:J_dd=α^a α^b ε_Zo^a ε_Zo^b (sinθ_o^a sinθ_o^b (1-3 sin^2 θ_E)+cosθ_o^a cosθ_o^b (1-3 cos^2 θ_E))/4 πϵR^3 (E_l^a-E_u^a) (E_l^b-E_u^b)This expression can be substituted for each caseCase (i): We getJ_dd=3/2ε_Z^2(1+sin(2ϕ))α^a α^b/8 πϵR^3 (E_l^a-E_u^a).(E_l^2-E_u^2) Case (ii): Omitting the complex dependence in the dynamic sweet spot of θ_o on ϕJ_dd=3ε_Z^2√(1+sin(2ϕ))α^a α^b |cos(2ϕ)|^2(3cos(3ϕ-θ_o^a)-cos(ϕ+θ_o^a)-3cos(2θ_E)(cos(ϕ-θ_o^a)-3cos(3ϕ+θ_o^a))]/16 πϵR^3 (E_l^a-E_u^a)(E_l^b-E_u^b)√(7+4cos 4ϕ-3cos 8ϕ) . Case (iii):J_dd=3ε_Z^2α^a α^b cos^2(2ϕ)[(1-3cos^2θ_E)cosθ_o^acosθ_o^b+(1-3sin^2θ_E)]sinθ_o^asinθ_o^b/8 πϵR^3 (E_l^a-E_u^a)(E_l^b-E_u^b)(3cos(4ϕ)-5) .The normalized angular distribution of cases (ii) and (iii) as a function of the relative orientation θ_E, and the magnetic field orientation can be seen in the supplementary movie two-qubit-coupling-distribution.mov. Where the blue and red curves correspond to case (ii) and (iii) respectively. § CHARGE NOISE EXPOSURE DURING THE ENTANGLEMENT PROTOCOLSSince the exact amount of charge noise is device dependent, we account for the charge noise exposure qualitatively. For ϕ=nπ/2 as in <cit.>, the electric field is adiabatically swept from F_z=0 to F_z at the isotropic sweet spot, hence the simplest qualitative way of comparing protocols is to account for the ratio T_2^*(ϕ)/T_2^*(0). Here T_2^*(ϕ) would be related to the amount of charge noise accumulated when going from the anisotropic sweet spot to the isotropic sweet spot for a given ϕ, and T_2^*(0) the exposure to charge noise for ϕ=0. Given a fluctuating charged defect with field δ F_z we compute the change in energy δ E_ Larmor for a given value of F_z in the qubit. How much the defect affects the qubit energy depends on the derivative of the Larmor energy on F_z. From <cit.> we know that 1/T_2^*∝δ E_ Larmor^2. Defining F_z^* and F̃_z as the values of the vertical electric field at the isotropic and anisotropic sweet spots respectively, we account for the total charge noise exposure by integrating the Larmor energy change along the path from F̃_z to F_z^*I(ϕ)=∫_F̃_z^F_z^*δ E_ Larmor^2(F_z) dF_zBy assuming a constant sweep rate, the time of exposure to charge noise is proportional to the difference between the initial and final electric fields. In total we getT_2^*(ϕ)/T_2^*(0)=|I(0)F_z^*/I(ϕ)(F_z^*-F̃_z)|Intuitively, this ratio is simply proportional to the charge noise sensitivity along the path and its length.The values of this ratio can be seen in Fig. <ref>.From Fig. <ref> it is clear that any ϕ≠ 0 reduces the charge noise exposure by sweeping between sweet spots. The explanation is simply that the closer the sweet spots are the less time the qubits are exposed to charge noise. Moreover, when the two sweet spots are closer, the derivative dE_ Larmor/dF_z also becomes smaller. As a result, the exposure to charge noise by sweeping between sweet spotsis minimal when the two sweet spots merge at ϕ=1/2arcsin(1/3) and ϕ=π/2-1/2arcsin(1/3). This implies that Protocol 2 is particularly robust against the charge noise exposure during the adiabatic sweep between sweet spots, since the value of ϕ for this protocol is close to the divergence of T_2^*(ϕ)/T_2^*(0). | http://arxiv.org/abs/1706.08858v2 | {
"authors": [
"J. C. Abadillo-Uriel",
"Joe Salfi",
"Xuedong Hu",
"Sven Rogge",
"M. J. Calderón",
"Dimitrie Culcer"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170627135831",
"title": "Entanglement control and magic angles for acceptor qubits in Si"
} |
[ Cognitive Psychology for Deep Neural Networks: A Shape Bias Case Study equal*Samuel Ritterequal,dm David G.T. Barrettequal,dm Adam Santorodm Matt M. BotvinickdmdmDeepMind, London, UK Samuel [email protected] David G.T. [email protected] one-shot, word learning, bias, cognitive psychology, deep learning 0.3in ] Deep neural networks (DNNs) have achieved unprecedented performance on a wide range of complex tasks, rapidly outpacing our understanding of the nature of their solutions. This has caused a recent surge of interest in methods for rendering modern neural systems more interpretable. In this work, we propose to address the interpretability problem in modern DNNs using the rich history of problem descriptions, theories and experimental methods developed by cognitive psychologists to study the human mind. To explore the potential value of these tools, we chose a well-established analysis from developmental psychology that explains how children learn word labels for objects, and applied that analysis to DNNs. Using datasets of stimuli inspired by the original cognitive psychology experiments, we find that state-of-the-art one shot learning models trained on ImageNet exhibit a similar bias to that observed in humans: they prefer to categorize objects according to shape rather than color. The magnitude of this shape bias varies greatly among architecturally identical, but differently seeded models, and even fluctuates within seeds throughout training, despite nearly equivalent classification performance. These results demonstrate the capability of tools from cognitive psychology for exposing hidden computational properties of DNNs, while concurrently providing us with a computational model for human word learning.§ INTRODUCTION During the last half-decade deep learning has significantly improved performance on a variety of tasks (for a review, see <cit.>). However, deep neural network (DNN) solutions remain poorly understood, leaving many to think of these models as black boxes, and to question whether they can be understood at all <cit.>. This opacity obstructs both basic research seeking to improve these models, and applications of these models to real world problems <cit.>.Recent pushes have aimed to better understand DNNs: tailor-made loss functions and architectures produce more interpretable features <cit.> while output-behavior analyses unveil previously opaque operations of these networks <cit.>. Parallel to this work, neuroscience-inspired methods such as activation visualization <cit.>,ablation analysis <cit.> and activation maximization <cit.> have also been applied. Altogether, this line of research developed a set of promising tools for understanding DNNs, each paper producing a glimmer of insight. Here, we propose another tool for the kit, leveraging methods inspired not by neuroscience, but instead by psychology. Cognitive psychologists have long wrestled with the problem of understanding another opaque intelligent system: the human mind. We contend that the search for a better understanding of DNNs may profit from the rich heritage of problem descriptions, theories, and experimental tools developed in cognitive psychology. To test this belief, we performed a proof-of-concept study on state-of-the-art DNNs that solve a particularly challenging task: one-shot word learning. Specifically, we investigate Matching Networks (MNs) <cit.>, which have state-of-the-art one-shot learning performance on ImageNet and we investigate an Inception Baseline model <cit.>.Following the approach used in cognitive psychology, we began by hypothesizing an inductive bias our model may use to solve a word learning task. Research in developmental psychology shows that when learning new words, humans tend to assign the same name to similarly shaped items rather than to items with similar color, texture, or size. To test the hypothesis that our DNNs discover this same “shape bias”, we probed our models using datasets and an experimental setup based on the original shape bias studies <cit.>.Our results are as follows: 1) Inception networks trained on ImageNet do indeed display a strong shape bias. 2) There is high variance in the bias between Inception networks initialized with different random seeds, demonstrating that otherwise identical networks converge to qualitatively different solutions. 3) MNs also have a strong shape bias, and this bias closely mimics the bias of the Inception model that provides input to the MN. 4) By emulating the shape bias observed in children, these models provide a candidate computational account for human one-shot word learning. Altogether, these results show that the technique of testing hypothesized biases using probe datasets can yield both expected and surprising insights about solutions discovered by trained DNNs. §.§ Related Work: Cognitive Modeling with Neural Networks The use of behavioral probes to understand neural network function has been extensively applied within psychology itself, where neural networks have been employed effectively as models of human cognitive function <cit.>. In contrast, in the present work we are advocating for the application of behavioral probes along with associated theories and hypotheses from cognitive psychology to address the interpretability problem in modern deep networks. In spite of the widespread adoption of deep learning methods in recent years, to our knowledge, work applying behavioral probes to DNNs in machine learning for this purpose has been quite limited; we only are aware of <cit.> and <cit.>, who used psychophysics-like experiments to better understand image processing models. § INDUCTIVE BIASES, STATISTICAL LEARNERS AND PROBE DATASETS Before we delve into the specifics of the shape bias and one-shot word learning, we will describe our approach in the general context of inductive biases, probe datasets, and statistical learning. Suppose we have some data {y_i, x_i }_i=1^N where y_i=f(x_i). Our goal is to build a model of the data g(.) to optimize some loss function L measuring the disparity between y and g(x), e.g., L=∑_i ||y_i - g(x_i)||^2. Perhaps this data x is images of ImageNet objects to be classified, images and histology of tumors to be classified as benign or malignant <cit.>, or medical history and vital measurements to be classified according to likely pneumonia outcomes <cit.>. A statistical learner such as a DNN will minimize L by discovering properties of the input x that are predictive of the labels y. These discovered predictive properties are, in effect, the properties of x for which the trained model has an inductive bias. Examples of such properties include the shape of ImageNet objects, the number of nodes of a tumor, or a particular constellation of blood test values that often precedes an exacerbation of pneumonia symptoms.Critically, in real-world datasets such as these, the discovered properties are unlikely to correspond to a single feature of the input x; instead they correspond to complex conjunctions of those features. We could describe one of these properties using a function h(x), which, for example, returns the shape of the focal object given an ImageNet image, or the number of nodes given a scan of tumor. Indeed, one way to articulate the difficulty in understanding DNNs is to say that we often can't intuitively describe these conjunctions of features h(x); although we often have numerical representations in intermediate DNN layers, they're often too arcane for us to interpret.We advocate for addressing this problem using the following hypothesis-driven approach: First, propose a property h_p(x) that the model may be using. Critically, it's not necessary that h_p(x) be a function that can be evaluated using an automated method. Instead, the intention is that h_p(x) is a function that humans (e.g. ML researchers and practitioners) can intuitively evaluate. h_p(x) should be a property that is believed to be relevant to the problem, such as object shape or number of tumor nodes.After proposing a property, the next step is to generate predictions about how the model should behave when given various inputs, if in fact it uses a bias with respect to the property h_p(x). Then, construct and carry out an experiment wherein those predictions are tested. In order to execute such an experiment, it typically will be necessary to craft a set of probe examples x that cover a relevant portion of the range of h_p(x), for example a variety of object shapes. The results of this experiment will either support or fail to support the hypothesis that the model uses h_p(x) to solve the task. This process can be especially valuable in situations where there is little or no training data available in important regions of the input space, and a practitioner needs to know how the trained model will behave in that region.Psychologists have developed a repertoire of such hypotheses and experiments in their effort to understand the human mind. Here we explore the application of one of these theory-experiment pairs to state of the art one-shot learning models. We will begin by describing the historical backdrop for the human one-shot word learning experiments that we will then apply to our DNNs. § THE PROBLEM OF WORD LEARNING; THE SOLUTION OF INDUCTIVE BIASES Discussions of one-shot word learning in the psychological literature inevitably begin with the philosopher W.V.O. Quine, who broke this problem down and described one of its most computationally challenging components: there are an enormous number of tenable hypotheses that a learner can use to explain a single observed example. To make this point, Quine penned his now-famous parable of the field linguist who has gone to visit a culture whose language is entirely different from our own <cit.>. The linguist is trying to learn some words from a helpful native, when a rabbit runs past. The native declares “gavagai", and the linguist is left to infer the meaning of this new word. Quine points out that the linguist is faced with an abundance of possible inferences, including that “gavagai" refers to rabbits, animals, white things, that specific rabbit, or “undetached parts of rabbits". Quine argues that indeed there is an infinity of possible inferences to be made, and uses this conclusion to bolster the assertion that meaning itself cannot be defined in terms of internal mental events.Contrary to Quine's intentions, when this example was introduced to the developmental psychology community by <cit.>, it spurred them not to give up on the idea of internal meaning, but instead to posit and test for cognitive biases that enable children to eliminate broad swaths of the hypothesis space <cit.>. A variety of hypothesis-eliminating biases were then proposed including the whole object bias, by which children assume that a word refers to an entire object and not its components <cit.>; the taxonomic bias, by which children assume a word refers to the basic level category an object belongs to <cit.>; the mutual exclusivity bias, by which children assume that a word only refers to one object category <cit.>; the shape bias, with which we are concerned here <cit.>; and a variety of others <cit.>. These biases were tested empirically in experiments wherein children or adults were given an object (or picture of an object) along with a novel name, then were asked whether the name should apply to various other objects. Taken as a whole, this work yielded a computational level <cit.> account of word learning whereby people make use of biases to eliminate unlikely hypotheses when inferring the meaning of new words. Other contrasting and complementary approaches to explaining word learning exist in the psychological literature, including association learning <cit.> and Bayesian inference <cit.>. We leave the application of these theories to deep learning models to future work, and focus on determining what insight can be gained by applying a hypothesis elimination theory and methodology.We begin the present work with the knowledge that part of the hypothesis elimination theory is correct: the models surely use some kind of inductive biases since they are statistical learning machines that successfully model the mapping between images and object labels. However, several questions remain open. What predictive properties did our DNNs find? Do all of them find the same properties? Are any of those properties interpretable to humans? Are they the same properties that children use? How do these biases change over the course of training?To address these questions, we carry out experiments analogous to those of <cit.>. This enables us to test whether the shape bias – a human interpretable feature used by children when learning language – is visible in the behavior of MNs and Inception networks. Furthermore we are able to test whether these two models, as well as different instances of each of them, display the same bias. In the next section we will describe in detail the one-shot word learning problem, and the MNs and Inception networks we use to solve it.Unlike Quine, we use a pragmatic definition of meaning - a human or model understands the meaning of a word if they assign that word to new instances of objects in the correct category.§ ONE-SHOT WORD LEARNING MODELS AND TRAINING §.§ One-shot word learning task The one-shot word learning task is to label a novel data example x̂ (e.g. a novel probe image) with a novel class label ŷ(e.g. a new word) after only a single example. More specifically, given a support set S={(x_i,y_i): i ∈ [1,k]}, of images x_i and their associated labels y_i, and an unlabelled probe image x̂, the one-shot learning task is to identify the true label of the probe image ŷ from the support set labels {y_i: i ∈ [1,k]}:ŷ = max_y P(y | x̂, S).We assume that the image labels y_i are represented using a one-hot encoding and that P(y | x̂, S) is parameterised by a DNN, allowing us to leverage the ability of deep networks to learn powerful representations.§.§ Inception: baseline one-shot learning model In our simplest baseline one-shot architecture, a probe image x̂ is given the label of the nearest neighbour from the support set: ŷ = y(x, y)= min_(x_i,y_i) ∈ Sd(h(x_i),h(x̂))where d is a distance function. The function h is parameterised by Inception – one of the best performing ImageNet classification models <cit.>. Specifically, h returns features from the last layer (the softmax input) of a pre-trained Inception classifier, where the Inception classifier is trained using rms-prop, as described in <cit.>, section 8. With these features as input and cosine distance as the distance function, the classifier in equation <ref> achieves 87.6% accuracy on one-shot classification on the ImageNet dataset <cit.>. Henceforth, we call the Inception classifier together with the nearest-neighbor component the Inception Baseline (IB) model. §.§ Matching Nets model architecture and trainingWe also investigate a state-of-the-art one-shot learning architecture called Matching Nets (MN) <cit.>. MNs are a fully differentiable neural network architecture with state-of-the-art one shot learning performance on ImageNet (93.2% one-shot labelling accuracy).MNs are trained to assign label ŷ to probe image x̂ according to equation <ref> using an attention mechanism a acting on image embeddings stored inthe support set S:a(x̂,x_i) = e^d(f(x̂,S),g(x_i,S))/∑_j e^d(f(x̂,S),g(x_j,S)) ,where d is a cosine distance and where f and g provide context-dependent embeddings of x̂ and x_i (with context S). The embedding g(x_i,S) is a bi-directional LSTM <cit.> with the support set S provided as an input sequence. The embedding f(x̂,S) is an LSTM with a read-attention mechanism operating over the entire embedded support set. The input to the LSTM is given by the penultimate layer features of a pre-trained deep convolutional network, specifically Inception, as in our baseline IB model described above <cit.>. The training procedure for the one-shot learning task is critical if we want MNs to classify a probe image x̂ after viewing only a single example of this new image class in its support set <cit.>. To train MNs we proceed as follows: (1) At each step of training, the model is given a small support set of images and associated labels. In addition to the support set, the model is fed an unlabelled probe image x̂; (2) The model parameters are then updated to improve classification accuracy of the probe image x̂ given the support set. Parameters are updated using stochastic gradient descent with a learning rate of 0.1; (3) After each update, the labels {y_i: i ∈ [1,k]} in the training set are randomly re-assigned to new image classes (the label indices are randomly permuted, but the image labels are not changed). This is a critical step. It prevents MNs from learning a consistent mapping between a category and a label. Usually, in classification, this is what we want, but in one-shot learning we want to train our model for classification after viewing a single in-class example from the support set. Formally, our objective function is:L = E_C∼ T[ E_S∼ C,B∼ C[∑_(x,y)∈ Blog P(y | x, S)]]where T is the set of all possible labelings of our classes, S is a support set sampled with a class labelling C ∼ T and B is a batch of probe images and labels, also with the same randomly chosen class labelling as the support set. Next we will describe the probe datasets we used to test for the shape bias in the IB and MNs after ImageNet training. § DATA FOR BIAS DISCOVERY§.§ Cognitive Psychology Probe Data The Cognitive Psychology Probe Data (CogPsyc data) that we use consists of 150 images of objects (Figure <ref>). The images are arranged in triples consisting of a probe image, a shape-match image (that matches the probe in colour but not shape), and a color-match image (that matches the probe in shape but not colour). In the dataset there are 10 triples, each shown on 5 different backgrounds, giving a total of 50 triples. The CogPsyc dataset is available at<http://www.indiana.edu/ cogdev/SB_testsets.html> The images were generously provided by cognitive psychologist Linda Smith. The images are photographs of stimuli used previously in shape bias experiments conducted in the Cognitive Development Lab at Indiana University. The potentially confounding variables of background content and object size are controlled in this dataset. §.§ Probe Data from the wild We have also assembled a real-world dataset consisting of 90 images of objects (30 triples) collected using Google Image Search. Again, the images are arranged in triples consisting of a probe, a shape-match and a colour-match. For the probe image, we chose images of real objects that are unlikely to appear in standard image datasets such as ImageNet. In this way, our data contains the irregularity of the real world while also probing our models' properties outside of the image space covered in our training data. For the shape-match image, we chose an object with a similar shape (but with a very different colour), and for the colour-match image, we chose an object with a similar colour (but with a very different shape). For example, one triple consists of a silver tuning fork as the probe, a silver guitar capo as the colour match, and a black tuning fork as the shape match. Each photo in the datasetcontains a single object on a white background.We collected this data to strengthen our confidence in the results obtained for the CogPsych dataset and to demonstrate the ease with which such probe datasets can be constructed. One of the authors crafted this dataset solely using Google Image Search in the span of roughly two days' work. Our results with this dataset, especially the fact that the bias pattern over time matches the results from the well established CogPsych dataset, support the contention that DNN practitioners can collect effective probe datasets with minimal time expenditure using readily available tools.§ RESULTS§.§ Shape bias in the Inception Baseline Model First, we measured the shape bias in IB: we used a pre-trained Inception classifier (with 94% top-5 accuracy) to provide features for our nearest-neighbour one-shot classifier, and probed the model using the CogPsyc dataset. Specifically, for a given probe image x̂, we loaded the shape-match image x_s and corresponding label y_s, along with the colour-match image x_c and corresponding label y_c into memory, as the support set S = {(x_s,y_s),(x_c,y_c)}. We then calculated ŷ using Equation <ref>. Our model assigned either y_c or y_s to the probe image. To estimate the shape bias B_s, we calculated the proportion of shape labels assigned to the probe:B_s = E(δ(ŷ - y_s)),where E is an expectation across probe images and δ is the Dirac delta function.We ran all IB experiments using both Euclidean and cosine distance as the distance function. We found that the results for the two distance functions were qualitatively similar, so we only report results for Euclidean distance.We found the shape bias of IB to be B_s = 0.68. Similarly, the shape bias of IB using our real-world dataset was B_s = 0.97. Together, these results strongly suggest that IB trained on ImageNet has a stronger bias towards shape than colour. Note that, as expected, the shape bias of this model is qualitatively similar across datasets while being quantitatively different - largely because the datasets themselves are quite different. Indeed, the datasets were chosen to be quite different so that we could explore a broad space of possibilities. In particular, our CogPsyc dataset backgrounds have much larger variability than our real-world dataset backgrounds, and our real-world dataset objects have much greater variability than the CogPsyc dataset objects.§.§ Shape bias in the Matching Nets ModelNext, we probed the MNs using a similar procedure. We used the IB trained in the previous section to provide the input features for the MN as described in section <ref>. Then, following the training procedure outlined in section <ref> we trained MNs for one-shot word learning on ImageNet, achieving state-of-the-art performance, as reported in <cit.>. Then, repeating the analysis above, we found that MNs have a shape of bias B_s = 0.7 using our CogPsyc dataset and a bias of B_s = 1 using the real-world dataset. It is interesting to note that these bias values are very similar to the IB bias values. §.§ Shape bias statistics: within models and across models The observation of a shape bias immediately raises some important questions. In particular: (1) Does this bias depend on the initial values of the parameters in our model? (2) Does the size of the shape bias depend on model performance? (3) When does shape bias emerge during training - before model convergence or afterwards? (4) How does shape bias compare between models, and within models?To answer these questions, we extended the shape bias analysis described above to calculate the shape bias in a population of IB models and in a population of MN models with different random initialization (Figs. <ref> and <ref>).(1) We first calculated the dependence of shape bias on the initialization of IB (Fig. <ref>). Surprisingly, we observed a strong variability, depending on the initialization. For the CogPsyc dataset, the average shape bias was B_s = 0.628 with standard deviation σ_B_s = 0.049 at the end of training and for the real-world dataset the average shape bias was B_s = 0.958 with σ_B_s = 0.037. (2) Next, we calculated the dependence of shape bias on model performance. For the CogPsych dataset, the correlation between bias and classification accuracy was ρ=0.15, with t_n=15=0.55, p_one_tail=0.29, and for the real-world dataset, the correlation was ρ=-0.06 with t_n=15=-0.22, p_one_tail=0.42. Therefore, fluctuations in the bias cannot be accounted for by fluctuations in classification accuracy. This is not surprising, because the classification accuracy of all models was similar at the end of training, while the shape bias was variable. This demonstrates that models can have variable behaviour along important dimensions (e.g., bias) while having the same performance measured by another (e.g., accuracy). (3) Next we explored the emergence of the shape bias during training (Fig. <ref>a,c; Fig. <ref>a,c). At the start of training, the average shape bias of these models was B_s = 0.448 with standard deviation σ_B_s = 0.0835 on the CogPsyc dataset and B_s = 0.593 with σ_B_s= 0.073 on the real-world dataset. We observe that a shape bias began to emerge very early during training, long before convergence. (4) Finally, we compare shape bias within models during training, and between models at the end of training. During training, the shape bias within IB fluctuates significantly (Fig. <ref> b; Fig. <ref>b). In contrast, the shape bias does not fluctuate during training of the MN. Instead, the MN model inherits its shape bias characteristics at the start of training from the IB that provides it with input embeddings (Fig. <ref>) and this shape-bias remains constant throughout training. Moreover, there is no evidence that the MN and corresponding IB bias values are different from each other (paired t-test, p = 0.167). Note that we do not fine-tune the Inception model providing input while training the MN. We do this so that we can observe the shape-bias properties of the MN independent of the IB model properties. § DISCUSSION§.§ A shape bias case study Our psychology-inspired approach to understanding DNNs produced a number of insights. Firstly, we found that both IB and MNs trained on ImageNet display a strong shape bias. This is an important result for practitioners who routinely use these models - especially for applications where it is known a priori that colour is more important than shape. As an illustrative example, if a practitioner planned to build a one-shot fruit classification system, they should proceed with caution if they plan to use pre-trained ImageNet models like Inception and MNs because fruit are often defined according to colour features rather than shape. In applications where a shape bias is desirable (as is more often the case than not), this result provides reassurance that the models are behaving sensibly in the presence of ambiguity. The second surprising finding was the large variability in shape bias, both within models during training and across models, depending on the randomly chosen initialisation of our model. This variability can arise because our models are not being explicitly optimised for shape biased categorisation. This is an important result because it shows that not all models are created equally - some models will have a stronger preference for shape than others, even though they are architecturally identical and have almost identical classification accuracy. Our third finding – that MNs retain the shape bias statistics of the downstream Inception network – demonstrates the possibility for biases to propagate across model components. In this case, the shape bias propagates from the Inception model through to the MN memory modules. This result is yet another cautionary observation; when combining multiple modules together, we must be aware of contamination by unknown properties across modules. Indeed, a bias that is benign in one module might only have a detrimental effect when combined later with other modules.A natural question immediately arises from these results - how can we remove an unwanted bias or induce a desirable bias? The biases under consideration are properties of an architecture and dataset synthesized together by an optimization procedure. As such, the observation of a shape-bias is partly a result of the statistics of natural image-labellings as captured in the ImageNet dataset, and partly a result of the architecture attempting to extract these statistics. Therefore, on discovering an unwanted bias, a practitioner can either attempt to change the model architecture to explicitly prevent the bias from emerging, or, they can attempt to manipulate the training data. If neither of these are possible - for example, if the appropriate data manipulation is too expensive, or, if the bias cannot be easily suppressed in the architecture, it may be possible to do zero-th order optimization of the models. For example, one may perform post-hoc model selection either using early stopping or by selecting a suitable model from the set of initial seeds.An important caveat to note is that behavioral tools often do not provide insight into the neural mechanisms. In our case, the DNN mechanism whereby model parameters and input images interact to give rise to a shape bias have not been elucidated, nor did we expect this to happen. Indeed, just as cognitive psychology often does for neuroscience, our new computational level insights can provide a starting point for research at the mechanistic level. For example, in future work it would be interesting to use gradient-based visualization or neuron ablation techniques to augment the current results by identifying the mechanisms underlying the shape bias. The convergence of evidence from such introspective methods with the current behavioral method would create a richer account of these models' solutions to the one-shot word learning problem. §.§ Modelling human word learning There have been previous attempts to model human word learning in the cognitive science literature <cit.>. However, none of these models are capable of one-shot word learning on the scale of real-world images. Because MNs both solve the task at scale and emulate hallmark experimental findings, we propose MNs as a computational-level account of human one-shot word learning. Another feature of our results supports this contention: in our model the shape bias increases dramatically early in training (Fig. <ref>a); similarly, humans show the shape bias much more strongly as adults than as children, and older children show the bias more strongly than younger children <cit.>.As a good cognitive model should, our DNNs make testable predictions about word-learning in humans. Specifically, the current results predict that the shape bias should vary across subjects as well as within a subject over the course of development. They also predict that for humans with adult-level one-shot word learning abilities, there should be no correlation between shape bias magnitude and one-shot-word learning capability.Another promising direction for future cognitive research would be to probe MNs for additional biases in order to predict novel computational properties in humans. Probing a model in this way is much faster than running human behavioural experiments, so a wider range of hypotheses for human word learning may be rapidly tested. §.§ Cognitive Psychology for Deep Neural NetworksThrough the one-shot learning case study, we demonstrated the utility of leveraging techniques from cognitive psychology for understanding the computational properties of DNNs. There is a wide ranging literature in cognitive psychology describing techniques for probing a spectrum of behaviours in humans. Our work here leads the way to the study of artificial cognitive psychology - the application of these techniques to better understand DNNs. For example, it would be useful to apply work from the massive literature on episodic memory <cit.> to the recent flurry of episodic memory architectures <cit.>, and to apply techniques from the semantic cognition literature <cit.> to recent models of concept formation <cit.>. More generally, the rich psychological literature will become increasingly useful for understanding deep reinforcement learning agents as they learn to solve increasingly complex tasks.§ CONCLUSION In this work, we have demonstrated how techniques from cognitive psychology can be leveraged to help us better understand DNNs. As a case study, we measured the shape bias in two powerful yet poorly understood DNNs - Inception and MNs. Our analysis revealed previously unknown properties of these models. More generally, our work leads the way for future exploration of DNNs using the rich body of techniques developed in cognitive psychology. § ACKNOWLEDGEMENTSWe would like to thank Linda Smith and Charlotte Wozniak for providing the Cognitive Psychology probe dataset; Charles Blundell for reviewing our paper prior to submission; Oriol Vinyals, Daan Wierstra, Peter Dayan, Daniel Zoran, Ian Osband and Karen Simonyan for helpful discussions; James Besley for legal assistance; and the DeepMind team for support. icml2017 | http://arxiv.org/abs/1706.08606v2 | {
"authors": [
"Samuel Ritter",
"David G. T. Barrett",
"Adam Santoro",
"Matt M. Botvinick"
],
"categories": [
"stat.ML",
"cs.CV",
"cs.LG"
],
"primary_category": "stat.ML",
"published": "20170626213118",
"title": "Cognitive Psychology for Deep Neural Networks: A Shape Bias Case Study"
} |
1.2 23.5cm 16cm 1ex 0pt 0pt -40pt .tifpng.png`convert #1 `dirname #1`/`basename #1 .tif`.png → ↔∂Δϵλϵ Phenomenology with F-theory SU(5) George K. Leontaris^a,b[E-mail: ] and Qaisar Shafi^c[E-mail: ] ^a LPTHE, UMR CNRS 7589, Sorbonne Universités, UPMC Paris 6, 75005 Paris, France^b Physics Department, Theory Division, University of Ioannina, GR-45110 Ioannina, Greece^c Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, DE 19716,Newark, USA We explore the low energy phenomenology of an F-theory based SU(5) model which, in addition to the known quarks and leptons, contains Standard Model (SM)singlets, and vector-like color triplets and SU(2) doublets. Depending on their masses and couplings, some of these new particles may be observed at the LHC and future colliders. We discuss the restrictions by CKM constraints on theirmixing with the ordinary down quarks of the three chiral familes. The model is consistent with gauge coupling unification at the usual supersymmetric GUT scale, dimension five proton decay is adequately suppressed, while dimension-six decay mediated by the superheavy gauge bosons is enhanced by a factor of 5-7. The third generation charged fermion Yukawa couplingsyield the correspondinglow-energy masses in reasonable agreement with observations. The hierarchical nature of the masses of lighter generations is accounted for via non-renormalisable interactions,with the perturbative vacuum expectation values (vevs) of the SM singlet fields playing an essential rôle. § INTRODUCTIONModels originating from string theory constructions often contain SM singlets and vector-like fields which can mix with the light spectrum and therefore are natural candidates for predicting rare processesthat might be discovered infutureexperiments at the LHC and elsewhere.F-theory models <cit.>, in particular, have the necessary ingredients to describe in a simple and convincing manner a complete picture of such new phenomena. One of the most appealing grand unified theories incorporating these features in an F-theory context, is SU(5) [ForF-theory model building reviews and early references see <cit.>. For an incomplete list including more recent research papers see <cit.>-<cit.>.]. Indeed, on breakingF-SU(5)to SM symmetry, one ends up with the MSSM spectrum augmented by scalar fields and vector-like states, which are remnants of the underlying GUT representations. In this framework, it is possible to retain gauge coupling unification even in the presence of some additional fields, provided that these form complete multiplets of SU(5).In view of the ongoing experimental searches and possible future signatures, in this work we reconsider some issues regarding the exotic part of these models.We start with a brief review of the basic features of an SU(5) model <cit.> derived in an F-theory frameworkand, in particular, in the context of the spectral cover.We derive an effectivetheory model by imposing a Z_2 monodromyand identify the complex surfaces where the chiral matter and Higgs can be accommodated in the quotient theory. We assume ahypercharge flux breaking of the SU(5) symmetry down to the SM one, and proceed with a specific assignment ofthe MSSM representations on these matter curves and then work out the spectrum and the superpotential.After fixing the necessary free parameters (such as flux units and singlet vevs), we proceed with the investigation of the exotic massless spectrum left over from higher dimensionalfields.We then derive their superpotential couplings and analyse the implications for baryonnumber violating decays as well as other rare processes. We examine the possibility that these states remainmassless at low energies being consistent with gauge couplingunification, and discuss the physics implications of the TeV scale exotic states.§ F-SU(5)We consider the elliptically fibred case wherethe highest smooth singularityin Kodaira's classification is associated with the exceptional group of E_8 <cit.>. We assume 7-branes wrapping an SU(5) divisor and interpretthis as the GUTsymmetry of the effective model. Under these assumptionsE_8⊃ SU(5)_GUT×SU(5)_⊥ , where the first factor is interpreted as the well known SU(5)_GUT and the second factor is usually denoted as SU(5)_. The MSSM spectrum and possible exotic fields descend from the decomposition of the E_8 adjoint which, under the assumed breaking pattern (<ref>), decomposes as follows:248 (24,1)+(1,24)+(10,5)+(5,10)+(5,10)+(10,5) . Thus, matter transforms in bi-fundamental representations, with the GUT10-pletslying in the fundamental ofSU(5)_⊥, and the 5̅, 5-plets lying in the antisymmetric representation of SU(5)_⊥.We choose to work in the Higgs bundle picture(the spectral cover approach). In this context the properties of the GUT representations with respect to the spectral cover are described bya degree-five polynomial <cit.> C_5: ∑_k=0^5 b_ks^5-k=0 , where the b_k coefficients carry the information of the internal geometry and their homologies, are given by [b_n]=η-nc_1, (withη=6c_1-t), where c_1=c_1(S) is the first Chern class of the tangent bundle and -t that of the normal to the surface S. Theroots of the equation are identified as the weight vectors t_1,...,5satisfying the standard SU(N) constraint (N=5 in the present case) ∑_i=1^5t_i = 0 . Under t_ithematter curves acquire specific topological and symmetry properties inherited by the fermion families and Higgs fields propagating there. We denote the matter curves accommodating the 10-plets, 5-pletsof SU(5) and singlets emerging from SU(5)_⊥ adjoint decomposition as Σ_10_t_i, Σ_5_t_i+t_j,Σ_1_t_i-t_j. Correspondingly, the possible representations residing on these matter curves are denoted byΣ_10_t_i: 10_t_i, 10_-t_i,Σ_5_t_i+t_j: 5_t_i+t_j ,5_-t_i-t_j,Σ_1_t_i-t_j:1_t_i-t_j ,where, as far as 5-plets and singlets are concerned, we must have t_i t_j. Working in the framework of spectral cover, while assumingdistinct roots t_i of (<ref>), one may further consider the breaking SU(5)_→ U(1)_^4. Then, the invariant tree-level superpotentialcouplings are of the formW⊃ h_1 10_t_i10_t_j 5_-t_i-t_j+h_2 10_t_i5̅_t_j+t_k5̅_t_l+t_m+ h_31_t_i-t_j 5_-t_i-t_k5̅_t_j+t_m +h_41_t_i-t_j1_t_j-t_k1_t_k-t_i,where h_1,2,3,4 represent the Yukawa strengths. In each of the above terms, the sum of the t_i `charges' should add up to zero. Hence, in the second termt_i+t_j+t_k+t_l+t_m=0, which unambiguously impliesthat all indices in the term proportional to Yukawa couplingh_2 should differ from each other (due to the fact that t_1+t_2+t_3+t_4+t_5=0). Returning to the polynomial (<ref>), although its coefficients b_n belong to a certain field (holomorphic functions), the roots t_i do not necessarily do so. Solutions, in general, imply branch cuts and, as a result,certain roots might be interrelated.The simplest case is if two of them are subject to a Z_2 monodromy,say, [For various choices of monodromies, see <cit.>.] Z_2:t_1=t_2 . From the point of view of theeffective field theory model, the appearanceof the monodromy is a welcome resultsince it implies rank-one mass matrices for the fermions. Indeed, under theZ_2 monodromy, the coupling W⊃ 10_t_110_t_2 5_-t_1-t_2Z_2⟶10_t_110_t_1 5_-2t_1 ensures a top-quark mass at tree-level, while the remaining mass matrix entries are expected to be generated from non-renormalisable terms.After this brief description of the basic features,we proceed in the next section with the analysis of the implications of the hypercharge flux on the symmetry breakingand the massless spectrum of SU(5). § HYPERCHARGE FLUXBREAKING OF SU(5)The Z_2 monodromy implies that the spectral cover polynomial factorises as follows:b_0s^5+b_2s^3+b_3s^2+b_4s+b_5= ( a_1+a_2s+a_3s^2)(a_4+a_5s)(a_6+a_7s)(a_8+a_9s), where all a_i are assumed in the same field as b_n's. Thus,while the roots of the three monomials on the right-hand side of (<ref>) are rational functions in this field, it is assumed that the two roots of the binomial ( a_1+a_2s+a_3s^2) cannot be written in terms of functions in the same field. The b_n(a_i)relations are easily extractedby identifying coefficients of the same powers in s and are of the form b_n=∑ a_ia_ja_ka_l, where the indices satisfy i+j+k+l+n=24. Therefore, given the homologies [b_n], the correspondingones for the a_i coefficients satisfy [a_i]+[a_j]+[a_k]+[a_l]=[b_n]. Solving the resulting simple linear system of equations, it turns out that these can be determined in terms of the known classes c_1, -tand three arbitrary ones (dubbed here χ_6,7,8), which will be treated as free parameters <cit.>. Each matter curve is associated with a defining equation involving products of a_i's and, as such, it belongs to a specific homological class which subsequently is used to determine the flux restriction on it. IfF_Y represents the hypercharge flux, we will requirethe vanishing of F_Y· c_1= F_Y·(-t)=0, so that allcan be expressed in terms of three free (integer parameters) defined by the restrictionsN_7= F_Y·χ_7,N_8= F_Y·χ_8,N_9= F_Y·χ_9 .To construct a specific model, we start by assuming that a suitable U(1)_X flux (where the abelian factor U(1)_X lies outside SU(5) GUT) generates chirality for the 10 and 5̅ representations.Next, the hyperchargeflux breaks SU(5)down to the SM and, at the same, time it splits the 10, 10 and 5,5̅'sinto different numbers of SM multiplets. If some integersM_10, M_5 areassociated with the U(1)_X flux, and some linear combinationN_y of N_7,8,9 represents the corresponding hyperflux piercinga givenmatter curve, the 10-plets and 5-pletssplit according to: 10_t_i= {[ Representationfluxunits;n_(3,2)_1/6-n_(3̅,2)_-1/6 = M_10; n_(3̅,1)_-2/3-n_( 3,1)_2/3 = M_10-N_y;n_(1,1)_+1-n_(1,1)_-1 = M_10+N_y;]. , 5_t_i= {[ Representationfluxunits; n_(3,1)_-1/3-n_(3̅,1)_+1/3= M_5;n_(1,2)_+1/2-n_(1,2)_-1/2= M_5+N_y ·;].As already discussed, depending on therestrictions of the flux on the matter curves Σ_j, there are certain conditions on the corresponding hypercharge flux,denoted as N_y_j (for the specific matter curveΣ_j). These are deduced from the topological properties of the coefficients a_i as well as the fluxes.For a given choice of the flux parameters M_i, N_y_j, the most general spectrum and its properties under the assumption of a Z_2 monodromy are exhibited in Table <ref>. The first column shows the available mattercurves and the assumed chiral state propagating on it. The chirality is fixed by the specific choice of M_i, N_y_jflux coefficients shown in the lasttwo columns of Table <ref>. The second column shows the`charge' assignments, ± t_i for the 10-plets, and ± (t_i+ t_j), ± (t_i- t_j) for 5-plets and singlets respectively. For this particular arrangement, the structure of the fermion mass matricesexhibits a hierarchical form, consistent withthe experimentally measured masses and mixings <cit.>. In the present work, we will explore other interestingphenomenological implications of this model. The defining equations are shown inthe fourth column where, forbrevity, the notation a_ijk...=a_ia_ja_k⋯ is used. The next column indicates the homologies, the sixth column their associated integers expressing the restrictions of fluxon the corresponding matter curves, and the last column listsa choice of M_i values consistent with a chiral SU(5) spectrum. Notice that the flux integers are subject to therestrictions <cit.>N=N_7+N_8+N_9 and ∑_iM_5_i+∑_jM_10_j=0.In the minimal casen=0 and there are no extra 5+5̅ pairs. Furthermore,the multiplicities M_ij, M_δ of singlet fields are notdetermined in the context of the spectral coverand are left arbitrary. § SPECTRUM OF THE EFFECTIVE LOW ENERGY THEORY A comprehensive classificationof the resulting spectrumis shown in Table <ref> where, in the first column, the SU(5) properties are shown. The third column shows the accommodation of the SM representations with their corresponding `charges' given in column 2. Column 4 includes the exotics which, for the specific choice of parameters, involves the triplet pair D+ D^c and, in principle, n copies of 5+5̅ representations. In the minimal case we set n=0, but perturbativity allows values up to n≤ 4. In the modified version of the modelwe allow for n 0 and explore the phenomenological implications. note that restrictions on the number of vector-like 5-pletsarise when the model is embedded in an E_6framework <cit.>-<cit.>.In the last column of theTable, we have also introduced a Z_2 matter parity to the MSSM field as well as the singlets.Before proceeding with the main part of our paper we present a few remarks about R-parity in supersymmetric models. A discrete Z_2 R-parity is often invoked in four dimensional supersymmetric SU(5) models in order to eliminate rapid proton decay mediated by the supesrymmetric partners of the SM quarks and leptons. If left unbroken, this discrete symmetry also yields an attractive candidate for cold dark matter, namely the lightest neutralino. It is perhaps worth noting that this Z_2 symmetry naturally appears if we employ an SO(10) GUT which is broken down to SU(3)_c ×U(1)_em by utilizing only tensor representations <cit.>.The question naturally arises: how do string theory based unified models avoid rapid proton decay? In the ten-dimensional E_8 ×E_8 heterotic string framework <cit.>, the compactification process utilizes Calabi-Yau manifolds which typically yields non-abelian discrete symmetriesthat may contain the desired R-parity (<cit.> and references therein.)In F-theorymodelsdiscrete symmetries including R-parity mayarisefrom a variety of sources. They can emerge fromHiggsingU(1) symmetries in F-theory compactifications,or from a non-trivial Mordell-Weil group associated with the rational sections of the elliptic fibration, first invoked in <cit.> and further discussed in several works including <cit.>. More generally, Z_n symmetries are associated with Calabi-Yaumanifolds whose geometries are associated with the Tate-Shafarevich group <cit.>.Finally, they may appear as geometric properties of the construction in the spectral cover picture <cit.>. Based on the existence of such possibilities, in the present model we implement the notion of R-parity assuming that it is associated with somesymmetry of geometric origin. §.§ Matter curves and Fermion masses Returning to the description of the emerging effective model, for further clarification we include a few more details. Initially, in the covering theory there are five matter curves [Recall from (<ref>), Σ_10_i, i=1,2,…,5 that the 10-plets transform in the fundamental and 5-plets in the antisymmetric representation of SU(5)_.] but due to monodromy Z_2: t_1=t_2, two of them are identified and thus they are reducedto four. Similarly, the ten Σ_5_t_i+t_j reduce to seven matter curves. Furthermore, there are 24singlets from the decomposition of the adjoint of SU(5)_⊥ denoted with θ_ij, i,j=1,2…, 5, and 20 of them live on matter curves defined by t_i-t_j while four are `chargeless'. However, because oftheZ_2 monodromyamong the various identifications, θ_i1≡θ_i2 and θ_1j≡θ_2j,the following two singlets: θ_12=θ_21→ S are equivalent to one singlet S withzero charge. The remaining singlets with non-zero `charges' areθ_13, θ_14, θ_15, θ_34, θ_35, θ_45,and θ_31, θ_41, θ_51, θ_43, θ_53, θ_54The following singlets acquirenon-zero vevs which help in realising the desired fermion mass textures: ⟨θ_14⟩≡ V_1≡ v_1 M_GUT 0, ⟨θ_15⟩≡ V_2 ≡ v_2 M_GUT 0, ⟨θ_43⟩≡ V_3 ≡ v_3 M_GUT 0 . All othersinglets(designated with θ^⊥_ij in Table <ref>) have zero vevs. Using theSM Higgs and singletvevs given by (<ref>), we obtainhierarchicalquark and charged masstexturesM_u∝( [ v_1^2 v_3^2 v_1^2 v_3 v_1 v_3; v_1^2 v_3 v_1^2 v_1; v_1 v_3 v_1 1; ])⟨ H_u⟩ ,M_d,ℓ=( [ v_1^2 v_3^2 v_1 v_3^2 v_1 v_3; v_1^2 v_3 v_1 v_3 v_1; v_1 v_3 v_3 1; ])⟨ H_d⟩ ,where, the Yukawa couplings are suppressed for simplicity.§.§.§ Neutrino sector Thetinymasses accompanied by the relativelylarge mixings of the neutrinos, as indicated by various experiments,can find a plausible solution in the context of the see-saw mechanism and the existence of family symmetries. In the present F-SU(5) GUTmodel, the SM singlet fields such as θ_ij form Yukawa terms invariant under the additional familysymmetries described above and could bethe natural candidates for the righthanded neutrinos.Furthermore, observing thatthe right-handed neutrino mass scale is of the order of the Kaluza-Klein scale in string compactifications, a minimal scenario would be to associate the right handed neutrinos with the KK-modes <cit.> of these singlet fields, θ_ij^KK→ N_R. An obstruction to this interpretationis that in the covering theory these singletsθ_ij transform in the complexrepresentation, so that θ_ij^KK= N_R, θ_ji^KK= N^c_R and the mass term becomes M_KKN_RN_R^c, but there are no corresponding Dirac mass terms for both N_R, N_R^c. However, in the quotient theory under the Z_2 monodromy t_1=t_2, theKK-modes θ_12^KK≡θ_21^KK transform in the real representation, so that for any KK-level the corresponding modes N_R_k=N_R_k^c→ν_k^c are identified and a see-saw mechanism is possible.Hence,the non-renormalisable term 5_-t_1-t_25̅_t_1+t_4θ_14θ_21^KK under the Z_2 monodromy is identified with 5_-2t_15̅_t_1+t_4θ_14θ_21^KK→ 5_h_u5̅_3θ_14ν^c and so on. Therefore, under the above assumptions, the KK-modes corresponding to right-handed neutrinos couple to the following combination of the left-handed neutrino components 5_H_u(5̅_1 θ_14^2 θ_43+5̅_2θ_14θ_43+5̅_3θ_14) .The interesting fact is thatthe right-handed neutrinos are associated with a specific class ofwavefuctions <cit.> such that the emerging mass hierarchy is milder than that of the charged leptons and quarks. It is shown that themass matrix obtained this way <cit.> can accommodate the two large mixing angles observed in atmospheric and solar neutrino experiments.§.§ Mass terms for the doublets and triplets Returning to the content of Table <ref>, we observe that there is still freedom to accommodate additional vector-like 5-plets which respect all the required conditions. Hence, aiming to accommodate potential diphoton resonances and other possible experimental signatures of exotic matter beyond the MSSM spectrum, in the present constructionwe assume the existence of 5+5̅ pairs and discusspossible implications of the exotic states. As already explained, the Z_2 monodromy allows a tree-level coupling for the top quark 10_310_3 5_H_u. Furthermore, from the specific accommodation of the fermion generations listed in Table 2, we observe that a tree-level coupling for the bottom quark is also available. A geometric perspective of the Yukawa couplings in the internal manifold is depicted in figure <ref>. All other mass entries aregenerated from non-renormalisable terms <cit.>. Regardingthe 5-plets accommodating the MSSM Higgs,we observe that theflux splits the doublet from the triplet in the Higgs sector.As a result, theMSSM μ term θ_14θ_43θ_15/M_GUT^2 5̅_t_3+t_55_-2 t_1→V_1V_2V_3/M_GUT^2H_uH_d→μ H_uH_d. does not involve masses for the triplet fields. Fermion mass hierarchies require at least that the singlet vev V_1=⟨θ_14⟩≳ O(10^-1)M_GUT,so that the MSSM μ parameter can be kept light for v_2· v_3≪ v_1.In the general case, we need to take into account the extra doublet pairs emerging from the 5-plets remaining in the zero-mode spectrum. As an illustrative example, we take only one additional vector-like pair of 5-plets, that is n=1. In this casethe available coupling are5_H_u5̅_H_dθ _14θ _43θ _15/M_GUT^2 +5_H_u5̅_x̅θ _14θ _15/M_GUT + 5_x5̅_H_dθ _14θ _43/M_GUT+ 5_x5̅_x̅θ _14 . The Higgs mass matrix in the basis L⊃ (H_d,H_d') M_H ([H_u; H_u' ]) isM_H ∝ V_1 ( [ v _3 v _2v _3; v_2 1; ]) ,where the Yukawa couplings are suppressed to avoid clutter.This implies a light Higgs mass term μ∼ V_1v_2 v_3 and a heavy one M_H∼ V_1.The triplet mass terms emerge from different couplingsθ_14θ_15 5̅_t_4+t_55_-2 t_1/M_GUT + ϵθ_14 5̅_t_4+t_55_- t_1-t_5→θ_14θ_15 5̅_x̅5_H_u/M_GUT + ϵθ_14 5̅_x̅5_x · Hence, written in a matrix form L_D ⊃( 5_H_u, 5_x) M_D ( [ 5̅_H_d;5̅_x̅;]), where the triplet mass matrix is M_D= V_1([ v _2 ; ' v _21;]), and the parameters ≃'≲ 1 stand for corrections when more than one matter multiplets are on the same matter curve.Theeigenmasses alsodepend on the singlet vevs and will be discussed in conjunction with proton decay in the subsequent sections.In addition to thesesuperpotential couplings, the vector pairs 5+5̅ generate superpotential terms with the matter fields10_3 5̅_x̅5̅_2, 10_3 5̅_x̅(5̅_1θ_14+5̅_3θ_34), 10_1 5̅_x̅(5̅_1θ_14θ_43+5̅_2θ_43+5̅_3)θ_14, 10_2 5̅_x̅(5̅_1θ_14+5̅_2+5̅_3θ_34)θ_14 · where the non-renormalisable terms are assumed to be scaled by appropriate powers of M_GUT.In the next sections we will explore possible phenomenological consequences of (<ref>). However, we note that it is feasible to eliminatesuch couplings from the lagrangian byintroducinga differentR-parity assignment for the colour triplets.§GAUGE COUPLING UNIFICATION The presence of additional vector-like pairs of colour triplets and higgsinos with masses in the TeV rangeaffect the renormalisation group runningof the gauge couplings and the fermion masses.The existence ofcomplete 5+5̅SU(5) multipletsat the TeV scale may enhance processes that could be observed in future searches, while they can be consistent with perturbativegaugecoupling unification as long as their number is less than four. Threshold corrections from Kaluza-Klein (KK) modes and fluxes play a significant rôle <cit.> too.Under certain circumstances <cit.>, (for example when the matter fields are localised on genus one surfaces) the KK threshold effects can be universal, resulting to a common shift of the gauge coupling constant at the GUT scale. This has been analysed in somedetail in ref <cit.> and will notbe elaborated further. However, in F-theory constructions, there are additional corrections associated withnon-trivial line bundles <cit.>). More precisely, assuming that the SU(5) is generated by D7-branes wrappinga del-Pezzo surface, gauge flux quantization condition <cit.> implies that D7-branes are associated with a non-trivial line bundle L_a. On the other hand,the breaking of SU(5) occurs with a non-trivialhypercharge flux L_Ysupported on the del Pezzo surface, (but with a trivial restriction on the Calabi-Yau fourfold so that the associatedgauge bosonremains massless). The flux threshold corrections to the gauge couplingsassociated with these two line bundles can be computed bydimensionally reducing the Chern-Simons action.If we define y=1/2 ReS ∫ c_1^2( L_a), x=-1/2 ReS ∫ c_1^2( L_Y) , where, c_1( L) denotes the first Chern class of the corresponding line bundle and S=e^-ϕ+iC_0 is the axion-dilaton field (and g__IIB=e^ϕ), the flux corrections to the gauge couplings areexpressed as follows1/a_3(M_U) = 1/a_U-y 1/a_2(M_U) = 1/a_U-y+x 1/a_1(M_U) = 1/a_U-y+3/5x , where a_U representsthe unified gauge coupling. From (<ref>-<ref>) we observe that the corrections from the L_a line bundle are universal and therefore y can be absorbed in a redefinition of a_U. On the other hand,hypercharge flux thresholds expressed in terms of x, are not universal and destroythegauge coupling unification at the GUT scaleM_U.Notice that in order to eliminate the exotic bulk states (3,2)_5+(5̅,2)_-5 emerging from the decomposition of 24, we need to impose ∫ c_1^2( L_Y)=-2, and therefore we find the simple form x= e^-ϕ=1/g__IIB. The value of the gauge couplingsplitting has important implications on the mass scale of the color triplets discussed in the previous section. In the following we will explore this relation within the matter and Higgs field context of the present model. Weassume that the color tripletsD+ D^c∈ 5_H+5̅_H receive masses at a scale M_X, whilethe complete 5+5̅ extra multiplets obtain masses at a few TeV. The renormalisation group equationstake the form1/a_i(M_U)=1/a_i(M_U)+b_i^x/2πlogM_U/M_X+ b_i/2πlogM_X/μ · It can be readily checked that the GUT values of the gauge coupling satisfy5/31/a_1(M_U)= 1/a_2(M_U)+2/31/a_2(M_U) ·Assuming n_D pairsof (D+D^c) and n_V vector-like 5-plets,the beta functions areb_3^x=-3+n_V+n_D,b_2^x=1+n_V,b_1^x=33/5+2/5n_D+n_V b_3=-3+n_V,b_2=1+n_V,b_1=33/5+n_V ·Using (<ref>,<ref>) and (<ref>) we findlogM_U/M_X= 2π/β_x1/ A-β/β_xlogM_X/μwhere we introduced the definitionsβ = 5/3(b_1-b_3)+(b_3-b_2) β_x = 5/3(b^x_1-b^x_3)+(b^x_3-b^x_2) 1/ A = 5/31/a_1-1/a_2-2/31/a_3=1-2sin^2θ_W/a_e-2/31/a_3 · Noticethat for the particular spectrum,β_x,β are equal, β_x=β=12,and independent of the number of multiplets n_D and n_V.Then, from (<ref>) we find that the unificationscale isM_U=e^2π/12 AM_Z≈ 2.04× 10^16 GeV , i.e., independent of n_V, n_D and the intermediate scale M_X. To unravel the relation between the scale M_X and theparameter x, we proceed as follows. First, we subtract (<ref>)from (<ref>)x = 1/a_2-1/a_3+b_3^x-b_2^x/2πM_U/M_X+b_3-b_2/2πM_X/μ= 1/a_2-1/a_3-4-n_D/2πM_U/M_X-4/2πM_X/μ ·Using (<ref>) and the fact that in our model n_D=1, we findlogM_X/μ= 2π(6sin^2θ_W-1/4a_e-5/61/a_3-x) ·This determines the relation between theparameter x=e^-ϕ and the scale M_X where the Higgstriplets become massive.We can use the expression for M_U to express the M_X scale as followslogM_X/M_U=2π(5sin^2θ_W-1/3a_e-7/91/a_3-x) · To determine the value ofthe GUT coupling a_U we use (<ref>,<ref>) and (<ref>) to find1/a_U+x= 1/a_2-b^x_2/β_x1/ A= 1/a_2-1+n_V/121/ A ·For the present application, we allow three pairs of 5-plets, n_V=3, and we obtain the relation1/a_U = 5sin^2θ_W-1/3 a_3+2/91/a_3-x ·Substitution of (<ref>) in (<ref>) gives an elegant and very suggestive formula:M_X =e^2π(1/a_U-1/a_3) M_U ·We observe that in order to have M_X≤ M_U, we always need a_U≥ a_3≈1/8.5.We depict the main results in the figures that follow. In fig.<ref> we show the variationof the color triplets' decoupling scale versusthe range of values of the dilaton and, in fig.<ref>,we plot the inverse SM gauge couplings taking into account the thresholds of the color-triplets. §.§ RGEs for Yukawa Couplings Thesemodifications in the gauge sector and,in particular, the large g_U value compared to that of the standard MSSM unification scenario(g_U∼ 1/25 in MSSM) are expected to have a significant impact on the evolutionof the Yukawa couplings.On the other hand, inF-theory constructions the Yukawa coupling strengths at the unification scale are computed analytically and can be expressed in terms of the geometric properties of the internal six-dimensional compact space and the fluxes of the particular construction. For the sake of argument, we assumethat all three 5+5̅surplus matter fields receive masses in the TeV range,with tanβ values ∼ 48-50 andM_GUT∼ 2× 10^16 GeV. Then, according to <cit.>, the top mass, in particular, is achievedfor Yukawa couplingh_t(M_GUT) ≳ 0.35 which is significantly lower than the value ∼ 0.6 obtained in thecase of RG running with the beta-functions for the MSSM spectrum. Turning now to F-theory predictions, as we have seen, the Yukawa couplings are realised at the intersections of three matter curves. The properties of the corresponding matter fields in a given representation R arecaptured by the wavefunction Ψ_R whose profile isobtained by solving the Equations of Motion (EoM) <cit.>. It is found that the solution exhibits a gaussian profile picked along the matter curve supporting the particular state, Ψ_R ∝ f(z_i) e^M_ij z_iz̅_j. Here z_1,2 are local complex coordinates, the `matrix' M_ij takes into account background fluxes,and f(z_j) is a holomorphic function. The value of the Yukawa coupling results from integrating overthe overlappingwavefunctions. Thus, for the up/down Yukawa couplings,h_t ∝∫Ψ_10Ψ_10Ψ_ 5_H_u dz_1∧ dz̅_1∧ dz_2∧ dz̅_2,h_b ∝∫Ψ_10Ψ_5̅Ψ_5̅_H_d dz_1∧ dz̅_1∧ dz_2∧ dz̅_2 .The top Yukawa coupling is realised at the intersection where the symmetry is enhanced to E_6, while the bottom and τ Yukawa couplings are associated with triple intersections of SO(12) enhancements. We note in passing that the corresponding solution of the EoM providing the wavefunction for the up-type quark coupling is rather involved because of the monodromyand must be solved in a non-trivialbackground where the notion ofT-brane is required <cit.>. Usingappropriate background fluxes, we can break E_6 to SU(5),while the latter can break down to the SM gauge group with the hypercharge flux. To estimate the top Yukawa coupling, one has to perform the corresponding integration (<ref>). Varying the various flux parameters involved in the corresponding wavefunctions, it is found that thetop quark Yukawa takes values in the interval h_t∼ [0.3-0.5], in agreement with previous computations <cit.>, and hence the desired value h_t∼ 0.35 can be accommodated.In the present approach, the bottom and τ Yukawa couplings are formed at a different point of the compact space where the symmetry enhancement is SO(12). Proceeding in analogy with the top Yukawa, one can adjust the flux breaking mechanism toachieve <cit.>) the successive breaking to SU(5) and SU(3)× SU(2)× U(1). Further, for certain regions of the parameter space, one can obtain h_b, τ values in agreement with those predicted by the renormalisation group evolution <cit.>. § DECAY OF VECTORLIKE TRIPLETS While analysing the spectrum insection 4, we have seen that the existence ofvector-like triplets is a frequently occurring phenomenon. They can be produced in pairs at LHC through their gaugecouplings to gluons.However, such exotic particles are not yet observedandmust decay through higher dimensional operators through mixing with the MSSM particles. We start with the minimal model by setting n=0, in which case the only states beyond the MSSM spectrum are D^c, D found in the 5̅_x̅ and 5_H_u respectively.We will consider the case of their mixing with the third family whichenhances theirdecays, due to the large Yukawa coupling compared to thetwo lighter generations.he available Yukawa couplings which mix the down-type triplets areW ⊃ λ 10_t_15̅_t_1+t_45̅_t_3+t_5+λ_1 10_t_15̅_t_4+t_55̅_t_3+t_5θ_15/M_GUT +λ_2 5̅_t_4+t_55_-2t_1θ_14θ_15/M_GUT→ λ 10_35̅_35̅_H_d+λ_1 10_35̅_H_d5̅_x̅v_2 +λ_2 5̅_x̅5_H_u v_1 v_2 ,where the non-renormalisable terms are scaled by the appropriate powers of the compactification scale or the GUT scale. These terms generate a mixing matrix of thethird generation down quark andD^c,D which can be cast in the formL_Y⊃(Q_3, D) M_D ( [ b^c; D^c; ]),M_D∝( [ λ/√(2)v_d λ _1/√(2) v_2 v_d; 0λ _2 v_1 v_2; ]) ,where v_d stands for the down Higgs vev scaled by the GUT scale. This non-symmetricmatrix M_D is diagonalised by utilizing the left and right unitary matricesM_D^δ= V_L^† M_D V_R,implyingM_D^δ^2= V_L^† M_D M_D^†V_L= V_R^† M_D^†M_D V_RwhereM_DM_D^†= ( [ 1/2λ ^2 v_d^2+1/2λ _1^2 v_2^2v_d^2 λ _1 λ _2/√(2)v_1 v_2^2v_d; λ _1 λ _2/√(2) v_1 v_2^2 v_d λ _2^2v_1^2v_2^2;])andM_D^†M_D= ( [ 1/2λ ^2 v_d^2 1/2λλ _1v_2 v_d^2; 1/2λλ _1v_2 v_d^2 1/2λ _1^2 v_2^2 v_d^2 +λ _2^2 v_1^2 v_2^2; ]) · Following standard diagonalisation procedures, in the limit v_1≫ v_2≫ v_d, we find that the left mixing angle is tan 2θ_L = √(2)λ _1 λ _2 v_1 v_2^2 v_d/-1/2λ ^2v_d^2-1/2λ _1^2 v_2^2 v_d^2+λ _2^2 v_1^2 v_2^2≈√(2)λ _1 v_d/λ _2 v_1, and for the right-handed mixing we obtain tan 2θ_R =λλ _1 v_2 v_d^2/-1/2λ ^2 v_d^2+1/2λ _1^2 v_2^2 v_d^2+λ _2^2 v_1^2 v_2^2≈λλ _1 v_d^2/λ _2^2 v_1^2 v_2 . From these, we findtan(2θ_R)≈λv_d/√(2)λ _2 v_1 v_2tan(2θ_L) · For the assumed hierarchy of vevs we see that the left-mixing prevails.The mixing is restricted by CKM constraints and the contributions of the heavytriplets to the oblique parameters S,T which have been measured with precisionin LEP experiments(Fordetailed computations see <cit.>.). A rough estimate would give the upper boundstan 2θ_L∼ 0.1, tan 2θ_R∼ 0.3which can be easily satisfied for thev_1, v_2 valuesused in this work. §.§ Proton decayIn this model the dimension-five proton decay R-parity violating tree-level couplings of the form10_f5̅_f5̅_f are absent due to the t_i charge assignments of matter fields. However, non-renormalisable terms that could lead tosuppressed baryon and lepton number violating processes may still appear.A classof these operatorshave the general structure λ_eff10_i 5̅_t_j+t_k5̅_t_l+t_m,; λ_eff∼⟨θ_pq^n⟩ , i,i,j,l,m 5 , where θ_pq^n represents products of singlet fields required to cancel thenon-vanishing combinations of t_i,j... charges. Notice, however, that for the particular family assignment in this model none of t_i,j,k,l,m in (<ref>) is t_5 and therefore, to fulfil the condition ∑_k=1^5t_k=0 some singlet θ_5s≡ 1_t_5-t_s, with s=1,2,3,4, always must beinvolved. [Notice however, that all possiblehigher order R-parity violatingterms10_1 5̅_1 ( 5̅_1θ_14θ_53+5̅_2θ_53+ 5̅_3θ_54) θ_13 +10_1 5̅_2 ( 5̅_2θ_43+5̅_3) θ_53 +10_25̅_35̅_3θ_54can be eliminated due tothe R-parity assignment of the singlets θ_ij shown in Table <ref>.] In the present model no singlet of this kind acquires a non-zero vev, namely ⟨θ_5s⟩≡ 0, and hence dimension four operators are suppressed. However, as already pointed out, additional Yukawa termsgive rise to new tree-level graphs mediated by color triplets. Such graphs induce dimension-5 operators of the form 1/M_effQQQℓ, 1/M_effu^cu^cd^ce^c, where M_eff is an effective colour triplet mass M_eff≥ M_GUT∼ 2.0× 10^16 GeV<cit.>. Here, because of the missing triplet mechanism described in the previous section, the D, D^c tripletsdevelop masses through mixing with other heavy triplets D_i, D_i' emerging from the decomposition of the additional 5+5̅-pairs. Besides, severalcouplings are realised as higher order non-renormalisable termsso that, in practice, an effective triplet mass M_eff is involved which, with suitable conditions on the triplet mixing, could be of the order ofthe GUT scale. For the case of the Higgsino exchang diagram, for example, with a Higgsino mass identified with the supersymmetry breaking scale M_S, the proton lifetime is estimated to be <cit.>τ_p ≈10^35 (√(2)sin 2β)^4 (0.1/C_R)^2 (M_S/10^2 TeV)^2(M_D_eff/10^16 GeV)^2 ,where the coefficientC_R≥ 0.1,taking into account the renormalisation group effects on the masses. From (<ref>) we infer that with an effective triplet mass ≳ M_GUT and a relatively high supersymmetry breaking scale, proton decay can be sufficiently suppressed in accordancewith the Super-Kamiokande bound on the proton lifetime.To estimate the effects of these operators in this model, we consider the triplet mass matrix derived in the previous sectionM_T = ([ λθ _14θ _15 'θ _14θ _15; θ _14 θ _14; ]) θ_14 → ( [λ v_2 ϵ' v_2; 1;])⟨θ_14⟩, with v_2 =⟨θ_15⟩/M_GUT andv_1 =⟨θ_14⟩/M_GUT as defined in (<ref>). As before, the left and right unitary matrices V_L,V_R, as well as the eigenmasses are determined by M_T^δ^2= V_L^† M_T M_T^†V_L= V_R^† M_T^†M_T V_R where, in general,M_T M_T^† andM_T^†M_T are Hermitian but, for simplicity, we will take to be symmetric, M^2∼([ a b; b d; ])⟨θ_14⟩^2,with real entries and triplet eigenmasses M^2_1,2=1/2(a+d±√(4 b^2+(a-d)^2))⟨θ_14⟩^2.In figure <ref> a representative graph is shownmediated by the colour triplets leading to the dominantproton decay mode p→ K^+ν̅.The mass insertion (red bullet)in the graph isλ⟨θ _14θ _15⟩/M_GUT ≡ ⟨Φ⟩ . Aftersumming over the eigenstates, one finds that the effective mass involved is 1/M^0_eff∝∑_j V_1j⟨Φ⟩/M_j^2V_j2^†→ (λv_2/v_11/M_GUT) b/ad-b^2 ,while there is an additional suppression factor v_1=⟨θ_14⟩/M from the non-renormalisable term (yellow bullet in the graph). Finally 1/M_eff∼v_1/M^0_eff. For the V_L mixing, assuming reasonable values for the parameters ϵ, ϵ'<1, while taking v_1∼ O(10^-1) and λ∼ 1 we findM_eff∼v_2/v_1 M_GUT .For the V_R case we findM_eff∼M_GUT/ϵ .For a supersymmetry breaking scale M_S in the TeV region, we conclude that the lifetime of the proton is consistent with the experimental boundsfor an effective mass M_effa few times largerthan M_GUT which can be satisfiedfor [Since the mass insertion⟨Φ⟩ 5_H5̅_x̅∝ v_2 one would expect thatfor v_2≪ 1 the contribution of the graph to Proton Decay would be small.However,the element bcancels the effect because it is also proportional to b∝ v_2.]v_2>v_1 and <1.Thereare implications for theμ term given by μ∼v_1v_2v_3 (see Eq. <ref>). Since v_1, v_2 cannot be small, in order to sufficiently suppress this term we must havev_3=⟨θ_43⟩/M_GUT≪ 1.On the other hand, the smallness of v_3 suppresses also the mass scale of the lighter generations andmight lead to inconsistences with the experimental values. We should recall, however, that there are significant contributions to the fermion masses fromone-loop gluino exchange diagrams <cit.> implying masses of the orderm_u/d∝a_3/4πA_qm_q̃m_t/b/m_g̃^2 for the up/down quarks, where A_q, m_g̃, m_q̃ are respectively the trilinear parameter, thegaugino and squark masses. We have already stressed thatthe presence of additional vector-like 5-plets in themodel under consideration, is compatible with a smaller value of the unified gauge coupling g_U∼ 10^-1 at the GUT scale. This has significant implications for the protondecay rate which occurs through the exchange of the gauge bosons (dimension six operators).For the well known casep→ e^+π^0 the life-time is estimated to be <cit.>τ(p→ e^+π^0)≈8× 10^34 years ×( a_U^0/a_U×2.5/A_R×0.015GeV^3/a_H)^2×(M_V/10^16 GeV)^4 · The various quantities in the above formula are as follows:a_U^0=1/25 is assumed to be the value of the unified gauge coupling in the minimal SU(5),while a_U stands for itsvalue for the present model which is taken to be a_U≈1/8.5 (seesection 5). The factor A_R takes into account the various renomalisation effects, a_H is the hadronic matrix element and M_V denotes the mass of the gauge boson mediating the process p→ e^+π^0.Comparing with the recent experimental limit <cit.>τ(p→ e^+π^0)≥ 1.6× 10^34 years, we find a lower bound on the mass of the gauge boson M_V≥ 1.14× 10^16 GeV, which is reciting since it just below the GUT scale predicted in this model M_U≈ 2.04× 10^16 GeV. §.§ Variation with new physics predictions accessible at LHCIn this section we consider the possibility of predicting new physics phenomena (such as diphoton events) from relatively light (∼ TeV) scalars and triplets.The model discussed so far cannot accommodate a process such as the diphoton event, since there is no direct coupling D^cD S with a light singlet S. Indeed, the only singlet coupled toD^c,D is θ_14 which acquires a large vev and decouples. To circumventthis we briefly present a modification of the above model by assuming the following non-zero vevs, v_1=⟨θ_13⟩ ,v_2=⟨θ_34⟩ ,v_3=⟨θ_43⟩ , and we maintain the same assignments for the fermion generations listed in Table <ref>. The mass matrices for the up, down quarks and charged leptons, are given bym_u∼([v_1^2 v_1^2v_2v_1; v_1^2v_2 v_1^2v_2^2 v_1v_2;v_1 v_1v_21 ])h_t⟨ H_u⟩ ,m_d,ℓ∼([v_1^2 v_1v_3v_1; v_1^2v_2v_1 v_1v_2;v_1v_31 ])h_b⟨ H_d⟩,where, as before, we have suppressed the Yukawa couplings expected to be of O(1). We observe that the matrices exhibit the expected hierarchical structure. Assuming a natural range of the vevs and Yukawa couplings weestimate that the fermion mass patterns are consistent with the observed mass spectrum. With this modification, the singletθ_14 is not required to acquire a large vev and it can remain as a light singlet θ_14=S'. Through its superpotential couplingθ_145̅_x̅5_x → S' ( D”^c D+ H_u'H_d') ,whereD”^c stands forthe linear combination D”^c = cosϕ D^c+ sinϕD'^c, S' could contribute to diphoton emission.§ SUMMARYF-theory appears to be a natural and promisingframework for constructing unified theories with predictive power. TheSU(5) GUT model in particular, appears to be the most economic unified group containing all those necessary ingredients to accommodate vectorlike fermions that might show up infuture experiments. Therefore, in the light of possible new physics at the LHC experiments, in this letter, we reconsidered a class of F-theory SU(5) models aiming to concentrate on the specific predictionsand low energyimplications.In the F-theory framework, after the SU(5) breaking down to the Standard Model gauge symmetry, we end up with the MSSM chiral mass spectrum, the Higgs doublet fields and usually a number of vector-likeexotics as well as neutral singlet fields. We point out that we dispense with the use of large Higgs representations for theSU(5) symmetry breaking since the latter takes place by implementingthe mechanism of the hypercharge flux. The corresponding U(1)_Y gauge field remains massless by requiring the hypercharge flux to be globally trivial. As a result of these requirements, the spectrum of the effective theory and the additional abelian symmetries accompanying the GUT group, are subject to certain constraints. In addition to the SU(5) GUT group, themodel is subject to additional symmetry restrictions emanating from theperpendicular `spectral cover' SU(5)_⊥ group, which in the effective theory reduces down to abelian factors according to the `breaking' chainSU(5)_⊥⊃ U(1)_⊥^4U(1)_⊥^3where Z_2 is the monodromy action, chosen for this particular class of models under discussion [For the SU(5)_⊥ spectral cover symmetry, the possible monodromies fall into a discrete subgroup of the Weyl group W(SU(5)_⊥)∼ S_5, with S_5 being the permutation symmetry of five objects.]. A suitable choice of fluxes along these additional abelian factors is responsible for the chirallity of the SU(5) GUT representations and their propagation on the specific matter curves presented in this paper.In practice,the effects of the remaining spectral cover symmetryin the low energy effective theoryare described by a few integers (associated with fluxes) and the `charges'-roots t_i, i=1,2,…, 5of thespectral cover fifth-degree polynomial where two of them, namely t_1,2, are identified under the action of themonodromy Z_2: t_1↔ t_2 applied in this work. The implementation of the hyperflux symmetry breaking mechanism has additional interesting effects. As is known, chiral matter and Higgs fields reside on the intersections (i.e., Riemann surfaces, dubbed here as matter curves and characterised by the remaining U(1) factors through the `charges' t_i) of seven braneswith those wrapping the SU(5) singularity. In general the various intersections are characterised by distinct geometric properties andas a consequence flux restricts differentlyon each of them, while implying splittings of the SU(5) representation content in certain cases.As a result, in the present model doublet Higgs fieldsare accommodated on matter curves which split the SU(5) representations realising an effective doublet-triplet splitting mechanism in a natural manner. More precisely,this ammounts to removing one tripletfrom the initial Higgs curve with the simultaneous appearance (excess) of another one on a different matter curve. This displacement however is enough toallow a light mass term for the Higgs doublets while heavytriplet-antitriplet mass terms originate from different terms leading to suppression of baryon number violating processes. Chiral fermion generations are chosen to be accommodated on different matter curves, so that a Froggatt-Nielsen type mechanism is implemented to generate the required hierarchy. Furthermore, certain Kaluza-Klein modes are associated with the right-handed neutrino fields implementing thesee-saw mechanism through appropriate mass terms with their left-handed counterparts. The additional spectrum in the present model consists of neutral singlet fields as well ascolour triplets and Higgs-like doublets comprising complete SU(5) vector-like pairs in 5+5̅ multiplets, characterised by non-trivial t_i-`charges'. Some singlet fieldsare allowed to acquire vevs at the TeV scale inducingmasses of the same order for the vector-like exotics through the superpotential terms. Such `light'exotics contribute to the formation ofresonances producing excess of diphoton events which could be discovered in future LHC experiments.A RGE analysis shows that the resulting spectrum is consistent with gauge coupling unification and the predictions of the third family Yukawa couplings.Acknowledgements.The authors would like to thank R. Blumenhagen, S.F. 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"authors": [
"George K. Leontaris",
"Qaisar Shafi"
],
"categories": [
"hep-ph",
"hep-th"
],
"primary_category": "hep-ph",
"published": "20170626135327",
"title": "Phenomenology with F-theory SU(5)"
} |
1Yale Center for Astronomy & Astrophysics, Physics Department, P.O. Box 208120, New Haven, CT 06520, USA, [email protected] 2Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA 3Department of Astronomy, University of Massachusetts, Amherst, MA 01002, USA 4 University of California, 501 Campbell Hall, Berkeley, CA 94720 Santa Cruz, USA 5INAF-Osservatorio di Radioastronomia, Via Piero Gobetti 101, I-40129 Bologna, Italy 6Department of Physics and Astronomy, Colby College, Waterville, ME 04901, USA 7Departamento de Astrofísica y CC. de la Atmósfera, Universidad Complutense de Madrid, E-28040 Madrid, Spain 8Dipartimento di Fisica e Astronomia, Università di Padova, vicolo dellOsservatorio 2, 35122 Padova, Italy 9Department of Physics & Astronomy, Tufts University, Medford, MA 02155, USA 10Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA 11Department of Physics and Astronomy, University of Missouri-Kansas City, 5110 Rockhill Road, Kansas City, MO 64110, USA 12UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA 13ESA/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands 14INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy The bulk of the stellar growth over cosmic time is dominated by IR luminous galaxies at cosmic noon (z=1-2), many of which harbor a hidden active galactic nucleus (AGN).We use state of the art infrared color diagnostics, combining Spitzer and Herschel observations, to separate dust-obscured AGN from dusty star forming galaxies (SFGs) in the CANDELS and COSMOS surveys. We calculate 24 μm counts of SFGs, AGN/star forming “Composites”, and AGN. AGN and Composites dominate the counts above 0.8 mJy at 24 μm, and Composites form at least 25% of an IR sample even to faint detection limits. We develop methods to use the Mid-Infrared Instrument (MIRI) on JWST to identify dust-obscured AGN and Composite galaxies from z∼1-2. With the sensitivity and spacing of MIRI filters, we will detect >4 times as many AGN hosts than with Spitzer/IRAC criteria. Any star formation rates based on the 7.7 μm PAH feature (likely to be applied to MIRI photometry) must be corrected for the contribution of the AGN, or the SFR will be overestimated by ∼35% for cases where the AGN provides half the IR luminosity and ∼ 50% when the AGN accounts for 90% of the luminosity. Finally, we demonstrate that our MIRI color technique can select AGN with an Eddington ratio of λ_ Edd∼0.01 and will identify AGN hosts with a higher sSFR than X-ray techniques alone. JWST/MIRI will enable critical steps forward in identifying and understanding dust-obscured AGN and the link to their host galaxies. § INTRODUCTIONThe galaxies most actively contributing to the buildup of stellar mass at cosmic noon (z∼1-2) contain large amounts of dust <cit.>. This dust obscures the majority of star formation, making it necessary to study these galaxies through their dust emission at infrared wavelengths <cit.>. Additionally, the majority of supermassive black hole growth at these redshifts is also heavily dust-obscured <cit.>. Many of the massive dusty galaxies contain a true mix of star formation and obscured black hole growth, the obscured signatures of which can be seen in their infrared spectral energy distribution (SED). These galaxies are then ideal laboratories for understanding the physical link between star formation and active galactic nuclei (AGN). The AGN-star formation connection is an open question, particularly whether AGN feedback is a key component of star formation quenching, and whether all galaxies have a distinct star formation phase followed by an AGN phase before ultimately quenching <cit.>. The nature of AGN within strongly star forming galaxies (what we term “Composites”) is even more uncertain. Do these objects represent a unique phase between star forming galaxies (SFGs) and AGN? Unfortunately, due to limitations of previous space telescopes, detailed studies of the energetics of these objects were severely restricted, but the James Webb Space Telescope (JWST) will reveal their true nature.Prior to JWST, the most reliable method for identifying Composites and disentangling AGN emission from star formation was mid-IR spectroscopy from the Spitzer Space Telescope. The low resolution spectra can be modeled as a combination of star formation features (most notably the polycyclic aromatic hydrocarbons, or PAHs, that exist in photodissociation regions and in stellar/Hii regions), and hot continuum emission primarily arising from a dusty torus surrounding the accreting black hole <cit.>. In this way, the division of IR luminosity between star formation and an AGN can be quantified. The medium-resolution spectrometer <cit.>, which is part of the Mid-Infrared Instrument (MIRI) on JWST, will enable separation of PAH emission from continuum in the same manner, but with higher resolution and on smaller spatial scales within host galaxies. It will also enable detection of high ionization gas lines excited by the AGN <cit.>, further improving our ability to detect and measure the physical properties (such as accretion rates and Eddington ratios) of dust-obscured black holes.As there are only a few hundred Spitzer IRS spectra available for distant galaxies <cit.>, color techniques were also developed to identify large samples of luminous dust-obscured AGN.The most popular color selection techniques are with Spitzer IRAC photometry <cit.>, which separate AGN using different combinations of the 3.6, 4.5, 5.8, and 8.0 μm filters. The original techniques presented in <cit.> and <cit.> were limited to the most luminous AGN and become increasingly contaminated with galaxies when deeper IR data are used <cit.>. Moreover, with increasing redshift, the rest wavelengths of these bands decrease, causing contamination of the AGN signatures by star forming galaxies to become significant such that the original IRAC-based criteria cannot be applied. <cit.> propose more conservative IRAC criteria that, at cosmic noon, essentially separate galaxies that exhibit a so-called stellar bump (emission from stars that peaks at ∼1.6 μm and then declines to a minimum around ∼5 μm) from those that do not, where the torus radiation is strong enough to fill in the dip in the star forming spectrum around 3-5 μm, producing power-law emission such as is typical of unobscured AGN <cit.>. The <cit.> criteria increase the reliability of AGN color selection, although they are less complete due to excluding Composites, where the IR emission of the AGN does not dominate over the star formation.For the purposes of probing the AGN-star formation connection, the limitation of IRAC techniques is that AGN within strongly star forming galaxies can have different levels of host contamination. Then, many galaxies containing AGN signatures at longer wavelengths will also include a stellar bump and therefore be missed <cit.>. To alleviate host contamination, <cit.> propose combining K-band with IRAC and 24 μm to separate AGN from host galaxies all the way out to z∼7.Going further, including mid-IR and far-IR colors can greatly improve the selection of Composite galaxies, since this will trace the contribution of warmer AGN-heated dust compared with cold dust from the diffuse interstellar medium in the host galaxy <cit.>. However, this requires observations from the Herschel Space Observatory, which have a large beam size and do not reach the same depths as Spitzer observations.MIRI will greatly improve color selection techniques due to the increased sensitivity and the number of transmission filters covering the mid-infrared <cit.>. Now, we will be able to separate AGN from SFGs by comparing PAH emission with the minimum emission from stars that occurs around 5 μm; in AGN, the stellar minimum is not visible due to strong torus emission, and Composites will lie in between strong AGN and pure SFGs in colorspace. In this paper, we build on the Herschel and Spitzer color selection techniques initially presented in <cit.> to identify Composite galaxies at z∼1-2 using the CANDELS and COSMOS surveys. We present galaxy counts of 24 μm sources classified as SFGs, AGN, or Composites based on their IR colors, making this the first identified statistical sample of Composites at cosmic noon. We use this sample to predict black hole and star formation properties of samples that JWST/MIRI will identify. We also present color diagnostics for identifying both AGN and Composites using JWST/MIRI filters in three redshift bins. Throughout this paper, we assume a standard cosmology withH_0=70 km s^-1 Mpc^-1, Ω_M=0.3, and Ω_Λ=0.7.§ CANDELS AND COSMOS CATALOGSTo calculate galaxy counts, we use Spitzer and Herschel photometry from the COSMOS, EGS, GOODS-S, and UDS fields from the Cosmic Assembly NearIR Deep Extragalactic Survey (CANDELS, P.I. S. Faber and H. Ferguson; GOODS-Herschel, P.I. D. Elbaz; CANDELS-Herschel, P.I. M. Dickinson). We do not include GOODS-N as, at the time of the writing of this paper, the IR catalog does not have uniquely identified optical counterparts. We also use photometric redshifts <cit.> and M_∗ <cit.>. The stellar masses are derived by fitting the CANDELS UV/Optical photometry in ten different ways, each fit using a different code, priors, grid sampling, and star formation histories (SFHs). The final M_∗ is the median from the different fits, and it is stable against the choice of SFH and the range of metallicity, extinction, and age parameter grid sampling. The CANDELS z_ phots are the median redshift determined through five separate codes that fit templates to the UV/optical/near-IR data <cit.>. Taking the median of several methods improves the accuracy, and comparison of z_ phots with spectroscopic redshifts for a limited sample gives σ = 0.03 where σ is the rms of (z_phot-z_spec)/(1+z_spec). As we sort sources into redshift bins of Δ z = 0.5, we do not expect the uncertainty on the photometric redshifts to be a dominant source of uncertainty in our results. We will be using the z_ phots to help classify sources as AGN, SFGs, or Composites. MIPS 24 μm and Herschel PACS and SPIRE catalogs were built following the prior-based PSF fitting method described in <cit.> and <cit.>. For additional details on the methods used for Herschel catalog building, see <cit.>. Briefly, the algorithm uses IRAC and MIPS data to extract photometry for sources in longer wavelength data using positional priors. Deblending is not possible when sources lie closer than 75% of the FWHM of the PSF in each band, making this value a minimum separation required to perform deblending.The final product of the cataloging method is a list of IRAC sources with possible counterparts in all longer wavelength data. In this sense, several IRAC sources might be identified with the same MIPS or Herschel source. This is what we call multiplicity. The multiplicity for MIPS and PACS is in more than 95% of the cases equal to 1 (i.e., only one IRAC source is identified with a single MIPS and PACS source), but it is higher for SPIRE (on average, 6 IRAC sources are found within the FWHM of the SPIRE 250 μm PSF). In order to identify the “right” IRAC counterpart for each far-IR sources, we follow the method described in Rodríguez-Muñóz et al. (2017, in prep). In practice, we choose the MIPS most probablecounterpart as the brightest IRAC candidate. Then, we shift this methodology to longer wavelength bands. We identify the most likely PACS counterpart as the brightest source in MIPS 24 μm among the different candidates. When MIPS is not available, we use the reddest IRAC band in which the source is detected. We note that using IRAC as a tracer of PACS emitters can lead to spuriousidentifications. For this reason, these cases are flagged to evaluate the possible impact in the results. Finally, we use the fluxes in PACS or MIPS (if PACS is not available) to find the counterparts of the SPIRE sources. The flux of each FIR source is assigned to a single IRAC counterpart. The FWHM of the PACS PSF is roughly the same as for MIPS, so the most serious concern in this work is matching to the SPIRE 250 μm sources. We are primarily using the IR photometric catalogs to calculate galaxy counts. As a check, we remove all classifications of galaxies (as SFG, AGN, and Composites) that were done with SPIRE data (described in the following section). Our main result, the galaxy counts at cosmic noon, are unchanged, giving confidence that any misidentification of a SPIRE sources with a MIPS and IRAC counterpart is not biasing our results. We have also added sources from the COSMOS survey <cit.> which are necessary to boost the bright end of the galaxy counts, due to the small survey area of CANDELS (0.22 deg^2). We use the public COSMOS2015 catalog in <cit.>, which presents multiwavelength data as well as stellar masses and photometric redshifts. The Spitzer IRAC data in this catalog originally comes from SPLASH COSMOS and S-COSMOS <cit.> while the MIPS 24 μm observations are described in <cit.>. The catalog also contains Herschel observations from the PEP guaranteed time program <cit.> and the HERMES consortium <cit.>. The counterpart identification and procedures for measuring stellar masses and photometric redshifts are fully described in <cit.>.The difficulty in matching MIPS, PACS, and SPIRE sources with their IRAC counterparts underscores the improvements that will be made by using MIRI color selection to identify AGN host galaxies, since the much smaller PSF (<1” for all filters) and smaller spectral range used will obviate the need for counterpart identification for robust color diagnostics. § IR IDENTIFICATION OF AGN AND COMPOSITES To identify SFGs, Composites, and AGN, we build on the color techniques in <cit.> that sample the full IR SED. At z∼1-2, the color S_8/S_3.6 separates sources with a strong stellar bump, present in SFGs, from those with hot torus emission, found in AGN. Composites span a range in this color, depending on the ratio of relative strengths of the AGN and host galaxy emission and the amount of obscuration of the AGN due to dust.[In fact, heavily obscured AGN such as Mrk 231, NGC 1068, the Circinus galaxy, and IRAS 08572+3915 have SEDs that drop rapidly from 10 μm toward shorter wavelengths and will show the near IR stellar spectral peak characteristic of SFGs. Hereafter, we refer to `AGN' with the understanding that the samples discussed may suffer from incompleteness of sources like these. This issue is discussed further in Section 4.1.] S_100 and S_250 trace the peak of the IR SED, which is generally dominated by the cold dust in the diffuse ISM. S_24 traces the PAH emission in SFGs or the warm dust emission heated by the AGN. Then, the color S_250/S_24 or S_100/S_24 will measure the relative amounts cold emission to warm dust or PAH emission, and this ratio is markedly higher in SFGs. However, significant scatter is introduced into color selection by redshift, since S_24 will move over different PAH features and silicate absorption at 9.7 μm, changing where SFGs lie in color space. We can more robustly identify SFGs, AGN, and Composites if we introduce a redshift criterion.The color diagnostics (S_250/S_24 vs. S_8/S_3.6 and S_100/S_24 vs. S_8/S_3.6) were calibrated with a sample of 343 galaxies with Spitzer IRS spectroscopy and S_24>0.1mJy spanning the range z∼0.5-4 and M_∗ >10^10 M_⊙. This sample is fully described in <cit.>, <cit.>, and <cit.>. We identified SFGs, Composites, and AGN through spectral decomposition, where we fit the mid-IR spectrum (5-18 μm restframe) with a model consisting of PAH features for star formation, a power-law continuum for the AGN, and extinction. We then quantified the AGN emission, , as the fraction of mid-IR luminosity (5-15 μm) due to the power-law continuum. We define three classes of galaxies based on , and we also report the fraction of MIR luminosity solely due to emission from the PAH features in the 5-15 μm range: (1) SFGs are dominated by PAH emission (<0.2, L_ PAH/L_ MIR > 0.6); (2) AGN have negligible PAH emission (>0.8, L_ PAH/L_ MIR < 0.15); (3) Composites have a mix of PAH and continuum emission (=0.2-0.8, L_ PAH/L_ MIR = 0.15 - 0.6). We note that below, we will redefine these thresholds for color selection. We relate the mid-IR classification to the full IR SED by creating empirical templates using data from Spitzer and Herschel. We sort sources into subsamples based on , and after normalization, determine the median L_ν in differential bin sizes of λ <cit.>. The <cit.> SEDs are the first comprehensive public library of IR templates specifically designed for high redshift galaxies that account for AGN emission.We create a redshift dependent color diagnostic through use of the empirical MIR-based template Library from <cit.>.[There are many AGN templates in the literature. In the 1-20 μm (rest wavelength) range critical for most of our color sorting the AGN templates agree well <cit.>. At wavelengths longer than 20 μm, there is considerable divergence; fortunately for our goals, the star forming output is so dominant by 100 and 250 μm that the range of possibilities for AGN output has little effect on our results.] We use a template library because our spectroscopic sample of 343 sources is not large enough to separate sources into multiple z bins. The MIR-based Library contains 11 templates created from our spectroscopic sources that demonstrate the change in IR spectral shape as the contribution of the AGN to the mid-IR luminosity increases, in steps of Δ=0.1. We randomly redshift each template 500 times, uniformly sampling a redshift distribution from z=0.75-2.25. We convolve each redshifted template with the observed frame IRAC, MIPS, PACS, and SPIRE transmission filters to create photometry, and then we resample the photometry within the template uncertainties at that particular wavelength, following a Gaussian distribution. We now have a catalog of 5500 synthetic galaxies, where we know the intrinsic AGN contribution, that represent the scatter in colorspace of real galaxies.Next, we create color diagrams in redshift bins of z=0.75-1.25, z=1.25-1.75, and z=1.75-2.25. Beyond this redshift, it becomes too difficult to reliably separate Composites from SFGs with these colors. Because only a fraction of CANDELS and COSMOS sources have a SPIRE or PACS detection, we also create a color diagnostic using the colors S_24/S_8 vs. S_8/S_3.6, although this is slightly less accurate. In each redshift bin, we divide the color space into regions of 0.2×0.2 dex and calculate the averageand standard deviation, σ_ AGN, of all the synthetic galaxies that lie in that region. In the Appendix, we show our three diagnostics: S_250/S_24 v. S_8/S_3.6 (used when a galaxy has the appropriate photometry, as it is the most complete at selecting Composite galaxies), S_100/S_24 v. S_8/S_3.6 (used when a galaxy does not have a 250 μm detection), and S_24/S_8 v. S_8/S_3.6 (used for all galaxies without a longer wavelength detection). Our color diagnostics assign sources anin bins of Δ=0.1, but the σ_ AGN of each region is often larger than this (see the Appendix for a visual representation). Therefore, it is more accurate to broadly group sources as SFGs, Composites, and AGN. We determine how to group sources by comparing theassigned to each synthetic galaxy by the three different color diagnostics. There is a one-to-one correlation between (250 ), (100 ), and (24 ), with a scatter of σ=0.15. Accordingly, we classify as SFGs sources with<0.30, while the AGN have >0.70, and Composites are everything in between. We assess the completeness and reliability of our color technique by determining how many of our synthetic galaxies are correctly identified as SFGs, Composites, and AGN in each diagnostic, and we list the completeness and reliability in Table <ref>. In the following definitions, we use N_ input to represent the total number of intrinsic objects (so N_ AGN,input is number of synthetic galaxies that are intrinsically AGN) and n_ sel to represent the number of objects recovered by our color criteria (so n_ AGN,sel is the number of intrinsic AGN that our color selection identifies as AGN). Completeness is defined as the fraction of AGN (for example) selected: n_ AGN,sel/N_ AGN,input. Reliability is the fraction of all the sources selected by the diagnostic as AGN (for example) that actually are, intrinsically, AGN: n_ AGN,sel/n_ all,sel. The lower completeness and reliability of the Composites and SFGs is due to these sources being more easily confused with each other when relying on the limited SED coverage (particularly of the mid-IR) provided by 8.0, 24, 100, and 250 μm. By adding more bands, MIRI will allow for a more nuanced measurement of the strength of the PAH emission compared with continuum and stellar bump emission. It is also important to note that we are missing AGN with extreme obscuration, whose IR colors could mimic those of SFGs. We discuss this issue more fully in Section 4.1.We assign each CANDELS or COSMOS source with z=0.75-2.25 anand associated uncertainty (σ_ AGN) and then broadly group sources into SFGs, Composites, and AGN.Overall, from CANDELS (COSMOS), 534 (6426) sources have been classified with S_250/S_24 v. S_8/S_3.6, 864 (175) with S_100/S_24 v. S_8/S_3.6, and 871 (5360) with S_24/S_8 v. S_8/S_3.6. From CANDELS, we also fit an additional 111 sources, which lie slightly beyond the regions (within 0.2 dex) in our color classification scheme, with the <cit.> template library to determine the classification.lccccc 3in Completeness (Reliability) of Redshift Dependent Color Selection Region z∼1 z∼1.5z∼2 S_250/S_24 v. S_8/S_3.6 AGN 93 (85)% 92 (89)% 97 (86)% Composite 67 (67)% 81 (63)% 56 (66)% SFG 66 (77)% 41 (75)% 64 (64)% S_100/S_24 v. S_8/S_3.6 AGN 97 (89)% 94 (89)% 98 (81)% Composite 69 (71)% 76 (64)% 42 (59)% SFG 67 (75)% 46 (69)% 62 (57)% S_24/S_8 v. S_8/S_3.6 AGN 93 (85)% 90 (88)% 97 (83)% Composite 69 (65)% 67 (66)% 48 (82)% SFG 60 (77)% 61 (65)% 87 (68)% Completeness is defined as the percentage of sources of a given intrinsic classification that are also selected by the color diagnostic. Reliability (shown in parenthesis) is the percentage of all sources classified in a given category where the intrinsic classification agrees. §.§ Galaxy and AGN CountsNow, we determine how traditional 24 μm number counts break down into the SFG, Composite, and AGN categories. We only consider sources with S_24>80 μJy, which is the 80% completeness limit <cit.>. We measure directly the EGS, COSMOS, GOODS-S, and UDS field sizes covered by our sources. We show the total CANDELS+COSMOS 24 μm number counts as the open grey stars in Figure <ref>, and these counts are in agreement with the counts from <cit.>. We plot the 24 μm counts at cosmic noon (z=0.75-2.25) as the filled black stars. There is a disagreement with the full counts that arises from applying the redshift cut, and this chiefly affects the bright end (S_24 > 1mJy), which is where AGN will dominate the counts <cit.>. The lack of bright sources is a result of the small field sizes of CANDELS (0.22 deg^2) and COSMOS (2 deg^2). We show how the cosmic noon counts break down into SFGs (blue), Composites (purple), and AGN (orange). We have calculated uncertainties on the counts using a Monte Carlo technique, where we vary thefor each source within its associated uncertainty and recount sources. We follow this procedure 1000 times. The counts in Figure <ref> represent the mean from the Monte Carlo simulations, and the error bars are standard deviation from the Monte Carlo trials and the standard Poisson errors, summed in quadrature. Below 0.8 mJy, SFGs dominate the counts, but AGN become more prevalent with increasing brightness. In the bottom panel of Figure <ref>, we show the percentage of sources above a given flux threshold. We find that AGN contribute ∼10% at 0.3 mJy and increase to ∼80% at 2 mJy, in good agreement with measurements in <cit.> in the Boötes field. Although AGN are frequently assumed not to be abundant in fainter IR samples, the presence of AGN hosts at S_24<100 μJy was also seen in a small Spitzer/IRS spectroscopic sample of lensed galaxies at z∼2, where the authors found that 30% of the sample had IR AGN signatures and 40% had X-ray AGN signatures <cit.>. The Composites comprise >25% of a sample down to the faintest flux threshold at 63% completeness, which we determined by applying the completeness estimates listed in Table <ref> to the number of sources classified with each method.Then, at least 25% of a JWST/MIRI sample will be Composite galaxies, providing a rich data set for probing the AGN/star formation connection at cosmic noon.§ JWST COLOR SELECTION Color selection is a powerful technique for identifying likely AGN, Composites, and SFGs. We have done an exhaustive search to identify the best MIRI filter combinations for separating galaxies into these three classes at cosmic noon by creating synthetic photometry in the JWST/MIRI filters from the <cit.> MIR based Library following the Monte Carlo technique outlined in Section <ref>.As many JWST/MIRI observations will be carried out in fields with available photometric redshifts, or in parallel with NIRcam and NIRspec observations, we include redshift information in our color diagnostics to improve reliability and completeness. We identify three diagnostics covering the ranges z∼1 (z=0.75-1.25), z∼1.5 (z=1.25-1.75), and z∼2 (z=1.75-2.25). These three diagnostics, shown in Figure <ref>, are different combinations of the S_21, S_18, S_15, S_12.8, S_10, and S_7.7 filters, which cover the 6.2 and 7.7 μm PAH complexes and the 3-5 μm stellar minimum at these redshifts. We present two methods for separating SFGs, Composites, and AGN. First, we have determined the optimal AGN, Composite, and SFG regions, labeled in Figure <ref>. The boundaries of each region are circles, with AGN lying inside the inner circle, SFGs lying outside the outer circle, and Composites lying in between.The z∼1 boundaries areinner : (logS_15/S_7.7-0.40)^2 + (logS_18/S_10-0.38)^2 = 0.25^2outer : (logS_15/S_7.7-0.35)^2 + (logS_18/S_10-0.45)^2 = 0.65^2 The z∼1.5 boundaries areinner : (logS_21/S_10-0.49)^2 + (logS_18/S_12.8-0.18)^2 = 0.21^2outer : (logS_21/S_10-0.60)^2 + (logS_18/S_12.8-0.03)^2 = 0.65^2 The z∼2 boundaries areinner : ( logS_18/S_10-0.43)^2 + (logS_21/S_15-0.18)^2 = 0.18^2outer : ( logS_18/S_10-0.50)^2 + (logS_21/S_15-0.12)^2 = 0.52^2 These regions are useful for broadly classifying large numbers of sources or identifying targets for follow-up observations. We use these regions to assess the reliability and completeness of our color diagnostic, where again, we classify all synthetic sources as SFGs when < 0.3, Composites where 0.3 ≤<0.7, and AGN when 0.7 ≥. Table <ref> lists these values for all three redshift regimes. Comparison with Table <ref> shows an improvement over what we were able to reliably classify with the Herschel and Spitzer diagnostics, particularly for separating Composites from SFGs. The spacing of the MIRI filters allows us to sensitively trace the strength of the PAH features relative to the stellar minimum, where the proportionate amount of PAH emission will be lower for Composite galaxies as the power-law emission from the AGN begins to outshine the stellar minimum (see the insets in Figure <ref> for a visual guide).Perhaps, instead of broad classifications, the reader would rather have an estimate of . Without mid-IR spectroscopy, robust decomposition into an AGN and star forming component still is not feasible, even with 6 photometry filters. However, we have determined how to linearly combine the colors in each redshift regime in order to estimate , and we also measure the standard deviation (σ_ AGN) of the residuals when each equation is applied to our synthetic sources so that the reader has a measure of the uncertainty. At z∼1=-0.97 × (logS_15/S_7.7) -0.10× (logS_18/S_10)+1.29and σ_ AGN = 0.15.At z∼1.5:= -0.56× (logS_21/S_10) -0.85 × (logS_18/S_12.8) +1.29and σ_ AGN = 0.13.At z∼2:= -0.55 × (logS_18/S_10) -1.01×(logS_21/S_15) +1.25and σ_ AGN=0.16.§.§ Mid-IR concerns: Metallicity and Obscuration At cosmic noon, the bulk of the star formation is occurring in massive, dusty galaxies with M_∗>10^10 M_⊙ <cit.>, which is the type of galaxies that our MIRI diagnostics were created from <cit.>. For studying the AGN-star formation connection, we expect these types of galaxies to form the most appealing targets. Nevertheless, the sensitivity of JWST/MIRI will enable studies of lower mass galaxies, which tend to have lower metallicities <cit.>. Decreasing gas phase metallicities have been linked with decreasing PAH strengths <cit.>, which is a source of concern since we are effectively detecting AGN hosts based on the strength of PAH features compared with the stellar minimum at 3-5 μm. <cit.> find that below Z<0.7 Z_⊙, PAH emission no longer scales linearly with L_ IR, which based on the mass-metallicity relation, could be a source of concern for contamination of our Composite regions at M_∗ < 3× 10^9 M_⊙, up to z∼2.3 <cit.>. Recently, using the MOSDEF optical spectroscopic survey, <cit.> found that at z∼2, L_ 7.7/L_ IR is lower for galaxies with M_∗ < 10^10 M_⊙ with a behavior similar to that seen for local galaxies <cit.>. lccc[ht!] Completeness (Reliability) of MIRI color selectionRegion z∼1 z∼1.5z∼2 AGN 87 (90)% 87 (89)% 87 (80)% Composite 77 (72)% 79 (74)% 71 (71)% SFG 76 (81)% 81 (86)% 79 (89)% At z∼2, a main sequence galaxy with M_∗ = 10^10 M_⊙ will have a SFR of ∼45 M_⊙/yr <cit.>. At z∼2, 21 μm is tracing the 7.7 μm PAH feature, so applying Equation 11 from <cit.> for this SFR gives S_21≈30 μJy, which is achievable in 7 minutes for a 10σ detection. An hour of integration time at 21 μm will produce 10σ detections of galaxies at roughly 8 μJy, corresponding to M_∗∼ 3×10^9 M_⊙, which is well below the threshold where we expect low metallicity galaxies might contaminate the Composite regime. As such, our color diagnostics may require recalibration for low metallicity galaxies when using observations below S_21≲30 μJy.As a visual check, we demonstrate in Figure <ref> where SFGs with different PAH strengths will lie in our z∼1.5 diagnostic. To accomplish this, we use the Small Magellanic Cloud (SMC) dust model (PAH fraction q_ PAH=0.10%) and a Milky Way dust model with q_ PAH=0.47% from <cit.>, which is included to show where a galaxy with a low SFR will lie. The <cit.> models are also parameterized in terms of the strength of the radiation field, U_ min and U_ max. We set these values to U_ min=1 and U_ max = 1e5, although these parameters have little effect on the final colors. Also, we note that we add in a stellar blackbody with T=5000K to complete the near-IR portion of the spectrum. Even with a low PAH fraction, the Milky Way template still lies in our SFG region, while the SMC template lies directly on the Composite/SFG border. Haro 11, another well studied low metallicity galaxy <cit.> in the nearby Universe has nearly identical MIRI colors as our plotted SMC data point, further confirming that low metallicity galaxies will likely lie around the Composite/SFG border. The reason is that even though low metallicity galaxies have diminished PAH features, they still have a deep and broad stellar minimum at 3-5 μm <cit.>, unlike Composites which begin to exhibit the warmer dust characteristic of the AGN torus. We also plot the template from <cit.> which corresponds to an L_ IR=10^10 L_ IR, as this is an order of magnitude less luminous than the <cit.> library. A galaxy of this luminosity also lies in the Composite region, although it is away from the locus of our Composite galaxies (purple distribution). We caution the reader to be prudent when classifying galaxies as Composites, particularly low mass sources that lie near the Composite and SFG border. If stellar masses of MIRI samples are known (possibly through NIRcam observations), low mass galaxies that lie in our Composite regions provide excellent targets for follow-up spectroscopy observations, to distinguish between AGN or metallicity as the underlying cause of the diminished PAH emission.The other prominent concern in a mid-IR diagnostic is how obscuration can affect the detection of AGN. Our template library was built assuming the AGN can be represented as a power law, and we empirically measure the power law component to have an average slope of F_ν∝λ^1.5, but individual sources will show a range of slopes, and a range of dust obscurations. The AGN templates in the <cit.> library are derived from AGN where 75% of the sample are also detected in the X-ray, implying that they are largely unobscured. Of the Composite sources in <cit.>, only 35% are X-ray detected, indicating that they contain more heavily obscured AGN. We now explore the effects of dust obscuration by examining where different galaxies will lie in the z∼1.5 colorspace (Figure <ref>).Arp 220 (orange bowtie) is a local Ultra Luminous Infrared Galaxy (ULIRG) that is heavily dust-obscured and may host an AGN <cit.>. Its position near the SMC and at the edge of the Composite region indicates another possible ambiguity, that the aromatic bands tend to be suppressed in the most luminous and compact infrared galaxies. How many such objects exist at cosmic noon is not well quantified, as most galaxies of the same luminosity as local ULIRGs (L_ IR>10^12 L_⊙) have extended ISMs <cit.>. NGC 1068 (red cross) is an archetypal local Compton thick Seyfert II AGN. Despite its extreme obscuration, it lies securely in our Composite region, close to the AGN boundary. We also use the AGN library of <cit.> to examine what extinction conditions would push an AGN into our SFG region. These AGN templates are calculated assuming the AGN IR emission arises from 2-phase dust region consisting of a torus and disk, a torus radius R, viewing angle, and cloud filling factor. The optical depth of the clouds in the torus is primarily what causes the AGN to move into the Composite and SFG regions, so we hold all other parameters fixed (for reference, we use the model with viewing angle=67^∘, R=1545×10^15cm, A_d=300,V_c=77.7%). This model is a pure AGN, with no star formation, but when the optical depth of the torus is A_V=13.5 (blue triangle), the AGN model lies in our Composite region, and when A_V=45 (yellow triangle), the AGN lies in the SFG region. Detecting such an obscured AGN at other wavelengths would also beextremely challenging, and identifying complete samples of true Type II obscured AGN remains an unsolved problem. <cit.> find that 30% of mid-IR luminous quasars at z∼1-3 in the GOODS-S field are not detected in the Chandra 6 Ms data. Of those that are detected, >65% are Compton thick.Beyond these estimates, it is difficult to say how many heavily obscured AGN there are that would not be selected as such in the X-ray or the mid-IR. Identifying these very obscured AGN will require detailed SED modeling using a full suite of NIRcam+MIRI observations, which is beyond the scope of this paper.§.§ AGN contributions in individual bandsIf we have a good understanding of the typical full IR SED of high redshift galaxies, as well as the scatter in the population, then a single photometric point can be used in conjunction with representative templates to estimate L_ IR and star formation rates (SFRs). Since PAH molecules are illuminated by the UV/optical photons from young stars, they are a natural SFR indicator and have been extensively used in the literature to probe SFR and L_ IR <cit.>. Given the coverage of the MIRI filters, we will now examine how an AGN can affect the 7.7 μm PAH feature for the <cit.> templates used in this work, as any AGN contribution will need to be corrected for before converting a PAH luminosity to a SFR. We remind the reader that for these templates, the AGN component is represented as a power-law with a slope of F_ν∝λ^1.5. We measure the intrinsic L_7.7 of each template using PAHFIT <cit.>. Then, we measure L_ MIRI, which is the photometry of the template through the following MIRI filters at the given redshifts:z=0,7.7 z=0.95,15.0 z=1.34,18.0 z=1.73,21.0 z=2.31,25.5The redshifts mark where the rest frame central wavelength of each filter is 7.7 μm.In the top panel of Figure <ref>, we demonstrate how much of the 7.7 μm feature each filter covers at the above listed redshifts. In the bottom panel, we show the relationship L_7.7/L_ MIRI as a function offor each filter at the listed redshifts. The decreasing fractions with increasingare due to the increased contribution of the warm dust continuum to the measured photometry. We fit a quadratic relationship to all the points and measureL_7.7/L_ MIRI= (-1.09±0.20)×^2 -(0.50±0.21)×+(1.86±0.04)This equation, in conjunction with estimatingfrom MIRI colors, can be used for first order corrections to L_7.7 before converting to a SFR. Similarly, in <cit.>, we demonstrated that there is a quadratic relationship betweenand the total contribution of an AGN to L_ IR that can be used to correct L_ IR for AGN emission:= 0.66 ×^2 - 0.035×whereis the fraction of L_ IR(8-1000 μm) due to AGN heating. Then, the portion of L_ IR due to star formation is L_ IR^ SF = L_ IR×(1-). Once the AGN contribution is accounted for, L_ IR can be converted to a SFR using standard equations <cit.>. For a strong AGN (≥ 0.9), at least 50% of L_ IR needs to be removed before converting to a SFR, and the same is true if using 7.7 μm to calculate SFR. Then, the strongest AGN will have SFRs that are overestimated by at least a factor of 2 if not properly accounted for. Of more concern is Composites, which are routinely misidentified as SFGs. For a Composite with =0.5, an L_ IR based SFR will be overestimated by ∼15%. But, if one uses L_7.7, then the resulting SFR will be overestimated by ∼35%. § DISCUSSION: PHYSICAL PROPERTIES OF A MIRI SAMPLEWe now return to our CANDELS+COSMOS sample to investigate the physical properties of galaxies that MIRI color selection will identify as being AGN hosts. First, we illustrate the predicted number counts at cosmic noon with the MIRI 10 μm filter, which is chosen for its sensitivity <cit.> and because we use it in all three color diagnostics. We calculate the 10 μm flux for all CANDELS+COSMOS galaxies at z=0.75-2.25 and with M_∗ > 10^8 M_⊙ by scaling the appropriate <cit.> template (based on the source'sdetermined through color classification) to the available IR photometry and convolving with the 10 μm transmission filter. By template fitting, we are also able to calculate L_ IR and . The total 10 μm counts are plotted as the black stars in the bottom panel Figure <ref>. By including lower mass galaxies, we push below the 80% completeness in Figure <ref> and down to the 20% completeness limit (corresponding to ∼40 μJy at 24 μm). For reference, the 80% completeness limit (measured at 24 μm) corresponds to S_10∼10 μJy. Our counts are in good agreement at the faint end with the published 8 μm galaxy counts in <cit.>. At the bright end, we have fewer sources due to the redshift cut we imposed and the small field sizes, similar to our 24 μm number counts in Figure <ref>. is strictly a measure of the dust heated by a AGN relative to that heated by star formation, so now we examine a more physically motivated quantity, the Eddington ratio. The Eddington ratio is defined as λ_ Edd= L_ bol/L_ Edd, where L_ bol is the bolometric luminosity of the AGN and L_ Edd is the Eddington luminosity. In this way, λ_ Edd is a measure of how efficiently a black hole is accreting material. L_ bol is commonly estimated from the hard X-ray luminosity, L_2-10 keV. Due to obscuration and varying depths of the Chandra catalogs in the CANDELS fields, we do not have L_2-10 keV for all of our IR identified AGN and Composites. As a first step towards calculating L_ bol, we estimate L_2-10 keV from L_ IR^ AGN for all sources. We empirically determine the scaling between these luminosities to be log(L_2-10 keV/L_ IR^ AGN) = (31.698±3.535) - (0.734±0.082)×log L_ IR^ AGN [ erg s^-1]measured directly using Chandra observations of the GOODS-S field, which is the only field where the Chandra data is complete down to L_2-10 keV= 10^42erg s^-1 out to z=2 <cit.>. Note that L_2-10 keV is the observed luminosity, as in most cases we do not have high enough counts to make a meaningful obscuration measurement. Figure <ref> shows this empirically derived relationship, along with the approximate conversion factors derived in <cit.>, using a local sample of AGN with L_2-10 keV∼10^43erg s^-1, and derived in <cit.> from quasars with L_2-10 keV>10^45erg s^-1. Our conversion is in line with the literature results for the brighter AGN.We then apply Equation <ref> to all sources in the CANDELS and COSMOS fields. Next, we convert L_2-10 keV to L_ bol using Equation 2 in <cit.>. This Equation results in L_2-10 keV / L_ bol∼0.06-0.01, in agreement with direct measurements in the literature <cit.>. Finally, we calculate L_ Edd [ erg s^-1]= 1.3× 10^38× M_ BH [M_⊙], where M_ BH = 0.002 M_∗ following the convention in <cit.> and <cit.>.With the techniques outlined in Section <ref>, we will be able to calculate λ_ Edd for samples with M_∗ or M_ BH measurements. The relationship between λ_ Edd andis not linear, since λ_ Edd depends not only on but also on L_ IR and M_∗. Then, eachcan have a range of λ_ Edd depending on the host galaxy properties. We show in the top panel of Figure <ref> the distribution ofλ_ Edd for each galaxy category. In the bottom panel of Figure <ref>, we break our 10 μm number counts into bins of λ_ Edd. Comparison with the top panel demonstrates that the λ_ Edd<0.01 curve (pink circles) is dominated by SFGs, while the λ_ Edd>0.1 curve (yellow) has accretion rates typical of sources identified as AGN at IR and X-ray wavelengths. The majority of the counts are λ_ Edd=0.01-0.1 (purple squares), and these are objects that could be classified as AGN, SFGs, or Composites. The MIRI field of view is 1.2'×1.9', so we also illustrate the counts in a MIRI FOV on the right axis of Figure <ref>. We expect nearly 100 objects per MIRIFOV down to 2 μJy at 10 μm, achievable at a SNR of 10 (5) in roughly 15 minutes (3.6 minutes). Of these objects, >50% may be AGN hosts where we can detect and measure the black hole accretion. BelowS_10=10 μJy, the counts become dominated by sources with M_∗<10^9 M_⊙. Of the galaxies with λ_ Edd>0.01, 30% have M_∗<10^9 M_⊙ and comprise a prime population for followup studies to more concretely pin down the AGN fraction in low mass galaxies at z∼1-2.The use of the λ_ Edd parameter highlights an area where MIRI will enable great strides forward–namely, understanding how the observable properties of AGN hosts correlate to their physical properties. The broad distributions of λ_ Edd in the top panel of Figure <ref> demonstrates the limitations of either broadly grouping sources into AGN, Composites, and SFGs based on observables, or using scaling relations to calculate physical properties, or very likely a combination of the two. But with the high resolution spectroscopy on MIRI, and the increased number of photometric filters, we will be able to classify galaxies on the relative strengths of PAH features, estimate and combine with M_∗ (attainable with NIRcam) to measure λ_ Edd, providing clearer insight into the relationship between galaxy dust emission and black hole accretion.Finally, we demonstrate the host galaxy properties of CANDELS AGN and Composites selected with different techniques at z∼1-2 in Figure <ref>. We calculate SFR for all galaxies by fitting templates from the <cit.> library, based on classification as a SFG, Composite, or AGN, and then removing the AGN contribution to L_ IR before converting to a SFR using Equation 3 in <cit.>. We combine SFR with M_∗ <cit.> to measure sSFR=SFR / M_∗, a common probe of galaxy evolution, as this ratio will be lower in galaxies that are quenching <cit.>. In red, we plot the distribution of sSFR for those galaxies identified as AGN in the hard X-ray band (L_2-10 kev≥ 10^43erg s^-1). Then, we plot the distribution of sSFR in blue for those galaxies that will be selected as either AGN or Composites by our MIRI color diagnostics, based on our estimation of their JWST colors through template fitting. For easier comparison, we normalize both distributions to have a peak at one, although the MIRI distribution actually has 20× more galaxies than the X-ray distribution. In practice, the relative numbers of X-ray and MIRI AGN will depend on the depth of the observations and the area covered, but the sensitivity of MIRI and our ability to select Composite sources will enable larger samples than X-ray selection alone. Crucially, our cosmic noon CANDELS AGN hosts have higher sSFR than the X-ray selected CANDELS galaxies <cit.>. Combining MIRI and X-ray samples will increase our dynamic range in sSFR, allowing us to explore how black hole accretion varies with star formation and main sequence location <cit.>. Prior to JWST, the most popular way of identifying large samples of IR AGN is with IRAC color techniques <cit.>. The <cit.> IRAC diagnosticis the most reliable, since it eliminates host galaxy contamination, but it is only sensitive to the most actively accreting AGN as it is based on a power-law selection criterion. Hence, it is likely to be significantly incomplete for Compton thick and other obscured AGN. Of theCANDELS sources selected by our MIRI diagnostics, we show in green the sources that are also selected as AGN by the <cit.> criteria. Clearly, due to the sensitivity and spacing of the MIRI filters, we will be able to detect >4 times as many AGN hosts as would be identified with IRAC alone. MIRI color selection will enable identification of statistical samples of AGN hosts in their star forming prime (as measured by sSFR), allowing astronomers to trace the star formation-AGN connection at the peak period of stellar and black hole growth in the Universe. § CONCLUSIONSWe identify SFGs, AGN, and Composites in four CANDELS fields and in the full COSMOS field using three different redshift dependent color identification techniques. We present the first 24 μm counts of star forming+AGN Composite galaxies at z∼1-2. We find that IR AGN and Composites dominate 24 μm samples at S_24>0.8mJy. Any 24 μm selected sample contains >25% of Composites.We use a library of SFG, AGN, and Composite templates to create synthetic galaxies, and we use these synthetic galaxies to create JWST/MIRI color selection techniques for three redshift bins, z∼1, z∼1.5, and z∼2. Our techniques can safely be applied to galaxies with M_∗>10^10 M_⊙. However, below this regime, metallicity may effect the strength of the PAH features, causing contamination of our Composite regime. MIRI can achieve 10σ detections of M_∗<10^10 M_⊙ galaxies out to z∼2 in a matter of minutes, so future JWST observations will prove crucial in separating differences in mid-IR emission due to metallicity rather than AGN in low mass galaxies.At these redshifts, our color selection techniques cover the 6.2 μm and 7.7 μm PAH features and the 3-5 μm stellar minimum, which are robust tracers of star formation. We demonstrate how to correct L_7.7 for AGN contamination before converting to a SFR, a crucial step or SFRs based on 7.7 μm PAH emission will be overestimated by >50% for AGN and 35% for Composites.Finally, we predict the Eddington ratios (λ_ Edd), a measure of black hole accretion efficiencies, that we will observe with MIRI imaging. Our MIRI color selection diagnostic can identify samples of AGN and Composite galaxies with λ_ Edd > 0.01 that are four times larger than samples of AGN selected by Spitzer/IRAC techniques. We also use our new 24 μm number counts to predict the number counts at 10 μm in different bins of λ_ Edd. With MIRI color identification, we will be able to probe the star formation - AGN connection in dusty galaxies at cosmic noon.A. K. thanks Sandy Faber for helpful conversations. A. K. gratefully acknowledges support from the YCAA Prize Postdoctoral Fellowship. A. P. and A. S. acknowledge NASA ADAP13-0054 and NSF AAG grants AST- 1312418 and AST-1313206. In this appendix, we show our redshift dependent color diagnostic to find SFGs, Composites, and AGN using Spitzer and Herschel photometry. We create a catalog of 5500 synthetic galaxies from 11 templates where we know the intrinsic AGN contribution. We resample each photometric point within the uncertainties of the template from which it was created, so that we can represent the scatter in colorspace of real galaxies, which is an improvement upon using so-called redshift tracks alone to explore where SFGs, Composites, and AGN lie in colorspace. We create color diagrams in redshift bins of z=0.75-1.25, z=1.25-1.75, and z=1.75-2.25. In each redshift bin, we divide the color space into regions of 0.2×0.2 dex and calculate the averageand standard deviation, σ_ AGN of all the synthetic galaxies that lie in that region. In Figure <ref>, <ref>, <ref> below, we show our three diagnostics: S_250/S_24 v. S_8/S_3.6, S_100/S_24 v. S_8/S_3.6, and S_24/S_8 v. 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"authors": [
"Allison Kirkpatrick",
"Stacey Alberts",
"Alexandra Pope",
"Guillermo Barro",
"Matteo Bonato",
"Dale D. Kocevski",
"Pablo Perez-Gonzalez",
"George H. Rieke",
"Lucia Rodriguez-Munoz",
"Anna Sajina",
"Norman A. Grogin",
"Kameswara Bharadwaj Mantha",
"Viraj Pandya",
"Janine Pforr",
"Paola Santini"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170627213025",
"title": "The AGN-Star Formation Connection: Future Prospects with JWST"
} |
We study the long-time behavior of the dynamics of interacting planar Brownian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann – Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincaré inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox – Ingersoll – Ross process, in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.The EBLM projectThe data is publicly available at the CDS Strasbourg and on demand from the first author. Alexander von Boetticher1,2 Amaury H.M.J. Triaud2 Didier Queloz1,3 Sam Gill4 Monika Lendl5,6 Laetitia Delrez1, 9 David R. Anderson4 Andrew Collier Cameron7 Francesca Faedi8 Michaël Gillon9 Yilen Gómez Maqueo Chew10 Leslie Hebb11 Coel Hellier4 Emmanuël Jehin9 Pierre F.L. Maxted4 David V. Martin3 Francesco Pepe 3 Don Pollacco8 Damien Ségransan 3 Barry Smalley4 Stéphane Udry 3 Richard West812. June, 2017 ===========================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION AND STATEMENT OF THE RESULTS§.§ The model and its well-posedness This work is concerned with the dynamics of N≥2 particles at positions x_1,…,x_N in ^d, d≥1, confined by an external field and experiencing a singular pair repulsion. The configuration space that we are interested in is the open subset D⊂(^d)^N defined byD:=(^d)^N∖∪_i≠ j{(x_1,…,x_N)∈(^d)^N:x_i=x_j}where i,j run over {1,…,N}. The boundary of D in the compactification of (^d)^N isD:={∞}∪∪_i≠ j{(x_1,…,x_N)∈(^d)^N:x_i=x_j}.The vector x=(x_1,…,x_N)∈ D encodes the position of the N particles, and the energy H(x) of this configurationis modeled byH(x):=1/N∑_i=1^N V(x_i)+1/2N^2∑_1 ≤ i≠ j ≤ NW(x_i-x_j) =:H_V(x)+H_W(x).Here, V:^d→∪{+∞} is an external confinement potential such that V(z)→+∞ as z→∞, and W:^d∖{0}→ is a pair or two-body interaction potential such that W(z)=W(-z) and W(z)→+∞ as z→0 (singularity). Unless otherwise stated, we consider particles in ^2≡, with quadratic confinement and Coulomb repulsion, namely:d=2,V(z)=z^2,W(z)=log1/z^2.Here z denotes the Euclidean norm of z∈^2 (modulus of the complex number z). With this notation, we study the system of N interacting particles in ^2 modeled by a diffusion process X^N=(X^N_t)_t≥0 on D, solution of the stochastic differential equationX^N_t=√(2_N/_N) B^N_t-_N H(X^N_t)t,for any choice of speed _N>0 and inverse temperature _N>0; here (B^N_t)_t≥0 is a standard Brownian motion of (^2)^N. In other words, letting X^N_t=(X^i,N_t)_1≤ i≤ N and B^N_t=(B^i,N_t)_1≤ i≤ N denote the components of X^N_t and B^N_t,X^i,N_t=√(2_N/_N) B^i,N_t-_N/N V(X^i,N_t)t -_N/N^2∑_j≠ i W(X^i,N_t-X^j,N_t)t,1≤ i≤ N.Since V(z)=z^2 and W(z)=-2logz we have more explicitlyX^i,N_t =√(2_N/_N) B^i,N_t-2_N/NX^i,N_tt -2_N/N^2∑_j≠ iX^j,N_t-X^i,N_t/|X^i,N_t-X^j,N_t|^2t,1≤ i≤ N.To lightweight the notations, we will very often drop the notation N in the superscript, writing in particular X_t, B_t, X^i_t, and B^i_t instead of X^N_t, B^N_t, X^i,N_t and B^i,N_t respectively. We shall see later that the cases _N=N and _N=N^2 are particularly interesting, the latter being related to the complex Ginibre Ensemble in random matrix theory.Global pathwise well posedness of a solution X to the stochastic differential equation (<ref>) is not automatically granted since W is singular. Nevertheless, the set D is path-connected (see Lemma <ref>) and, given an initial condition X_0 in D, one can resort to classic stochastic differential equations properties to define, in a unique pathwise way, the process X^N up to the explosion timeT_ D:=sup_>0T_∈[0,+ ∞].Here, T_ =inf{t≥0:max_1≤ i≤ N|X^i_t|≥^-1 or min_1≤ i≤ N|X^i_t-X^j_t|≤}is the first exit time of a typical compact set in D. Then, one can show that explosion never occurs: For any X_0=x∈ D, pathwise uniqueness and strong existence on [0, + ∞) hold for the stochastic differential equation (<ref>) on [0,+∞), and we have T_ D= + ∞ a.s. The absence of explosion provided by Theorem <ref> is remarkably independent of the choice of the inverse temperature, and this is in contrast with the behavior of the Dyson Brownian motion associated with the one-dimensional log-gas, see for instance <cit.>. The proof of Theorem <ref> is given in Section <ref>. It uses the fact that W is the fundamental solution of the Poisson – Laplace equation. The main idea is similar to the one used for other singular repulsion models, such as in <cit.>, or for vortices such as in <cit.>, but the result ultimately relies on quite specific properties of our model (<ref>). Note also that our particles will never collide and in particular never collide at the same time, in contrast with for instance the singular attractive model studied in <cit.> – see also <cit.> for the control of explosion using the Fukushima technology.Hence there exists a unique Markov process X=(X_t)_t≥0 solution of (<ref>). Its infinitesimal generator L is given for a smooth enough f:D→ byL f=_N/_N f-_N H f.Hereandare understood in (^2)^N≡^2N and u· v=⟨ u,v⟩ denotes the Euclidean scalar product. By symmetry of the evolution, the law of X_t is exchangeable for every t≥0, as soon as it is exchangeable for t=0. Recall that the law of a random vector is exchangeable when it is invariant by any permutation of the coordinates of the vector. It is then natural to encode the particle system with its empirical measureμ^N_t=1/N∑_i=1^Nδ_X^i_t.§.§ Second moment dynamics Theorem <ref> gives the evolution of the second momentH_V(X_t) =1/N∑_i=1^N|X_t^i|^2 =∫_^2x^2 μ^N_t(dx)of μ^N_t. This evolution depends on the choices for _N and _N, for which meaningful choices are discussed in Section <ref>. We let W_1 denote the (Kantorovich –) Wasserstein transportation distance of order one defined by W_1(μ,ν)=inf {[|X-Y|]:X∼μ,Y∼ ν} for every probability measures μ and ν onwith finite first moment. The process (H_V(X_t))_t≥0 is an ergodic Markov process, equal in law to the Cox – Ingersoll – Ross process (R_t)_t≥0 given by the unique solution in [0,∞) of the stochastic differential equationR_t = √(8 α_N/ N β_N R_t)b_t+4_N/N[ N/β_N+ N-1/2N-R_t ] t,where (b_t)_t≥0 is a real standard Brownian motion. In particular, its invariant distribution is the Gamma law _N on _+ with shape parameter N+N-1/2N_N and scale parameter _N, and density with respect to the Lebesgue measure ongiven byr∈↦γ_N(r):= β_N^N+β_N(N-1)/2 N/Γ(N+β_N(N-1)/2 N )r^(N-1)(1+β_N/2 N)^-rβ_N1_r≥0.Moreover, for any t ≥ 0 we haveW_1(Law(H_V(X_t)),_N) ≤^-4 _N/NtW_1(Law(H_V(X_0)), _N).Furthermore for any x∈ D and t≥0, we have[H_V(X_t)| X_0=x]=H_V(x) ^-4_Nt/N +1/2 +N/_N-1/2N1-^-4_Nt/N.In particular, as t→∞, the left-hand sides in (<ref>) and (<ref>) converge to 0 and 1/2+N/_N-1/(2N) respectively with a speed independent of N as soon as _N is linear in N. A Cox – Ingersoll – Ross (CIR) process also naturally arises as the dynamics of the second empirical moment of the vortex system studied in <cit.>. Theorem <ref> is proved in Section <ref>. §.§ Invariant probability measure and long-time behavior Despite the repulsive interaction, the confinement is strong enough to give rise to an equilibrium. Namely, the Markov process (X_t)_t≥0 admits a unique invariant probability measure which is reversible. It is the Boltzmann – Gibbs measure P^N on D⊂ (^2)^N with densityP^N(x_1,…,x_N)/ x_1⋯ x_N =^-_NH(x_1,…,x_N)/Z_N =^-_N/N∑_i=1^N | x_i |^2/Z_N∏_1 ≤ i<j ≤ Nx_i-x_j^2_N/N^2where Z_N:=∫_D^-_NH(x_1,…,x_N) x_1⋯ x_Nis a normalizing constant known as the partition function. Such a Boltzmann – Gibbs measure with a Coulomb interaction is called a Coulomb gas. Actually H(x)→+∞ when x→∂ D and ^- H is Lebesgue integrable on D for any >0, see Lemma <ref>. Moreover the density of P^N does not vanish on D. One can extend it on (^2)^N by zero, seeing P^N as a probability measure on (^2)^N. Since the domain D and the function H are both invariant by permutation of the N particles, the law P^N is exchangeable. The behavior of P^N relies crucially on the “inverse temperature” _N. The choice _N=N^2 gives a determinantal structure to P^N which is known in this case as the complex Ginibre ensemble in random matrix theory. As we will see in Section <ref>, there is another interesting regime which is _N=N.In Theorem <ref> below we quantify the long time behavior of our Markov process X^N via a Poincaré inequality for its invariant measure P^N. Recall that if S is an open subset of ^n andis a class of smooth functions on S, then a probability measure μ on S satisfies a Poincaré inequality onwith constant c>0 if for every f∈,Var_μ(f):=_μ(f^2)-(_μ f)^2 ≤ c _μ(∇ f^2) where_μ(f):=∫f μ,see <cit.> for instance. If f is the density with respect to μ of a probability measure ν then the quantity Var_μ(f)=_μ(|f-1|^2) is nothing else but the chi-square divergence χ(νμ). Letbe the set of ^∞ functions f:D→ with compact support in D, in the sense that the closure of {x∈ D:f(x)≠0} is compact and is included in D. Then for any N, the probability measure P^N on (^2)^N satisfies a Poincaré inequality onwith a constant which may depend on N. By (<ref>), the invariance of P^N gives-_P^N(fL f)=_N/_N_P^N( f^2)where f^2=∑_i=1^n_x_if^2 for f=∇_x_if_1≤ i≤ N in (^2)^N. Let P^N_t be the law of X^N_t in (^2)^N. Up to determining a dense class of test functions stable by the dynamics, it is classical, see <cit.> or <cit.>, that the Poincaré inequality (<ref>) for P^N with constant c=c_N imply the exponential convergence of P^N_t to P^N, namelyχ(P^N_t P^N) ≤^-2t/c_N_N/_Nχ(P^N_0 P^N).More precisely, provided we already know that P^N_t has a smooth density f^N_t, we have/ tVar_P^N(f^N_t) =/ t∫(f^N_t)^2P^N =2∫f^N_tLf^N_tP^N ≤ -2_N/_Nc_NVar_P^N(f^N_t).Theorem <ref> is proved in Section <ref>. Poincaré inequalities can classically be proved by spectral decomposition, tensorization, convexity, perturbation, or Lipschitz deformation arguments, see <cit.>. None of these approaches seem to be available for P^N. It turns out that H_V is up to an additive constant an eigenvector of L. Namely, from (<ref>) we getLU=-4_N/NU where U:=H_V-N/_N-N-1/2N.This fact is the key of the proof of Theorem <ref>. However, due to the varying sign of U, we do not know how to use U with the Lyapunov method to get a Poincaré inequality.The invariant measure P^N of X is not product, in contrast for instance with the case of vortex models with constant intensity studied in <cit.>.Neither the domain D nor the energy H:D→ are convex, see Proposition <ref>, and thus the law P^N is not log-concave. Remarkably, for one-dimensional log-gases, one can order the particles, which has the effect of producing a convex domain instead of D on which H is convex, and in this case P^N satisfies in fact a logarithmic Sobolev inequality which is stronger, see for instance the forthcoming book <cit.> and also <cit.> for the optimal Poincaré constant. Here d=2 and the one dimensional trick is not available.The law P^N is not a Lipschitz deformation of the Gaussian law on _N(). Actually, the map which to M∈_N() associates its eigenvalues inis not Lipschitz. To see it take M,M'∈_n() with M_j,j+1=1 for j=1,…,n-1 and M_jk=0 otherwise, and (M'-M)_jk= if (j,k)=(n,1) and (M'-M)_jk=0 otherwise. Then the eigenvalues of M'-M are {^1/n^2ikπ/n:0≤ k≤ n-1},while the Hilbert – Schmidt norm and operator norm of M'-M are both equal to . Note that in contrast, this map is Lipschitz for Hermitian matrices and more generally for normal matrices; this statement is known as the Hoffman – Wielandt inequality <cit.>.The proof of Theorem <ref> is based on a Lyapunov function and as usual this does not provide in general a good dependence on N. Of course it is natural to ask about the dependence in N and in _N and _N of the best constant in Theorem <ref> and, specifically, if convergence to equilibrium can be expected to hold at a rate that does not depend on N, as in <cit.>. Theorem <ref>below and the previous Theorem <ref> and Remark <ref> constitute steps in that direction. If _N=N^2 then the one-particle marginal law P^1,N of P^N on ^2 satisfies a Poincaré inequality, with a constant which does not depend on N. In particular, the smallest (i.e. best) constant for P^N is bounded below uniformly in N. Theorem <ref> is proved in Section <ref>. Although the measure P^N is not product, at least in the regime _N=N^2 a product structure arises asymptotically as N goes to infinity. More precisely, for k ≤ N, let P^k,N be the k-th dimensional marginal distribution of the exchangeable probability measure P^N, as in (<ref>); then, in the regime _N=N^2, we have P^k,N-(P^1,N)^⊗ k→ 0,N →∞weakly with respect to continuous bounded functions. It follows from Theorem <ref> below. Let β_N = N^2 and let μ_∞ be the uniform distribution on the unit disc {z∈:|z|≤1} with density _∞(z) = π^-11_z≤1. For every fixed k≥1,P^k,N→μ_∞^⊗ k,N→∞weakly with respect to continuous and bounded functions. Moreover, denoting ^k,N the density of the marginal distribution P^k,N, as defined in (<ref>), we have^1,N→_∞and^2,N→_∞^⊗ 2,N→∞uniformly on compact subsets of respectively{z ∈: |z|≠1}and{(z_1,z_2)∈^2:|z_1|≠1,|z_2|≠1,z_1≠ z_2}.Theorem <ref> is proved in Section <ref>. Note that the convergence of ^1,N cannot hold uniformly on arbitrary compact sets ofsince the pointwise limit is not continuous on the unit circle. Moreover the convergence of ^2,N cannot hold on {(z,z):z∈, |z|<1} since, by (<ref>), ^2,N(z,z)=0 for any N≥2 and z∈ while ^1,N(z)^1,N(z) →_N 1/π^2≠0 when |z|<1, and this phenomenon is due to the singularity of the interaction.The case _N=N^2 is related to random matrix theory, see Section <ref>. To our knowledge, Theorem <ref> and Theorem <ref> have not appeared previously in this domain. §.§ Comments and open problems §.§.§ Inverse temperature Following <cit.>, there are two natural regimes _N=N and _N=N^2.* Random matrix theory regime: _N=N^2. This is natural from the point of view of random matrices. Namely let M be a random N× N complex matrix with independent and identically distributed Gaussian entries onwith mean 0 and variance 1/N with density z∈↦π^-1Nexp(-Nz^2). The variance scaling is chosen so that by the law of large numbers, asymptotically as N→∞, the rows and the columns of M are stabilized: they have unit norm and are orthogonal. The density of the random matrix M is proportional toM↦∏_1≤ j,k≤ Nexp(-NM_jk^2) =exp-NTr(MM^*).The spectral change of variables M=U(D+N)U^*, which is the Schur unitary decomposition, gives that the joint law of the eigenvalues of M has density^N,N(z_1,…,z_n) :=N^N(N+1)/2/1!2!⋯ N!^-∑_i=1^N Nz_i^2/π^N∏_i<jz_i-z_j^2with respect to the Lebesgue measure on ^N. This law is usually referred to as the “complex Ginibre Ensemble”, see <cit.>. This matches P^N with (<ref>) with _N=N^2 so that the density of P^N on (^2)^N=^N can be written asP^N(z_1,…,z_N)/ z_1⋯ z_N =^N,N(z_1,…,z_N).It is a well known fact – see <cit.>, <cit.>, or <cit.> – that for every 1≤ k≤ N, the k-th dimensional marginal distribution P^k,N of P^N has density^k,N(z_1,…,z_k)=∫_^N-k^N,N(z_1,…,z_N)z_k+1⋯ z_N =(N-k)!/N!^-N(|z_1|^2+⋯+|z_k|^2)/π^kN^-k(_N(Nz_iz_j))_1≤ i,j≤ k,where _N(w):=∑_ℓ=0^N-1w^ℓ/ℓ! is the truncated exponential series. The energy H is a quadratic functional of the empirical measure μ^N:=1/N∑_i=1^N_x_i of the particles:H(x_1,…,x_N)=∫V(x) μ^N( x) +1/2∬_≠W(x-y) μ^N( x)μ^N( y)=:_≠(μ^N)where “≠” indicates integration outside the diagonal. Since _N≫ N as N→∞, under (P^N)_N, the sequence of empirical measures (μ^N)_N satisfies a large deviation principle with speed (_N)_N and good rate function -inf whereis given for nice probability measures μ on ^2 by(μ):=∫V(x) μ( x)+1/2∬W(x-y) μ( x)μ( y).See for instance <cit.> and references therein. The functionalis strictly convex where it is finite, lower semi-continuous with compact level sets, and it achieves its global minimum for a unique probability measure μ_∞ on ^2, which is the uniform distribution on the unit disc with density z∈↦π^-11_z≤1. From the large deviation principle it follows that almost surelyμ^NN→∞⟶μ_∞:=infweakly, regardless of the way we put (P^N)_N in the same probability space.* Crossover regime: _N=N. In this case P^N has density proportional to(x_1,…,x_N)∈ D ↦^-∑_i=1^Nx_i^2∏_1 ≤ i<j ≤ Nx_i-x_j^2/N.We do not have a determinantal formula as in (<ref>), and this gas is not associated with a standard random matrix ensemble. It is a two dimensional analogue of the one dimensional gas studied in <cit.> leading to a Gauss-Wigner crossover. Following <cit.>, we can expect that under (P^N)_N the sequence of empirical measures (μ^N)_N satisfies a large deviation principle with speed (_N)_N and rate function -inf; hereis given for every probability measure μ on ^2 by(μ):=(μ)-(μ) where(μ):=-∫μ/ xlogμ/ xxwhen μ is absolutely continuous with respect to the Lebesgue measure, while (μ):=+∞ otherwise.is the so-called Boltzmann – Shannon entropy. The minimizer ofis no longer compactly supported but can still be characterized by Euler – Lagrange equations, and is a crossover between the uniform law on the disc and the standard Gaussian law on ^2. See <cit.> for the link with Sanov's large deviation principle. §.§.§ Dyson Brownian Motion If we start with an N× N random matrix M_t=(M_t^j,k)_1≤ j,k≤ Nwith i.i.d. entries following the diffusion M_t^j,k= B_t^j,k-N V(M_t^j,k)t then the eigenvalues in =^2 of M_t will not match our diffusion X solution of (<ref>). This is due to the fact that M_t is not a normal matrix in the sense that M_tM_t^*≠ M_t^*M_t with probability one as soon as M_t has a density. In fact the Schur unitary decomposition of M_t writes M_t=U_tT_tU_t^* where U_t is unitary and T_t=D_t+N_t is upper triangular, D_t is diagonal, and N_t is nilpotent. The dynamics of D_t is perturbed by N_t. The dynamics (<ref>) is not the analogue of the Dyson Brownian motion, the process of the eigenvalues associated with the Gaussian Unitary Ensemble, the one-dimensional log-gas studied in <cit.>. We refer to <cit.> and references therein for more information on this topic.§.§.§ Initial conditions In the case of the one-dimensional log-gas known as the Dyson Brownian Motion, the stochastic differential equation still admits a unique strong solution when the particles coincide initially. This is proved in <cit.> by crucially using the ordered particle system. Unfortunately, it does not seem possible to extend such an argument to higher dimensions. But it is likely that at least weak well-posedness should still hold for our model.§.§.§ Arbitrary dimension, confinement, and interaction As in <cit.>, many aspects should remain valid in arbitrary dimension d≥2, with a Coulomb repulsion and a more general confinement V. For instance, by analogy with the case without interaction studied in <cit.>, it is natural to expect that Theorem <ref> remains valid beyond the quadratic confinement case, for example in the quadratic “dispersive” case V(x)=-|x|^2, and in confined cases for which V(x)→+∞ as x→∞ with polynomial growth. Nevertheless, our choice is to entirely devote the present article to the two-dimensional quadratic confinement case: this model is probably the richest in structure, notably due to its link with the Ginibre Coulomb gas, which is a remarkable exactly solvable model.The model with non-singular interaction has extensively been studied in arbitrary dimension, in relation with McKean – Vlasov equations, see <cit.> and references therein. The model in dimension d=1 with logarithmic singular interaction has also extensively been studied, see for instance <cit.> and references therein. See also <cit.>.§.§.§ Logarithmic Sobolev inequality and other functional inequalities It is natural to ask whether P^N satisfies a logarithmic Sobolev inequality, which is stronger than the Poincaré inequality with half the same constant, see <cit.>. Indeed, for P^N, a Lyapunov approach is probably usable by following the lines of <cit.>, see also <cit.>, but there are technical problems due to the shape of D which comes from the singularity of the interaction. Observe that the one-particle marginal P^1,N satisfies indeed a logarithmic Sobolev inequality with a constant uniform in N, as mentioned in Remark <ref> after the proof of Theorem <ref>.Still about functional inequalities, the study of concentration of measure for Coulomb gases in relation with Coulomb transport inequalities is considered in the recent work <cit.>.§.§.§ Mean-field limit In the regime β_N = N^2, by (<ref>) the empirical measure μ^N under P^N tends to μ_∞ as N→∞.More generally, when the law of X_0 is exchangeable and for general β_N, one can ask about the behavior of the empirical measure of the particles μ^N_t:=1/N∑_i=1^N_X_t^i as N→∞ and as t→∞. This corresponds to study the following scheme:P^N_t t→∞⟶ P^N and[ μ^N_tt→∞⟶ μ^N; ↓ ↓; μ_tt→∞⟶ μ_∞ ]for a suitable deterministic limit μ_∞.At fixed N, the limit lim_t→∞P^N_t=P^N, valid for an arbitrary initial condition X_0=x, corresponds to the ergodicity phenomenon for the Markov process X, quantified by the Poincaré inequality of Theorem <ref>. By the mean-field structure of (<ref>) and (<ref>), it is natural to expect that if:=lim_N→∞_N/_N∈[0,+ ∞)then the sequence ((μ^N_t)_t≥0)_N converges, as a continuous process with values in the space of probability measures in ^2, to a solution of the following McKean – Vlasov partial differential equation with singular interaction:_tμ_t =μ_t+·(( V+ W*μ_t)μ_t).The convergence of ((μ_t^N)_t≥0)_N can be thought of as a sort of law of large numbers. This is well understood in the one-dimensional case with logarithmic interaction, see for instance <cit.>, using tightness and characterization of the limiting laws. However the uniqueness arguments used in one-dimension are no longer valid for our model, and different ideas need to be developed, see <cit.>. We also refer to <cit.> and references therein for the analysis of similar evolution equations without noise and confinement. Theorem <ref> suggests to take _N=N. Let us comment on the couple of special cases already considered in our large deviation principle analysis of (P^N)_N: _N=N^2 and _N=N, when _N=N.* Random matrix theory regime with vanishing noise: _N=N and _N=N^2. In this case =0 and the limiting McKean – Vlasov equation (<ref>) does not have a diffusive part.Since _N=N we have a constant speed for the second moment evolution. Since _N=N^2 we have explicit determinantal formulas for P^N from the complex Ginibre Ensemble (<ref>). The absence of diffusion implies that if we start from an initial state μ_0 which is supported in a line, then μ_t will still be supported in this line for any t∈[0,∞), and will thus never converge as t→∞ to the uniform distribution on the unit disc of the complex plane. In particular, the long time equilibrium depends clearly on the initial condition.* Crossover regime with non-vanishing noise: _N=N and _N=N. In this case =1 and the McKean – Vlasov equation (<ref>) has a diffusive term. This regime is also considered in <cit.> for instance, see also <cit.>. The Keller – Segel model studied in <cit.> is the analogue with an attractive interaction instead of repulsive. § USEFUL FORMULASIn this section we gather several useful formulas related to the energy H = H_V + H_W and the operator L, defined in (<ref>) and (<ref>) respectively. Recall that V(z) = | z |^2 and W(z)=-2logz on ^2 ∖{0}, giving V(z)=2z and W(z)=-2z/z^2.Moreover we let | x |^2 = ∑_i=1^N | x_i |^2 for x = (x_1, …, x_N) ∈ (^2)^N.Gradient.By (<ref>), for any x∈ D and i∈{1,…,N},_x_i H_V(x) = 1/N V(x_i) = 2/N x_iand_x_i H_W(x) = 1/N^2∑_j≠ i W(x_i-x_j) = -2/N^2∑_j≠ ix_i-x_j/x_i-x_j^2. Hessian. By (<ref>)-(<ref>), for any x∈ D and i,j∈{1,…,N},^2_x_i,x_j H_V(x)=1/N^2 V(x_i) if i=j0if i≠ jand^2_x_i,x_j H_W(x)= +1/N^2∑_k≠ i^2 W(x_i-x_k)if i=j-1/N^2^2 W(x_i-x_j)if i≠ j.This gives^2 H_V=2/NI_2N and ^2 H_W = 1/N^2Awhere I_2N is the 2N× 2N identity matrix and A is a N× N bloc matrix with diagonal and off-diagonal 2× 2 blocsA_i,i=∑_k≠ i^2W(x_i-x_k),A_i,j=-^2W(x_i-x_j),i≠ j. Operator. The generator L defined in (<ref>) on functions f: D → is given by L f(x)= _N/_N f(x) -_N/N∑_i=1^N V(x_i)·_x_if(x) -_N/N^2∑_1 ≤ i≠ j ≤ N W(x_i-x_j)·_x_i f(x)= _N/_N f(x) -2_N/N∑_i=1^Nx_i·_x_if(x) +2_N/N^2∑_1 ≤ i≠ j ≤ N(x_i-x_j)·_x_i f(x)/|x_i-x_j|^2. Let us compute now LH_V, LH_W, and LH. First of all, since ∇ W is odd, we get by symmetrization from (<ref>)-(<ref>) thatH^2(x) =1/N^2∑_i=1^N| V(x_i)|^2 +1/N^4∑_i=1^N|∑_j≠ i W(x_i-x_j)|^2+1/N^3∑_i≠ jV(x_i)- V(x_j)· W(x_i-x_j)=4/N^2x^2 +4/N^4∑_i=1^N| ∑_j≠ ix_i-x_j/x_i-x_j^2|^2 -4N-1/N^2.Moreover, from (<ref>) and W=0 on D, we getH_W(x)=0 and H(x) = H_V(x) =∑_i=1^N1/N V(x_i)=4.By (<ref>)-(<ref>) and by symmetry we also haveL H_V(x)=_N/_NH_V(x)-_N H(x)·H_V(x)=4_N/_N-4_N/N^2x^2 +4_N/N^3 ∑_i=1^N∑_j≠ ix_i-x_j/x_i-x_j^2· x_i=4_N/_N-4_N/N^2x^2 +2_N/N^3 ∑_1 ≤ i≠ j ≤ Nx_i-x_j/x_i-x_j^2·(x_i-x_j)=4_N/_N+ 2 _NN-1/N^2 -4_N/NH_V(x)and likewiseLH_W(x)=_N/_N H_W(x)-_N∇ H(x)·∇ H_W(x)=2_N N-1/N^2 -4_N/N^4∑_i=1^N|∑_j≠ ix_i-x_j/x_i-x_j^2|^2From (<ref>) and (<ref>) we finally getLH(x)= LH_V(x)+LH_W(x)= 4_N/_N +4_N(N-1/N^2-1/NH_V(x)-1/N^2∑_i=1^N |1/N∑_j≠ ix_i-x_j/x_i-x_j^2|^2). Note that the fact that the singular repulsion potential W is the fundamental solution of the diffusion partsimplifies the expression of LH, in contrast with the situation in dimension 1 studied in <cit.>, see also <cit.>.§ PROOF OF THEOREM <REF> The set D defined by (<ref>) is path-connected in (^2)^N.It suffices to show that for any x:=(x_1,…,x_N)∈ D and y:=(y_1,…,y_N)∈ D, there exists a continuous map γ:=(γ_1,…,γ_N):[0,1]↦ D such that γ(0)=x and γ(1)=y, which must be understood as the position in time of N moving particles in space. This corresponds to move a cloud of N distinct and distinguishable particles into another cloud of N distinct and distinguishable particles. Let us proceed by induction on N. The property is immediate for N=1. Suppose that N≥1 and assume that one has already constructed t∈[0,1]↦(γ_1(t),…,γ_N(t)). One can first construct γ_N+1 in such a way that {t∈[0,1]:γ_N+1(t)∈{γ_1(t),…,γ_N(t)}} is a finite set. Second, one may modify the path γ_N+1, locally at the intersection times by varying the speed, in order to make this set empty. This is possible since d=2, and possibly impossible if d=1 since a particle cannot bypass another one.For any fixed N, we have H≥ 0, lim_x→∂ DH(x)=+∞,and ^- H is Lebesgue integrable on D for any >0.Let x = (x_1, …, x_N) in D. Then1/2∑_i ≠ j| x_i - x_j |^2= 1/2∑_i, j=1^N | x_i - x_j |^2= N ∑_i=1^N | x_i |^2 - |∑_i=1^N x_i |^2≤ N|x|^2so for u_ij = | x_i - x_j |^2 it holds2N^2 H(x)= N | x |^2+ N | x |^2- ∑_i ≠ jlog u_ij≥ N | x |^2+ ∑_i ≠ j( u_ij/2- log u_ij).But u/2 - log u ≥ 1 - log(2) ≥ 1/4 for all u>0, soH(x)≥| x |^2/2N + 1/2N^2N(N-1)/4 ≥| x |^2/2N + 1/16·In particular H ≥ 0 and ^- H is Lebesgue integrable on D for any >0.We now prove that H(x)→+∞ as x→∂ D. It suffices to show that for any R>0 there exists A>0 and >0 such that H(x)≥ R as soon as max_1≤ i≤ Nx_i≥ A or min_1≤ i≠ j≤ Nx_i-x_j≤. First, let us fix R>0. Then, by (<ref>), H(x) ≥ R as soon as | x |^2 ≥ 2NR, giving such an A.Then, for >0 to be chosen later, assume that for some i≠ j we have x_i-x_j≤. Then, by definition of H(x),N^2H(x)≥ 2log1/x_i-x_j +∑_1 ≤ k≠ l ≤ N {k,l}≠{i,j}log1/x_k-x_l.We can assume that max_1≤ r≤ Nx_r≤ A otherwise we have already seen that H(x)≥ R. Hence, for any (k,l) with k≠ l we have log1/x_k-x_l≥-log(1+x_k)-log(1+x_l)≥ -2log(1+A)using the inequality |a-b|≤(1+|a|)(1+|b|) for a,b∈. As a consequenceN^2H(x)≥ -2log-2N^2log(1+A),which is ≥ R for a small enough . In the sequel we use the notation _x=[ ·| X_0=x] and _x = [ ·| X_0=x]. We first construct the process X starting in D up to its explosion time. Given an initial condition x∈ D, for each ε∈ (0, min_1≤ i≤ N|x^i-x^j| ) we consider a smooth function W^ε on ^2 coinciding with W on {z∈^2: |z|≥ε} and we setH^ε=H_V+H_W^ε.Given a Brownian motion B in a fixed probability space we let X^ε denote the unique pathwise solution to the stochastic differential equationX^ε_t =√(2_N/_N) B_t -_N H^ε(X^ε_t)t,X^ε_0 = x.Notice that for ε'∈ (0,ε], the processes X^ε and X^ε' coincide up to the stopping timeT^ε,ε' =inf{s≥ 0:min_i≠ j|(X^ε'_s)^i - (X^ε'_s)^j|≤ε}.For each ε∈ (0, min_1≤ i≤ N|x^i-x^j| ) we can thus unambiguously define a stopping time T^ε and a process X on [0,T^ε], setting T^ε=T^ε,ε' and X=X^ε' for any ε'∈ (0,ε). By continuity, we have T^ε'>T^ε a.s., and so X is uniquely defined up to the stopping time T_ D defined in (<ref>). On the other hand, the process X satisfies equation (<ref>) on each interval [0, T^ε ) and hence on [0, T_ D) too. Thus, we just have to prove that T_ D=∞ a.s.Given R>0, define the stopping timesT'_R:=inf{t≥0:H(X_t)>R}∈[0,∞], and T':=lim_R→∞T'_R=sup_R>0T'_R∈[0,∞].Lemma <ref> gives {T'=∞}⊂{T_∂ D=∞}: indeed on {T'=∞}, for every t≥0 we have sup_s∈[0,t]H(X_s)<∞ ; by Lemma <ref> this means that T_∂ D=∞.Let us now show that _x (T' = ∞) = 1. Thanks to (<ref>), we have LH≤ c on D for c= 4 α_N (1/β_N + 1/N). Moreover, given R ≥ 1 and proceeding as in the end of the proof of Lemma <ref> we can choose ε <e^- CR N^2 log N for a numerical constant C such that the function H (respectively LH) coincides with H^ε (respectively LH^ε) along the trajectory of X on the interval [0, T'_R]; we can therefore apply the Itô formula to X_t∧ T'_R and H^ε to get that_x(H(X_t∧ T'_R))-H(x) =_x∫_0^t∧ T'_RLH(X_s)s≤_x∫_0^t∧ T'_R cs≤ ct,for each t≥ 0.In particularsup_R >0_x(H(X_t∧ T'_R))< ∞.On the other hand, since H is everywhere nonnegative by Lemma <ref>, we haveR_T'_R ≤ t≤ H(X_t∧ T'_R),from which it follows that_x (T'_R ≤ t) ≤1/Rsup_R >0_x(H(X_t∧ T'_R)).Finally _x (T' ≤ t) = lim_R →∞_x (T'_R ≤ t) =0, for any t≥0, and thus _x (T'=∞) = 1. Note that our proof of non-explosion notably differs from the one of<cit.> and <cit.>: we deal with ∂ D at once,instead of handling separately ∞ and x_i≠ x_j, thanks to thegeometric Lemma <ref>.a) From the previous proof we see that the process X and the process X^ε as in (<ref>) coincide up to the stopping time T^ε. Moreover, T^ε→∞ a.s. as ε→ 0. This readily implies that X^ε→ X a.s. uniformly on each finite time interval [0,T] and, in particular that Law(X^ε)→Law(X) in C([0,T],(^2)^N).b) Since H is bounded from below and LH is bounded from above, letting R→∞ in the first equality in (<ref>) and using twice Fatou's Lemma we get that_x(H(X_t))-H(x) ≤_x∫_0^tLH(X_s)swithboth sides finite, for all t≥ 0. § PROOF OF THEOREM <REF>By the Itô formula and (<ref>), N^-1X_t^2 = H_V(X_t) evolves according to the stochastic differential equationH_V(X_t)=LH_V(X_t) t+√(2_N/_N)∇H_V(X_t)B_t =4_N/_N +2_NN-1/N^2-4_N/NH_V(X_t) t +√(2_N/_N)2/N X_tB_t.The process H_V(X_t) thus satisfies, until the first time it hits 0, the stochastic differential equation (<ref>) with the Brownian motion b_t defined by b_t= X_t ·B_t/|X_t|. Standard properties of the CIR process (see <cit.>) and the fact that 4_N/_N +2_NN-1/N^2≥ 4 _N/N _N, imply this stopping time is ∞ a.s. Pathwise uniqueness for (<ref>) ensures that the law of H_V(X_t) is the same as for the CIR process (in particular, its invariant distribution is given in <cit.>).Ergodicity of the solution R to (<ref>) is proved in <cit.> by a non quantitative approach. Let us prove the long time convergence bound (<ref>) in Wasserstein-1 distance. By standard arguments, it is enough to show that for any pair (R^x_t,R^y_t) of solutions to (<ref>) driven by the same (fixed) Brownian motion b_t, and such that (R^x_0,R^y_0)=(x, y), one has[|R^x_t-R^y_t|]≤^-4_N/Nt|x-y|.This can be done adapting classical uniqueness argument for square root diffusions found in <cit.>. Indeed, consider the functionx∈_+↦ρ(x):= √(8 α_N/ N β_N x)and the sequence {a_ℓ}_ℓ≥ 1 defined as a_0 = 1anda_ℓ= a_ℓ-1^-ℓ8 α_N/N β_N, ℓ≥ 1.Note that a_ℓ↘ 0 and ∫_a_ℓ^a_ℓ-1ρ(z)^-2z = ℓ. For each ℓ≥ 1, let moreover z ↦ψ_ℓ(z) be a non-negative continuous function supported on (a_ℓ , a_ℓ-1) such that ∫_a_ℓ^a_ℓ-1ψ_ℓ(z) z = 1 and 0 ≤ψ_ℓ(z)≤ 2ℓ^-1ρ(z)^-2 for a_ℓ < z < a_ℓ-1. Consider also the even non-negative and twice continuously differentiable function ϕ_ℓ defined byϕ_ℓ(x)=∫_0^|x|d y∫_0^yψ_ℓ(z) z,x ∈For all x∈ it satisfies : ϕ_ℓ(x) ↗ |x| ,ϕ'_ℓ(x) →(x) as ℓ→∞, 0 ≤ϕ'_ℓ(x)x ≤ |x| and 0≤ϕ”_ℓ(x) 8 α_N/ N β_N |x| ≤ 2ℓ^-1. Applying the Itô formula to ϕ_ℓ and ζ_t:= R^x_t-R^y_t we getϕ_ℓ(ζ_t)= M^ℓ_t-4_N/N∫_0^tϕ'_ℓ(ζ_s) ζ_ss +4_N/N _N∫_0^tϕ”_ℓ(ζ_s) ζ_ssfor some martingale M_t^ℓ. Taking expectation, letting ℓ→∞ and applying Gronwall's lemma, the desired inequality is obtained. Assertion (<ref>) follows from (<ref>), noting that the function f(t) = [H_V(X_t)| X_0=x] solvesf(t) = f(0)+ ∫_0^t ( 4_N/_N+2_NN-1/N^2- 4_N/N f(s) )sfor all t>0, and integrating this equation. § PROOF OF THEOREM <REF> The set D defined by (<ref>) is not convex. Moreover, the Hessian matrix of the function H is not always positive definite on D.The set D is not convex since 0∈[-x,x]∩ D^c for any x∈ D.The convexity of H could be studied using a bloc version of the Ghershgorin theorem, see <cit.>, if W were convex. Unfortunately it turns out that W is nowhere convex. More precisely, setting z=(a,b)^⊤∈^2∖{(0,0)}, we getW(z) =-log(z^2) =-log(a^2+b^2)andW(z) =-2z/z^2 =-2(a,b)^⊤/a^2+b^2and^2W(z) = 2[ a^2-b^2 2ab; 2ab b^2-a^2 ]/(a^2+b^2)^2. ThusTr(^2 W(z))=0 and(^2 W(z)) =-4/z^4.Consequently the two eigenvalues _±(z) of ^2W(z) satisfy_-(z)=-_+(z)=-2/a^2+b^2=-2/z^2z→0⟶-∞,and have respective eigenvectors (-b, a) and (a,b). In particular W is not convex.Now, by (<ref>), if we fix x_1,…,x_N-1 and let x_N tend to x_1, then ∇^2W(x_1-x_j) will remain bounded for any j∈{2,…,N-1} while the smallest eigenvalue of ∇^2W(x_1-x_N) blows down to -∞. Therefore ∇^2_x_1,x_1H(x_1,…,x_N) and thus ∇^2H(x_1,…,x_N) is not positive definite for such points.Note however that we may also use (<ref>) to get that ∇^2H(x_1,…,x_N) is positive definite at points of D for which all the differences x_i-x_j are large enough. The following Lemma is the gradient version of Lemma <ref>. For any N and x = (x_1, …, x_N) in D we have|∇ H (x) |^2 ≥4/N^2| x |^2+ 4/N^4∑_i≠ j1/|x_i-x_j|^2 - 4 N-1/N^2· In particular lim_x→∂ D∇ H(x)=+∞. This is a consequence of (<ref>) and the fact that for any N and any distinct x_1,…,x_N∈^2,S_N:=∑_i=1^N|∑_j≠ ix_i-x_j/|x_i-x_j|^2|^2 -∑_i=1^N∑_j≠ i1/|x_i-x_j|^2 ≥0.For the proof of (<ref>), we first observe thatS_2=0and we now consider N≥ 3 for whichS_N=2∑_i=1^N∑_1≤ j<k≤ N j,k≠i(x_i-x_j) · (x_i-x_k)/| x_i-x_j|^2| x_i-x_k|^2.Decomposing∑_i=1^N∑_1≤ j<k≤ N j,k≠i·=∑_1≤ i<j<k≤ N· +∑_1≤ j<i<k ≤ N·+∑_1 ≤ j<k<i ≤ N·and letting I=j,J=i and K=k in the second sum on the right-hand side and I=j,J=k and K=i in the third sum, we see that S_N/2 is equal to∑_1≤ i<j<k≤ N| x_j-x_k|^2 (x_i-x_j)· (x_i-x_k)/| x_i-x_j|^2| x_j-x_k|^2| x_k-x_i|^2+| x_k-x_i|^2 (x_j-x_i)·(x_j-x_k)/| x_i-x_j|^2| x_j-x_k|^2| x_k-x_i|^2 +| x_i-x_j|^2 (x_k-x_i)·(x_k-x_j)/| x_i-x_j|^2| x_j-x_k|^2| x_k-x_i|^2·But| x_j-x_k|^2= | x_i-x_j|^2+| x_i-x_k|^2-2 ( x_i-x_j) · (x_i-x_k)soS_N=4∑_1≤ i<j<k ≤ N| x_i-x_j|^2 | x_i-x_k|^2-(x_i-x_j)· (x_i-x_k)^2/| x_i-x_j|^2| x_j-x_k|^2| x_k-x_i|^2·Hence S_N≥ 0 by the Schwarz inequality. This shows also that equality is achieved when x_i-x_j and x_i-x_k are parallel for any i,j,k for instance when x_i=(i,0) for any i, thanks to the equality case in the Schwarz inequality. Let us observe from the proof that the same bound would hold in any Hilbert space. The following lemma is the counterpart on H_W of Theorem <ref> for H_V. It is likely that the bounds in the lemma are not optimal, as we would expect bounds independent of N. This is probably due to our use of the bound (<ref>). The lemma is not used but has its own interest as we see that the particular speed α_N =N naturally appears in the upper bounds, as in Theorem <ref>.For every x∈ D and t≥0, let us defineη_x(t):=2N/N-1_x [H_W(X_t)] whereH_W(x):=1/2N^2∑_i≠ jW(x_i-x_j).Then, for every x∈ D and t≥0,η_x(t)≤-log(^-η_x(0)-4 α_Nt/N+2/N(1-^-4 α_Nt/N))and in particularη_x(t)≤logN/2(1-^-4 α_Nt/N)andη_x(t)≤max(η_x(0),logN/2). Taking expectation to the first line in equation (<ref>) and subtracting the obtained identity from the inequality in Remark <ref> b), we get_x(H_W(X_t))-H_W(x) ≤_x∫_0^tLH_W(X_s)sfor all t≥ 0. But from (<ref>) and (<ref>) we getLH_W(x)=2_N/N^2(N-1) -4_N/N^4∑_i=1^N|∑_j≠ ix_i-x_j/x_i-x_j^2|^2≤ 2_N/N^2(N-1) -4_N/N^4∑_i≠ j1/x_i-x_j^2= 4_N/N^2N-1/NN/2-1/N(N-1)∑_i≠ j1/x_i-x_j^2.On the other hand, by the Jensen inequality,H_W(x) =N-1/2N1/N(N-1)∑_i≠ jlog1/x_i-x_j^2≤N-1/2Nlog(1/N(N-1)∑_i≠ j1/x_i-x_j^2).Therefore, we getLH_W(x) ≤ 4_N/N^2N-1/NN/2-^2N/N-1H_W(x).Usingagain the Jensen inequality, it follows thatη_x(t)≤η_x(0) + 2N/N-1∫_0^t _x L H_W (X_s) s ≤η_x(0)+ 8 α_N/N^2 ∫_0^tN/2-_x [ ^2N/N-1H_W(X_s)] s≤η_x(0)+8α_N/N^2∫_0^tN/2-e^η_x(s) s.Therefore^-η_x(t)≥^-η_x(0)-4 α_Nt /N +2/N(1-^-4 α_Nt/N) ≥min{2/N, ^-η_x(0)}by time integration for the first bound and then, for the second bound, by writing the obtained expression as the interpolation between 2/N and ^-η_x(0). Dropping the ^-η_x(0)-4 α_Nt/N term gives the second upper bound in the lemma.In order to prove that P^N satisfies a Poincaré inequality, we follow the approach developed in <cit.> based on a Lyapunov function together with a local Poincaré inequality (see also the proof of <cit.>). This approach amounts to find a positive ^2 function ϕ on D, a compact set K⊂ D and positive constants c,c', such that on DL ϕ≤ -cϕ+c'_K.Such a ϕ is called a Lyapunov function. Indeed, for a centered f∈ this gives∫f^2P^N ≤∫_Kc'/cϕf^2P^N +∫-Lϕ/cϕf^2 P^N.The first term of the right-hand side can be controlled using a local Poincaré inequality, in other words a Poincaré inequality on every ball included in D, by comparison to the uniform measure. The second one can be handled using an integration by parts which is allowed since f∈. See <cit.> and <cit.> for the details.For our model P^N we take the ^∞ functionϕ=^ Hfor some >0. This function is larger than or equal to 1 by Lemma <ref>, and the probability measure P^N has a smooth positive density on D, which provides a local Poincaré constant that may depend on N however. Let us check that ϕ is a Lyapunov function. To this end, let us show that there exist constants c, c”>0 and a compact set K⊂ D such that, on D,L ϕ/ϕ≤ -c+c”_K.Indeed, since ϕ is positive and bounded on the compact set K, this gives, on D,L ϕ≤ -cϕ+c”sup_x∈ Kϕ(x)_K = -cϕ+c'_K.In order to compute L ϕ/ϕ, we observe thatϕ=ϕ Handϕ =^2 ϕ H^2+ϕ H.Therefore, by (<ref>), β_N/_NLϕ/ϕ =ϕ-_N H·ϕ/ϕ = H+(-_N) H^2.Now Δ H = 4 on D by (<ref>). Moreover by Lemma <ref>, for <_N there exists a compact set K⊂ D such that(β_N - γ) inf_(x_1,…,x_N)∈ K^c H^2(x_1,…,x_N)>5.One can take for instanceK={x∈(^2)^N:|x|≤ R and min_i≠ jx_i-x_j≥}for R>0 large enough and >0 small enough.Then β_N /(α_N γ) Lϕ / ϕ≤ -1 on D ∖ K and the Poincaré inequality is proved. Note that we can take =1 if _N≥ N.Let us give an alternative direct proof of the Poincaré inequality for the probability measure P^2. Consider indeed the change of variable (u,v)=((x_1+x_2)/2,(x_1-x_2)/2) on ^2 ×^2, which has the advantage to decouple the variables (this miracle is available only in the two particle case N=2). Letting β = β_2, we get a probability density function on ^2×^2 proportional to(u,v) ∈^2 ×^2↦ ^-βu^2-βv^2-+β/2 logv =^-βu^2v^β/2^-βv^2.This probability measure is the tensor product of the Gaussian measure, which satisfies a Poincaré inequality, and of the measure μ with density^-βΨ(v)/ZwithΨ(v) =| v |^2 - 1/2log| v |.The measure μ is not log-concave at all (singularity at zero notably) but Ψ(v) is a convex function of the norm r = | v |. Hence <cit.> ensures that μ satisfies a Poincaré inequality, and then so does our product measure by tensorization.Note that one can prove Poincaré for μ by using a Lyapunov function as in the proof of Theorem <ref>, instead of <cit.>: namely if L' :=Δ- βΨ· in dimension two and ϕ=^βΨ/2, thenL' ϕ/ϕ = β/2Ψ - β^2/4Ψ^2, Ψ (v) =2v-v/2 v^2, Ψ=4for v≠0 (recall that logv is harmonic in dimension two). ThereforeL' ϕ/ϕ = 2 β^2 - β^2 ( x-1/4x)^2≤ -c + c'_Kfor the compact setK:={x∈^2:r≤x≤ R}with 0<r<R well chosen. § PROOF OF THEOREM <REF>Recall that if μ^N is the random empirical measure under P^N then for any continuous and bounded test function f:^2→, using exchangeability and (<ref>),_P^N∫_^2f(x) μ^N( x)=∫_(^2)^N1/N∑_k=1^Nf(x_k) ^N,N(x_1,…,x_N) x_1⋯ x_N=∫_^2f(x) ^1,N(x)x =_P^1,N(f)where P^1,N is the 1-dimensional marginal of P^N. By Theorem <ref>, as N→∞, the density ^1,N of P^1,N tends to the density of the uniform distribution μ_∞ on the unit disc of ^2. The probability measure μ_∞ satisfies a Poincaré inequality for the Euclidean gradient, since for instance it is a Lipschitz contraction of the standard Gaussian on ^2. Unfortunately, the convergence of densities above is not enough to deduce that P^1,N satisfies a Poincaré inequality (uniformly in N or not). The idea is to view P^1,N as a Boltzmann – Gibbs measure and to use some hidden convexity. Namely, from (<ref>) its density is given on ^2 by ^1,N(x)=^-Nx^2-ψ(√(N)x)/πwithψ(x):=-log∑_ℓ=0^N-1x^2ℓ/ℓ!.If we now write f(x)=x^2+ψ(x)=g(r^2) with r=x and g(t)=t-log∑_ℓ=0^N-1t^ℓ/ℓ! thenf(x)=2g'(r^2)x and^2 f(x)=4g”(r^2)x⊗ x+2g'(r^2)I_2andg'(t) =t^N-1/(N-1)!/∑_ℓ=0^N-1t^ℓ/ℓ!≥0 and g”(t) =t^N-2/(N-2)!∑_ℓ=0^N-1t^ℓ/ℓ! -t/N-1∑_ℓ=0^N-2 t^ℓ/ℓ!/∑_ℓ=0^N-1t^ℓ/ℓ!^2≥0.It follows that f is convex (note that its Hessian vanishes at the origin), and in other words P^1,N is log-concave. Therefore, according to a criterion stated in <cit.> and essentially due to Kannan, Lovász and Simonovits, it suffices to show that the second moment of P^1,N is uniformly bounded in N.But, using the density (<ref>) of P^1,N, this moment is∫_^2x^2 P^1,N ( x) =∑_ℓ=0^N-1N^ℓ/ℓ! ∫_0^∞r^2(ℓ+1)2r^-Nr^2 dr =∑_ℓ=0^N-1N^ℓ/ℓ!(ℓ+1)!/N^ℓ+2 =N+1/2N≤1/2· This concludes the argument thanks to the Bobkov criterion.With _N=N^2 and since P^1,N = μ^N, (<ref>) is consistent with (<ref>) since in this caselim_t→∞[H_V(X_t) | X_0=x] =1/2+1/N-1/2N =N+1/2N·Note also that, by (<ref>), the second moment of P^1,N = μ^N tends to 1/2 as N→∞; this turns out to be the second moment of its weak limit μ_∞ since∫_^2x^2 μ_∞( x) =2π/π∫_0^1r^3 r =1/2·Observe finally that a bound on the second moment of P^1,N = μ^N can be obtained as follows. Let M be a N× N random matrix with i.i.d. entries of Gaussian law (0,1/2NI_2) (in other words an element of the Complex Ginibre Ensemble); then, by Weyl's inequality <cit.> on the eigenvalues,∫_^2x^2 μ^N( x) =1/N∑_k=1^N_k(M)^2 ≤1/N∑_k=1^N_k(MM^*) =1/NTr(MM^*) =1.The probability measure P^1,N is also spherically symmetric, or rotationally invariant, as in Bobkov <cit.> (see also <cit.>). Namely, in the notation f(x)= g( r^2) with r = | x | for the “potential” of the density of P^1,N, as in the proof of Theorem <ref>, let h( r)=g( r^2). Thenf(x) = h( r),f(x)=h'(r)x/xand^2f(x) =h”(r)x⊗ x/x^2 +h'(r)[ x_2^2 -x_1x_2; -x_1x_2 x_1^2 ]/x^3·The matrix on the right-hand side has non-negative trace and null determinant, and is thus positive semi-definite (it is the Hessian of the norm x↦x=r). Moreoverh'(t)=2g'(t^2)t,h”(t)=4g”(t)t^2+2g'(t^2)≥0.It follows that P^1,N is a spherically symmetric probability measure on ^2, and its density is a log-concave function of the norm (and it vanishes at the origin). Now according to <cit.>, it follows that the probability measure P^1,N satisfies a Poincaré inequality with a constant which depends only on the second moment, which again is bounded in N.According to Bobkov's result <cit.>, we even get for P^1,N a logarithmic Sobolev inequality with a uniform constant in N provided that P^1,N has a sub-Gaussian tail uniformly in N (which is stronger than the second moment control). This is indeed the case. Namely, if Z∼ P^1,N, then for any real R≥0,(Z≥ R) =∫_0^2π∫_R^+∞^-Nr^2/π∑_ℓ=0^N-1(Nr^2)^ℓ/ℓ! r r =1/N ∫_NR^2^+∞^-s∑_ℓ=0^N-1s^ℓ/ℓ!s.Moreover1/N∑_ℓ=0^N-1s^ℓ/ℓ!≤s^N/N!≤ 2^N^1/2sfor s≥ N. Hence, for R≥2,(Z≥ R)≤∫_NR^2^+∞ 2^N^-1/2ss =2^N+1^-1/2NR^2≤ 4^-1/2R^2. § PROOF OF THEOREM <REF> Proof of the first part of Theorem <ref>. It is a consequence of (<ref>) and of the following theorem. Indeed, by Lebesgue's dominated convergence, (<ref>) implies that F(μ^N) tends to F(μ_∞) for every continuous and bounded function F : (E) →ℝ. In other words, (i) holds in Theorem <ref>, whence (ii), which is exactly the first part of Theorem <ref>. Let E be a Polish space and (E) be the Polish space of Borel probability measures on E endowed with the weak convergence topology. Let μ be an element of (E) and let (P^N)_N a sequence of exchangeable probability measures on E^N. Let us define the random empirical measureμ^N= 1/N∑_i=1^N δ_X_iwhere (X_1, …, X_N) has law P^N. Then the following properties are equivalent: (i) the law of μ^N converges to δ_μ weakly in ((E));(ii) for any fixed k ≤ N the k-th dimensional marginal distribution P^k,N of P^N converges weakly in (E^k) to the product probability measure μ^⊗ k;(iii) the 2-nd dimensional marginal P^2,N of P^N converges to μ^⊗ 2 weakly in (E^2). Theorem <ref> is stated for instance in <cit.>, <cit.> and <cit.>, but with a sketchy proof that (iii) implies (i). For the reader's convenience, we detail this proof when E satisfies the following property : there exists a countable subset D of the set C_b(E) of continuous and bounded functions E →ℝ, such that for (μ_n)_n, μ in (E), it holds ∫ϕdμ_n →∫ϕdμ for any ϕ in C_b(E) as soon as it holds for any ϕ in D. For instance this property holds when E is the Euclidean space.Since ((E)) is metrizable, it is enough to check that for any sequence (N_k)_k there exists a subsequence (N_k_j)_j such that the law of μ^N_k_j converges to δ_μ. But, by expanding the square, exchangeability and (iii),(|∫_E ϕd μ^N_k - ∫_E ϕdμ|^2) → 0,k → + ∞for any ϕ in C_b(E) and hence in D. Hence for any such ϕ there exists a subsequence still denoted (N_k_j)_j such that ∫ϕd μ^N_k_j→∫ϕdμ almost surely. Now, by a diagonal extraction argument, we can build another subsequence (N_k_j)_j such that, almost surely, ∫ϕdμ^N_k_j→∫ϕdμ for any ϕ in D. By definition of D, this implies that, almost surely, μ^N_k_j converges to μ in the metric space (E). It follows that the law of μ^N_k_j converges to δ_μ by the Lebesgue dominated convergence theorem. Hence (i) since ((E)) is metrizable. Proof of the second part of Theorem <ref>. We first describethe behavior of the one-marginal density function ^1,N. From (<ref>) it is given by^1,N(z)= ^-N|z|^2/π_N(N|z|^2), z∈,where _N(w):=∑_ℓ=0^N-1w^ℓ/ℓ! is the truncated exponential series. Then, pointwise in ℂ,^1,N(z)→ 1/π(1_|z|<1 + 1/21_|z|=1),N →∞. Namely, by rotational invariance, it suffices to consider the case z = r>0. Next, if Y_1, …, Y_N are i.i.d. random variables following thePoisson distribution of mean r^2, thene^-N r^2e_N(Nr^2) =ℙ(Y_1+⋯+Y_N<N) =ℙ(Y_1+⋯+Y_N/N<1).Now, as N→∞, Y_1 + … + Y_N/N→ r^2 almost surely by the law of large numbers, and thus the right-hand side above tends to 0 if r>1 and to 1 if r<1. In other wordse^-N r^2e_N(Nr^2) →1_r < 1provided r ≠ 1. For r=1 by the central limit theorem we getℙ(Y_1+⋯+Y_N/N<1) = ℙ(Y_1+⋯+Y_N - N/√(N)<0) →1/2. In fact, the convergence in (<ref>) holds uniformly on compact sets outside the unit circle | z | = 1, as shown in Lemma <ref> below. It cannot hold uniformly on arbitrary compact sets ofsince the pointwise limit is not continuous on the unit circle.We now turn to the two-marginal density function ^2,N. By (<ref>) it is given by^2,N(z_1,z_2)= N/N-1^-N(|z_1|^2+|z_2|^2)/π^2( _N(N|z_1|^2)_N(N|z_2|^2)-|_N(Nz_1z_2)|^2 )=N/N-1^1,N(z_1)^1,N(z_2) - N/N-1^-N(|z_1|^2+|z_2|^2)/π^2 |_N(Nz_1z_2)|^2for every z_1,z_2∈.It follows that for any N≥2 and z_1,z_2∈,_N(z_1,z_2):= ^2,N(z_1,z_2)-^1,N(z_1)^1,N(z_2)= 1/N-1^1,N(z_1)^1,N(z_2)-N/N-1^-N(|z_1|^2+|z_2|^2)/π^2|_N(Nz_1z_2)|^2. In particular, using ^2,N≥0 for the lower bound,-^1,N(z_1)^1,N(z_2) ≤_N(z_1,z_2) ≤1/N-1^1,N(z_1)^1,N(z_2).From this and Lemma <ref> we first deduce that for any compact subset K of {z∈:|z|>1}lim_N→∞sup_z_1∈ z_2∈ K|_N(z_1,z_2)| = lim_N→∞sup_z_1∈ K z_2∈|_N(z_1,z_2)| =0. To conclude the proof ofTheorem<ref> it remains to show that _N(z_1,z_2)→0 as N→∞ when z_1 and z_2 are in compact subsetsof |z_1| < 1, |z_2| < 1. In this case |z_1z_2|≤1, and Lemma <ref> gives|_N(Nz_1z_2)|^2 ≤ 2^2N(z_1z_2) + 2r_N^2(z_1z_2).Next, using the elementary identity 2(z_1z_2)=|z_1|^2+|z_2|^2-|z_1-z_2|^2, we get^-N(|z_1|^2+|z_2|^2)| _N(Nz_1z_2)|^2 ≤ 2^-N|z_1-z_2|^2 +2^-N(|z_1|^2+|z_2|^2)r_N^2(z_1z_2).Since |z_1z_2|≤1, the formula for r_N in Lemma <ref> gives^-N(|z_1|^2+|z_2|^2)r^2_N(z_1z_2) ≤^-N(|z_1|^2+|z_2|^2-2-log|z_1|^2-log|z_2|^2)(N+1)^2/2π N.Using (<ref>), (<ref>) and the bounds φ^1,N≤ 1/π and u-1 - log u >0 for 0<u<1, it follows that Δ_N(z_1, z_2) tends to 0 as N→∞ uniformly in z_1,z_2 on compact subsets of{(z_1,z_2)∈^2:|z_1|<1,|z_2|<1,z_1≠ z_2}.This achieves the proof of Theorem <ref>. Let _N(w):=∑_ℓ=0^N-1w^ℓ/ℓ! denote the truncated exponential series. For every N≥1 and z∈,|_N(Nz)-^Nz1_|z|≤1|≤ r_N(z)wherer_N(z):= ^N/√(2π N)|z|^NN+1/N(1-|z|)+11_|z|≤1 +N/N(|z|-1)+11_|z|>1.In particular, for any compact subset K⊂∖{z∈:|z|=1},lim_N→∞sup_z∈ K^1,N(z)-1_|z|≤1/π = π^-1lim_N→∞sup_z∈ K^-N|z|^2_N(N|z|^2)-1_|z|≤1 =0. As in Mehta <cit.>, for every N≥1, z∈, if |z|≤ N then^z-_N(z) =∑_ℓ=N^∞z^ℓ/ℓ!≤|z|^N/N!∑_ℓ=0^∞|z|^ℓ/(N+1)^ℓ =|z|^N/N!N+1/N+1-|z|,while if |z|>N then|_N(z)| ≤∑_ℓ=0^N-1|z|^ℓ/ℓ!≤|z|^N-1/(N-1)!∑_ℓ=0^N-1(N-1)^ℓ/|z|^ℓ≤|z|^N-1/(N-1)!|z|/|z|-N+1.Therefore, for every N≥1 and z∈,|_N(Nz)-^Nz1_|z|≤1| ≤N^N/N!|z|^NN+1/N+1-|Nz|1_|z|≤1 +|z|^N-1|Nz|/|Nz|-N+11_|z|>1.It remains to use the Stirling bound √(2π N)N^N≤ N!^N to get the first result. §.§ AcknowledgmentsJ.F. thanks the hospitality and support of Université Paris-Dauphine via an invited professor position. This work was partly carried out during a visit to CIRM in Marseille; it is a pleasure for the authors to thank this institution for its kind hospitality and participants for discussions on this and related topics, notably Joseph Lehec and Camille Tardif. The article benefited from a very useful and relevant anonymous report. The authors acknowledge partial support from the STAB ANR-12-BS01-0019, Fondecyt 1150570, Basal-Conicyt CMM, Millennium Nucleus NC120062, and EFI ANR-17-CE40-0030 grants. smfplain | http://arxiv.org/abs/1706.08776v3 | {
"authors": [
"François Bolley",
"Djalil Chafai",
"Joaquín Fontbona"
],
"categories": [
"math-ph",
"math.FA",
"math.MP",
"math.PR"
],
"primary_category": "math-ph",
"published": "20170627110505",
"title": "Dynamics of a planar Coulomb gas"
} |
Second-Order Moment-Closure for Tighter Epidemic Thresholds [ December 30, 2023 =========================================================== A mean-field treatment is presented of a square lattice two-orbital-model for BiS_2taking into account intra- and inter-orbital superconductivity. A rich phase diagram involving bothtypes of superconductivity is presented as a function of the ratiobetween the couplings of electrons in the same and different orbitals(η= V_XX/V_XY) and electron doping .With the help of a quantity we call orbital-mixing ratio, denoted as R(ϕ),the phase diagram is analyzed using a simple and intuitive picture based onhow R(ϕ) varies as electron doping increases. The predictive power of R(ϕ)suggests that it could be a useful tool in qualitatively (or even semi-quantitatively)analyzing multiband superconductivity in BCS-like superconductors.§ INTRODUCTIONThe study of two-band superconductivity (SC) <cit.> (or, more generally, multiband SC)has become of increasing relevance as superconducting materials with overlapping bands at the Fermi surface,like, for example, MgB_2 <cit.>, are discovered.What distinguishes these systems from the more traditional single-band case is the coexistence,at the Fermi level, of electrons from different bands (originating from different orbitals).These electrons, which are directly involvedin the superconducting ground state, can, in principle, pair in a variety of ways.The large class of multiband superconductors includes heavy-fermion systems <cit.>,the well-studied MgB_2 <cit.>, thepnictides <cit.> and, more recently, the layered sulfides BiS_2 <cit.>.The types of pairing in multiband systems can be categorizedin two main groups, namely intraband and interband pairing, depending onthe predominant paring interaction in the system. These two types of pairings are not mutually exclusive, theymay coexist and even `compete' in the same material, changing in relative importanceas some externalparameter, as pressure or doping, is varied <cit.>. The superconducting state resulting fromthe addition of a second band <cit.> to the traditional single-band BCSstate <cit.> shows many interesting new features, like the possibility of formationof two superconducting gaps, which may then be observed by eitherAngle-Resolved Photoemission Spectroscopy (ARPES)<cit.>, Scanning Tunneling Spectroscopy (STS) <cit.>,or thermal transport measurements under magnetic field <cit.>, for example; the possibility of pairing evenwhen the electron-electron interaction in one of the bands is repulsive, in which case, when an interaction betweenthe bands is introduced, T_c increases in comparison to the single-band attractive case<cit.>.In addition, the isotope effect vanishes when the interband interaction is large, explainingthe behavior of superconductors like Nb_3Sn <cit.>.Note that the motivation for Suhl et al. <cit.> to introduce the `extra' band was to tryand explain the (relatively) high-T_c observed in transition metal superconducting compounds <cit.>.An indication that a second band had to be taken into account to treat SC in the transition metal elementswas that s-d electron scattering seemed to be important to explain their resistivity inthe normal state. Suhl et al. <cit.>analyzed three different situations (denoting the intraband pairing interactions as V_ss and V_dd,and the interband as V_sd): (i) finite V_sd and V_ss=V_dd=0, obtaining two different gaps(unless the density of states ρ_s=ρ_d, in which case the gaps are equal) whose dependence on temperatureis BCS-like, but that, nonetheless, have the same T_c; (ii) V_sd=0, where there are two gaps as well,with a BCS-like temperature dependence, however, with two different T_c values; and (iii) ifa small V_sd≪√(V_ssV_dd) is turned on, a single T_c is obtained that is closebut always above the larger T_cin (ii), as well as a gap with a dependence on temperature that is an interpolation between the gaps obtained in (ii).Two clear examples of case (i) can be observed first in MgB_2trough STS data as a function of temperature <cit.>,and second in the pnictide compound Ba_0.6K_0.4Fe_2As_2through ARPES <cit.>.In this work, using an orbital basis, we will study the contribution of different types ofpairings, intra- and inter-orbital,to the superconducting phase of BiS_2 systems as a function of electron doping. Reference <cit.>, where 32 classes of superconductorswere studied, placed the BiS_2 family of superconductors in the `possiblyunconventional' column. Reference <cit.> describes the experiments that suggest the possibility of these materials exhibiting unconventional superconductivity.The fact that these results come from polycrystalline samples, which are prone to inhomogeneitiesand random orientation of crystallites (which becomes relevant for measurements depending onthe application of a magnetic field) warrants the cautious approach taken by the communityworking on BiS_2. Thus, in the present work, we do not assume any specific pairing mechanism,although we briefly refer to `phonon pair-scattering', for the sake of argument, when discussing theresults.As to the SC gap, muon-spin rotation (μSR) experiments <cit.>, for example, supportmultiband SC in the BiS_2-based layered compound Bi_4O_4S_3, pointingto two s-wave-type energy gaps, although the authorsdo not rule out the possibility of fitting the data with a single s-wave gap. Therefore, we will consider these materials as s-wave (k⃗-independent) singlet superconductorsand use a mean-field approach to analyze its double-gap properties.In general, interband pairing between bands which cross theFermi surface at different wave-vectors may favor the appearanceof inhomogeneous superconducting states characterized by awave-vectorQ corresponding to the difference between the different band wave-vectors <cit.>. No evidence of such phenomenon has been experimentally observed in BiS_2 compounds, therefore we do nottake this possibility into account in our model.Features associated with low dimensionality <cit.> are important to determinethe electronic structure of these materials, but, close to the SC transition, fluctuations are averaged outas indicated by the large coherence length measured for these materials <cit.>,thus justifying the use of a BCS (mean-field) treatment of the problem, as undertaken here.Aside from the controversy regarding the pairing mechanism, superconductors based on BiS_2 layershave revealed complex and surprising properties. For example, recently, coexistence of magnetism and SC hasbeen reported <cit.> in Bi_4-xMn_xO_4S_3. These phenomenaare observed in different layersof thesystem and appear as rather independent of each other. The substitution by Co and Ni instead of Mn suggeststhat the increase in T_c due to the latter can be attributed to its mixed valence, which allows for aneffective charge transfer to the superconducting layers.This work is divided as follows: In section <ref> we present the tight-binding two-orbital model for BiS_2,showing in detail how does the Fermi surface changes with electron doping.Section <ref> presents the pairing interactions we are considering, while section <ref> developsthe gap equations at themean-field level. Section <ref> closes with some simplifying assumptions regarding the paring interactions, which reduce thenumber of gap equations from 4 to 2. We close section II by presenting the solution to the gap equations as afunction of η=V_XX/V_XY (the ratio of the relevant pairing couplings) and the electron doping .In Section <ref>, we clearly define what is meant by orbital-mixing, by introducing the quantity R(ϕ)to measure it along the Fermi surface, and describe its relevance to multiband SC. Section <ref>describes how the structures seen in both gap functions below the Lifshitz transition can be understoodthrough the way R(ϕ) changes with doping. In section <ref> the same is done above the Lifshitz transition.In addition, section <ref> presents results for the superconducting critical temperature T_c, whichare qualitatively in agreement with those for BiS_2 compounds.The paper closes with section <ref>, where Summary and Conclusions are given. The main message of this work is that the systematic application of the orbital-mixing concept to systems showingBCS-like multiband SC can pinpoint regions of the phase diagram where one of the possible superconducting orderparameters may dominate over the others, or where, for example, a competition between different order parameters may occur.The concept is illustratedthrough its detailed application to a two-orbital model for BiS_2, which, due to the marked dependenceof its Fermi surface on electron doping and the well defined variation of orbital-mixing along the BZ,provides a particularly convincing connection between orbital-mixing and specific superconducting order parameters. § MODEL§.§ Tight-binding two-orbital model Theelectronic structure of BiS_2 layers, close to the Fermi energy, is described by a two-dimensional, two-orbitaltight-binding model, which is extracted from first principles Density Functional Theory calculationsby using maximally localized Wannier orbitals centered at the Bismuth sites <cit.>.These Wannier states originate from the Bismuth 6p_X and 6p_Y orbitals. In reciprocal space, thetight-binding Hamiltonian can be written as: H_0= ∑_𝐤,σ=↑↓∑_α,β=X,Y T^αβ ( 𝐤) p^†_α 𝐤σ p_β 𝐤σ,where T^XX =2t_x^X(cos k_x +cos k_y) +2t_x ∓ y^Xcos(k_x ± k_y)+2t_2x ∓ y^X[cos(2k_x ± k_y) + cos(k_x ± 2k_y) ]+ϵ_X - μ,T^YY =2t_x^Y(cos k_x +cosk_y) +2t_x ± y^Ycos(k_x ∓ k_y)2t_2x ± y^Y[cos(2k_x ∓ k_y) + cos(k_x ∓ 2k_y) ]+ϵ_Y - μ,T^XY =T^YX=2t_x^XY(cos k_x -cosk_y) + 4t_2x^XY(cos 2k_x -cos2k_y)+4t_2x+y^XY(cos 2k_xcosk_y - cosk_x cos 2k_y).The operator p^†_α 𝐤σ (p_α 𝐤σ) ineq. (<ref>) creates (annihilates) an electron in a Bloch state with orbital character α=X,Y,with spin σ=↑↓, and momentum 𝐤.The values for the hopping parameters are those from Ref. <cit.>, and are reproducedin Table 1 for convenience. Note that the choice of an upper or lower sign in the ± and ∓in the arguments of the trigonometric functions in the equations above will determine the choice of thecorresponding hopping parameters that also have ± and ∓ in their subindexes.It is important to note that, following Ref. <cit.>, we denote the p_X and p_Y Wannierorbitals using uppercase letters (X,Y) and the crystallographic axes by lowercase ones (x,y) as theyare rotated in relation to each other by π/4 <cit.>, i.e., the Wannier orbitals p_X and p_Yare oriented along the diagonals of the square lattice defined by the crystallographic axes. The chemical potential μ varies with electron dopingand will control the filling of the bands, where =0indicates that the bands are empty and =1 represents quarter-filling (i.e., 1 electron, out of a maximum of 4, per site).Figure <ref> shows the Fermi surface for two different values of doping, =0.4 in panel (a),where we see two electron pockets (red) around [π,0] and [0,π], which grow with . At ≈ 0.45they will touch and the Fermi surface will undergo a Lifshitz transition to two hole pockets centeredaround [0,0] and [π,π]. These are shown (blue) in panel (b) for =0.6. In addition, for ≈ 0.5two electron pockets (red) around [π,0] and [0,π] will emerge and grow with , while the holepockets decrease.§.§ Interacting Hamiltonian For our model ofBiS_2-based superconductors, we will assume that attractive interactionsmediatedifferent types of intra- and inter-orbital pairings <cit.>. The relevantorbitals, as discussed above, are thep_X and p_Y Wannier orbitals of the Bismuth atomsin a BiS plane.The total Hamiltonian of the system can be written as H=H_0+H_I,where H_0 is given by eqs. (<ref>) and (<ref>) above and the interacting part of the Hamiltonian can be written asasum of intra- and inter-orbital components,H_I=H_ intra+H_ inter,whereH_ intra= -1/N∑_k,k^',α( V_αααα p_α k^'↑^†p_αk̅^'↓^†p_αk̅↓p_α k ↑) ,H_ inter= -1/N∑_k,k^',α≠β(V_ααββ p_α k^'↑^† p_αk̅^'↓^† p_βk̅↓p_β k ↑ +V_αββα p_α k^'↑^† p_βk̅^'↓^† p_βk̅↓p_α k ↑+ V_αβαβ p_α k^'↑^† p_βk̅^'↓^† p_αk̅↓p_β k ↑),whereα,β=X,Y, k̅ = -k, and N=L^2 is the number of sites in a L × L square lattice.To be accurate, as we chose the orbital states p_X,Y 𝐤σ to write the pair operators, wewill use the terminology intra- and inter-orbital to refer to the associated pairing, in opposition to intra and interband.The main reason for using the p_X,Y 𝐤σ-orbital states to write the pair operators is that theactual bands [obtained by diagonalizing H_0] show weak X-Y hybridization, because of the smallvalue of t_x^XY=0.107 in comparison to t_x^X,Y=0.880 (see Table 1). In addition, in systems where many-body termsoriginating from intra-site interactions may influence superconductivity (see, for example, Ref. <cit.>),which could be the case for BiS_2 compounds <cit.>, it is advantageous to analyze pairing in the orbital basis.In the equations above, all the coupling terms V are positive, therefore all pairing interactions considered are attractive.In addition, experimental findings for BiS_2 compounds <cit.>, up to now, support s-wave SC,therefore, we take all the coupling terms as being 𝐤-independent.Given that the origin of the pairing interaction in BiS_2 compounds has not been settled yet <cit.>,those are the only general assumptions we will make. In the next two sections we will use symmetryarguments to decrease the number of V terms in eqs. (<ref>) and (<ref>) when applied toBiS_2 compounds. The inter-orbital terms in eq. (<ref>)may be listed through the associated couplings asV_XXYY (V_YYXX), where an YY (XX) pair is scattered into an XX(YY) pair; V_XYYX (V_YXXY), where an YX (XY) pair is scattered into anXY (YX) pair; and V_XYXY (V_YXYX), where an XY (YX) pair isscattered into an XY (YX) pair.We will see in what follows that, by treating H_ intra and H_ interat the mean-field level, and applying symmetries present in the BiS planes, will allow usto reduce these couplings to just two, which we will denote as V_XX and V_XY.§.§ Mean-field theory and gap equations The interacting Hamiltonian in eqs. (<ref>) and (<ref>) will be solvedat the mean-field level, through the usual approximationABCD ≈⟨ AB ⟩ CD + AB ⟨ CD ⟩ - ⟨ AB ⟩⟨ CD ⟩: H_I≈ H_MF =H_1+H_2+C,where H_1 =-∑_k(Δ_1 p^†_Xk↑p^†_Xk̅↓ +Δ_2 p^†_Yk↑p^†_Yk̅↓+h.c.)andH_2 = -∑_k( Δ_3 p^†_Xk↑p^†_Yk̅↓+ Δ_4 p^†_Yk↑p^†_Xk̅↓+h.c.),where C is a constant and the Δ_i, which are order parameters of thesuperconducting phases <cit.>, are given byΔ_1 = 1/N∑_k^'V_XXXX⟨ p_Xk̅^'↓p_Xk^'↑⟩ + V_XXYY⟨ p_Yk̅^'↓p_Yk^'↑⟩,Δ_2 = 1/N∑_k^'V_YYYY⟨ p_Yk̅^'↓p_Yk^'↑⟩ + V_YYXX⟨ p_Xk̅^'↓p_Xk^'↑⟩, Δ_3 = 1/N∑_k^'V_XYYX⟨ p_Yk̅^'↓p_Xk^'↑⟩ + V_XYXY⟨ p_Xk̅^'↓p_Yk^'↑⟩, Δ_4 = 1/N∑_k^'V_YXXY⟨ p_Xk̅^'↓p_Yk^'↑⟩ + V_YXYX⟨ p_Yk̅^'↓p_Xk^'↑⟩. §.§ Applying symmetries We start our analysis from the fact that V_XXYY=V_YYXXand V_XYYX=V_YXXY.Now, to simplify eqs. (<ref>) to (<ref>), we will apply some symmetry propertiesrelated to the Bismuth p_X and p_Y orbitals, which lead to relations betweenthe remaining couplings V and between the expectation values in those equations. Given that both orbitals have the same energy andare related by a C_4 rotation <cit.>, we expect that V_XXXX= V_YYYYand ⟨ p_Xk̅^'↓p_Xk^'↑⟩ =⟨ p_Yk̅^'↓p_Yk^'↑⟩. Therefore, Δ_1 = Δ_2 (which we now denote as Δ_XX), and, if we define V_XX≡ V_XXXX + V_XXYY, we obtainΔ_XX = -V_XX/N∑_k^'⟨ p_Xk̅^'↓ p_Xk^'↑⟩.Note that the above equation replaces eqs. (<ref>) and (<ref>). The same symmetry arguments lead to V_XYXY=V_YXYXand ⟨ p_Yk̅^'↓p_Xk^'↑⟩ =⟨ p_Xk̅^'↓p_Yk^'↑⟩, which resultin Δ_3 = Δ_4 (which we now denote as Δ_XY),and, if we define V_XY≡ V_XYYX + V_XYXY, we obtainΔ_XY = -V_XY/N∑_k^'⟨ p_Yk̅^'↓ p_Xk^'↑⟩.Note that the above equation replaces eqs. (<ref>) and (<ref>).The simplified gap equations (<ref>) and (<ref>) determine the systemof self-consistent equations to be solved, where the effective interactions V_XX and V_XY are parameters that controlpairing of same-orbital electrons (XX or YY), or dissimilar electrons (XY or YX), respectively.We stress that the order parameter Δ_XX involves both intra-orbital (XX ↔ XXand YY ↔ YY) as well as inter-orbital (XX ↔ YY) processes, while the orderparameter Δ_XY involves only inter-orbital processes (XY ↔ YX, XY ↔ XY,and YX ↔ YX). Figure <ref> shows the results obtained by self-consistently solving eqs. (<ref>) and (<ref>) forboth Δ_XY, in panel (a), and Δ_XX, in panel (b).The phase diagram presents a color map plot of both gap functions in the η vsplane,where η=V_XX/V_XY measures the ratio between thecouplings (V_XY = 0.19 eV was kept constant while V_XX varied).The color scale is the same for both panels and it is given in eV units.Details of the self-consistent numerical solution of the gap equations can be found in Ref. [<cit.>]. § RESULTS AND DISCUSSION§.§ Orbital-mixing and multiband superconductivity The superconducting state emerges from an instability of the Fermi sea (metallic normal state)to an attractive effective interaction. This interaction forms Cooper pairs that scatter againsteach other, always conserving total momentum and individual spin, while stayingin a shell around the Fermi surface. It is then expected that many properties of the superconducting state,like the gap function and, in a multiband system, the possibility of the existence ofdifferent types of Cooper pairs, will be directly associated to the properties of the Fermi surfacein the normal state. Thus, in a system like BiS_2, whose Fermi surface varies widely withelectron doping, even showing a Lifshitz transition, as illustrated in Fig. <ref>, onewould expect that the superconducting gap function shouldalso show a marked variation with electron doping. Indeed, the gap function results in Fig. <ref>clearly confirm this expectation by showing very marked variations when the systemgoes through the Lifshitz transition (for ≈ 0.45). However, as described in this section, our results,when analyzed more carefully, also show that there is a moresubtle aspect relating multiband SC with the nature of the band states at the Fermi surface.This aspect, once properly quantified, can be directly linked to the veryparticular electronic structure of BiS_2 compounds.As will be illustrated below, the orbital-mixing character of a band state changes from point to point in the BZ of BiS_2, varying continuously from pure-X to pure-Y(and back, passing by completely-XY-mixed) in accordance to symmetry requirements.As a consequence, the degree of mixing of the X- and Y-orbital at the Fermi surface, for a particularelectron doping, may change between different regions of the Fermi surface. This is not surprising in itself.What is interesting in the case of BiS_2 is that the systematic variation ofthe orbital-mixing along the BiS_2 Fermi surface can besemi-quantitatively connected to the Δ_XY and Δ_XX results in Fig. <ref>. Therefore, we will use the idea of orbital-mixing, as defined below, as well as the way the Fermi surface changes with doping, to explain the mainstructures seen in the superconducting gap functions shown in Fig. <ref>, as,for example, the position of the maxima and minima of Δ_XY andΔ_XX as a function of electron doping .Note that we assume a rigid band situation (i.e., doping does notchange the band structure); ARPES results <cit.>have shown that this is a good approximation for BiS_2 compounds.Consider a band state, at a generic 𝐤=[k_x,k_y] in the first BZ,written as |𝐤⟩ = C_X(k_x,k_y)|p_X,k⟩ + C_Y(k_x,k_y)|p_Y,k⟩, where |p_α,k⟩ = p^†_α𝐤 |vac ⟩,for α = X, Y (where the spin index was omitted for the sake of brevity).To quantify the degree of orbital-mixing, we define R(k_x,k_y) = |C_α(k_x,k_y)|^2/|C_β(k_x,k_y)|^2, where α and β take values X or Y such that 0.0 ≤ R(k_x,k_y) ≤ 1.0 for all [k_x,k_y].Thus, we refer to R(k_x,k_y) as the orbital-mixing ratio between the X-orbital and Y-orbitalfor each point of the BZ, where R(k_x,k_y)=1.0 indicates maximum orbital mixing, where theband state does not have a well defined X- or Y-orbital character, being an equal mix of both;and R(k_x,k_y)=0.0 indicates no orbital mixing at all,i.e., the band state has a well defined (either X- or Y-) orbital character. We will refer to theformer as a orbital-mixed band state and to the latter as a zero-mixing band state. Figure <ref> shows a color-map plot of R(k_x,k_y) for the lower energy band in the first BZ.As indicated by the labels, a well defined orbital character can be associated to the band states(denoted as either |k,X ⟩, when |C_X(k_x,k_y)| ≈ 1, or |k,Y ⟩, when |C_Y(k_x,k_y)| ≈ 1)in a wide range around the Γ M symmetry lines (where R(k_x,k_y) ≈ 0)in each quadrant (alternating from |k,X ⟩ to |k,Y ⟩ from one quadrant to the next),while the band states in a narrow region around the Γ X and XM symmetry lines are orbital-mixed (R(k_x,k_y) ≈ 1).Results for the higher energy band are identical, but for the swapping of X and Y.Based on these results, parts of the Fermi surface that are formed by large hole pocketsaround the Γ and M points in the BZ [see Fig. <ref>(b)] contain mostlyband states with zero-mixing, while parts of the Fermi surface formed by smaller electron pocketsaround the X points in the BZ will contain band states with a larger degree of orbital mixing.In what follows, we will analyze the orbital-mixing ratio R(k_x,k_y) over Fermi surface pockets, i.e.,we will be interested on the variation of R(ϕ) as we move around the pocket's edge, where we have parametrizedk⃗_F as [k_F,ϕ], where ϕ is the polar angle measured around the pocket's center. To clarify the connection between orbital-mixing and Cooper pair formation in a multiband system and thereforedevelop an intuitive picture of the relation between the Δ_XY,XX superconducting order parametersand orbital-mixing, we show in Fig. <ref>(a) the zero-mixing case for a hole pocket around [0,0]obtained for =0.75, with the corresponding values for |C_X|^2 [(red) squares],|C_Y|^2 [(blue) circles], and R(ϕ) [(green) solid curve] shown in Fig. <ref>(b),where we see that R(ϕ) ≈ 0 for the whole pocket with the exception of small regions around ϕ values that are multiples of π/2,where symmetry imposes an X ↔ Y swap in the |C_X,Y|^2 coefficients <cit.>.Therefore, in a zero-mixing pocket, if states at the Fermi surface have very well definedorbital character in one specific quadrant[X-orbital, for example, in the first quadrant, as shown in Fig. <ref>(a)], they will have the oppositecharacter in the next quadrant [Y-orbital in the second quadrant in Fig. <ref>(a)], and so on.In that case, Cooper pairs will be formed by X-orbital electrons only [XX pairs, like the one depictedin panel (a)] or Y-orbital electrons only (YY pairs). Exchange ofphonons that scatter electrons between opposing quadrants will lead to XX ↔ XX andYY ↔ YY pair scattering, while phonons that scatter electrons between adjacent quadrants will lead to XX ↔ YY pair scattering.Therefore, zero-mixing pockets are associated to the SC order parameter Δ_XX.On the other hand, for electron pockets around the X points in the BZ, where orbital-mixing dominates i.e., |C_X|^2 ≈ |C_Y|^2 ≈ 1/2, formation ofp^†_X,k, ↑ p^†_Y,-k, ↓ pairs becomes possible,therefore, the order parameter Δ_XY is connected to orbital-mixed pockets. It is important to realize at this point that the results shown in Fig. <ref>, for R(k_x, k_y), are obviouslyindependent of the electron doping . However, since we are interested in whathappens at the Fermi surface, as the pockets continuously expand orcontract asvaries (see Fig. <ref>), causing their edges to sweep through the BZ,R(ϕ) at the edge of each pocket will change substantially with electron doping forelectron pockets centered around X points in the BZ(see Figs. <ref> and <ref>), while it will change very little for hole pockets centeredaround the Γ and M points in the BZ [see Fig. <ref>(a)]. Therefore, in what follows, when werefer to changes in R(ϕ) with electron doping, that is what is meant.§.§ Orbital-mixing and multiband SC below the Lifshitz transition Now, using R(ϕ) as a measure of orbital-mixing, we will qualitatively connectthe structures seen in the gap function results in Fig. <ref> to the way R(ϕ)varies with doping. Let us start atvalues below the Lifshitz transition, which occurs for ≈ 0.45, where the Fermi surface changes from electron pockets centered around [π,0] and [0,π] to hole pocketscentered around [0,0] and [π,π] (see Fig. <ref>).In Fig. <ref>(a) we show R(ϕ) results for the first quadrant of the electron pocket around [π,0] for 5 different values of doping 0.05 ≤≤ 0.4. In Fig. <ref>(b) it is shown how the size of the [π,0] electron pocket increases with electron doping for the samevalues as in panel (a). Given the C_4 symmetry of BiS_2, the pattern shown in Fig. <ref>(a) repeats itself for allquadrants, with the appropriate X ↔ Y swap in the definition of R(ϕ) <cit.>. The results in Fig. <ref>(a) show a decrease in mixing as the doping increases.Referring to Fig. <ref>, it is easy to see that this is due to the increasein size of the electron pockets centered around the X points in the BZ [see Fig.<ref>(b)]:as these pockets increase, larger parts of the Fermi surface will be in R(ϕ) ≈ 0regions of the BZ. Keeping in mind, as discussed above, that Δ_XY is associated to orbital-mixing (R(ϕ) ≈ 1) and Δ_XX with the absence of it (R(ϕ) ≈ 0),one should then expect, asvaries, a maximum (at low doping) in Δ_XY and asteady increase in Δ_XX. Indeed, if one takes a fixed η = 0.4, for example,in Fig. <ref> (see dashed line in both panels), and contrasts how Δ_XY andΔ_XX vary asincreases from zero, that is exactly what happens.§.§ Orbital-mixing and multiband SC above the Lifshitz transition Now, using the same ideas as in the previous section,we will explain the main structures of Δ_XY and Δ_XX at, and above, the Lifshitztransition. As mentioned above, as one approaches the Lifshitz transition (at ≈ 0.45)from below (and the [π,0] and [0,π]electron pockets are about to touch and become hole pockets around [0,0] and [π,π])R(ϕ) ≈ 0 over the full extension of the Fermi surface, causing Δ_XY to vanish.In reality, as the R(ϕ) results in Fig. <ref> show, Δ_XY should haveessentially vanished for ≈ 0.2, which agrees with the results in Fig. <ref>.Figure <ref> explains the behavior ofΔ_XY for > 0.55, where, as shown in Fig. <ref>(b), an electronpocket around [π,0] forms again and increases with . Fig. <ref>(a) shows thatthis pocket initially presents strong orbital-mixing (R(ϕ) ≈ 1.0 for the whole pocket),which slowly decreases as the pocket increases, leading to the broad maximumin Δ_XY around ≈ 0.6, as seen in Fig. <ref>(a). As to Δ_XX, at the Lifshitz transition it should reach a maximum, since, above it,the hole pockets [0,0] and [π,π] will start decreasing (askeeps increasing). This is shown in Fig. <ref>(a) for =0.46 [(black) squares], =0.6 [(green) up triangle],and =0.75 [(cyan) diamonds]. The behavior of R(ϕ) for =0.46 and =0.75 is shown in Fig. <ref>(b) for the [0,0] hole pocket and <ref>(c) for [π,π].As expected, based on the results shown in Fig. <ref>, the orbital-mixing ratio R(ϕ) is very small for both pockets at =0.46 [(black] squares) and it barely changes between =0.55 (not shown) and =0.75 [(red) circles], indicating that these pockets only contribute to Δ_XX, as discussed above. Therefore, a somewhat broad maximum in Δ_XX, as shown in Fig. <ref>(b), occurs around the Lifshitz transition at ≈ 0.46 and it is associated to the larger Fermi surface at this doping.Incidentally, the largest gap value in Fig. <ref> is that for Δ_XX right after theLifshitz transition, when the [0,0] and [π,π] hole pockets have maximum size and basicallyno mixture.§.§ Critical Temperature Results Figure <ref> shows results for the superconducting critical temperature T_c vsobtained from aplot (not shown) very similar to the one in Fig. <ref>. The parameters used for calculatingT_c were V_XY=0.16 eV, ω_D=5 meV, and η=V_XX/V_XY=0.7.We used slightly smaller parameter values from the ones used in Fig. <ref>so that the maximum T_c ≈ 10 K is similar to that measured for BiS_2compounds <cit.>. A comparison with a compilation of T_c values forLnO_1-xF_xBiS_2 (where Ln = La, Ce, Pr, and Nd) [see Fig. 2 in Ref. <cit.>],shows an overall similarity with the results in Fig. <ref>: T_c ≈ 2 K for =0.1,T_c ≈ 6 K for ≈ 0.6, and (for Ln = Nd) a dip in T_c for larger values of .Our results, therefore, are in good qualitative agreement with experimental results for BiS_2 compounds.The T_c values were obtained by solving, for a fixed value of ,eqs. <ref> and <ref> self-consistently for temperaturesT ≥ 0; then, a critical temperature associated to each diferent gap, Δ_XY [(blue) circles]or Δ_XX [(red) squares], is determined once the corresponding gap value falls below 10^-6.§ SUMMARY AND CONCLUSIONSIn summary, we have presented results for a mean-field treatment of a modelfor multiband SC of BiS_2-based layered compounds, using as starting point a minimal two-orbital tight-binding model,known to reproduce the main properties of the Fermi surface of LaO_(1-x)F_xBiS_2,including its variation with electron doping. In this minimal model, the bands crossing the Fermi surfaceoriginate from the 6p_X and 6p_Y Bismuth orbitals (labeled X- and Y-orbital).The attractive pair-scattering part of the Hamiltonian allows for the formation ofall three types of Cooper pairs (XX, YY, and XY), resulting in(after symmetry considerations) two gap equations involving superconducting order parameters Δ_XX(associated to pair scatterings of the type XX ↔ XX, YY ↔ YY,and XX ↔ YY) and Δ_XY (XY ↔ XY).The self-consistent numerical solution of the gap equations was presented as a functionof η=V_XX/V_XY, the ratio between the pairing couplings, and theelectron doping . We then defined the quantity R(k_x,k_y), which measures thedegree of X- and Y-orbital mixing of a band state, and used its value 0 ≤ R(ϕ) ≤ 1to classify the Fermi surface pockets (parametrized through the polar angle ϕ) aszero-mixing (R(ϕ) ≈ 0) or orbital-mixed (R(ϕ) ≈ 1).The definition of R(ϕ) allowed us to identify twodistinct situations regarding SC: (i) zero-mixing pockets allow the formation of XXand YY pairs only, promoting XX ↔ XX, YY ↔ YY,and XX ↔ YY pair scattering, and therefore strengthens the Δ_XX order parameter(or, as we call it, XX-type SC),while (ii) orbital-mixed pockets result in XY pairs, promoting XY ↔ XY pair scatteringand strengthening the Δ_XY order parameter (XY-type SC).Calculating R(k_x,k_y) in the first BZ we couldassert that hole pockets around [0,0] and [π,π] are mostly zero-mixing andsmall electron pockets around[0,π] and [π,0] are mostly orbital-mixed, becoming gradually zero-mixing asthey increase in size.Based on that, and knowing how the Fermi surface pockets evolve with doping, we could semi-quantitatively predictthe main structures observed in the Δ_XY and Δ_XX phase diagrams.In regions of the parameter space where both order parameters are present, we have, in general, thatΔ_XX≫Δ_XY, unless η≪ 1.This can be explained by the relative size ofthe areas in the first BZ where R(k_x,k_y) ≈ 0 or R(k_x,k_y) ≈ 1, with the former taking amuch larger share of the first BZ. This implies that, unless the Fermi surface is restricted to smallelectron pockets around the X points, which only occurs at very low doping, XX-type SC will alwaysdominate (unless η≪ 1). Finally, we also showed results for the superconducting critical temperatureT_c, as a function of doping , which are in qualitative agreement with those measured for BiS_2 compounds. In conclusion, in this work we present results for a particularly simple model of multiband SC, describingBiS_2 compounds, where the two bands crossing the Fermi surface originate fromsymmetry-related orbitals, and for which all types of Cooper pairs are allowed. The resultsfor the two superconducting order parameters obtained can be semi-quantitatively linkedto the way R(ϕ), the orbital-mixing ratio, changes as the Fermi surface evolves withelectron doping. Given the current importance of multiband SCand the availability of computational techniques to produceeffective models, at the tight-binding level, that describe the normal phase Fermi surfacewith relative accuracy, we envisage the use of the ideas here presented tospot favorable regions in the phase diagram (controlled mostly by carrier doping) toanalyze specific aspects of multiband SC. For example, as clearly shown in this work,the use of the orbital-mixing ratio R(ϕ) allows us to correctly infer that Δ_XYdominates at low electron doping, while Δ_XX dominates close to the Lifshitz transition.This type of information could guide experimentalists into where to investigate for phenomena associated to each differentorder parameter. We hope that this approach would be appealing to experimental research groupsinterested in pinpointing favorable scenarios to observe such elusive phenomena as Leggett modes <cit.> or the Fulde-Ferrell-Larkin-Ovchinnikov state, <cit.> which are associated to multiband SC.Finally, we also speculate that one could usethese ideas to propose simple effective multiband models with the appropriate Fermi surfacesand orbital mixing, which will lead, at the appropriate band filling, to the dominance of one, or the other,of the superconducting order parameters. This could lead to proposals for real materialsthat could be described by these effective simple models, completing a reverse engineering strategy. § ACKNOWLEDGEMENTSThe authors thank MB Maple for comments about experimental resultson BiS_2 compounds. MAG and TOP acknowledge CNPq, MAC acknowledges CNPq and FAPERJ,and GBM acknowledges the Brazilian Government for financial support through a Pesquisador Visitante Especialgrant from the Ciências Sem Fronteiras Program, from the Ministério da Ciência,Tecnologia e Inovação. iopart-num | http://arxiv.org/abs/1706.08600v1 | {
"authors": [
"M. A. Griffith",
"T. O. Puel",
"M. A. Continentino",
"G. B. Martins"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170626211113",
"title": "Multiband superconductivity in ${\\rm BiS_2}$-based layered compounds"
} |
#1, #1#1#1, #1#1Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia ψLaboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, CH-5232 Villigen, Switzerland Department of Quantum Matter Physics, University of Geneva, CH-1211 Geneva, Switzerland Department of Chemistry and Biochemistry, University of Bern, CH-3012 Bern, Switzerlandψ ψ[email protected] Quantum spin liquid is a disordered magnetic state with fractional spin excitations. Its clearest example is found in an exactly solved Kitaev honeycomb model where a spin flip fractionalizes into two types of anyons, quasiparticles that are neither fermions nor bosons: a pair of gauge fluxes and a Majorana fermion. Here we demonstrate this kind of fractionalization in the Kitaev paramagnetic state of the honeycomb magnet α-RuCl_3. The spin-excitation gap measured by nuclear magnetic resonance consists of the predicted Majorana fermion contribution following the cube of the applied magnetic field, and a finite zero-field contribution matching the predicted size of the gauge-flux gap. The observed fractionalization into gapped anyons survives in a broad range of temperatures and magnetic fields despite inevitable non-Kitaev interactions between the spins, which are predicted to drive the system towards a gapless ground state. The gapped character of both anyons is crucial for their potential application in topological quantum computing.Observation of two types of anyons in the Kitaev honeycomb magnet M. Klanjšek December 30, 2023 ================================================================= In many-body systems dominated by strong fluctuations, an excitation with an integer quantum number can break up into exotic quasiparticles with fractional quantum numbers. Well known examples include fractionally charged quasiparticles in fractional quantum Hall effect <cit.>, spin-charge separation in one-dimensional conductors <cit.>, and magnetic monopoles in spin ice <cit.>. A major hunting ground for novel fractional quasiparticles are disordered magnetic states of interacting spin-1/2 systems governed by strong quantum fluctuations, called quantum spin liquids (QSLs). Most of their models predict that a spin-flip excitation fractionalizes into a pair of spinons, each carrying spin 1/2 <cit.>. Even more interesting in this respect is the Kitaev model <cit.> of S=1/2 spins on a two-dimensional (2D) honeycomb lattice with nearest neighbors interacting through an Ising exchange, whose axis depends on the bond direction, as shown in Fig. <ref>(a). This is one of a few exactly solved 2D models supporting a QSL ground state. According to the solution, a spin flip fractionalizes into a pair of bosonic gauge fluxes and a Majorana fermion <cit.>. As both types of quasiparticles behave as anyons, i.e., neither bosons nor fermions, under exchange, they could be used for decoherence-free topological quantum computation <cit.>. The experimental detection of such anyons is thus the primary goal of current QSL research.As fractional quasiparticles are always created in groups, their common signature is a continuous spin-excitation spectrum, observed in recent QSL candidates on the kagome and triangular lattices <cit.>, instead of sharp magnon modes found in ordered magnets. A Kitaev QSL also exhibits this feature <cit.>, as well as additional, specific signatures, all related to the fact that fractionalization in this case leads to different quasiparticles. First, the fractionalization proceeds in two steps, with both types of quasiparticles releasing their entropy at different temperatures <cit.>. Second, although Majorana fermions themselves are gapless in zero magnetic field, the response of the QSL to a spin flip is gapped due to the inevitable simultaneous creation of a pair of gapped gauge fluxes <cit.>. And third, in the presence of an external magnetic field, the Majorana fermions also acquire a gap, which is predicted to grow with the characteristic third power of the field in the low-field region <cit.>. Currently, α-RuCl_3 stands as the most promising candidate for the realization of the Kitaev QSL <cit.>. Among the listed signatures, a spin-excitation continuum was observed by Raman spectroscopy <cit.> and inelastic neutron scattering <cit.>, and the two-step thermal fractionalization was confirmed by specific-heat measurements <cit.>, all in zero field. However, an application of a finite field, which should affect the gaps of both types of quasiparticles differently, is crucial to identify them. Using nuclear magnetic resonance (NMR), we determine the field dependence of the spin-excitation gap Δ shown in Fig. <ref>(c), which indeed exhibits a finite zero-field value predicted for gauge fluxes and the cubic growth predicted for Majorana fermions. This result clearly demonstrates the fractionalization of a spin flip into two types of anyons in α-RuCl_3.α-RuCl_3 is structurally related to the other two Kitaev QSL candidates, Na_2IrO_3 <cit.> and α-Li_2IrO_3 <cit.>. All three are layered Mott insulators based on the edge-sharing octahedral units, RuCl_6 and IrO_6 [Fig. <ref>(a)], respectively, and driven by strong spin-orbit coupling <cit.>, which together lead to a dominant Kitaev exchange coupling between the effective S=1/2 spins of Ru^3+ and Ir^4+ ions, respectively <cit.>. A monoclinic distortion of the IrO_6 octahedra in both iridate compounds results in the presence of non-Kitaev exchange interactions between the spins, which lead to the low-temperature magnetic ordering and thus prevent the realization of the QSL ground state. Judging by the lower transition temperature, these interactions are smaller in α-RuCl_3 <cit.>. Signatures of fractional quasiparticles should thus be sought in a region of the phase diagram outside the magnetically ordered phase, at temperatures where the Kitaev physics is not yet destroyed by thermal fluctuations. This is the Kitaev paramagnetic region [Fig. <ref>(b)] extending to a relatively high temperature around 100 K where the nearest-neighbor spin correlations vanish <cit.>.The boundary of the magnetically ordered phase measured in a large α-RuCl_3 single crystal <cit.> using ^35Cl NMR is displayed in Fig. <ref>(b). Magnetic properties of α-RuCl_3 are known to be highly anisotropic <cit.>, mainly because of the anisotropic Ru^3+ g-tensor [Fig. <ref>(a)] with g_ab=2.5 and g_c^*=1.1 <cit.>. We exploit this anisotropy to scan the phase diagram by varying the direction of the applied fixed field, instead of varying the magnitude of the field applied in the ab plane <cit.>. Namely, as the Zeeman term contains the product gB, a magnetic field B applied at an angle ϑ from the ab plane is equivalent to the effective field B_ab=g(ϑ)B/g_ab applied in the ab plane, where g(ϑ)=√(g_ab^2cos^2ϑ+g_c^*^2sin^2ϑ) is the direction-dependent g-factor. This is valid if the studied underlying physics is close to isotropic, a condition to be verified at the end. As shown in the inset of Fig. <ref>(b), we determine the transition temperature T_N2 as the onset of NMR line broadening <cit.> monitored on the dominant NMR peak [inset of Fig. <ref>(b)]. The obtained phase boundary extending up to the critical field B_c≈ 8 T matches the result of a recent reference study <cit.>. The observed transition temperature T_N2 of around 14 K near zero field is consistent with a considerable presence of the two-layer AB stacking in the monoclinic C2/m crystal structure [Fig. <ref>(a)], in addition to the three-layer ABC stacking, which is characterized by a lower transition temperature T_N1 of around 7 K in zero field <cit.>. As our study is focused on the Kitaev paramagnetic region [Fig. <ref>(b)] governed by the physics of individual layers, it is not affected by the particular stacking type.To detect and monitor the spin-excitation gap as a function of the magnetic field, we use the NMR spin-lattice relaxation rate T_1^-1, which directly probes the low-energy limit of the local spin-spin correlation function and thus offers a direct access to the spin-excitation gap. Fig. <ref>(a) shows the ^35Cl T_1^-1(T) datasets taken on the dominant NMR peak [inset of Fig. <ref>(b)] in 9.4 T for two magnetic field orientations. A noticeable feature of the T_1^-1(T) dataset for B⊥ c^* (i.e., in the ab plane, B_ab=9.4 T) is a broad maximum around 30 K, followed by a steep decrease towards lower temperatures. In the T_1^-1(T) dataset for B∥ c^*, such a feature would apparently develop at a lower temperature, if the dataset was not disrupted by the phase transition at T_N2=12 K [in a field of B_ab=4.1 T, Fig. <ref>(b)]. Instead, two T_1^-1 components develop below T_N2, both exhibiting a steep drop, one below T_N2 and the other one below T_N1=8 K. These two phase transitions were observed before and ascribed to the presence of AB and ABC stackings, respectively <cit.>. The analysis of the data below T_N2 and T_N1 using the expression T_1^-1∝ T^2exp(-Δ_m/T) valid for gapped magnon excitations in the 3D ordered state <cit.> gives comparable values of the magnon gap Δ_m=32 K and 35 K, respectively, implying the same low-energy physics in both cases. The obtained values are compatible with the gap of 29 K determined by inelastic neutron scattering <cit.>.To access the key information held by the T_1^-1(T) datasets in the Kitaev paramagnetic state, we first observe in Fig. <ref>(a) that the dataset for B⊥ c^* below 100 K exhibits the same shape as the theoretical dataset numerically calculated for the ferromagnetic Kitaev model in zero field <cit.>. A characteristic broad maximum of the latter is a sign of thermally excited pairs of gauge fluxes over the two-flux gap <cit.>, whose exact value amounts to Δ_0=0.065J_K <cit.> where J_K is the Kitaev exchange coupling. As shown in Fig. <ref>(a), a large part of the theoretical dataset, up to around 0.2J_K, well above the maximum, can be excellently described by the phenomenological expressionT_1^-1∝1/Texp(-nΔ/T),where n is set to 0.61 in order for Δ to match the required value of Δ_0. This expression has a useful property: its maximum appears at a temperature Δ/n, which allows for a simple estimate of Δ directly from the T_1^-1(T) dataset. The B⊥ c^* (B_ab=9.4 T) dataset up to 50 K is excellently reproduced by Eq. (<ref>) using Δ=51 K [Fig. <ref>(a)]. The validity of Eq. (<ref>) in this case and in the case of the theoretical dataset is clearly demonstrated in the inset of Fig. <ref>(a), which shows the resulting linear dependence of ln(T_1^-1T) on T^-1 below 50 K. Meanwhile, even in the absence of the characteristic maximum, the B∥ c^* (B_ab=4.1 T) dataset above 17 K, i.e., slightly above T_N2, and up to high temperatures matches the theoretical zero-field dataset using the value J_K=190 K determined by inelastic neutron scattering, also based on the ferromagnetic Kitaev model <cit.>. This means that the gap for 4.1 T is already close to the zero-field value Δ_0=0.065J_K=12.4 K. A large difference between the two determined gaps points to a significant Δ(B_ab) variation in the Kitaev paramagnetic state. Finally, the temperature-independent part of both T_1^-1(T) datasets above 100 K indicates a crossover into the classical paramagnetic state <cit.>, in line with the result of Ref. <cit.>.The expression given by Eq. (<ref>) is not merely phenomenological, but reveals the presence of gapped fractional spin excitations. Similar expressions are obtained for the T_1 relaxation due to gapped magnons in magnetic insulators at low temperatures T≪Δ <cit.>. In this case, the prefactor T^-1 is replaced by a more general T^p originating from the magnon density of states g(E), which depends on the dimensionality D, while n is generally the number of magnons involved in the process. For n=1 (single-magnon scattering) and a quadratic dispersion relation for magnons, one obtains g(E)∝ E^D/2-1 and thus p=D-1≥ 0, while higher n (multi-magnon scattering) lead to even higher powers p <cit.>. At higher temperatures T∼Δ, the effective p changes, but always remains positive. As the very unusual p=-1 in Eq. (<ref>) valid for T≲Δ cannot be obtained for magnons, fractional spin excitations should be involved. This is furthermore supported by a fractional n in Eq. (<ref>), implying that fractions of a spin-flip excitation are involved in the relaxation process. In contrast to this unusual gapped T_1^-1(T) behavior, the temperature dependence of the local susceptibility monitored by the ^35Cl NMR shift in Fig. <ref>(b) is monotonic over the whole covered temperature range, as predicted for the ferromagnetic Kitaev model <cit.>. Such a dichotomy between the two observables is a direct sign of spin fractionalization, as different fractional quasiparticles enter the two observables in different ways <cit.>.To obtain the spin-excitation gap Δ as a function of B_ab in Fig. <ref>(c), the T_1^-1(T) datasets in Fig. <ref> taken in magnetic fields of different directions and magnitudes are fitted to Eq. (<ref>) in the temperature range of the Kitaev paramagnetic phase. As the curve T_1^-1∝ T^-1 defined by Δ=0 is steeper than any dataset in this range, the obtained excitation gaps are apparently all nonzero. The inset of Fig. <ref>(c) showing the symmetric Δ(ϑ) dependence around 90^∘ in 9.4 T, where ϑ traverses nonequivalent directions with respect to the Kitaev axes on both sides [inset of Fig. <ref>(a)], demonstrates that the underlying physics is indeed isotropic as assumed when introducing B_ab. The obtained Δ(B_ab) in Fig. <ref>(c) can be perfectly reproduced as a sum of two terms: the two-flux gap Δ_0 and the gap acquired by Majorana fermions in a weak magnetic field, predicted to be proportional to the cube of the field <cit.>,Δ=Δ_0+α/3B^3/Δ_0^2,using J_K=190 K <cit.> to evaluate Δ_0, as before, while B=g_abμ_BB_ab/k_B is the field in kelvin units, k_B is the Boltzmann constant, μ_B the Bohr magneton, and α=4.5 (leading to the best fit) accounts for the sum over the excited states in the third-order perturbation theory, which is the origin of the B^3 term <cit.>. This result demonstrates that a spin-flip excitation in α-RuCl_3 indeed fractionalizes into a gauge-flux pair and a Majorana fermion.Focusing on the Kitaev paramagnetic region in the phase diagram of α-RuCl_3 in Fig. <ref>(b) is essential for our observation of two types of anyons. Instead, other recent experimental studies focused on the low-temperature region above B_c, observing the spin-excitation continuum <cit.> with either a gapless behavior <cit.> or the gap opening linearly with B-B_c <cit.>, but without a definite conclusion about the identity of the involved quasiparticles. Such an ambiguous behavior likely originates in the presence of additional, smaller non-Kitaev interactions between the spins <cit.>, whose role should be pronounced particularly at low temperatures and which are indeed predicted to drive the system towards a gapless QSL ground state <cit.>. 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Abragam, Principles of Nuclear Magnetism, Oxford University Press, Oxford (2011). § SUPPLEMENTAL MATERIAL §.§ Crystal growth Crystals of α-RuCl_3 were synthesized from anhydrous RuCl_3 (Strem Chemicals). The starting material was heated in vacuum to 200 ^∘C for one day to remove volatile impurities. In the next step, the powder was sealed in a silica ampoule under vacuum and heated to 650 ^∘C in a tubular furnace. The tip of the ampoule was kept at lower temperature and the material sublimed to the colder end during one week. Phase pure α-RuCl_3 (with a high-temperature phase of C2/m crystal structure) was obtained as thin crystalline plates. The residual in the hot part of the ampoule was black RuO_2 powder. The purified α-RuCl_3 was sublimed for the second time in order to obtain bigger crystal plates. The phase and purity of the compounds was verified by powder X-ray diffraction. All handling of the material was done under strictly anhydrous and oxygen-free conditions in glove boxes or sealed ampoules. Special care has to be taken when the material is heated in sealed-off ampoules. If gas evolves from the material, this may result in the explosion of the ampoule.§.§ Magnetic susceptibility Magnetic susceptibility measurements of α-RuCl_3 were performed using a Quantum Design MPMS. A powdered sample of the mass 22.5 mg was placed into a plastic capsule, in a glovebox to avoid contact with air, and then quickly transferred into the MPMS. Fig. <ref> shows the measured susceptibility taken with cooling in field and in zero field. The obtained curve with the magnetic transition at T_N2=14 K (inset of Fig. <ref>) is almost identical to the corresponding curve in Ref. <cit.>.§.§ Nuclear magnetic resonance General. The ^35Cl nuclear magnetic resonance (NMR) experiments were performed on a foil-like α-RuCl_3 single crystal of approximate dimensions 5× 5× 0.1 mm^3 in a continuous-flow cryostat allowing us to reach temperatures down to 4.2 K. When handling the sample, we took extreme care to minimize its exposure to air. A thin NMR coil fitting the sample size was made from a thin copper wire with 20-40 turns, depending on the required tuning frequency determined by the external magnetic field. The coil was covered with a mixture of epoxy and ZrO_2 powder, which was allowed to harden, in order to ensure the rigidity of the coil. The coil was then mounted on a teflon holder attached to a rotator, which allowed us to vary the orientation of the sample with respect to the external magnetic field. In order to reduce the noise of an already weak ^35Cl NMR signal, a consequence of the extremely broad ^35Cl NMR spectrum, we used a bottom-tuning scheme. With the output radio-frequency power of around 20 W, the typical π/2 pulse length amounted to 2 μs. The NMR signals were recorded using the standard spin-echo, π/2-τ_d-π pulse sequence with a typical delay of τ_d=70 μs (much shorter than the spin-spin relaxation time T_2) between the π/2 and π pulses.T_1 relaxation. The spin-lattice relaxation (i.e., T_1) experiment was carried out using an inversion recovery pulse sequence, φ_i-τ-π/2-τ_d-π, with an inversion pulse φ_i<π (suitable for broad NMR lines) and a variable delay τ before the read-out spin-echo sequence. The spin-lattice relaxation datasets were typically taken at 20 increasing values of τ. The datasets were analyzed using the model of magnetic relaxation for I=3/2 spin monitored on the central -1/2⟷ 1/2 transition:m(τ)=1-(1+s)[0.1exp(-τ/T_1)+0.9exp(-6τ/T_1)],where T_1 is the spin-lattice relaxation time and s is the inversion factor. In the region of the phase diagram outside the magnetically ordered phase [Fig. 2(b)], this expression reproduces the experimental relaxation curves perfectly. In the magnetically ordered phase, two T_1 components appear, and the relaxation curves are reproduced as a sum of two terms of the form given by Eq. (<ref>). For instance, the temperature dependence of the corresponding two T_1 values for B=9.4 T with B∥ c^* is given in Fig. 2(a). In cases where only a narrow temperature region below the transition was covered, the two components in the relaxation curves were hard to identify, and we used Eq. (<ref>) with a stretched exponent instead.Orientation dependence of the NMR spectrum. The ^35Cl NMR spectra were recorded point by point in frequency steps of 50 or 100 kHz, so that the Fourier transform of the signal was integrated at each step to arrive at the individual spectral point. The covered NMR frequency range was from 34 MHz, the lower limit of our setup, up to 50 MHz. The dependence of the corresponding part of the NMR spectrum on the direction of the external magnetic field of 9.4 T (described by the angle ϑ from the crystal ab plane) at a temperature of 20 K is shown in Fig. <ref>. The spectra are extremely broad because of large ^35Cl (with I=3/2 spin) quadrupole interaction. As concluded in the following, a large portion of the covered frequency range is associated with the central, 1/2⟷ -1/2 ^35Cl NMR transition. As this transition is observed to consist of at least three peaks (Fig. <ref>), even for the symmetric B∥ c^* orientation (with ϑ=90^∘), while there are only two inequivalent Cl sites in the crystal structure, the splitting of the central line is likely a consequence of stacking faults in the layered crystal structure or crystal twinning, or both.Relation bewteen orientation and field dependence of T_1. Measuring the T_1 dependence on the direction of the magnetic field (described by the angle ϑ from the crystal ab plane) instead of on its magnitude in the ab plane allows us to cover low B_ab values, while keeping the applied magnetic field B high. This is beneficial for two reasons related to the strong quadrupole broadening of the ^35Cl NMR spectrum (Fig. <ref>): to minimize an already large NMR linewidth and to keep the Larmor frequency well above the quadrupole splitting, which is of the order of 10 MHz as concluded in the following. The validity of this approach is supported by the fact that Δ(B_ab) data points for various angles ϑ and field values 2.35, 4.7 and 9.4 T in Fig. 1(c) all collapse on a smooth experimental curve. The Δ(B_ab) data points taken in lower fields apparently exhibit much larger error bars. Namely, the corresponding T_1^-1(T) datasets in Fig. 3(b) are more scattered than the datasets taken in 9.4 T despite a much longer averaging for noise reduction.Temperature dependence of the NMR line. We measured the temperature dependence of the dominant ^35Cl NMR peak in a field of 9.4 T for various sample orientations. From these measurements, we determined the temperature dependence of the frequency width [Fig. <ref>(a)] and the NMR shift of the peak with respect to the Larmor frequency [Fig. <ref>(b)]. For B_ab<8 T, the width exhibits a clear kink as a function of temperature, which indicates the onset of NMR line broadening at the phase transition into the magnetically ordered state. Plotting the temperature of the kink as a function of B_ab in the inset of Fig. <ref>, we obtain the phase boundary of the magnetically ordered state, which perfectly matches the result of the reference study <cit.>. In contrast, the NMR shift does not exhibit any signs of a magnetic transition, except for the ϑ=90^∘ (B∥ c^*) dataset. We find the NMR shift to be a monotonic function of temperature T, empirically following a log T dependence over a broad temperature range. Contributions to the NMR shift. To separate the magnetic contribution to the NMR frequency shift from the temperature-independent quadrupole contribution, we plot the relative NMR shift (i.e., the NMR shift divided by the ^35Cl Larmor frequency ν_L=39.18 MHz in a field of 9.4 T) measured on the dominant NMR peak for B⊥ c^*, i.e., in the ab plane [the ϑ=0^∘ dataset in Fig. <ref>(b)], against the rescaled magnetic susceptibility χ_ab in the inset of Fig. <ref>. In Ref. <cit.>, an experimental ratio between the susceptibility χ of the powdered sample and the susceptibility χ_ab of the single crystal with a field applied in the ab plane is obtained as (2+r)/3 with r=0.157, leading to χ_ab=3χ/(2+r). We use this empirical relation to evaluate χ_ab(T) from our field-cooled χ(T) dataset shown in Fig. <ref>. As we did not measure susceptibility in high magnetic fields, we rely on the dataset taken in 1.0 T. This is valid in a broad temperature range, except at low temperatures where this dataset starts to deviate from the high-field susceptibility <cit.>. From the observed linear relation between the relative shift and χ_ab up to 20· 10^-3 emu/mol (i.e., down to 35 K), we obtain the hyperfine coupling constant A=2.2 T/μ_B and the zero-temperature relative shift -0.039 that, when multiplied by ν_L, gives the quadrupole shift Δν_Q=-1.53 MHz.From the obtained quadrupole shift Δν_Q, we can estimate the quadrupole splitting ν_Q between the successive ^35Cl NMR transitions. For the case of an axially symmetric EFG tensor and the field applied at an angle ϑ' from the principal EFG axis v_ZZ with the largest EFG eigenvalue, the second-order quadrupole shift is given by Δν_Q=-3ν_Q^2(1-cosϑ'^2)(9cosϑ'^2-1)/(16ν_L) for the I=3/2 nucleus <cit.>. As the axes of the EFG tensor are not known, we assume a typical tilt 45^∘ of v_ZZ from c^*, so that ϑ'∼ 45^∘. From the previously evaluated Δν_Q we then obtain ν_Q∼ 14.2 MHz. This is an estimate of the quadrupole splitting between the central ^35Cl NMR transition and the satellite transitions. We can thus conclude that the NMR peaks in the covered frequency range of Fig. <ref> all belong to the central transition.§.§ Theory Theoretical T_1^-1(T) curve. The theoretical temperature dependence of T_1^-1 is numerically calculated for the Kitaev model in zero field <cit.>. T_1^-1 contains two contributions, one coming from a single fluctuating spin (i.e., on-site) and the other one coming from fluctuating nearest-neighboring (NN) spins in the Kitaev honeycomb lattice. As the ^35Cl nucleus in α-RuCl_3 is located at equal distances from the closest two Ru^3+ S=1/2 spins [Fig. 1(a)], T_1^-1 contains both contributions with equal weights. Namely, as the hyperfine coupling constant A of ^35Cl to both spins is the same, the relevant spin-spin correlation function can be generally written as< A {S_1(t)± S_2(t)}· A(S_1± S_2)> = = A^2[ < S_1(t)S_1> + < S_2(t)S_2>±±(< S_1(t)S_2> + < S_2(t)S_1>) ],for the involved components S_1 and S_2 of both Ru^3+ spins, where the plus (minus) sign is valid for ferromagnetic (antiferromagnetic) fluctuations. The first two terms on the right side of Eq. (<ref>) represent the on-site contributions, while the last two represent the NN-sites contributions, both with apparently equal weights. Accordingly, the theoretical curve for the ferromagnetic case, plotted in Fig. 2(a), is the average of the on-site and NN-sites contributions. T_1 relaxation due to gapped magnons. When spin fluctuations in the magnetic lattice are due to excited magnons, the corresponding spin-lattice relaxation rate for a single-magnon process is given by <cit.>T_1^-1∝∫ g^2(E)n(E)[1+n(E)] dE,where E is the energy of magnons, g(E) is their density of states, n(E)=[exp(β E)-1]^-1 is the Bose-Einstein distribution function, β=1/(k_B T), and k_B is the Boltzmann constant. Denoting the magnon gap by Δ (in kelvin units), we define ε=E-k_BΔ as the energy measured from the bottom of the magnon band. The power-law dispersion relation ε∝ k^s in D dimensions, which includes the standard parabolic dispersion (s=2) and the Dirac dispersion (s=1) as special cases, leads to g(E)∝ε^D/s-1. For low temperatures T≪Δ, the distribution function n(E) can be approximated by the Boltzmann distribution, n(E)≈exp(-β E)=exp(-Δ/T)exp(-βε). Plugging these expressions for g(E) and n(E) into Eq. (<ref>), we obtainT_1^-1∝ T^2D/s-1exp(-Δ/T)∫_0^∞exp(-x) x^2(D/s-1)dx.The integral on the right side of Eq. (<ref>) converges if s<2D and evaluates to Γ(2D/s-1) where Γ is the gamma function. We can thus rewrite Eq. (<ref>) asT_1^-1∝ T^pexp(-Δ/T)with the power of the prefactor p=2D/s-1. In case of D=2, which is relevant for the Kitaev honeycomb magnet, p=1 for s=2 and p=3 for s=1, so that the power p cannot be negative. Even in case of D=1, p can only reach the lowest value of 0 precisely for s=2 [although care should be taken in this case, as the integral in Eq. (<ref>) then formally diverges]. If more than a single magnon is involved in the T_1 process, the power p is also positive and becomes even higher <cit.>. Gapped magnons thus cannot lead to the T_1 relaxation described by Eq. (<ref>) with p<0.Instead, we can use Eq. (<ref>) in the 3D magnetically ordered state, when the elementary excitations are magnons with a gap Δ_m. In this case D=3 and s=2, and this leads to T_1^-1∝ T^2exp(-Δ_m/T). We use this expression to analyze the T_1^-1(T) data [Fig. 2(a)] in the low-temperature ordered state of α-RuCl_3.As a side observation, all these examples show that a frequently used simple gapped model T_1^-1∝exp(-Δ_s/T) with the gap Δ_s, which was used before to analyze the T_1^-1(T) datasets in α-RuCl_3 <cit.>, is actually not justified in any region of the phase diagram of α-RuCl_3. Majorana fermion gap. In the Kitaev model, Majorana fermions acquire a gap in the presence of an external magnetic field <cit.>. This is shown for a field applied perpendicularly to the honeycomb plane, i.e., in the (1,1,1) direction in the coordinate system defined by the Kitaev axes x, y and z. The corresponding Zeeman term then reads ℋ_Z=-h∑_j (S_j^x+S_j^y+S_j^z), where h=gμ_BB/√(3) is a single component of the magnetic field B in energy units, g is the g-factor and μ_B is the Bohr magneton. When treated as a perturbation to the Kitaev Hamiltonian, the Zeeman term contributes to the Majorana fermion gap only at third order <cit.>. The corresponding effective Hamiltonian is thus proportional to h^3 and can be written as <cit.>ℋ_ eff^(3)=-αh^3/k_B^2Δ_0^2∑_jklS_j^xS_k^yS_l^z,where Δ_0 is a two-flux gap (in kelvin units), while α (of the order of unity) accounts for the sum over the excited states, and its exact value is not known. The Kitaev model extended with such a three-spin exchange term -κ∑_jklS_j^xS_k^yS_l^z with κ=α h^3/(k_B^2Δ_0^2) is still exactly solvable and the dispersion relation of the Majorana fermions is calculated as <cit.>E_ k=2√(k_B^2J_K^2| 1+e^i k· a_1+e^i k· a_2|^2 +κ^2sin^2( k· a_1)),where J_K is the Kitaev coupling (in kelvin units), while a_1 and a_2 are the unit vectors of the honeycomb lattice. The dispersion relation given by Eq. (<ref>) is gapped for κ≠ 0, and the corresponding gap Δ_f can be calculated numerically as a function of κ and thus as a function of the magnetic field. For small magnetic fields, i.e., for κ≪ k_BJ_K, the Majorana fermion gap (in kelvin units) simplifies toΔ_f=√(3)κ/k_B=α/3Δ_0^2(gμ_BB/k_B)^3,while for high magnetic fields it saturates to Δ_f=2J_K. The total spin-excitation gap Δ is obtained by adding Δ_f to the two-flux gap Δ_0. The field dependence of Δ is shown in Fig. <ref> for J_K=190 K (taken from Ref. <cit.> and used in this work), g=g_ab and α=4.5 [leading to the best fit of our Δ(B_ab) data points]. The cubic approximation given by Eq. (<ref>), which is also plotted in Fig. <ref>, is apparently valid up to 15 T, well beyond the field range covered in this work.§.§ Comparison with recent works Recent NMR works. Very recently, two ^35Cl NMR studies of α-RuCl_3 appeared <cit.>. The analysis of both studies is focused on the low-temperature region below 15 K. Using a simple exponential model T_1^-1∝exp(-Δ_s/T) in this region, Ref. <cit.> finds that the excitation gap Δ_s opens linearly with the field above the critical field around 10 T. On the other hand, Ref. <cit.> extends the covered temperature range down to 1.5 K and finds a low-temperature gapless, power-law behavior of T_1^-1(T) in the covered high-field region above the critical field around 8 T.As these results are very different from our results, also because of quite different analysis, we analyze the T_1^-1(T) datasets obtained in these two works also with our model given by Eq. (1). As in our work, we focus on the Kitaev paramagnetic region, to the temperature range from 50 K down to slightly above the transition temperature below 8 T, and down to 4.2 K above 8 T (or a bit higher at higher fields), including the characteristic maximum in T_1^-1(T) as a main feature. The data in Ref. <cit.> were taken with a field applied in the crystal ab plane, while the data in Ref. <cit.> were taken with a field applied at 30^∘ and -60^∘ with respect to the ab plane. In this case, we calculate the effective field values B_ab in the same way as in our work. The obtained field dependence of the excitation gap Δ(B_ab) for both works is shown in Fig. <ref> together with our result. Applying our analysis to the data in all three works apparently leads to relatively consistent results. Nevertheless, the results for the data from Refs. <cit.> alone do not allow to conclude on the cubic field dependence of Δ, mostly due to the lack of important low-field data points. Regarding Ref. <cit.>, the two data points at the highest fields seem to deviate from the trend set by the other points. The corresponding two T_1^-1(T) datasets exhibit a suspicious plateau at low temperatures, not observed in any other dataset of the three works, which casts some doubts on their credibility. Regarding Ref. <cit.>, the obtained Δ(B_ab) data points, which complement our field region, nicely continue the trend set by our data points. The obtained trend is approximately linear in field, consistent with the theoretical prediction in this intermediate-field region, although with a different slope. However, the theoretical prediction is based on a perturbative treatment, which may not give reliable results outside the low-field region.In light of our conclusions, the reported findings of these two works should be understood in the following way. While the true signs of fractionalization into Majorana fermions and gauge fluxes can be found in the Kitaev paramagnetic region covering a broad temperature range up to around 100 K [Fig. 1(a)], the physics at low temperatures of the order of 10 K and below is apparently obscured, most likely due to the effect of inevitable non-Kitaev interactions, as predicted already in Ref. <cit.>. | http://arxiv.org/abs/1706.08455v2 | {
"authors": [
"N. Janša",
"A. Zorko",
"M. Gomilšek",
"M. Pregelj",
"K. W. Krämer",
"D. Biner",
"A. Biffin",
"Ch. Rüegg",
"M. Klanjšek"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170626160709",
"title": "Observation of two types of anyons in the Kitaev honeycomb magnet"
} |
Max Planck Institute for Astrophysics, Karl-Schwarzschildstr.1, 85741 Garching, Germany; Ludwig-Maximilians-Universitt Mnchen, Geschwister-Scholl-Platz 1, 80539 Munich, Germany Max Planck Institute for Astrophysics, Karl-Schwarzschildstr.1, 85741 Garching, Germany; Ludwig-Maximilians-Universitt Mnchen, Geschwister-Scholl-Platz 1, 80539 Munich, Germany Osservatorio Astronomico Cagliari, Via della Scienza 5 - 09047 Selargius, Italy Canadian Institute for Theoretical Astrophysics; 60 St. George Street; Toronto, ON M5S 3H8; Canada Dunlap Institute for Astronomy and Astrophysics; 50 St. George Street; Toronto, ON M5S 3H4; CanadaThe diffuse Galactic synchrotron emission should exhibit a low level of diffuse circular polarization (CP) due to the circular motions of the emitting relativistic electrons. This probes the Galactic magnetic field in a similar way as the product of total Galactic synchrotron intensity times Faraday depth. We use this to construct an all sky prediction of the so far unexplored Galactic CP from existing measurements. This map can be used to search for this CP signal in low frequency radio data even prior to imaging. If detected as predicted, it would confirm the expectation that relativistic electrons, and not positrons, are responsible for the Galactic radio emission. Furthermore, the strength of real to predicted circular polarization would provide statistical information on magnetic structures along the line-of-sights. The Galaxy in circular polarization: all-sky radio prediction, detection strategy, and the charge of the leptonic cosmic raysNiels Oppermann December 30, 2023 ==================================================================================================================================§ INTRODUCTION§.§ Circular polarisation emission The radio synchrotron emission of the Milky Way should be circularly polarized due to the circular motions of relativistic electrons in the Galactic magnetic field (GMF). Because of the relativistic beaming effect of the electron's motion on the emitted radiation we see mostly the electrons that spiral around fields oriented perpendicular to the line-of-sight (LOS) and therefore predominantly linearly polarized emission. The magnetic fields that point towards us could be a source of circular polarization (CP), reflecting the circular motions of the relativistic electrons visible in this geometry. However, the aforementioned beaming effect diminishes any radiation parallel to the magnetic field. The largest CP emission should therefore result from magnetic fields with an inclination in between parallel and perpendicular to the LOS. The field component parallel to the LOS, B_ , ensures that a circular component of the electron gyration is visible to the observer, and determines thereby the sign of Stokes-V. The field component perpendicular to the LOS, B_⊥, enables the gyrating electrons to send some beamed flux into the direction of the observer and therefore largely determines the strength of the CP. So far only linear polarization has been detected and imaged in the diffuse radio-synchtron emission of the Milky Way <cit.>.[CP from compact Galactic objects like Sagrittarius A^*, GRS 1915, SS 433 has been detected <cit.>, which seems to result from a different process as discussed here, namely Faraday conversion operating in the much stronger magnetic fields of these objects <cit.>.] CP should be much weaker and therefore harder to be detected and charted. Nevertheless, Galactic CP emission should exist and therefore should in principle be observable. Since the CP signal is weak, and has to be discriminated from instrumental polarization leakage effects, it would be very helpful to have a prediction not only on the magnitude of this emission, but also on its detailed morphology on the sky. This paper provides such a prediction. §.§ Predicting circular polarisation To predict the CP emission accurately knowledge of the GMF strength and orientation is necessary throughout the Galactic volume, as well on the number density and the energy spectrum of the relativistic electron population. Currently we are lacking this information, despite substantial efforts to model the GMF <cit.>, the Galactic thermal electrons <cit.>, and relativistic electrons <cit.> from various observables. The observables informing us about the perpendicular GMF component (times the relativistic electron density) are the linear polarization and total emission of the synchrotron emission. The parallel GMF component imprints onto the Galactic Faraday rotation measures of extra-galactic sources, however modulated by the thermal electron density. Instead of using these observables to construct a 3D GMF model from which CP can be predicted <cit.>, here we exploit that a certain combination of these observables should be linearly correlated with the CP signal. Exploiting this correlation without the detour of building a simplified 3D GMF model should permit to predict more small-scale structures of the Galactic CP signal than by usage of a coarse 3D GMF model. The CP sky prediction by <cit.> is based on such a 3D model and exhibits small scale structures. The latter are, however, due to a random magnetic field added to the 3D model, and therefore will not represent the real small-scale structures of the CP sky.The small scale-structures of our predicted CP signal will be more realistic. However, they only represent a statistical guess for the real CP signal. The pieces of information that are put together, the Faraday signal as a tracer of B_ and the total synchrotron intensity as a tracer of B_⊥, might report about different locations along the LOS, whereas the combination of both components at the locations of CP emission would be needed. The former signal results predominately from locations of high thermal and the latter of high relativistic electron density, and these do not need to coincide spatially. Fortunately, the GMF exhibits some spatial correlation as the observables are correlated as a function of sky direction and this correlation should also hold in the LOS direction. Therefore, information on field components resulting from slightly different locations might still provide a good guess at a position. Some of the structures imprinted onto the observables are caused by structures in the underlying thermal or – to a lesser degree – relativistic electron population and might, however, be misleading and lead to spurious structures in a CP sky predicted this way. Anyhow, a CP prediction constructed directly form such observables will be mostly model independent and therefore ideal for template-based CP detection efforts. It will have small angular scale structures that should also permit the usage of interferometer data that usually lack large angular scale sensitivity. While using this CP template map, it should just be kept in mind that it resembles an educated guess for the galactic CP morphology, and is certainly not accurate in all details. It should, however, be a very helpful template for the extraction of a detectable signal out of the probably noisy CP data and help to verify the detection of the Galactic CP signal by discriminating this from instrumental systematics that plague the measurement of weak polarization signals. §.§ Testing the charge of the emitters The rotational sense of the CP flux should typically have the opposite sign of that of the Faraday rotation if both are measured with the same convention and if the relativistic and thermal particles involved have the same charge sign, e.g. are both electrons. The reason is that the CP sense should directly reflect the gyro-motion of the relativistic particles emitting the radio emission. Faraday rotation is caused by the different phase speeds of left and right polarised electromagnetic waves in a magnetized plasma. The waves that co-rotates with the lightest thermal charge carriers – usually the electrons – can interact most strongly with them and gets the largest delay. Consequently, a linear wave that can be regarded as a superposition of left and right circular waves gets rotated in the sense of the faster wave, and therefore counter rotates with respect to the gyro-motion of the light charge carriers. Thus, if the involved thermal and relativistic particles have the same charge sign, CP and Faraday rotation produced in the same magnetic field counter rotate. This opens the possibility to test for the existence of regions with positrons dominating the radio synchrotron emission. §.§ Structure of the paper The structure of the paper is the following. Sect. <ref> presents the theoretical derivation of the CP prediction map construction. Sect. <ref> provides the predicted CP sky and discusses its remaining model uncertainties. Sect. <ref> investigates the detectability of the predicted signal and Sect. <ref> concludes.§ CIRCULAR POLARIZATION SKY§.§ Observables The CP intensity as characterized by Stokes V for a given LOS is approximately given byV=α_V∫dl n_relB_|| B_⊥^3/2, whereα_V=-0.342·e^9/2/π √(2π) ν^3/2 m_ec^7/2(γ_min^-2-γ_max^-2)is a constant, which depends only on natural constants, model parameters and the CP-observational frequency ν <cit.>. The symbols e, m_e and c denote the elementary charge, the electron mass, and the speed of light, respectively. The relativistic electrons with density n_rel are assumed here to have the same power law-like spectrum with cut offs γ_min and γ_max and spectral index of p_e=-3 everywhere. B_|| is the LOS-parallel, and B_⊥ the LOS-perpendicular magnetic field component and we wrote ∫dl=∫_LOSdl for the LOS integration. Also by writing B_⊥^3/2 we assumed implicitly an electron power law index of p_e=-3, which is not too far from the one observed. However, this simplification could be dropped if needed by replacing B_⊥^3/2 with B_⊥^-p_e/2 everywhere, for the price of more contrived calculations. As we only strive for a rough estimate of the CP signal, we will continue with the simpler B_⊥^3/2 scaling. The building blocks of the CP signal n_rel, B_||, and B_⊥ appear in nearly the same combination in the Galactic total synchrotron intensity I and the Faraday depth ϕ,I =α_I∫dl n_relB_⊥^2, ϕ=α_ϕ∫dl n_thB_||.Here, α_I=e^4/6π m_e^2c^3ν (γ_min^-2-γ_max^-2)<cit.> and α_ϕ=e^3/2π m_e^2c^4<cit.> are other constants, which depend on similar natural constants and model parameters as α_V, and n_th is the density of free thermal electrons. In particular, the data combinationd=ϕ I=α_ϕα_I∫dl∫dl' n_th(l) n_rel(l') B_||(l) B_⊥^2(l')contains the same magnetic field components as V=α_V∫ dl n_rel(l) B_||(l) B_⊥^3/2(l), although with slightly different spatial dependence and a slightly different B_⊥ dependence. If the magnetic field would be spatially constant along a LOS, d and V would be correlated according toV/d=α_V/α_ϕα_I1/∫dl n_th B_^1/2 ,so that knowing d would allow us to predict V ap art from the weak B_ dependence, assuming we know the LOS integrated thermal electron density from other measurements like pulsar dispersions. In reality, d and V will not be perfectly correlated as there are unknown magnetic structures on the LOS. The ratioV/d=α_V/α_ϕα_I ∫ dl n_relB_|| B_⊥^3/2/(∫dl n_thB_||)(∫dl n_relB_⊥^2)therefore encodes information on magnetic structures along the LOS, in particular on the co-spatiality of Faraday rotating and synchrotron emitting regions. This information would be interesting to obtain in order to improve our GMF models. Before CP observations can be exploited for studying Galactic magnetism, the CP signal has to be detected. For this, a rough model of the CP sky would be extremely helpful, as it can be used to build optimal detection templates to be applied to the noisy CP data. In the following, we construct such a predictive CP-polarization all sky map for this purpose. As d is already an observable today, it can be used to predict V to some degree. V and d will in general be correlated. The production of CP is inevitably associated with total intensity emission and the sign of the produced V is determined by the sign of B_||, which always also imprints into the Faraday depth (for emission locations with thermal electrons). This correlation might be weak, in case the synchrotron emission and Faraday depth signals are mostly created at distinct locations with mostly uncorrelated magnetic LOS component B_||. If, on the other hand, synchrotron emissions and Faraday rotation are mainly co-spatial, a strong correlation between V and d can be expected. The fact that the Galactic radio emission exhibits strong signatures of Faraday depolarization <cit.> supports the idea of an intermixed Faraday rotating and synchrotron emitting medium, which promises a large cross-correlation of d and V. Thus the prospects for predicting the CP sky signal to some degree are good. §.§ Model All three observables under consideration here, I, ϕ, and V, could be predicted for a given Galactic model in n=(n_th,n_rel) and B⃗=(B_||,B⃗_⊥), where we have chosen the LOS direction to be always our first coordinate. Although we have rough models for the 3D Galactic electron distributions n, the full 3D GMF configuration is currently poorly known. The existing GMF models <cit.> largely exploit the available Faraday and synchrotron data and therefore do not contain too much in addition to what these data-sets have to offer. The additional information of these models is due to the usage of parametric models of the GMF spiral structure, which are inspired from the observations of other galaxies. Although this is certainly helpful information, the price to be paid for it is a loss of small-scale structure in the model prediction as the parametric models do not capture all complexity of the data sets they are fitted to. These small-scale structures are, however, extremely important for detecting the Galactic CP signal, as many radio telescopes and in particular radio interferometers are insensitive to large-scale angular structures. Furthermore, a GMF model based prediction is only superior on large scales if the included additional assumptions were correct. Although, this might well be the case, to have a more model independent prediction is certainly healthy. For these reasons, we will try to predict the CP sky from existing I and ϕ sky maps directly, using only a minimal set of absolutely necessary model assumptions, which we describe now. The inclusion of more information and assumptions is in principle possible and would to lead more sophisticated V-map predictions as we are aiming for here. As the fluctuations in our observables are mainly caused by magnetic field structures and to a lesser degree by structures in the electron densities n=(n_th, n_rel), for which rough, but sufficiently accurate models exist, we will assume n to be known along any given LOS. For n_th we adopt the large-scale structure of the popular NE2001 model <cit.> and n_rel is modeled as a thick exponential discs, with parameters as specified in detail in Sec. <ref>. Adapting a simplistic model for the electron densities means that any structure in the RM sky, which is a consequence of not modeled structures in the thermal electron density, will be attributed to magnetic field structures and imprints on the resulting CP sky. Thus, the predicted CP sky will show some features not being present in the real CP sky. Not modeled structures in the relativistic electron density will imprint to both, the total intensity map and the CP map. Therefore, those will imprint on the CP prediction despite the fact that the inference model assigns them to magnetic sub-structures internally.Although the detailed GMF is still a matter of research, reasonable guesses for how the magnetic energy density scales typically with Galactic locations as expressed through n exist and will be adopted here. This means, we assume that the GMF energy density is largely a function of the electron density. We therefore need an expression for B^2(n)=⟨B⃗^2⟩_(B⃗|n)with ⟨ f(x,y)⟩_(x|y)=∫dx 𝒫(x|y) f(x,y) expressing the probabilistic expectation value of a function f(x,y) (here B⃗^2) averaged over the conditional probability 𝒫(x|y) of an unknown variable x (here B⃗) given a known variable y (here n to characterize the different typical environments in the Galaxy).In this work, a simple parametrization of the form B^2(n)=B_0^2/n_th0^β_thn_rel0^β_reln_th^β_thn_rel^β_rel=B_0^2 x_th^β_thx_rel^β_relwill be used, with x_i≡n_i/n_i0 and plausible scaling indices of β=(β_th,β_rel)∈[0,1]^2 . To be definitive, we adopt β_th=0 and β_rel=1 to model our intuition that the observed thick synchrotron disk of the Milky Way and other galaxies probably require magnetic fields which have a thick disk as well as the relativistic electrons causing this thick disk emission. This is in line with the expectation that the relativistic fluid in galaxies, consisting of mainly of relativistic protons, other ions, and electrons, drags magnetic fields with it when it streams out of galactic disks.In order to show to which degree our CP sky prediction depends on this assumption we also show results for the complementary case β=(1,0). It will turn out that β has only a marginal effect on our prediction, indicating also that the 3D modeling of the electron distributions is not the most essential input to our calculation. The exact normalization of the scaling relation Eq. <ref> is given by the parameters B_0^2,n_th0^β_th and n_rel0^β_rel. In the explicit calculation later on we use B_0≈6 μG and n_th0≈5·10^-2cm^-3. The parameter for the relativistic electron density n_rel0 drops out later on in the course of the calculation and is therefore left unspecified. The reason for this is that it affects the observable I in exactly the same way as the predicted quantity V, and therefore becomes irrelevant when conditioning our prediction on the observable I, which contains the necessary information on n_rel0 . We will exploit the correlation of V with the quantity d=ϕ I to predict the former. These quantities depend on the magnetic field structure along a LOS in different ways. Their cross-correlation depends on the magnetic field correlation tensor M_ij(x⃗,y⃗)=⟨ B_i(x⃗) B_j(y⃗)⟩_(B⃗)as well as on higher correlations functions. A priori, we have no reason to assume that within a roughly homogeneous Galactic environment (as defined by roughly constant n) any direction or location to be singled out. Thus, a statistical homogeneous, isotropic, and mirror-symmetric correlation tensor should model our a priori knowledge about the field, which then is of the form <cit.>M_ij(x⃗,y⃗) = M_ij(r⃗) = M_N(r) δ_ij+(M_L(r)-M_N(r)) r̂_ir̂_̂ĵ,with M_N(r) and M_L(r) normal and longitudinal scalar correlation functions, which depend only on the magnitude r of the distance vector r⃗=x⃗-y⃗ with normalized components r̂_i=r_i/r. These functions describe the correlation of the field at one location with that at another location shifted in a normal or longitudinal direction with respect to the local magnetic field orientation. These correlation functions are connected due to ∇⃗·B⃗=0 viaM_N(r)=1/2rd/dr[r^2 M_L(r)]and can be combined into the magnetic scalar correlation w(r)=⟨B⃗(x⃗)·B⃗(x⃗+r⃗)⟩_(B⃗)=2M_N(r)+M_L(r) so that B^2=w(0)=2M_N(0)+M_L(0) <cit.>. In our calculations, only correlations along of LOSs are needed, leading to the restriction r⃗=(r,0,0) if we identify the LOS direction with the first coordinate axis. This implies a component-wise diagonal correlation structure.M_ij(r⃗)|_r⃗=(r,0,0)=[M_N(r)+(M_L(r)-M_N(r)) δ_i1] δ_ij =[ M_L 0 0; 0 M_N 0; 0 0 M_N ]_ij(r)and therefore no a priori expectation of any cross-correlation of B_|| and B_⊥ along a given LOS. This simplifies the calculation of higher order magnetic correlation functions. For such we will use the Wick theorem, e.g. ⟨ B_iB_jB_kB_l⟩_(B⃗)=M_ijM_kl+M_ikM_jl+M_ilM_jk,and therefore implicitly a Gaussian probability for the magnetic field components. The real magnetic field statistics is most likely non-Gaussian, leading to differences between our estimated higher order correlates and the real ones. However, since we do not know how to model this non-Gaussianity correctly as we do not know even the sign of its effect on higher order correlations, and as we also like to keep the complexity of our calculations moderate we accept this simplification. We expect only a moderate and global multiplicative change of order unity on our predicted CP sky if the nature of non-Gaussianity would be known and taken into account in the prediction, as non-Gaussianity corrections would roughly affect all LOSs more or less similarly. Furthermore, we assume the longitudinal and normal magnetic correlation lengths (defined here differently to match our later needs)λ_L=∫dr M_L(r)/M_L(0) λ_N=∫dr M_N^2(r)/M_N^2(0)to be much smaller than typical variations in the underlying electron density profiles, so that e.g. the expected Farday dispersion can be calculated via ⟨ϕ^2⟩_(B⃗|n)=α_ϕ^2∫_0^∞dl ∫_0^∞dl' n_th(l) n_th(l')⟨ B_||(l) B_||(l')⟩_(B⃗|n)≈ α_ϕ^2∫_0^∞dl ∫_-∞^∞dr n_th(l) n_th(l+r) M_L(r)≈ 1/3α_ϕ^2λ_L∫_0^∞dl n_th^2 B^2(n).We introduced the notation f(l)=f(l r_LOS) for the value of the 3D field f(r⃗) along the LOS coordinate l in direction r_LOS . Here, and in the following we will treat the individual LOSs separately. Furthermore, we assumed that magnetic structures are smaller than the part of the LOS that resides in the Galaxy as expressed in terms of the structure of the adopted thermal electron model, so that a negligible error is implied by extending the integration over the relative distances r=l'-l from minus to plus infinity or by using the same thermal electron density for both locations, l and l+r. Furthermore, we used M_L(0)=M_N(0)=1/3B^2, which follows from isotropy and Eq. <ref>.Finally, we assume the observed Faraday and total intensity skies to be noiseless. This approximation will simplify the CP sky estimator and make it independent of the normalization of the scaling relation Eq. <ref> and the actual value of the correlation length λ_L as long this does not vary (strongly) along a given LOS. The assumed correlations length λ_N will have some small impact on our result, however, of sub-dominant order and therefore it is also not necessary to specify it if only a rough CP sky prediction is required. §.§ Estimator We want to exploit the correlation of V with d=ϕ I to construct an optimal linear estimator for V given d. This is given byV=⟨ V d⟩_(B⃗|n)⟨ d^2⟩_(B⃗|n)^-1dirrespectively the underlying statistics, since one can easily show that the quadratic error expectation ϵ^2=⟨[V-V(d)]^2⟩_(B⃗|n) is always minimized for linear estimators of the form V(d)=v d for v=⟨ V d⟩_(B⃗|n)⟨ d^2⟩_(B⃗|n)^-1:dϵ^2/dv= -2⟨[V-v d] d⟩_(B⃗|n) = 2 [v⟨ d^2⟩_(B⃗|n)-⟨ V d⟩_(B⃗|n)]=0. All remaining analytical work is to calculate the correlates which compose v. The simpler one is ⟨ d^2⟩_(B⃗|n)=⟨ϕ^2I^2⟩_(B⃗|n) =α_ϕ^2α_I^2∫dl_1…∫dl_4 n_th1 n_th2 n_rel3 n_rel4× ⟨ B_||1B_||2B_⊥3^2B_⊥4^2⟩_(B|n) =α_ϕ^2α_I^2∫dl_1…∫dl_4 n_th1 n_th2 n_rel3 n_rel4×M_L12 [M_N33 M_N44+2 M_N34^2]≈ 1/27 λ_Lα_ϕ^2α_I^2 [∫dl n_th^2B^2]× [(∫dl n_rel B^2)^2+2λ_N∫dl n_rel^2 B^4].Here, we used the abbreviations n_th1=n_th(l_1), B_||2=B_||(l_2), M_N34=M_N(l_3-l_4), and the like, exploited the diagonal structure of the magnetic correlations along the LOS as expressed by Eq. <ref> while applying the Wick theorem, and inserted the correlation lengths λ_L and λ_N as defined in Eq. <ref> while applying the short correlation length approximation as previously used in Eq. <ref>.The calculation of ⟨ V d⟩_(B⃗|n) is slightly more complicated. To handle the B_^3/2 dependence of V, we Taylor expand it in terms of B_^2 around B_0^2=2/3B_0^2 viaB_^3/2 =(B_^2)^3/4=n=0∑^∞3/4n B_0^2(3/4-n)(B_^2-B_0^2)^n =3/40 B_0^3/2+3/41 B_0^-1/2(B_^2-B_0^2)+𝒪(𝐵_^4) ≈1/4B_0^3/2+3/4 B_0^-1/2 B_^2. We choose to expand in B_⊥^2 rather than B_⊥, as the linear terms would vanish anyway during the application of the Wick theorem. We then find: ⟨ V d⟩_(B⃗|n)=⟨ V ϕ I⟩_(B⃗|n) =α_Vα_ϕα_I∫dl_1…∫dl_3 n_th1 n_rel2 n_rel3× ⟨ B_||1B_||2B_⊥2^3/2B_⊥3^2⟩_(B|n)≈ α_Vα_ϕα_I∫dl_1…∫dl_3 n_th1 n_rel2 n_rel3× ⟨ B_||1B_||2 (1/4B_0^3/2+3/4 B_0^-1/2 B_2^2)B_⊥3^2⟩ _(B|n) =α_Vα_ϕα_I∫dl_1…∫dl_3 n_th1 n_rel2 n_rel3× B_0^-1/2/4 M_L12 (B_0^2M_N33+ +3 [M_N22 M_N33+2 M_N23^2])≈ B_0^-1/2/36 λ_Lα_Vα_ϕα_I × [B_0^2(∫dl n_thn_rel B^2)(∫dl n_rel B^2)+(∫dl n_th n_rel B^4)(∫dl n_rel B^2)+2λ_N∫dl n_th n_rel^2 B^6]. Again we used M_L(0)=M_N(0)=1/3B^2 and λ_L and λ_N as defined in Eq. <ref>. This gives us in Gaussian units V=α σ ϕ I, α=3 α_V/4 α_ϕ α_I B_0^1/2≈-4.269·√(m_e^3 c^7/e^5 ν B_0)≈ ·10^18(ν/408 MHz)^-1/2(B_0/6 μG)^-1/2being a LOS-independent dimensionless quantity andσ=(∫dl n_th^2B^2)^-1×[(∫dl n_rel B^2)^2+2λ_N∫dl n_rel^2 B^4]^-1×[2/3B_0^2(∫dl n_rel B^2)(∫dl n_thn_rel B^2). + (∫dl n_rel B^2)(∫dl n_th n_rel B^4). + 2λ_N∫dl n_th n_rel^2 B^6]a LOS-dependent constant with dimension of an area. The unknown λ_L canceled out and the unknown λ_N affects only sub-dominant terms, as it is e.g. compared in the denominator to the Galactic dimension L=(∫dl n_rel B^2)^2/(∫dl n_rel^2 B^4)≫λ_N. We therefore neglect terms proportional to λ_N in the following and calculateσ ≈ 2/3 B_0^2 ∫dl n_thn_rel B^2+∫dl n_th n_rel B^4/(∫dl n_th^2B^2) (∫dl n_rel B^2)≈ 2/3 ∫dl x_th^1+β_thx_rel^1+β_rel+∫dl x_th^1+2β_th x_rel^1+2β_rel/n_th0(∫dl x_th^2+β_thx_rel^β_rel) (∫dl x_th^β_thx_rel^1+β_rel)for each LOS to translate d=ϕ I into V there. § PREDICTION To give an estimate for the CP sky, we need maps of the total synchrotron intensity and the Faraday rotation of the Milky Way. We use the 408 MHz map provided by <cit.>, which is based on the data of <cit.>, and the Faraday rotation map provided by <cit.>, which is largely based on the data of <cit.>. These are shown in Fig. <ref>We further need to quantify the σ parameter given in Eq. <ref>. For this we need the thermal and relativistic electron distribution of the galaxy and thereby x_rel and x_th. For the 3D distribution of the thermal electron density in the Milky Way we use the NE2001 model <cit.> without its local features. The spatial and the energy distribution of relativistic electrons in the Galaxy are more uncertain as we have only direct measurements of the cosmic ray electrons near the Earth. Considerable effort to infer these distributions have been made <cit.>. As we have shown in Eq. <ref>, we only need the spatial dependence and not the actual normalisation of n_rel, which means that this quantity only effects the relative strength of different structures in the CP map and not the overall strength of the predicted CP intensity itself. For this reason, and since we only aim for a rough estimate, we are content with a simplistic large-scale relativistic electron model. Given the distribution of matter in the galaxy, a exponential model for the spatial structure of cosmic ray electrons may make sense, as already adopted by other authors (<cit.>), at least in a similar way. In our case, we can use Eqs. <ref> and <ref> to give an estimate of the of total synchrotron map given our relativistic electron model and the scaling parameters of Eq. <ref>, where we adopt β=(0,1) and try to reproduce the large scale pattern of the 408 MHz map shown in Fig. <ref>. We thereby choose the following model for the spatial dependence of the relativistic electrons:x_rel=e^-|⃗r⃗|⃗/r_0·cosh^-2(|z⃗|⃗/z_0)The vector r⃗ points in the radial direction in the galactic plane, the vector z⃗ points out of the plane. As mentioned before, the parameters r_0 and z_0 are estimated via a naive comparison of the observed and estimated synchrotron maps at 408 MHz shown in Figs. <ref> and <ref>, respectively. The parameters adapted in this work are r_0=12 kpc and z_0=1.5 kpc. Given the morphological complexity of the map in relation to the simplicity of the model and the poorly understood nature of the origin and evolution of electron cosmic rays we acknowledge that the parameters of this model are highly uncertain. Also completely different parametrization of x_rel might lead to the same estimate for I because of the projection involved. The conversion factor α σ implied by our rough 3D model at 408 MHz is also shown in <ref> for β=(0,1).The resulting estimate of the circular polarisation intensity of the Milky Way is depicted in Figs. <ref> for the two cases β=(0,1) and β=(1,0). The morphology of the resulting maps is dominated by the morphology of the Faraday and the synchrotron map, what seems natural given our formalism. The influence of the dependence of the magnetic field on the different electron densities seems to be small, as the difference between between the two complementary cases is negligible, as we show for the predicted V/I ratio in Fig. <ref>. We predict a signal of up to 5·10^-4 Jansky per square arcminute at 408 MHz and more at lower frequencies. The CP is strongest in the center plane of the Galaxy. The relative strength of the CP intensity to the total synchrotron intensity up to V/I∼3·10^-4 as depicted in Fig. <ref>. The V/I ratio is largest just above and below the disc, as well as in some spots in the outer disc. We expect this ratio to increase with ν^-0.5, approaching 10^-3 at 40 MHz, which might be a detectable level for current instrumentation<cit.>. The frequency scaling of V/I∝ν^-0.5 was already predicted by <cit.> for the GHz range.The diffusion length of relativistic electrons depends on energy, therefore, the radio sky at different frequencies is not just a rescaled version of the 408 MHz map used as a template here. The V/I map provided by this work, however, should – within its own limitations – be valid at others frequencies as well. Therefore, it can be used after scaling by (ν/408 MHz)^-0.5 to translate total intensity templates at other frequencies into CP expectation maps at the same frequency, which then incorporate any difference of the radio sky due to spatially varying relativistic electron spectra.Anyhow, even if a total intensity template is not available at the measurement frequency, the main structure of the CP prediction, which are the sign changes induced by the sign changes of the Faraday sky, will be robust with respect to a change in frequency. Therefore, the CP template should be used as a structure expected on the sky, while allowing the real sky to deviate by some factor from it due to errors induced by the assumed frequency scaling and other simplifications. A template search method that is robust in this respect, is discussed below.The assumed scaling of the magnetic field energy density with the electron densities, β has only a minor impact on the result. The difference between the β=(0,1) and the β=(1,0) scenarios is less than 10%, as Fig. <ref> shows. Together with Fig. <ref> this is indeed evidence for the robustness of our results, as the profiles of relativistic and thermal electrons used in this work are quite different, nonetheless the different scaling does not lead to significantly different CP maps. § DETECTION STRATEGY§.§ Traditional imaging Now, we investigate the possibility to detect this CP signal with single-dish and interferometric observations, by requiring a signal-to-noise ratio of 10 in CP and by assuming a Stokes-V to I sensitivity ratio of σ_V/σ_I≈√(2) and a power-law total intensity frequency spectrum (I_ν∝ν^α, with α=-0.8). For example, the Sardinia Radio Telescope has the capability to observe in the low portion of the electro-magnetic spectrum: in P-band (305-410 MHz) and in L-band (1.3-1.8 GHz). By using the specifications given in <cit.>, an observing time of ≲1s per beam is required to reach the requested sensitivity in both frequency bands. One of the largest surveys of the sky at the moment available is the NRAO VLA Sky Survey (NVSS, <cit.>) characterized by a sensitivity in Stokes U and Q of 0.29 mJy/beam. The sensitivity at this frequency and resolution to detect the signal we are interested in is ≈6 μJy/beam. In principle, such a sensitivity can be reached by stacking all the 2326 fields of the survey if one can assume σ_ V≈σ_ Q≈σ_ U.We performed a similar evaluation for the LOFAR and the SKA, by referring to the lowest frequency band available for these instruments. For LOFAR, we use 45 MHz, where we expect V/I≈0.001. At this frequency, the required sensitivity is reached in less than 1 s. For the SKA, we considered the SKA-Low specifications given after the re-baselining in the frequency range 50-350 MHz, with a central frequency of 200 MHz and a bandwidth of 300 MHz. The required sensitivity can be reached in 3 h of observing time, if a resolution of ≈7 arcsec is considered. The Effelsberg telescope should obtain enough sensitivity within 20 minutes observation in its 400 MHz band and the GMRT within 30 min in its 200 MHz band. Thus, the prospects to detect the predicted CP signal are good from a pure signal to noise perspective. However, the polarization accuracy after calibration of the new generation of radio telescopes is typically of 0.1-1 % <cit.>. This instrumental limitation will make the imaging of the CP signal extremely hard as contamination of the CP signal by polarization leakage will be in the best case as strong as the signal we predict, in many cases one or two orders of magnitude stronger. To overcome this, we propose to cross-correlate the measured CP sky with our predicted one, as such instrumental effects are not present in our prediction and therefore should statistically averaged out in the comparison.§.§ Template search The CP all sky prediction constructed in the previous section can be used to search for the weak Galactic CP signal even in strongly contaminated data. Although CP sky images are usually not available, a number of radio telescopes take circular polarization data d_V=∫_𝒮^2dn̂R(n̂) V(n̂)+ξ.Here, d_V=(d_V1, … d_Vu)∈ℂ^u is the data vector of length u. n̂ is a direction on the celestial sphere 𝒮^2. R:𝒮^2→ℂ^u is the CP instrument response encoding the primary beam, the Fourier transform of the sky and subsequent sampling in case of interferometers, and any gain factors of the telescope. V(n̂) is the CP sky and ξ∈ℂ^u is the noise vector of the observation including the cross talk from other Stokes parameters. Here, we assume ξ to be generated by a zero-mean stochastic process with known covariance Ξ=⟨ξ ξ^†⟩_(ξ), which has to be obtained by careful studying the instrumental properties.Some part of the observed data vector can now be predicted using the CP predictionV(n̂), namely d_V=R V=α ∫_𝒮^2dn̂R(n̂) σ(n̂) ϕ(n̂) I(n̂).Since our prediction might be off by some multiplicative factor due to the various approximations involved in its derivation, and since we did not attempt to calculate the model uncertainty, a comparison via a likelihood function 𝒫(d_V|d_V) is out of reach. However, a simple, but sensitive indicator function (or test statistics) for the presence of the predicted CP signal is the inversely noise-weighted scalar-product of observed and predicted data,t=d_V^†Ξ^-1 d_V.If V=γ V+δ V is the correct CP sky, with γ∼1 the factor necessary to correct for our approximations, δ V the CP structures missed by our prediction due to imperfect correlation of V with d=ϕ I, and U=⟨δ V δ V^†⟩_(B⃗,n) the imperfection covariance , we expectt=⟨ t⟩_(B⃗,ξ|n)=γ V^†R^†Ξ^-1R V>0 σ_t^2=⟨(t-t)^2⟩_(B⃗,ξ|n)=u+Tr[U R^†Ξ^-1R]and therefore a signal-to-noise ratio (SNR) of S/N=t^2/σ_t^2=γ^2 (Tr[R V V^†R^†Ξ^-1])^2/u+Tr[R U R^†Ξ^-1],where we used Tr[Ξ Ξ^-1]=u, the number of data points. If we only reconstructed f=10% of the intensity of the true celestial CP signal, so that γ^2⟨V V^†⟩≈ f^2 U=10^-2 U, and if the CP data is 99% (=1-p) noise and cross leakage dominated, so that R ⟨ V V^†⟩ R^†≈ R U R^†≈ p^2 Ξ≈10^-4 Ξ, we get a SNR of S/N≈ f^2 p^2 u=10^-12 u and enter the detection range (S/N∼1) in the terabyte regime (u∼ f^-2 p^-2=10^12). § CONCLUSIONS Using the observational information on magnetic fields along and perpendicular to the LOS from Faraday rotation and synchrotron total emission we provided a detailed map of the expected diffuse Galactic CP emission[The V and V/I maps for the two scenarios discussed are available at <http://wwwmpa.mpa-garching.mpg.de/ift/data/CPol/>.], which is at a level of 3·10^-3 of the total intensity at 408 MHz and higher at lower frequencies. This prediction relies on assumptions about the magnetic field statistics, and the three dimensional distributions of thermal and relativistic electrons throughout the Milky Way. As these assumptions are not certain, the real Galactic CP sky can and will differ from our prediction. Nevertheless, the provided CP prediction can be used for template based searches for the elusive CP signal. Our model shows similarities and differences to a CP prediction based on a 3D models of the Milky Way <cit.>. We expect our model to capture more details of the real CP sky, as its construction is based directly on observed data sets, without the detour of using those to construct a parametrized, and therefore coarse, 3D model. However, which model is more accurate should certainly be answered by observations.A confirmation of the celestial CP signal we predict would indicate a co-location of the origin of the observed Faraday rotation signal and synchrotron emission. In case the predicted signal is not detectable with a strength comparable to the prediction, this would indicate a spatial separation of these regions along the LOSs and therefore important information on the Galactic magnetic field structure and its correlation with thermal and relativistic electrons. Finally, we like to point out that the hypothetical possibility exist that the observed CP signal has the opposing sign compared to our prediction (even after potential confusions of the used CP conventions are eliminated). This would happen in case the synchrotron emission of the Milky Way would predominantly result from relativistic positrons, which gyrate in the opposite direction compared to the electrons. This is – however – very unlikely given that the observed local density of relativistic electrons is much higher than that of the positrons and given that the relativistic particles in the Milky Way are believed to be accelerated out of the thermal particle pool. Nevertheless, it shows that the charge of the Galactic synchrotron emitters can actually be tested by sensitive CP observations. This research was supported by the DFG Forschengruppe 1254 “Magnetisation of Interstellar and Intergalactic Media: The Prospects of Low-Frequency Radio Observations”. We thank Henrik Junklewitz and Rick Perley for discussions and an anonymous referee for constructive feedback. apsrev | http://arxiv.org/abs/1706.08539v3 | {
"authors": [
"Torsten A. Enßlin",
"Sebastian Hutschenreuter",
"Valentina Vacca",
"Niels Oppermann"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170626180010",
"title": "The Galaxy in circular polarization: all-sky radio prediction, detection strategy, and the charge of the leptonic cosmic rays"
} |
Optimal charge-to-spin conversion in graphene on transition metal dichalcogenidesAires Ferreira==================================================================================Recent technological changes have increased connectivity between individuals around the world leading to higher frequency interactions between members of communities that would be otherwise distant and disconnected.This paper examines a model of opinion dynamics in interacting communities and studies how increasing interaction frequency affects the ability for communities to retain distinct identities versus falling into consensus or polarized states in which community identity is lost.We also study the effect (if any) of opinion noise related to a tendency for individuals to assert their individuality in homogenous populations.Our work builds on a model we developed previously <cit.> where the dynamics of opinion change is based on individual interactions that seek to minimize some energy potential based on the differences between opinions across the population.§ INTRODUCTION During the past several years, scientists have built up different modelsto explain dynamics of opinion evolution in a society <cit.>.They are called opinion models and have been used to study the dynamics of a set of individuals who communicate their opinion on one or more topics and adjust them based on these interactions.Early models typically reach a steady state withina small number of interactions (e.g. <cit.>) such that the populationreaches consensus, polarization, or a state in which a set of distinct opinions are held indefinitely by subpopulations.Subpopulations with similar opinions are referred to as clusters. In the case in which a set of subpopulations exist where each subpopulation reaches a distinct equilibrium opinion state, it is useful to study the effect of parameters on the system that affectcommunities' ability to maintain these distinctopinion states such that they do not collapse together. Collapse implies that these subpopulations are no longer distinguishable by having distinct opinion sets.In this work, we consider the consensus equilibrium opinion states to be the identity of each population. These identities reflect shared opinions on any topic, be it volatile topics (such as politics or religion), or benign topics (such as locally popular makes and models of cars).The stability of identity of such subpopulations is of relevance in social systems where subpopulations correspond to meaningful groups, such as families, townships, political groups, religious groups, and so on.Under what conditions can such subpopulations coexist and maintain their distinct identities with some degree of cross-group interaction?What factors cause them to either merge to a consensus state or diverge to a state in which they adopt polar opposite opinions on a given topic?This is particularly relevant to study in the current world in which subpopulations of people that traditionally have coexisted and maintained different, but non-polarizing opinions, may find it difficult to maintain these distinct identities with respect to specific topics.In particular, the emergence of high-frequency, high-reach communication mechanisms that did not exist prior to the 21st century (e.g., social media) fundamentally change the interaction dynamics between individuals and communities compared to lower frequency, lower reach mechanisms in the past.The variety of factors which play differentroles in real social life where the invention ofInternet and technology has brought people together and closer than ever, make the dynamics complex.Therefore, to keep it simple we study two such factors. We would like to investigate: * Stability of communities as a function ofinter-community interaction rate. * Effect ofsymmetric and asymmetric noise on stability of communities in presence of inter-community interactions.§.§ Opinion models For the purpose of formal modeling and analysis, we adopt an idealized numerical model of opinions.An opinion is considered to be a numerical value within a range (e.g., [0,1]) called opinion space.Individuals are simple agents that hold opinion values on one or more distinct topics. Agents interact in a pairwise fashion to exchange and update their opinion state over time.In the model that we adopt for this work (detailed in <cit.>), individuals update their opinions after interactions in a manner that reduces the energy of some potential function.A brief summary of the relevant components of the model for this paper are described in Section <ref>. §.§ Stability and robustness of subpopulations In this work we will consider a single large population of individuals in which a set of subpopulations exist, each of which contains a subset of the population whose collective opinion state is in equilibrium.An equilibrium opinion state is reached when interactions between members of the subpopulation do not reduce the overall interaction energy (e.g., the individuals are in a consensus state, (Figs. <ref> and <ref>[Agents in Fig. <ref>follow the bounded confidence model rule.]). We consider a stable subpopulation to be one in which the introduction of a new opinion by one individual that is sufficiently small anddifferent from the equilibrium state of thesubpopulation will not cause the overall opinionof the population to change by more than some ϵ. After introduction, the individual holding thenew opinion will eventually be driven towardsthe equilibrium state of the subpopulationsuch that the resulting new equilibrium statewill be within ϵ of the state before thenew opinion arrived. We would say that thisdeviation from the stable point will be damped out within a finite number of interactions (Fig. <ref>).If this deviant opinion retains its difference from the stable state of the subpopulation from which it emerged, such as by additional deviation before it can be damped out, we could see a difficulty in the subpopulation to damp the opinion out (Fig. <ref>) leading to an overall effect on the equilibrium state of the subpopulation.A reinforcement effect emerges when more than one member of the subpopulation makes opinion changes at a frequency above the time scale by which the population can damp out the deviations and return to their previous stable state.When such events occur, we expect that previously stable subpopulations may destabilize and fracture, driving them to new equilibrium states. Under an interaction potential function that supports multiple stable minima (such as a potential supporting both consensus and polarization states), we may observe a rapid disintegration of the subpopulation at equilibrium into a set of polarized subpopulations due to a frequency of introduction of divergent opinions that exceeds the timescale necessary to damp out deviations from equilibrium.When this threshold is passedwe would see a loss of identity for subpopulations. They may be driven to a polarized state in which individuals adopt opinions at the extrema of the opinion space (Fig. <ref>), or a homogenous consensus state in which no distinct subpopulations exist anymore (Fig. <ref>).Our hypothesis is that this frequency of deviation is the cause of fracturing of otherwise stable communities into those in which homogenous consensus or polarized opinion states dominate.The interesting result is that populations that would otherwise agree may rapidly split due to high-frequency introduction of opinions outside their consensus state, which is precisely what emerges in social networks where interaction frequency with large populations outside one's own community is commonplace. §.§ Individualization tendency In real life people would like to be unique and different from others, as such our model includes opinion noise representing this natural desire.It is shown that opinion noise is an important contributing factor to the existence of opinion clusters <cit.>.In this paper we also study its effect on the stability of interacting subcommunities and whether there exists a relationship between the frequency of interactions and properties such as bias or magnitude of noise. Noise can have different effects in terms of the response of a subcommunity to the introduction of new opinions away from equilibrium.Noise may cause an individual to counterbalance the effect of the new opinion, providing an opposing force that cancels out the deviation.Similarly, noise may reinforce the deviation if it is in the same direction, compounding its effect.In Fig. <ref> we see two agents decide to be different from the rest of the group.In Fig. <ref> they go their way too frequently and cause the group to have a wider range of opinion about previous equilibrium state that in a long run could divide the group into two smaller groups.But as can be seen in Fig. <ref> if the abrupt change is not too frequent, the community could save its identity.We study two types of noise:noise in which the change in opinion is symmetric about the equilibrium state, and noise where an individual is more likely to move their opinion in the opposite direction of their subpopulation peers.§.§ Concepts in the paper §.§.§ Damping perturbations from equilibriumCommunities do not losetheir identity when a single member makes a single (sufficiently small) abrupt change (Δ) in their opinion on a given topic. This is due to a damping process that occurs where subsequent repeated interactions within the community with the individual who made the change gradually pulls that individual back to the consensus opinion held by that community.This process is not instantaneous, as each interaction changes the opinions of individuals by a small amount δ << Δ. If Δ≈ nδ, and we expect one interaction with the individual who made the large jump every t time units, then we would say that the deviation of size Δ has a damping period of nt over which time the subpopulation absorbs the deviation and recovers its collective identity with respect to the given topic.The frequency of deviations is critical.If another deviation occurs before the damping period has elapsed, the deviations may reinforce each other (if they went in the same direction relative to the consensus state) and make it take longer for the entire subpopulation to return to its collective consensus state.When opinion change of nodes are due to interactions across subpopulations, then we must consider the relationship of the damping time necessary to recover from an individual deviation and the frequency of interaction between subpopulations that cause deviations to occur.§.§.§ Frequency of inter-group interactionProbability of interaction between two nodes is used as a surrogate for frequency of interaction between them. (See Section <ref>). These probabilitiesare stored in adjacency matrix as edge weights between nodes. Higher probability of interactions causes more frequent interactions.§.§.§ Noise via individualization tendency A somewhat different type of opinion change,an abrupt change, is introduced tothe system by adding the individualization tendency to the model.The individualization tendency we use here is an “adaptive” one, i.e. agents desire to leave their community increases according to the two following factors:(a) the difference between a given agent's opinion and other agents' opinion decreases and,(b) number of agents whose opinions are close to agiven agent increases. In other words, let opinion of agent ibe given by o, and let (o-ϵ, o+ϵ) be aneighborhood of o, then individualization tendency of agent ibecomes more intense as the number of other agentsin this neighborhood increases or as ϵ decreases. This is given by Eq. (<ref>).This noise is drawn from a normal distributionwith a zero mean and adaptive variance.§.§ ContributionsThe model we present in this paper makes somenoteworthy contributions to the study of opinion dynamic. We conduct computer experiments to show that: * Increased frequency of interactionbetween subcommunities causes identity loss. * Individualization tendency makesthe identity loss of communities to take placemore often and faster. § MODEL First let us start with some definitions and notation that will be used later in the paper.definitionLet G = ( V , E ) represent the fully connected network under consideration where V is the set of nodes and E is the set of edges in which all distinct nodes are connected via an edge with a probability of interaction weight assigned to each edge.The set of all spatial neighbors of node i ∈ V is denoted by N(i) contains all nodes in G that are connected to i via an edge with an interaction probability weight of more than zero. definitionThe set of all possible (numerical) opinions, denoted by 𝒪,is called opinion space, and in this paper it will be the interval [0,1]. definitionA social group is the set of all nodes in G with high probability of interactions. These nodes are located on diagonal blocks of the adjacency matrix of G.definitionA opinion cluster or a (opinion) community is the set of all nodes which agree about a topic. In other words, set of all nodes whose opinion belong to (o - ϵ, o + ϵ ) for some ϵ.Opinion o is called the identity of such community. In the experiments, members of a given social group hold the same opinion at time t=0.definitionDefine the δ-opinion-neighbor of node iat time t, denoted by 𝒫_δ^t(i), to be the set of all nodes whose opinion are in (o_i^(t) - δ, o_i^(t) + δ ),for some δ at time t .definitionIndividualization tendency, ξ, is a noise randomly chosen from a normal distribution with zero mean and some varianceσ (ξ∼ N(0, σ)).Notation used throughout the paper 2 [ ] * Matrices will be shown by bold letters.* I_k identity matrix of size k.* 1_k matrix of ones of size k.* n_s number of social groups.* n_p population of each social groups.* N = n_s × n_p total population of the network.* N(i) set of spatial neighbors of agent i.* 𝒪 opinion space.* o_i^(t) opinion of node i at time t.* α learning rate.* ψ potential function.* ξ_i(t) agent i's noise at time t.* N(μ, σ) normal distribution with mean μ and standard deviation σ.* s_p skewness parameter.* N(s_p, μ, σ) skew normal distributionwith skewness parameter s_p, mean μ and standard deviation σ. §.§ Network connectivityIn this paper, we work with a fully connected graph in which all individuals are connected via an edge with aweight assigned to it. The weight corresponds to theprobability that agent i will talk to j in a single step. The network adjacency matrix A is a block symmetric doubly-stochasticmatrix whose entries, a_ij, determine the probabilityof agent i choosing agent j foran interaction, i.e. weights assigned to edges in G are stored in A. Lets denote the number of social groups byn_s where each of them have equal population of size n_p.Then the matrix A would be of the size N × N where N = n_p × n_s is the population of the network.A = [A_11A_12 … A_1,n_s;A_21A_22 … A_2,n_s; ⋮ ⋮ ⋱ ⋮; A_n_s,1 A_n_s,2 … A_n_s,n_s ] The entries in diagonal blocks A_kk, 1 ≤ k ≤ n_sare interaction probabilities of agents within a social group whilethe entries in A_lk, 1 ≤ l ≠ k ≤ n_s areprobabilities of inter-group interactions.The adjacency matrix will be generated in two different ways, deterministically and randomly. Consequently two different sets of experiments and analysis will be represented.§.§.§ Deterministic adjacency matrix In the first case scenario, the adjacency matrix is generated deterministically byAlgorithm <ref> in which the probabilityof interaction between members of a social group are identical, and the interaction probability between any two agents of different groups are the same.§.§.§ Random adjacency matrixIn this method entries of off-diagonal blocks are randomly chosen from an interval bounded above by some upper bound u_b ∈ [0,1]. So, when u_b = 0 then there would not be anyinteraction between two different communities. We use the iterative method defined by Sinkhorn <cit.> to generate the doubly-stochasticmatrices with a little modification to make them symmetric (Algorithm <ref>).At the end of this process, the entries ofdiagonal blocks A_kk will not be equal,which means that, each pair of nodes in the samesocial group will not have the same frequency,i.e. probability, of interaction. Note that A_ij is a submatrix and a_ij is a single entry of A and we use a^k_ijto denote a single entry of the block A_kk.Figure <ref> is an (heat map) example ofadjacency matrix with n_s = 3 groups, each groupis consist of n_p = 9 nodes. As u_b increases the frequency of inter-groupinteraction goes up in expense of local group interactionsto the point that inter-group connections are tighter thangroup connections (u_b = 1). When u_b is larger than (n_p - 1) ^-1,it is possible that some entries in the off-diagonal blocks to be larger than the entries within the diagonal blocks. In (<ref>) the upper boundis set to be 1/(n_p-1) = 1/8, and we see there are placeswhere frequency of inter-group interactions are higherthan that of internal ones.The internal group interactions are diluted.The reason that we can still see subpopulations arerelatively knitted tightly is that the entries of off-diagonalblocks are chosen randomly from [0, 1/8) to begin with. So, lots of them start with values smallerthan 1/8. We generated 1000 of such matrices for n_s = 3,n_p = 9, and u_b = 1/8.Therefore, 3000 blocks of size 9 of subgroups.The mean and standard deviation of entries of diagonal blocks, mostly belong to the intervals(0.056, 0.062) and (0.004, 0.008), respectively. §.§ Micro dynamicsPairwise interaction rule of the model is borrowed from <cit.>. Let opinion of agent i at time t be given by o_i^(t)∈𝒪.Then the update rule is given by: {[o_i^(t+1)= o_i^(t)- α/2 ψ'(|d^(t)_ij|) d^(t)_ij + ξ_i(t);o_j^(t+1) = o_j^(t)+α/2 ψ'(|d^(t)_ij|) d^(t)_ij + ξ_j(t) ].where α is called learning rate,ψ is called potential function which governs the update rule, andd_ij^(t) = o_i^(t) - o_j^(t).If boundary condition of opinion space is violated, then opinions will be clamped. ξ_i(t) ∼ N(0, σ_i(t)) adapted from <cit.>, is individualization tendency of agent i at time t. It incorporates thenatural instinct of people wanting to be different <cit.>. ξ_i(t) is randomly sampled from a normal distributionN(0,σ_i(t)) where σ_i(t) is given by: (The opinion space in <cit.> is [-250, 250]. So, we had to scale the variance to fit our opinion space which is [0,1].) σ_i(t) = s/e-1( - |N(i)| + ∑_j ∈ N(i) e^1-|d_ij(t)|)where N(i) is set of (spatial) neighbors of node i,the parameter s is used to manipulate the strengthof individualization tendency.Individualization tendency increaseswhen there is high uniformity. The individualization tendency mentioned above does not take into account direction of movement of a given opinion cluster. It is equally probable that an individual makes anabrupt change in any direction.Therefore, we also considera systemin which agents have a memory in the sense that they will consider direction of movement of the opinion-cluster they belong to, so that it is more probable for them to move inthe opposite direction of cluster's movement.In order to increase probability of individual tendencies to be in the opposite direction of cluster movements, we sample ξ_i(t) from a skew normal distribution( ξ_i(t) ∼ N(s_p, 0, σ_i(t))), defined by <cit.>,where s_p is skewness parameter and variance is defined as before. The skew parameter of such a distribution is given by s_p = - c_m s_s, where s_sis a constant called skewness strength, and c_m, defined below determinesthe direction of movement of the cluster to which agent i belongs to.definitionLet 𝒫_δ^t(i), be the set of δ-opinion-neighbor of agent i at time t for some δ. Then define c_m to be the direction that majority of 𝒫_δ^t(i) moves towards (sgn(x) is the sign function): c_m = sgn [ ∑_j∈𝒫_δ^t(i) sgn ( o_j^(t) - o_j^(t-1)) ] §.§ Equilibrium response to abrupt individual changes Let Δ̂:= Δ_i^(t) be the deviationfrom consensus made by individual i on the giventopic at time t.During subsequentinteractions between i and membersj ∈𝒫_0^t(i) ∖{i}, we expectthat o_i will move closer to o_𝒫_0^t(i) ∖{i}.Let 𝒫_0^0(i) = {i, i_2, i_3,… , i_k} andsuppose there is no inter-community interaction and let the opinion of group to be ô≡ o_𝒫_0^0(i)at t=0.Let the potential function for all agents to be the same and suppose agent i makes an abrupt change: o_i = ô + Δ̂∈𝒪 (WLOG assume Δ̂> 0), so that it is in attraction domain of the subgroup, thenthe equilibrium state of the members of subgroup will drift by Δ̂/k.since Δ̂ is small enough so that other agents in the group will attract agent i, for all agents in 𝒫_0^0(i) define the energy e_jto be the height from ô. Then total energy of the given community is E:= Δ̂ at t=0. Since all agents are using the same potential function, the step size that a pair of node makes in a single interactionis the same.Therefore, for example, in the first interaction between i and i_j we have o_i^(1) = ô + Δ̂- ϵ_1 and o_i_j^(1) = ô + ϵ_1. There will not be loss of energy.Hence, after t = m steps, where m is large enough, we must haveE = Δ̂+ 0 + … + 0 = Δ̂/k + Δ̂/k + … + Δ̂/k.Therefore, all agents will come to consensus at õ = ô + Δ̂/k and the drift size is Δ̂/k. Note that in the Prop. <ref> it is assumed that Δ̂ is sufficiently small so that agent i will be attracted to the subgroup it belongs to. But, for example if a tent potential given by Eq. (<ref>) is used and Δ̂> τ, then agent i will repel the rest of the group , agent i and the subgroup will end up in opposite extreme pointsof opinion space. Or, if a bounded confidence potential is used, where agents will not interact if they are far enough, then after the abrupt change agent i and the rest of the group might ignore each other and everyone would stay where they are.In any interaction, both nodes involvedwould take steps of the same size.Hence, average of opinions of all nodeshas to be the same all the time. ∀ t: 1/N∑_i o_i^(0) = 1/N∑_i o_i^(t) However, for example in Fig. <ref> we see that the average of all opinions at t=30,000 is 0.75 which is different from average of opinions at t=0 which is 0.5. The reason is that the boundaries of opinion space, does not let the nodes to take steps of the right size and therefore, system loses energy and average of opinion of all nodes would not be the same. § EXPERIMENTSExperiments are done to fairly small populations.They can be extrapolated to larger populations.However, there would be some differences. For example, by <ref> for a large population, the drift would be smaller, therefore, longer time is needed for communities to collapse. Or, for a fixed i_p (u_b), as population N increases, the probability of interaction inside a group gets smaller and smaller relative to inter-group interaction probabilities. Therefore, more time is needed to reach a steady state. §.§ MethodsIn the experiments[The codes for this experimentsand that of <cit.> can be found here: https://github.com/HNoorazar/]pairwise interactions follow the update rule given by Eq. (<ref>) with learning rate α = 0.1, and a tent potential function defined by Eq. (<ref>) with τ = 0.63.Initial opinion of communities are 0.1, 0.5and 0.9 respectively so that they are in an equilibrium state.ψ(x) =1/τ x,x ≤τ,1/τ - 1x, o.w. Individualization tendency will be sampledfrom a normal and skewed normal distribution with thesame mean and variance.Note that a time step is equal to N pairwise interactions.§.§ Deterministic adjacency experiments§.§.§ Population size and inter-group interaction rateSuppose there are n_s subgroups in the network where each group is consist of n_p nodes.Then each column of the matrix has Nentries, of which n_p = 1/n_sN of them lie on a diagonal block submatrix and n_s-1/n_sN of themlie on the off-diagonal blocks. In the first step of generating the adjacency matrix, 1/n_p-1 is assigned to diagonal blocks, and i_p is assigned to off-diagonal blocks (Algorithm <ref>), denote such a matrix by A_0. For example, in the case of n_s=2 and n_p=2 the adjacency matrix(at the beginning of the process) looks like: A_0 = [ 0 1 i_p i_p; 1 0 i_p i_p; i_p i_p 0 1; i_p i_p 1 0; ] In such a setting we are interested in knowing whatis the relationship between the population size N and the smallest i_p for which the social groups losetheir identity and collapse.Hence, in order to find a relation between the two parameters we do the following. For a given column of the matrix A_0, we look at the ratio of sum of entries that lie on off-diagonal blocks to thesum of entries lying on diagonal blocks which is given by r_i_p = n_s-1/n_sNi_p. For example, in the matrix given by Eq. (<ref>) the ratio is r_i_p=2i_p. Let us denote the smallest i_p that causes community collapse by i_pc and the corresponding ratio by r_i_pc. Experiments with n_s=3, different n_p's and i_p's are run and its results are shown by Fig. <ref> andTable <ref>. For most experiments, the ratio is about r_i_pc≈0.34.§.§.§ Individuality tendency causes identity lossIn order to see effect of individuality tendency on identity loss, we pick up an small inter-group probability of interaction, i_p=0.0025 and vary s. We can see in Fig. <ref> that average of number of opinion clusters goes down as individuality tendency goes up, and causes the communities to lose their identities. However, after some point when s grows, individuality tendencycauses the social groups to earn a new identity and hence, we see after s =0.0023 where average number of clustersreaches its minimum at 2.24, the average number of clusters start to grow again. For example, in Fig. <ref>the parameters are u_b = 0.01 and s = 0.0047(This figure has used random adjacency matrix).The plot shows how the individuality tendency causes communitieslose their identities and then try to be different. The same result holds for the random adjacency matrix as well.Figure <ref> also shows,the more people want to be different, the sooner thecommunities will lose their identity. Please note that these are averages of the times for which the communities lose their identity first over time. §.§ Random adjacency experiments§.§.§ No individualization tendency First lets look at the case in which there is no individualization tendency, i.e.s=0, and u_b is varied (Fig. <ref>). Having two clusters means communities have merged,since there was not an experiment in which all three communities come to consensus,due to the choice of initialization of opinions and the fact that the tent potential function has two minima. We can see as frequency of interaction goes up,frequency of vanishing the community in the middlegoes up as well and the middle community loses itsidentity and is combined with the other two. Furthermore, the average time needed for community collapse has an inverse relation with inter-community interaction rate. The more inter-community interaction frequency, the less the time needed for communities to merge into one(Fig. <ref>).In some of the experiments, the community whoseopinions are in [ 0.49, 0.51] at t = 0 collapsesand merges with the other two communities. Average of collapsing time is taken just over the collapsed cases.§.§.§ Symmetric tendency of individualization In this section we sample individualization tendency from a (symmetric) normal distribution. In this case the noise can be in any direction and cluster's movement direction is not taken into account. An individual can either make an abrupt change in the direction that its neighbors are moving towards, or in the opposite direction.In the experiment whose result is represented byFig. <ref> we see that for u_b = 0.01we get approximately50% of identity loss(i.e. 50% of the experiments the community in the middleis merged with the other two and in 50% of experiments itmaintained its own identity).We set u_b = 0.01 andvary the individualization tendency strength. The result is shown inFig. <ref>. Individualization tendency will cause theidentity loss to takeplace more frequently. Figure <ref> and <ref> have an interesting message.The more one wants to be unique, the less s/he can be. Figure <ref> is illuminating. At time t=0 people in middle community have the urge to be different fromother members of their own. They deviate themselves and get close to the other two communities at the extreme points,and then interaction forces, bind them to the other communities. Afterwards, any interaction they make is either from the membersof their opinion-neighbors or they interact with agents in the otherside of the boundary, which in both cases, they are forcedto sit where they are!Lets look at different combinations of u_b and s. Average number of clusters for each pairs are computed.We see as both u_b and s increase, the community in the middle is effected by the other two and will merge into them. And change of u_b has more effective consequences.The missing pieces in Fig. <ref> correspond to the experiments in which all communities kept their identity.In Fig. <ref>, which isone slice of Fig. <ref> we see that as individualization tendency and collapse time have an inverse relation.§.§.§ Asymmetric individualization tendency In Fig. <ref>,where game is replicated 1000 times per s,the average time needed for community collapsefor both deterministic and random adjacency matrices are compared with four different skewness strength.Individualization tendency distributions have the same mean and variance in all cases, however, the skewness parametersare different. In one case there is no skewness, i.e.distribution is symmetric, and three asymmetric distributions of individualization sampleswith different skewness strengths are experimented so that the individualization tendency is more probable to be in the direction opposite to direction of movement of cluster to which each agent belongs to. It is interesting that the collapse times are almost identical in almost all cases.§ COMMUNITY MATTERSIn Fig. <ref>, there are 3 communities where each community consists of 4 people (N = 12) and the probability of interaction can be at most u_b = 0.01. In the process of making the adjacency matrix, communitiesstart with an interaction probability of 1/3. Hence, u_b = 0.01 is fairly small. However, we can see when the time is about 130, the middle community is merged with the community at the bottom, and consequently, the individuality tendency is very high. The community that began in the middle separates itself from the bottom community and its members stick together. The integrating force, i.e. interaction rule,overcomes the individuality tendency and glues the agents of middle community together.§ IDENTITY LOSS IS INEVITABLE With no external force countering pressure from other communities, the slightest probability of interaction between social groups causes loss of identity. Figure <ref> shows n_s = 3 social groups where each of which is consist ofn_p = 4 nodes where probability of interaction between them is very small, and there is no noise in the system. The game is run for 10^6 pairwise interactions. One experiment, Fig. <ref>, uses the deterministic adjacency matrix with i_p = 0.00001 and the other, Fig. <ref>, uses randomly generated matrix with u_p = 0.00001. However, in order to reduce the stochastic error, 100 adjacency matrices were generated and average of those is used in this experiment. § DISCUSSION AND FUTURE WORK Results presented here shows that as individualization tendency(in the symmetric case) is increased the communitiescollapse more often and faster.However, results in <cit.> indicates that individualization tendency can cause emergence of new clusters. Please note that the result presented here does notcontradict results of <cit.> for the following reasons:(a) we did not increase the individualization tendency parameter s very much,since its effect is not the primary subject of our study,(b) we did not give the system a very long time so thatindividuals have the chance of forming a new subcommunityeven for such small individualization tendency parameters. It can be seen in Fig. (2) of <cit.> that over time, communities collapse and break again and again and alsoit depends on individualization tendency parameter s. Moreover, their interacting update rule, which they refer to as integrating forces, is different from that of ours. Topology of the network, also, plays an important role in time evolution of opinions. Humans naturallytend to talk more frequently to those whom are moresimilar to, i.e. Homophily, and so, finding friendsis an alive creature that has dynamics. Hence, coevolution of network forboth discrete and continuous opinionsare studied <cit.>. It would be interesting to apply dynamic topologies to our model.apa | http://arxiv.org/abs/1708.03317v4 | {
"authors": [
"Hossein Noorazar",
"Matthew Sottile",
"Kevin Vixie"
],
"categories": [
"cs.SI",
"physics.soc-ph",
"q-bio.PE"
],
"primary_category": "cs.SI",
"published": "20170627175356",
"title": "Loss of community identity in opinion dynamics models as a function of inter-group interaction strength"
} |
Topometric Localization with Deep Learning^*These authors contributed equally. All authors are with the Department of Computer Science, University ofFreiburg, Germany. Corresponding author's [email protected]* Gabriel L. Oliveira^* Noha Radwan^* Wolfram Burgard ThomasBrox today ====================================================================* § INTRODUCTION Robot localization is essential for navigation, planning, and autonomous operation. There are vision-based approaches addressingthe robot localization and mapping problem in various environments <cit.>. Constant changes in the environment, such as varying weather conditions and seasonal changes, and the need of expert knowledge to properly lay out a set of domain specific features makes it hard to develop a robust and generic solution for the problem. Visual localization can be distinguished according to two main types:metric and topological localization. Metric localizationconsists of computing the coordinates of the location of the observer. Thecoordinates of the vehicle pose are usually obtained by visual odometrymethods <cit.>.Visual metric approaches can provide accurate localization values, butsuffer from drift accumulation as the trajectory length increases. In general, theydo not reach the same accuracy as LiDAR-based localization techniques.Topological localization detects the observer's approximate position from afinite set of possible locations <cit.>. This class of localization methodsprovides coarse localization results, for instance, if a robot is in frontof a specific building or room. Due to its limited state space, topologicalapproaches provide reliable localization results without drift, but only rough position measurements.Accurate, drift-free localization can be obtained by combining both approaches, which is known as topometric localization <cit.>.In this paper, we present a novel topometric localization technique that formulates metric and topological mapping as learning problems via a deep network. The first network computes the relative visual odometry between consecutive images,while the second network estimates the topological location. Bothnetworks are trained independently and their outputs are provided to ourtopometric optimization technique. By fusing the output of the visual odometrynetwork with the predicted topological location, we are able to produce anaccurate estimate that is robust to the trajectory length and has minimum driftaccumulation. An overview of the proposed method is shown in fig:SystemOverview. We introduce a real-world dataset collected from the Freiburg University Campus over the course ofsix month. We compare the proposed approach on the dataset, to state-of-the-art visual odometry methods. The experimentalevaluation shows that the proposed approach yields basically the same accuracy as LiDAR-based localization methods. The dataset will be madepublicly available to simplify comparisons and foster research in visual localization.§ RELATED WORKOne of the seminal deep learning approaches for visual odometry was proposed byKondaet al. <cit.>. They proposed a CNN architecture whichinfers odometry based on classification. A set of prior velocities anddirections are classified through a softmax layer to infer thetransformation between images from a stereo camera. A major drawback of this approach lies in modeling a regression problem as aclassification one which reduces the representational capabilities of the learned model. Other approaches <cit.> tackle the problem ofvisual ddometry as a regression problem. Nicolaiet al. <cit.> proposed a CNN architecturefor depth images based on LiDAR scans. They proposed a simplearchitecture that yields real-time capability.Mohantyet al. <cit.> proposed a Siamese AlexNet <cit.>based approach called DeepVO, where the translation and rotation outputs of the network are regressed through an L2-loss layer with equal weight values. Choosing weight values for the translational and rotational components of the regression output is explored by Melekhovet al. <cit.>. They propose a SiameseAlexNet <cit.>network similar to DeepVO <cit.>, where they add a weight term to balance the translational and rotational errors. They additionally use a spatial pyramid pooling (SPP) layer <cit.> which allows for arbitrary input image resolutions. Our metric localization approach shares similarities with <cit.>, since we use SPP and a loss function whichbalances translation and rotation losses. Our contribution to this problem is a new densely connected architecture along with adifferent angular representation. Another part of our approach is topological localization. With increasing focus on the long-term autonomy of mobile agents inchallenging environments, the need for life-long visual place recognition has become more crucial than before <cit.>. In <cit.>, the authors present an end-to-end approach for large-scale visual place recognition. Their network aggregates mid-level convolutional features extracted from the entire imageinto a compact vector representation using VLAD <cit.>, resulting in a compact and robust image descriptor. Chen et al. <cit.> combine a CNN with spatial and sequential filtering. Using spatio-temporal filtering and spatial continuity checks ensures consecutive first ranked hypotheses to occur in close indices to the query image. In the context of this work, the problem of visual place recognition can beconsidered as a generalized form of topological localization.Similar to the visual place recognition approaches presented above, the authors in <cit.> present a topological localization approach that is robust to seasonal changes using a CNN. They fuse information from convolutional layers at several depths, finally compressing the output into a single feature vector. Image matching is doneby computing the Hamming distance between the feature vectors after binarization, thus improving the speed of the whole approach. Similar to these approaches, we use a CNN architecture that aggregates information from convolutional layers to learn a compactfeature representation. However, instead of using some distance heuristic, the output of our network is a probabilitydistributionover a discretized set of locations. Moreover, whereas previous methods rely on either distance or visual similarity to create the topological set of locations, we introduce a technique that takes both factors into account.Topometric localization was explored by the works of Badinoet al. <cit.> and Mazuranet al.<cit.>. Badino et al. <cit.> proposed one of the first methods totopometric localization. They represent visual featuresusing a topometric map and localize them using a discrete Bayes filter. The topometricrepresentation is a topological map where each node is linked to a pose in theenvironment. Mazuranet al. <cit.> extended the previousmethod to relative topometric localization. They introduced a topometricapproach which does not assume the graph to be the result of an optimizationalgorithm and relaxed the assumption of a globally consistent metric map.While Mazuran et al. <cit.> use LiDAR measurements and Badinoet al. <cit.> relyon a multi-sensory approach, which employs camera, GPS, and an inertial sensor unit, our approach is based on two CNNs for metricand topological estimation and only requires visual data.§ METHODOLOGY This paper proposes a topometric localization method using image sequences from a camera with a deep learning approach. The topometric localizationproblem consists of estimating the robot pose𝐱_t ∈SE(2) and its topological node 𝐧_t ∈SE(2), given a map encoded as a set of globally referenced nodes equipped withsensor readings <cit.>.We propose a deep CNN to estimate the relative motion between two consecutiveimage sequences, which is accumulated across the traversed path to provide avisual odometry solution. In order to reduce the drift often encountered byvisual odometry, we propose a second deep CNN for visual place recognition. Theoutputs of the two networks are fused to an accurate location estimate. In the remainder of this section, we describe the two networks and the fusion approach in detail. §.§ Metric Localization The goal of our metric localization system is to estimate the relative camerapose from images. We design a novel architecture using Dense-blocks <cit.> as base, which given a pair of images in asequence (I_t,I_t-1) will predict a 4-dimensional relative camera pose𝐩_t: 𝐩_t := [Δ𝐱_t^tr, Δ𝐫_t] where Δ𝐱_t^tr:=𝐱^tr_t-𝐱^tr_t-1∈ℝ^2 is the x and y relative translation values andΔ𝐫_t:=[sin(𝐱_t^θ-𝐱_t-1^θ),cos(𝐱_t^θ-𝐱_t-1^θ)] ∈ℝ^2is the relative rotation estimation. We represent the rotation using Euler6 notation, i.e., the rotation angle θ is represented by two components [sin(θ),cos(θ)].§.§.§ Loss Function The proposed metric localization network is designed to regress the relativetranslation and orientation of an input set (I_t,I_t-1). We train the network based onthe Euclidean loss between the estimated vectors and the ground truth.Having a loss function that deals with both the translation and orientation in the samemanner was found inadequate, due to the difference in the scale between them. Instead, we define the following loss function: ℒ := Δ𝐱^tr - Δ𝐱^tr_2 + βΔ𝐫 - Δ𝐫_2,whereΔ𝐱^tr and Δ𝐫 are respectively therelative ground-truth translation and rotation vectors and Δ𝐱^tr andΔ𝐫 theirestimated counterparts. Similar to Kendallet al. <cit.>, we use the parameter β>0 to balancetheloss for the translation and orientation error. §.§.§ Network Architecture To estimate the visual odometry or relative camera pose we propose a Siamesearchitecture built upon dense blocks <cit.> and spatialpyramid pooling (SPP) <cit.>. The direct connections between multiple layers of a dense block yielded state-of-the-artresultson other tasks, and as we show in this paper, also yields excellent performance for the task at hand. fig:VONet showstheproposed VONetarchitecture. The network consist of two parts: time feature representation andregression, respectively. The time representation streams are built upon denseblocks with intermediate transition blocks. Each dense block contains multiple dense layers with direct connections from each layer to all subsequent layers. Consequently, each dense layer receives as input thefeature maps of all preceding layers. A dense layer is composed of four consecutive operations; namely batch normalization (Batch Norm) <cit.>, rectified linear unit (ReLU) <cit.>, a 3×3 convolution (conv) and a drop-out. The structure of the transition layer is very similar to that of a dense layer, with the addition of a 2×2 poolingoperation after the drop-out and a 1×1 convolution instead of the 3×3 one. Furthermore, we alter the first two convolutional operations in the time representation streams by increasing their kernel sizes to 7×7 and 5×5, respectively.These layers serve the purpose of observing larger image areas, thusproviding better motion prediction. We also modifythe dense blocks by replacing ReLUs with exponential linear units (ELUs), which proved to speed up training and provided betterresults <cit.>. Both network streams are identical and learn feature representations to each of the images I_tand I_t-1.We fuse both branches from VONet through concatenation. We also tried a fully connected layer, but extracting and fusing featuresfromconvolutional layers produced better results. The fused features are passed toanother dense block, which is followed by a Spatial Pyramid Pooling (SPP) layer. SPPs are another main building block ofour approach. Using SPP layers has two main advantagesfor the specific task we are tackling. First, it allows the use of the presented architecture with arbitrary image resolutions. The second advantage is the layer's ability to maintain part of the spatial information bypooling within local spatial bins. The final layers of our network are fully connected layers used as a regressor estimating two2-dimensional vectors. §.§ Topological LocalizationGiven a set of images acquired on a path during navigation, the task of topological localization can be considered as the problem of identifying the location among a set of previously visited ones. To this end, we first pre-process the acquired images to create the distinct key-frames, then we train a CNN that learns theprobability distribution over the likelihood of the key-frames given the input image.To extract visually distinct locations from a given path, we cluster poses by computing an image-to-image correlation score, so that similar images are grouped together in one cluster. We select clusters that are within a certain distance threshold 𝑑_𝑡ℎ to represent the distinct key-frames. We chose the value of 𝑑_𝑡ℎso that there is a small visual aliasing between the generated key-frames.Similar to our VONet, we introduce a network architecture based on DenseNet <cit.>, namely LocNet. Our core network consists of four dense blocks with intermediate transition blocks. The proposed architecture differs from the DenseNet architecture by the addition of an extra fully connected layer before the prediction, and the addition of extra connections between thedense blocks fusing information from earlier layers to later ones. Similar to the experiments of Huang et al. <cit.> on ImageNet, we experiment with the differentdepths and growth rates for the proposed architecture, using the same configurations as the ones reported by Huang et al.. fig:LocNetArch illustrates the network architecture for LocNet-121.Given an input image, the network estimates the probabilitydistribution over thediscrete set of locations. §.§ Topometric Localization Our topometric approach aims to refine metric localization given topologicalpriors. The topological localization network provides a set of valuescorresponding to locations and the probability of prediction confidence. Forthis purpose we need to fuse metric and topological network predictions into asingle representation C. The proposed topometric fusion approach isoptimized as below:C := F + B + λ Swhere F is the forward drift correction, B the backward path optimization, and λ >0 the smooth parameter of the S smoothness term. At timet≥0, 𝐱_t ∈SE(2) is the pose and 𝐧_t ∈SE(2) is the matched topological node with probability higher than a thresholdδ. F := 𝐱_t^tr - 𝐧_t^tr_2^2 + | 𝐱_t^θ - 𝐧_t^θ |^2, B := ∑_t-t_w≤τ≤ t-1 e^-λ_tr(t-τ)𝐱_τ^tr - 𝐧_τ^tr_2^2 +e^-λ_θ(t-τ)| 𝐱_τ^θ - 𝐧_τ^θ |^2,S := _α∈ℝ^3∑_0≤τ≤ t𝐱_τ^tr - P_α (τ ) _2^2 W ( τ ),where P_α(τ ) is a quadratic polynomandW(τ ) is the weight function described in<cit.>.(<ref>) presents our forward drift correction approach, which we model in the following manner: Given a high probability topological node matching, we computethe translation and rotation errors and propagate it through the next metriclocalization predictions until another topological node isdetected. Whereas the forward drift correction component is responsible formitigating future error accumulation, the backward and smoothness terms are designed tofurther correct the obtained trajectory. The backward path optimizationis introduced in (<ref>). The backward optimization term worksas follows: Given a confident topological node it calculates its error toit and, using an exponential decay, corrects previous predictions interms of translation and rotation values, until it reaches a predefined timewindow t_w. We also treat the exponential decay functions separately fortranslation and orientation because of their different scale values.The final term, which compromises smoothing the trajectory, is presented in(<ref>). It corresponds to alocal regression approach, similar to the that used by Cleveland <cit.>. Using aquadratic polynomial model with α∈ℝ^3we locally fit asmooth surface to our current trajectory. Onedifference from this term to the others is that such term is only applied totranslation. Rotation is not optimized in this term given the angles arenormalized. We choose smoothing using local regression due to the flexibilityof the technique, which does not require any specification of a function to fitthe model, only requiring a smoothing parameter and the degree of the localpolymonial.§ EXPERIMENTS We evaluated our topometric localization approach on a dataset collected fromFreiburg campus across different seasons. The dataset is split into two parts; RGB data and RGB-D data. We perform a separate evaluation for each the proposedVisual Odometry and Topological Localization networks, as well as thefused approach. The implementation was based on the publicly availableTensorflow learning toolbox <cit.>, and allexperiments were carried out with a system containing an NVIDIA Titan X GPU. §.§ Experimental setup - DatasetIn order to evaluate the performance of the suggested approach, we introduce the Freiburg Localization(FLOC) Dataset, where we use our robotic platform, Obelix <cit.> for the data collection. Obelix is equipped with several sensors, however in this work, we relied onthree laser scanners, a Velodyne HDL-32E scanner, the Bumblebee camera and a vertically mounted SICK scanner. Additionally, we mounted a ZED stereo camera to obtain depth data. As previously mentioned, the dataset was split into two parts; RGB and RGB-D data. We used images from the Bumblebee camera to collect the former, and the ZED camera for the latter.The dataset collection procedure went as follows; we navigate the robot along a chosen path twice, starting and ending at the same location. One run is chosen for training and the other for testing. The procedure is repeated several times for different paths on campus, at different times of the day throughout a period of six months. The collected dataset has ahigh degree of noise incurred by pedestrians and cyclists walking by in different directions rendering it more challenging to estimate the relative change in motion between frames. We use the output of the SLAM system of Obelix as a source of ground-truth information for the traversed trajectory. We select nodes that areat a minimum distance of 1 m away from each other, along with the corresponding camera frame. Each node provides the 3𝐷 position of the robot, and the rotation in the form of a quaternion.As mentioned previously in sec:metricMeth, we opt for representing the rotations in Euler6 notation. Accordingly, we convert the poses obtained from the SLAM output of Obelix to Euler6. Furthermore, we disregard translation motion along the 𝑧-axis, and rotations along the 𝑥- and 𝑦- axes, as they are very unlikely in our setup.For the remainder of this section, we focus our attention on two sequences of the FLOC dataset, namely Seq-1 and Seq-2. Seq-1 is an RGB-D sequence with262 meters of total length captured by the ZED camera, while Seq-2is comprised of a longer trajectory of 446 meters of RGB only data captured by theBumblebee camera. We favored those two sequences from the dataset as they are representativeof the challenges faced by vision-based localization approaches. Seq-1 represents a case where most vision-based localization systems are likely to perform well as the trajectory length is short. Moreover the presence of depth information facilitates the translation estimation from the input images. On the other hand, Seq-2 is more challenging with almost double the trajectory length and no depth information. §.§ Network Training The networks were training on a single stage manner. VONet was trained usingAdam solver <cit.>, with a mini-batch of 2 for 100 epochs. Theinitial learning rate is set to 0.001, and is multiplied by 0.99 every twoepochs. The input resolution image is downscale to 512×384 due to memorylimitations. We adopt the same weight initilization as in<cit.>. The loss function balance variable β is set to10. The training time for VONet for 100 epochs took around 12 hours on a singleGPU.LocNet was trained using Nesterov Momentum <cit.>, with a base learning rate of 0.01 for 10 epochs with a batch size of 10. The learning rate was fixed throughout the evaluation.§.§ Metric LocalizationWe evaluate our VO based metric localization approach over multiple sequencesof FLOC. For these experiments we comparedour approach with Nicolaiet al. <cit.>,DeepVO <cit.> and cnnBspp <cit.>. For eachsequence, two metrics are provided: average translation error and averagerotation error as a function of the sequence length.fig:shortpath shows the computed trajectories of the comparedmethods for Seq-1. tab:shortpath depicts the averagetranslation and rotation error as a function of sequence 1 length. Ourapproach outperforms the compared methods with almost two timessmaller error for translation and rotation inference making it the closest to the ground-truth.Seq-1 shows that our VO approach can achieve state-of-the-artperformance. However the cumulative error characteristic of the problem makes it harder for longer trajectories. fig:longpath presentsresults for Seq-2. For this experimentthe trajectories have a bigger error,especially for translation. tab:longpath quantifies the obtained values, confirming the difficulty ofthis sequence for all tested techniques. The results show that our approach isstill capable of largely outperforming the compared methods, with a translationand rotation error almost twice as low as the other methods.Despite the performance of our approach, it is still far from being competitive with LiDAR based approaches, likethe one used to generate our ground-truth <cit.>. With this goal in mind, we exploit the topological localization method to refine our metricapproach providing an even more precise topometric approach. §.§ Topological Localization In this section, we evaluate the performance of the proposed LocNet architecture. To get an estimate of the suitability of the proposed architecture to the problem at hand, we used the Places2 dataset <cit.> for scene recognition. The dataset contains over ten million scene images divided into 365 classes. We use the pretrained DenseNet model on ImageNet to initialize the weights for our LocNet-121 architecture, as our architecture is quite similar to DenseNet aside from using a different activation function. Using the pretrained model, we are able to achieve comparable performance to the Places365-GoogLeNet architecture as reported by the authors. Additionally, we compare the performance of our LocNet architecture with that of Residual Networks (ResNet) <cit.> given its recent performance in image recognition. We evaluate the performance of both architectures over multiple sequences of the FLOC dataset and the Cambridge Landmarks dataset <cit.>. For both datasets, we report the accuracy in terms of the number of images where the predicted location is within a1 m radius of the ground-truth pose.tab:performanceCampus illustrates the performance results on Seq-1 of the FLOC dataset. We investigate the effect of the depth of the network on the accuracy of the predicted poses, while comparing the number of parameters. The best performance was achieved using LocNet-169 with an accuracy of 90.4% with approximately 3× less parameters than its best performing counterpart in ResNet. tab:performancePoseNet illustrates the performance on the different scenes from theCambridge Landmarks dataset. On this dataset, LocNet-201 achieves the best performance with the exception of King's College scene. It is worth noting that LocNet-169 achieves the second highest accuracy in four out of the five remaining scenes, providing further evidence to the suitability of this architecture to the problem at hand. For the remainder of the experimental evaluation, we use the prediction output from LocNet-201.§.§ Topometric Localization This section presents the results of fusing both topological and metric localization techniques.fig:shortpathTopometric presents both themetric and topometric results for Seq-1. As can be noticed thetrajectory difference between ground truth and our topometric approach isalmost not visually distinguishable. tab:topometric shows animprovement of 7× in the translation inference and superior to 6× for orientation. Such values provide competitive results even to theLiDAR system utilized to provide ground-truth to FLOC. We also evaluated topometric localization using Seq-2.fig:longpathTopometric depicts the obtained results. While the results for this sequence are not as accurate as those of Seq-1, the gainin translation is more than 10× the metric counterpart. Fororientation, even though our metric approach already presents good results, the error isreduced by half using our topometric technique, as shownin tab:topometric.One important characteristic of our topometric approach is that the error is bounded in between consecutive key-frames and does not grow unboundedly over time like with the metric localization method. The presented results show that based on thefrequency of the topological nodes we can expect a maximum cumulative errorbased on the corrected topometric error and not on the pure metric cumulativeerror. § CONCLUSIONS In this paper, we have presented a novel deep learning based topometric localization approach. We have proposed a new Siamese architecture, which we refer to as VONet, to regress the translational and rotational relative motion between twoconsecutive camera images. The output of the proposed network provides the visual odometry information along the traversed path. Additionally, we have discretized the trajectory into a finite set of locations and have trained a convolutional neural network architecture, denoted as LocNet, to learn the probability distribution over the locations. We have proposed a topometric optimization technique that corrects thedrift accumulated in the visual odometry and further corrects the traversed path. We evaluated our approach on the new FreiburgLocalization (FLOC) dataset, which we collected over the course of six months in adverse weather conditions using differentmodalities and which we will provide to the research community. The extensive experimental evaluation shows that our proposedVONet and LocNet architectures surpass current state-of-the-art methods for their respective problem domain. Furthermore, usingthe proposed topometric approach we improve the localization accuracy by one order of magnitude. This work has been partially supported by the European Commission under thegrant numbers H2020-645403-ROBDREAM, ERC-StG-PE7-279401-VideoLearn, the Freiburg Graduate School of Robotics.spmpsci | http://arxiv.org/abs/1706.08775v1 | {
"authors": [
"Gabriel L. Oliveira",
"Noha Radwan",
"Wolfram Burgard",
"Thomas Brox"
],
"categories": [
"cs.CV",
"cs.RO"
],
"primary_category": "cs.CV",
"published": "20170627110331",
"title": "Topometric Localization with Deep Learning"
} |
ieeetr The IoT energy challenge: A software perspective K. Georgiou et al. University of Bristol, UK Universidade Federal do Rio Grande do Norte, Brazil Authors' Instructions The IoT energy challenge: A software perspective Kyriakos Georgiou1, Samuel Xavier-de-Souza2, Kerstin Eder1 December 30, 2023 ============================================================== The Internet of Things (IoT) sparks a whole new world of embedded applications. Most of these applications are based on deeply embedded systems that have to operate on limited or unreliable sources of energy, such as batteries or energy harvesters. Meeting the energy requirements for such applications is a hard challenge, which threatens the future growth of the IoT. Software has the ultimate control over hardware. Therefore, its role is significant in optimizing the energy consumption of a system. Currently, programmers have no feedback on how their software affects the energy consumption of a system. Such feedback can be enabled by energy transparency, a concept that makes a program's energy consumption visible, from hardware to software. This paper discusses the need for energy transparency in software development and emphasizes on how such transparency can be realized to help tackling the IoT energy challenge.§ INTRODUCTION The IoT is no longer just a buzzword in the media; it is becoming a reality. The emergence of IoT led us into a new era of innovation and creativity. This sets high expectations on both the research community and industry for delivering the necessary technological advancements, that will allow for the materialization of new IoT applications.Powering billions of embedded devices deployed into the environment is one of the biggest challenges that IoT faces. Battery-based solutions tend to be impractical and costly due to the need of recharging or replacement. Energy harvesting appears as a viable option for many IoT applications, but it comes with two caveats. Firstly, it is often an unreliable source of energy. Secondly, there is still a large gap between the energy it can deliver and the required energy budget for many IoT applications. For mission critical IoT applications, such as health-care, completing a task before running out of energy budget is vital.Traditionally, hardware innovation has been the safe heaven to achieve sufficiently large savings of energy in Information and Communication Technology (ICT). Similarly, hardware innovation is currently the prominent response to tackle the energy challenge IoT faces. New ultra-low-energy embedded devices were introduced, and existing technologies were customized to create new, more energy efficient versions, such as the Bluetooth Low Energy (BLE). These are well suited for energy-critical applications. But, is that all we can do to tackle this challenge? It is estimated that up to 80% of the total energy consumption of an embedded system is due to software-related activities <cit.>. Inefficient software can drive energy-efficient hardware to waste the system's energy budget. Steve Furber, principal designer of the ARM microprocessor, gave an interview in 2010 <cit.> and stated: “Programmers will not be able to afford to be ignorant about the energy cost of the programs they write ... You need tools that give you feedback and tell you how good your decisions are. Currently the tools don't give you that kind of feedback.” These tools are now needed more than ever to overcome the IoT energy challenge. Programmers have very limited information on how much energy their programs consume, and which parts use the most energy. This has two important implications for the development of IoT applications: * Much guesswork is needed, and thus bad energy-related choices are only identified at a late stage when the system malfunctions due to a failure to meet the system's energy requirements.* The high level of expertise needed and the lack of energy-aware development toolssignificantly reduce the number of embedded developers who are able to deliver energy-critical systems. Because of this, the whole process of developing energy-constrained applications becomes difficult and costly. There is a need for tools that expose the software's effect on the energy consumption of a system. Such energy transparency will allow programmers, toolchains and runtime systems to make energy-aware decisions in order to meet the strict energy constraints of the IoT. § ENABLING ENERGY TRANSPARENCY IS DIFFICULTVarious layers have been introduced in the system stack, abstracting away complex details to make programming easier. This prevents software developers from understanding the impact of their coding choices on the way hardware is utilized at runtime. The task of enabling the required energy transparency can be seen as equivalent to the task of reverse engineering the code transformations that take place between and at each software abstraction layer. This will allow the propagation of resource usage information from hardware to software. Such a reverse engineering process is a hard challenge and highly dependent on the architecture and compiler choice. Instead, novel energy transparency techniques are needed which can approximate the energy consumption of a program at different software abstraction levels, without the need for reverse engineering. For these techniques to enable energy-aware software development, the following requirements must be met:* Both the actual energy consumption and bounds must be provided: Energy consumption bounds will guide developers in meeting strict energy budget requirements. Actual energy consumption estimates are necessary to serve as a benchmark for potential energy optimizations.* Energy transparency at multiple levels of software abstraction: Software developers and toolchains can significantly influence the energy consumption of a program mainly at three software abstraction levels; the source code, the compiler's Intermediate Representation (IR), and the Instruction Set Architecture (ISA).* Fine-grained energy characterization of software: Identifying energy hot-spotsrequires the ability to attribute the energy consumption estimates to the various basic software components, such as Control Flow Graph (CFG) basic blocks. * Target and programming language independence: Target and programming language agnostic techniques must be provided in a common framework to enable practical and cost effective energy-aware development for a large number of embedded architectures.* The multi-threaded and multi-core case must be considered and explored: Novel multi-core, multi-threaded embedded architectures emerged over the last decade, driven by the increasing demand for more computing power. As this trend is expected to grow in the future, it is important to consider such architectures. Also, they must be explored for parallel codes because they offer potentially large energy savings when the number of cores increases and the cores' voltage and frequency decrease. * Enable design space exploration: Developers and toolchains need to apply multi-objective optimizations to find the optimum balance between the available resources, such as execution time, energy, code size, number of cores and threads used. To enable this, energy transparency techniques have to provide sufficiently accurate feedback on the effect that each different configuration has on the resources of interest.* Fast and easy to deploy: Energy consumption estimation speed is critical to the iterative software optimization process. Furthermore, energy transparency techniques should be easy to deploy and use. § EXISTING APPROACHES AND LIMITATIONS This section examines the state of the art of energy transparency techniques and their limitations in regards of the above requirements. §.§ Measuring energy consumptionWhile physical measurements are potentially most accurate to determine the energy consumption of a program, they fail to meet many of the requirements set in <Ref>. Firstly, measurements typically require sophisticated equipment and hardware knowledge that most software developers lack. Secondly, most hardware components have no provisions for energy measurements; thus custom modifications are needed to probe their power supply. This makes physical measurements difficult to deploy and use. Furthermore, extracting energy consumption bounds with end-to-end measurements is inappropriate in most cases, as the whole input space would need to be exhaustively searched. Finally, fine-grained energy characterization of software can be challenging. Usually, this requires expensive measuring equipment such as oscilloscopes, that can support a high sampling frequency. §.§ Estimating energy consumptionEstimating the energy consumption of a program for a particular hardware platform requires two main elements: an analysis technique and a way to convey energy information to the analysis. The latter is typically done via energy modeling. An energy model statically captures the dynamic behavior of a processor in regards to its energy consumption characteristics; for example, it can associate energy costs with atomic units in a program, such as ISA instructions, or to various events, such as a cache miss.To perform energy estimation at a software level of abstraction, an energy model is required at the same level. An energy analysis technique automatically inherits any precision loss existing in the energy model that the technique utilizes. <Ref> demonstrates the trade-off between the energy consumption estimation accuracy and the level of software abstraction.Modeling at a lower level is always more accurate as it is closer to the hardware, where the actual power dissipation occurs. On the other hand, when moving to higher levels of abstraction, the amount of program information, such as types and loop structures, increases. Such information can be crucial for static code analysis and optimization. However, when moving from source code to the ISA level, much of this information is lost due to the various transformation and optimization passes that occur at the respective levels of the software stack.In the next sections, the state of the art of energy consumption estimation techniques is examined.§.§.§ Profiling-based energy consumption estimation In this case, estimation is performed by collecting execution statistics and utilizing them with an appropriate energy model. Such a model needs to provide energy information for the various entities that occur in the execution statistics. Three main techniques are used to collect execution statistics for energy consumption estimation:Simulation: Typically performed at low hardware design levels, such as theRegister-Transfer Level (RTL) <cit.>, thus it is difficult to achieve fine-grained energy consumption attribution to the various software components. Moreover, modeling and profiling at such low levels is impractical for most commercial embedded processors since essential circuit information, such as the effective capacitance of major architectural blocks, is not available.To account for these issues, simulation-based energy estimation has been performed at the ISA level <cit.>. Energy models at this level can be constructed for deeply embedded commercial processors by treating the hardware as a black box <cit.>. ISA energy models are less accurate than lower-level models, but ISA simulation is considerably faster than hardware simulation and allows for fine-grained energy characterization of software. Energy modeling and profiling at the ISA level are insufficient for more complex architectures or system-level energy consumption estimation. This is because it is difficult to statically capture the behavior of performance-enhancing hardware components, such as caches.Generally, simulation-based energy consumption estimation tends to be slow. This makes it difficult for the technique to be incorporated in software development tools, were instant feedback is required for an iterative process of optimizing energy consumption. Code Instrumentation: This is performed by instrumenting the code with instructions that extract execution statistics at runtime. The main challenge is to extract the statistics out of the hardware and to minimize the overhead of the instrumentation that can significantly impact the estimation's accuracy. Therefore, the amount of execution statistics collected are typically less compared to simulation. This makes the retrieved energy consumption estimations less accurate than simulation-based estimations. A major advantage of this method is that it is significantly faster than simulation-based energy estimation. Recent work <cit.> demonstrated a new profiling technique that collects execution statistics at the compiler's IR level. This was combined with a dynamic mapping technique thatlifts an ISA energy model to the compiler's IR, to retrieve energy estimations at the IR level. The technique guarantees no energy overhead in the estimations due to instrumentation code, and achieved an average accuracy of 2.5%.Performance Monitoring Counters (PMC): Statistical PMC-based estimation is preferable for more complex architectures were ISA-level modeling and analysis is insufficient to capture their complexity. PMC execution statistics can be used to construct energy models and estimate the energy consumption of multi-threaded/core architectures <cit.>. Run-time power estimation can be enabled using PMC <cit.>. This allows for energy-aware decisions to be made at runtime.For a given processor, energy modeling and profiling using PMCs can be challenging due to the limited types of the events that can be monitored and the restricted number of counters that can be sampled simultaneously. The same constraints apply when trying to port the PMC-based modeling and estimation techniques to a new target. Moreover, runtime use of such PMC-based energy estimation methods could cause a significant overhead on the system's performance <cit.>.§.§.§ SRA-based energy consumption estimation Similar to physical measurements, it is impractical to capture energy consumption bounds using profile-based estimation. Static Resource Analysis (SRA) offers a better alternative. The following two techniques have been used for statically estimating energy consumption bounds:Automatic Complexity Analysis: This has been used to create cost relations that capture energy in terms of program-input size at both the ISA and the compiler's IR levels <cit.>. The technique can be fully automatic and programming-language independent. The main drawback is the difficulty to extract a closed-form solution for the cost relations of a program <cit.>. Therefore, the approach does not scale well to large programs with complex structure.Implicit Path Enumeration Technique (IPET): This is the most popular method for Worst Case Execution Time (WCET) analysis <cit.>. In <cit.>, the technique was used to extract energy consumption bounds on a simulated processor. More recently <cit.>, IPET was applied at the ISA level of a multi-threaded/core embedded architecture, using an ISA energy model, and at the compiler's IR level, using a mapping technique that lifts ISA energy models to the compiler's IR level. The authors also demonstrated how the technique can be used for design space exploration for two concurrency patterns: task-farms and pipelined programs. Like WCET estimation, SRA-based energy estimation works best with predictable architectures and software. Using, SRA to analyze multi-threaded programs with complex communication patterns is a hard challenge. Furthermore, currently, there is no practical method to perform average-case static analysis <cit.>. § OUTSTANDING CHALLENGES Techniques that are based on energy modeling, code-instrumentation and SRA avoid the need for difficult to deploy physical energy measurements and expensive simulations. Moreover, they can provide fine-grained energy characterization of software at multiple levels of abstraction with good precision. A combination of SRA and code-instrumentation techniques provides both the bounds and the actual energy consumption. Furthermore, new energy estimation techniques, such as those presented in <cit.>, are compiler and architecture agnostic, provided an ISA energy model exists. Therefore, such techniques can be relatively easily integrated into development toolchains to provide feedback-directed energy optimization. However, there is still a number of outstanding challenges that need to be addressed. Currently there is no practical solution to provide tight upper energy consumption bounds. SRA approaches combined with worst case energy models can lead to significant overestimation <cit.>. Symbolic simulation at the RTL could retrieve tighter bounds, <cit.>, but such approaches require sensitive architectural information, typically not available for commercial processors. To retrieve tight energy bounds at the ISA level, data-sensitive energy models and static analysis would be required. These have to take into account the inter-instruction effects, caused by the operand values used for each instruction, and identify the worst case data input. Recent work demonstrated that finding the data that will trigger the worst case energy consumption is an NP-hard problem and that no practical method can approximate (within reasonable time in general) tight energy consumption upper bounds within any level of confidence <cit.>.Energy models are typically characterized while using a constant power source. This is an ideal condition, as there are no significant variations in the power supplied to the processor. This is not an issue when the energy estimations retrieved are used to optimize the energy consumption at development time. However, IoT applications typically run on a battery or on an energy harvester. To be able to use the energy estimations for making real-time decisions, the power profile of the power source for a specific application has to be taken into consideration by both the energy modeling and the energy consumption estimation techniques.Activities external to the processor, such as sensing and communicating, can consume significantly more energy than the computation part of an embedded system. Computation usually controls such activity. Therefore, the energy usage of peripherals and I/O operations can be profiled and included as part of the energy cost of the computation that triggers them. This will allow existing energy estimation techniques to provide system-wide energy estimates. Such an approach is more feasible for systems with predictable behavior.While existing SRA-based estimation techniques can handle the complexity of deeply embedded architectures, they generally do not scale well to new multi-core/threaded architectures. Concurrent software introduces complexities over traditional sequential variants, which SRA is inherently limited to cope with. The task is even harder than estimating WCET since any computation contributes to the worst-case energy scenario, while, for WCET analysis, only the computation that causes the worst case needs to be considered. Therefore, SRA support for multi-threaded and multi-core software is currently limited to a range of simpler concurrent software patterns. Both hardware and toolchain vendors need to work closely together to provide novel architectures and methods that will allow energy transparency by design.The inter-thread core activity must be considered, to enable the interplay between performance and energy on many-, multi-core and multi-threaded architectures that support voltage and frequency scaling. Finding optimal configurations based on runtime execution statistics and a power model for the architecture is more practical, rather than using simulation. This is because simulation is typically several orders of magnitude slower than hardware execution and thus difficult to support an interactive optimization software development process. Furthermore, an energy modeling that accounts for voltage and frequency scaling and parallelism can answer optimization questions such as how many cores and at what frequency they should run for a given performance target<cit.>. Hardware vendors should provide more support to enable such techniques; for example more PMCs. Finally, the combination of both static and runtime analysis must be explored to realize energy savings beyond thosethancan be achieved when these approaches are used independently.Perhaps, the biggest challenge of enabling energy-aware software development lies in the tool-vendor's and programmer's perception. Traditionally, optimizing energy consumption has been treated as a side effect of improving execution time. Toolchains have been long focused on improving execution time, and developers are left assured that a compiler will do the best possible job on optimizing their code for both time and energy. The IoT energy challenge is an opportunity to start treating energy consumption as a first class citizen while developing software. § CONCLUSION Hardware offers energy saving capabilities for the software to exploit. The responsibility lies on the system engineer to program and configure the selected device in the most energy efficient way for the task at hand. Development toolchains need to be enhanced with energy transparency that provides the necessary feedback to enable energy-aware software development. Hardware vendors need to provide architectures and information, such as energy models, that will support energy transparency. Finally, tool vendors and programmers need to take a more active role in delivering energy efficient systems. We can't keep ignoring the software role in the energy consumption of a system. | http://arxiv.org/abs/1706.08817v1 | {
"authors": [
"Kyriakos Georgiou",
"Samuel Xavier-de-Souza",
"Kerstin Eder"
],
"categories": [
"cs.SE"
],
"primary_category": "cs.SE",
"published": "20170627123830",
"title": "The IoT energy challenge: A software perspective"
} |
1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany Institut za fiziku - P.O. Box 304, HR-10001 Zagreb, Croatia1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, GermanyInstitut za fiziku - P.O. Box 304, HR-10001 Zagreb, CroatiaInstitut za fiziku - P.O. Box 304, HR-10001 Zagreb, Croatia Division of Materials Research, National Science Foundation, Arlington, VA 22230, and Material Science Division, Argonne National Laboratory, Argonne, Illinois 60439-4831, U.S.A.Institute of Problems of Chemical Physics, Russian Academy of Sciences, RU-142 432 Chernogolovka, Moscow oblast, RussiaInstitute of Problems of Chemical Physics, Russian Academy of Sciences, RU-142 432 Chernogolovka, Moscow oblast, Russia1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany The organic charge-transfer salt κ-(BEDT-TTF)_2Hg(SCN)_2Br is a quasi two-dimensional metal with a half-filled conduction band at ambient conditions. When cooled below T=80 K it undergoes a pronounced transition to an insulating phase where the resistivity increases many orders of magnitude. In order to elucidate the nature of this metal-insulator transition we have performed comprehensive transport, dielectric and optical investigations. The findings are compared with other dimerized κ-(BEDT-TTF) salts, in particular the Cl-analogue, where a charge-order transition takes place at T_ CO=30 K.71.30.+h,75.25.Dk, 74.70.Kn,78.30.JwMetal-Insulator Transition in the Dimerized Organic Conductor κ-(BEDT-TTF)_2Hg(SCN)_2Br Martin Dressel December 30, 2023 =======================================================================================§ INTRODUCTIONThe current interest in dimerized two-dimensional organic conductors mainly focusses on κ-(BEDT-TTF)_2X salts built with extended polymeric anions X^- such as Cu(SCN)_2^-, Cu[N(CN)_2]Br^-, Cu[N(CN)_2]Cl^-, or Cu_2(CN)_3^-, all containing copper ions. These systems are characterized by half-filled conduction bands and are located in proximity to correlated insulating states. Depending on the strength of the on-site Coulomb repulsion U with respect to the bandwidth W, a metallic and superconducting ground state develops at low temperatures, or a Mott insulator, which might be antiferromagnetically ordered or behaves like a spin liquid due to strong frustrations.<cit.>Subsequent theoretical approaches <cit.> suggest that also the Coulomb interaction V between the dimers has to be included in the description of possible charge-ordering phenomena in these dimerized (BEDT-TTF) salts. Although numerous experimental results,<cit.> rule out appreciable charge disproportionation in ,and other copper-based κ-(BEDT-TTF)_2X salts, these considerations might be relevant for dimerized salts in general. Unfortunately, charge-order phenomena in κ-phase salts with an effectively half-filled band are scarce and only a few compounds are reported exhibiting charge order, such as κ-(BEDT-TTF)_4PtCl_6·C_6H_5CN, the triclinic κ-(BEDT-TTF)_4[M(CN)_6][N(C_2H_5)_4]·3H_2O and the monoclinic κ-(BEDT-TTF)_4[M(CN)_6][N(C_2H_5)_4]·2H_2O (with M =Co^ III, Fe^ III, and Cr^ III) salts.<cit.> Here the structure is rather complex as the phase transition includes the deformation of the molecule and the interaction with the anions; accordingly, details of their physical properties and their electronic states are not well known. Certainly electronic correlations as well as coupling to the lattice are important. Recently it was demonstrated thatundergoes a pronounced charge order transition at T_ CO=30 K,<cit.> as illustrated in Fig. <ref>. The insulating state is rapidly suppressed by hydrostatic pressure of p=4.2 kbar without any signs of superconductivity;<cit.> making this compound distinct from , where a tiny amount of pressure drives the Mott insulator superconducting, but also from the canonical charge-ordered system α-(BEDT-TTF)_2I_3.<cit.> The present investigation now addresses the question in which way κ-(BEDT-TTF)_2Hg(SCN)_2X is modified when replacing X = Cl by Br in the anion sheet. Let us recall that in the case ofthe corresponding substitution pushes the antiferromagnetic Mott insulator into a superconducting phase with T_c ≈ 12 K.<cit.> The family of κ-(BEDT-TTF)_2X salts with mercury-based anions have a structure similar to the Cu-salts,<cit.> but different ratios of transfer integrals, which result in modified parameters used to map them on the Hubbard model.<cit.> In particular the orbital overlap t_d within the dimers of the Hg-compounds is weaker compared to the the Cu-family, causing a reducedon-site repulsion U with respect to the inter-dimer interaction. This makes it necessary to go beyondthe simple Hubbard model, which allowed to treat the Cu-based κ-(BEDT-TTF) compounds successfully as half-filled system with one electron per dimer.<cit.> Now the description has to start from individual BEDT-TTF molecules, leading to a quarter-filled conduction band; in addition to U the inter-molecular interaction V has to be included in the extended Hubbard model.In an effective dipolar-spin model Hotta suggests <cit.> that quantum electric dipoles are formed on the dimers, which interact with each other and thus modify the exchange coupling. Since they fluctuate by t_d, for large orbital overlap the dimer Mott insulator is stable, forming a dipolar liquid; if V is large compared to t_d, however, charge order emerges (dipolar solid). Similar considerations have been put forward by other groups.<cit.> Mazumdar, Clay and collaborators <cit.> could show that in these system a frustration-induced transition occurs from a Néel antiferromagnetism to a spin-singlet state in the interacting quarter-filled band on an anisotropic triangular lattice. In the spin-singlet state the charge on the molecules becomes unequal: a paired-electron crystal is formed with pairs of charge-rich sites separated by pairs of charge-poor sites. This spin-singlet formation is driven by quantum effects at any value of nearest neighbor interaction V.Here we present the first comprehensive transport, dielectric and optical investigations of and discuss the findings within the framework of dimerized Mott insulators composed by these Hg- and Cu-families. § EXPERIMENTAL DETAILSAseries of κ-(BEDT-TTF)_2Hg(SCN)_3-nX_n salts with X = F, Br, I (n = 1) and X = Cl (n = 1, 2) has been synthesized by Lyubovskii and collaborators already twenty years ago,<cit.> ranging from insulators to metals and possible superconductors;<cit.> a thorough investigation of their physical properties, however, is still lacking. Here we have turned our attention to , an isostructural sister compound of the charge-ordering ; as demonstrated in Fig. <ref>, it also exhibits a metal-insulator transition upon cooling, but at somewhat higher temperatures.Based an recent x-ray scattering experiments at ambient conditions, the crystal structure ofis depicted in Fig. <ref> along two crystallographic directions. The BEDT-TTF molecules are all crystallographically equivalent and within the bc-plane form dimers according to the κ-pattern<cit.> that are rotated with respect to each other. As seen from Fig. <ref>(b) the stacking direction b is more pronounced than known from other κ-salts leading to a rather isotropic response in our transport and optical measurements. It should be noted that during the electrochemical synthesis, single crystals of the β^''-phase tend to grow as well to a considerable size (1–2 mm),<cit.> and subsequently have to be separated from the desired κ-phase specimens with the help of x-ray or infrared optical inspection.In the temperature-dependent dc resistivityofdisplayed in Fig. <ref> a sharp metal-insulator transition is identified at T=80 K, confirming the early characterization.<cit.> While fora saturation of ρ(T) is observed at very low temperatures, for the Br-analogue the slope | dρ(T)/ dT| continues to rise as the temperature is lowered. For a more complete study we have measured the charge transport within the highly-conducting (bc) plane, as well as perpendicular to it, using the standard four-probe technique. The sample was slowly cooled (up to 0.5 K/h) from room temperature down to T=4 K using a home-made helium bath cryostat.In addition, the spectra of the complex dielectric function were obtained from measuring the two-contact complex conductance in the frequency range from 20 Hz to 10 MHz for various temperatures. The Hewlett-Packard 4284A LCR meter and Agilent 4294A impedance analyzer with virtual ground method were used. The capacitive open-loop contribution of the sample holder was always subtracted.The optical properties ofhave been investigated by infrared reflectivity measurements from room temperature down to T=4 K for both polarizations along the main axes of the highly-conducting (bc) plane. To this end a Bruker Hyperion infrared microscope was attached to the Fourier-transform spectrometer Bruker IFS 66v/s or Vertex 80v. The sample was cooled down to helium temperatures by a Cryovac Microstat cold-finger cryostat. The far-infrared data were taken by a Bruker IFS 113v equipped with a cold-finger cyostat and in-situ gold evaporation as reference.<cit.> In order to perform the Kramers-Kronig analysis, a constant reflectivity extrapolation was used at low frequencies and temperatures T_ MI<80 K, while a Hagen-Rubens behavior was assumed for elevated temperatures.<cit.> In addition, the vibrational features were measured perpendicular to the plane (E∥ a) using an infrared microscope. In particular the ungerade C=C vibration ν_27( b_1u) of the BEDT-TTF molecules are utilized as a local probe to determine the charge per molecule <cit.> and follow any possible charge disproportionation. § RESULTS AND ANALYSIS§.§ Transport Properties In order to gain more information on the transport mechanisms ofabove and below the phase transition at T_ MI = 80 K, we have performed dc resistivity measurements along different crystal directions and present the resultsdown to T=20 K in an Arrhenius plotin Fig. <ref>(a) as a function of inverse temperature. At ambient conditions, the resistivity within the (bc)-plane is approximately 0.5 Ωcm with an anisotropy of less than a factor of 2. Perpendicular to the highly-conducting plane, i.e., parallel to the a-axis, ρ_ dc is about 20 times higher. For all orientations a very similar temperature dependent metallic resistivity is observed down to 100 K. At T_ MI a metal-insulator transition occurs in all three directions with a more or less steep rise in ρ(T) by an order of magnitude or more. It is interesting to note that this temperature-behavior very much resembles the metal-insulator transition observed in other ET-based compounds such as α-(BEDT-TTF)_2I_3 and θ-(BEDT-TTF)_2RbZn(SCN)_4,<cit.> where the gap at the density of states rapidly opens at the transition. The insulating state is characterized by an activated behavior ρ(T)∝exp{E_g/2k_BT} down to lowest temperatures; no saturation is observed up to our limit of 10^9 Ωcm. Within the plane (E∥ bc) the activation energy of E_g/2=(22± 3) meV is basically constant down to T= 20 K. In the out-of-plane direction (E∥ a), we extract a similar activation energy of E_g/2=(15.5± 2) meV, which means it is rather isotropic. If we consider E_g as the full gap in the density of states, the ratio E_g/k_B T_ MI≈ 4-5.6 is not much higher than 3.53 expected from a conventional mean-field transition. Thus we can consider this system mean-field-like with relatively weak electronic correlations. §.§ Dielectric Properties Like in many other charge-ordered (BEDT-TTF) compounds,<cit.> the dielectric properties ofcan be described by a generalized Debye expressionε(ω) - ε_ HF = Δε/1 + ( iωτ_0)^1-αat low temperatures, where Δε= ε_0 - ε_ HF. corresponds to the strength of the dielectric mode;ε_0 and ε_ HF are the static and high-frequency dielectric constants, respectively; τ_0 is the mean relaxation time; and 1-α is the symmetric broadening of the relaxation time distribution. The temperature dependences of the extracted parameters Δε, 1-α, and τ_0 are plotted in Fig. <ref> as a function of inverse temperature 1/T. The dielectric relaxation can be detected only at temperatures below 60 K. Above T≈ 35 K we can determine only the dielectric relaxation strength by measuring the capacitance at 1 MHz. At first glance the relaxation appears rather broad. The dielectric strength perpendicular to planes (E ∥a) is only of the order of 10 and less while the strength of the dielectric response E ∥ bc is on the order of 100 (not shown), which mirrors the similarly weak anisotropy found in dc data. Most important, for both orientations the intensity of the modes becomes smaller by about an order of magnitude as the temperature is reduced to 10 K; these findings are in contrast to the behavior commonly observed in low-dimensional organic charge transfer salts.<cit.> It clearly indicates thatdoes not exhibit indications of charge order and shows only very weak relaxor-like properties, further underlining the absence of strong electronic correlations.For both orientations, the temperature behavior of τ_0(T) shows no saturation and basically follows an Arrhenius law between 40 and 12 K, as plotted in Fig. <ref>(c). The activation energy is comparable with the dc resistivity. This indicates that electronic screening by quasi-free charge carriers is the dominant relaxation mechanism. Around T≈ 20 - 30 K we find the most pronounced change in the dielectric behavior that can be seen even better by directly looking at the temperature dependence of the dielectric constant. In Fig. <ref>, we present ε'(T) measured at different frequencies for the orientations parallel and perpendicular to the (bc)-plane. A clear step can be seen around T=20 K that becomes more pronounced as the frequency increases to 1 MHz. This is accompanied by a change in 1-α and Δε at the same temperatures in Fig. <ref>, which suggests the entities responsible for the dielectric response are not correlated as temperature is lowered.§.§ Infrared-active Molecular VibrationsMicroscopic information on the charge distribution and its temperature behavior can be obtained by optical spectroscopy. To this end we performed infrared reflectance measurements for the polarization perpendicular to the layers, i.e., E∥ a. In this orientation the optical response resembles an insulator with low reflectivity and correspondingly low values of conductivity, in agreement with the quasi-two-dimensional character of these materials, deduced from the dc conductivity presented in Fig. <ref>.Of primary interest, however, are the superimposed infrared-active vibrational features of the BEDT-TTF molecule and lattice. The out-of-phase molecular vibrations of the two C=C bonds of the inner rings provide a powerful and well-established method to identify the amount of charge on molecular lattice sites.<cit.> The frequency dependence of the ν_27( b_1u) mode allows for a quantitative determination of the charge per BEDT-TTF molecule and thus the most accurate way to monitor the temperature-dependence of the charge disproportionation. The resulting conductivity spectra in the region of the ν_27( b_1u) mode are presented in Fig. <ref> forin comparison with spectra of the Cl-analogue.is known <cit.> to undergo a charge-order transition aroundT_ CO=30 K, as can be clearly seen from the splitting of the ν_27( b_1u) vibration, which peaks around1455at room temperature and narrows on cooling down. A higher-frequency shoulder can be identified that becomes more pronounced at low temperatures; this double features indicates the two crystallographically different sites per unit cell. For a quantitative analysis we fit the bands by two Fano resonances for T>30 K and four modes below T_ CO; the results are plotted in the inset and perfectly agree with our previous observations <cit.> where we concluded a charge difference of 2δ_ρ = 0.2e between two different molecular sites. Löhle et al. studied the pressure dependence of the charge ordered phase <cit.> and found a rapid suppression of T_ CO by about 0.7 kbar and the absence of charge disproportionation for p>4 kbar.For the present case ofnothing like that is observed in the lower panel of Fig. <ref>. At room temperature the broad ν_27( b_1u) feature occurs at 1457 with a comparable width and temperature dependence down to T≈ 100 K. No drastic change is observed when the sample is cooled further below T_ MI, in a way similar to Cu-based κ-BEDT-TTF_2X systems.<cit.> At the lowest temperature, the double peak structure around 1460has slightly changed in so far as the higher-frequency peak is more pronounced. In addition, a second feature develops at 1445 at below temperatures as high as 150 K, meaning it is not related to the metal-insulator transition. Even though it is tempting to ascribe its temperature evolution to the establishing of the low-temperature phase, it appears to be governed by thermal broadening: the width and strength behave in a manner similar to the dominant 1460peak as well as the normal-phase ν_27( b_1u) peak of theanalogue.It is interesting to note that the ν_27 mode observed inis much broader than in the Cl-analogue and comparable to the ones found in the spin-liquid candidatesand ,<cit.> where charge inhomogeneities or fluctuations might be of relevance.<cit.> These findings provide evidence that the metal-insulator transition observed inat 80 K is not due to static charge ordering. Recent measurements of the Raman spectra as a function of temperature support this conclusion: no indications of charge imbalance has been observed <cit.> as the metal-insulator transition is passed.§.§ In-Plane Infrared Properties The electrodynamic properties ofwere investigated by measuring the reflectivity at different temperatures for4K≤ T < 300 K; in Fig. <ref> we plot R(ω)for the two polarizations E∥ c and E∥ b together with the optical conductivity obtained by the Kramers-Kronig relations. The quasi-two-dimensional conductor displays a rather weak anisotropy in its optical response, very similar to its Cl-analogueand to κ-(BEDT-TTF) salts with Cu-containing anions. From the crystal structure [Fig. <ref>(b)] we can identify the b-axis as preferred stacking direction of the BEDT-TTF molecules.Most obvious, the mid-infrared band is more intense along the b-direction [Fig. <ref>(c,d)], leading to a drop in R(ω) around 3500that resembles the plasma edge of regular metals. We associate this optical anisotropy to the arrangement of the BEDT-TTF dimers as depicted in Fig. <ref>(b): the intra-dimer charge transfer is in particular excited by E∥ b. The low-frequency reflectivity is rather small, between 0.4 and 0.6. Only when cooled down to T=100 K a pronounced Hagen-Rubens behavior is measured. This corresponds to the increase in conductivity by a factor of 3 to 4 (see insets of Fig. <ref>), in good agreement with the metallic temperature dependence seen in dc resistivity (Figs. <ref> and <ref>) above the metal-to-insulator transition.From the mid-infrared conduction band we can extract the electronic parameters,<cit.> in particular the maximum ν_ max is proportional to the Coulomb repulsion U. At T=100 K we find approximately 2200for E∥ c, while for E∥ b it is only 1950 . In both cases the values increase by about 10% as the temperature rises to room temperature. The overall behavior is similar to the observations reported for , however, the absolute values are reduced, indicating smaller effect of electronic correlations.We describe the zero-frequency contributions by a Drude-like conductivity. In accord to the dc data, the spectral weight ∫σ(ω) dω = ω_p/8 obtained for both directions differs by a factor of 2, independent on temperature. For most-conducting direction (E∥ b) the plasma frequency ω_p/(2π c) = 1500at ambient temperature increases to 4000at T=100 K. Here the width increases by a slightly more than a factor of 2 when the temperature is reduced to T_ MI. Within the uncertainty, no change is observed along the c-direction.Forwe observe a larger spectral weight of the Drude component compared to the one extracted for the Cl-analogue, corroborating our conclusion drawn above from the mid-infrared band, that the title compound is less-correlated than . A similar order was observed for the copper-based salts, asis a Mott insulator while the Br-analogue becomes superconducting.For T<T_ MI the low-frequency reflectivity drops significantly in both polarizations. Although strong phonon bands around 200 - 450cause considerable intensity in σ_1(ω) even at T=4 K,we might identify an isotropic gap around 500for both directions. Within experimental errors, this value agrees nicely with the transport gap extracted from dc data. Additionally, we note that frequencies up to 1000the optical conductivity decreases when cooling from 70 to 4 K which indicates a sort of soft gap behavior. Accordingly the spectral weight is shifted to the mid-infrared band, which grow considerably. § DISCUSSIONBy now it is well established thatundergoes a metal-insulator transition due to charge ordering at T_ CO=30 K: the resistivity ρ_ dc(T) continuously rises by orders of magnitude (Fig. <ref>) and the charge-sensitive vibrational ν_27( b_1u)-mode splits (Fig. <ref>) indicating a charge disproportionation 2δ_ρ=0.2e at low temperatures. In contrast, no clear splitting of the ν_27( b_1u)-mode is observed in the sibling , ruling out any significant charge disproportionation. Hence,does not enter a charge-ordered ground state.Applying minor pressure, the Mott insulator with Cu-based anionsturns metallic and superconducting, very much likeat ambient pressure. In the case of the Hg-basedthe relation to the Br-analogue is rather different: pressure rapidly suppresses the charge-ordered state in , which becomes metallic at all temperatures.<cit.> This is in striking contrast towhich shows no charge ordering but still features a metal-insulator transition of a different kind. It has been suggested <cit.> that charge fluctuations within the dimers may couple to neighboring entities; and in particular in the frustrated case of a triangular lattice it may give rise to the spin liquid state. Pressure enhances the interdimer interaction and may explain whyexhibits a strong and static charge order, while only charge fluctuations may be present in .For the organic Mott insulatorsandthe temperature evolution of dielectric spectra resembles relaxor ferroelectrics.<cit.> In the case ofthe behavior is similar to other charge-ordered compounds, such as α-(BEDT-TTF)_2I_3,<cit.> where two dielectric relaxation modes appeared in the kHz-MHz frequency range. Nothing like that however is observed in , supporting our conclusion that the compound is not a charge-order insulator, but also that it is not clear whether the disorder in the molecular and anion layers is an issue here. The temperature-dependence of the dielectric parameters indicate some relaxational behavior screened by the conduction electrons remaining below the phase transition. It is 30 times more pronounced within the (bc)-plane compared to the perpendicular direction. Below 30 K the mode vanishes rapidly and is basically absent at T=10 K, although in this temperature range no additional transition can be identified in the transport or optical properties. It might be tempting to attribute the dielectric response to dipolar fluctuations,<cit.> forming a kind of non-Barrett quantum electric-dipole liquid.<cit.> Since pure dipolar coupling does not cause frustration in a two-dimensional triangular lattice, the transfer integrals are significant and do increase when going fromto the Cl-analogue.The low-temperature optical conductivity ofexhibits a shape similar to the room-temperature spectra of other κ-(BEDT-TTF) salts, such as , with a gap at low frequencies, strong electron-molecular vibrational (emv) coupled features, and a band at about 2000 . The electronic properties of these compounds can be well described by assuming a lattice of dimers with one electron per site. The simple Hubbard model of a half-filled system describes the main physics rather well. In comparison toand , the effect of electronic correlations seems to be less important in . This is concluded from the mean-field-like dc transport and maximum of the mid-infrared conductivity band, but also from the spectral weight of the Drude component (Fig. <ref>). The zero-frequency peak is higher than typically observed in the Cu-family and it shows up at high temperatures, as a result of smaller Hubbard on-site repulsion U.<cit.>With the metal-insulator transition prominently at 80 K, we do not observe a gradual increase in resistivity of(Fig. <ref>) as known from the dimerized Mott insulators κ-(BEDT-TTF)_2X with Cu-ions in the anion layers, such asor .<cit.> This leads us to conclude that the T=80 K transition is not a regular Mott transition. This is supported by the analysis of our optical data and structural considerations. The rapid change in resistivity must therefore have other reasons, most likely some subtle magnetic ordering. In this regard it is of interest to recall ESR investigations by Yudanova et al., who measured hydrogenated and deuteratedanddown to low temperature.<cit.> After subtracting a Curie contribution, the spin susceptibility vanishes at the metal-insulator transition, following χ(T)=C/T exp{-Δ/T}, with Δ≈ 500 K which agrees nicely with the charge gap extracted from our optical and dc transport data. They conclude a first-order structural phase transition with a considerable localization of the electrons on the BEDT-TTF molecules. Recent low-temperature x-ray diffraction studies, however, did not reveal any evidence of a unit cell doubling or other structural changes in . Thus we cannot explain the change in transport and magnetic properties by a change in the crystal structure. Comprehensive investigations of magnetic properties are on its way in order to clarify the magnetic ground state ofas well as the Cl-analogue. Additionally, conventional density-wave instabilities are excluded by the absence of structural changes as seen by x-ray and infrared vibrations. Future low-temperature x-ray experiments are needed to explore if subtler structural effects play a role in the fluctuating charge present in this compound. § CONCLUSIONS AND OUTLOOKFrom our comprehensive transport, dielectric and optical investigations of the dimerized organic charge transfer salt , we can rule out that the metal-insulator transition at T=80 K is due to a charge ordering. No pronounced structural change takes place as well. We can estimate the effective correlations to be weaker compared to the κ-(BEDT-TTF) salts containing Cu-based anions. The system is also more metallic than the sister compound . Although electronic correlations do play a decisive role, the phase below the 80 K transition cannot be explained as a simple Mott insulator. 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"authors": [
"Tomislav Ivek",
"Rebecca Beyer",
"Sabuhi Badalov",
"Matija Čulo",
"Silvia Tomić",
"John A. Schlueter",
"Elena I. Zhilyaeva",
"Rimma N. Lyubovskaya",
"Martin Dressel"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170626212828",
"title": "Metal-Insulator Transition in the Dimerized Organic Conductor $κ$-(BEDT-TTF)$_2$Hg(SCN)$_2$Br"
} |
Journal ofClass Files, Vol. 14, No. 8, August 2017 Shell et al.: Bare Demo of IEEEtran.cls for IEEE JournalsA Unified approach for Conventional Zero-shot, Generalized Zero-shot and Few-shot Learning Shafin Rahman,Salman H. Khanand Fatih PorikliReceived *** ; accepted *** ========================================================================================== Prevalent techniques in zero-shot learning do not generalize well to other related problem scenarios. Here, we present a unified approach for conventional zero-shot, generalized zero-shot and few-shot learning problems. Our approach is based on a novel Class Adapting Principal Directions (CAPD) concept that allows multiple embeddings of image features into a semantic space. Given an image, our method produces one principal direction for each seen class. Then, it learns how to combine these directions to obtain the principal direction for each unseen class such that the CAPD of the test image is aligned with the semantic embedding of the true class, and opposite to the other classes. This allows efficient and class-adaptive information transfer from seen to unseen classes. In addition, we propose an automatic process for selection of the most useful seen classes for each unseen class to achieve robustness in zero-shot learning. Our method can update the unseen CAPD taking the advantages of few unseen images to work in a few-shot learning scenario. Furthermore, our method can generalize the seen CAPDs by estimating seen-unseen diversity that significantly improves the performance of generalized zero-shot learning. Our extensive evaluations demonstrate that the proposed approach consistently achieves superior performance in zero-shot, generalized zero-shot and few/one-shot learning problems. Zero-Shot learning, Few-shot learning, Generalized Zero-Shot learning, Class Adaptive Principal Direction§ INTRODUCTION Being one of the most fundamental tasks in visual understanding, object classification has long been the focus of attention in computer vision. Recently, significant advances have been reported, in particular for supervised learning using deep learning based techniques that are driven by the emergence of large-scale annotated datasets, fast computational platforms, and efficient optimization methods <cit.>. Towards an ultimate visual object classification, this paper addresses three inherent handicaps of supervised learning approaches. The first one is the dependence on the availability of labeled training data. When object categories grow in number, sufficient annotations cannot be guaranteed for all objects beyond simpler and frequent single-noun classes. For composite and exotic concepts (such as American crow and auto racing paddock) not only the available images do not suffice as the number of combinations would be unbounded, but often the annotations can be made only by experts <cit.>. The second challenge is the appearance of new classes after the learning stage. In real world situations, we often need to deal with an ever-growing set of classes without representative images. Conventional approaches, in general, cannot tackle such recognition tasks in the wild. The last shortcoming is that supervised learning, in its customarily contrived forms, disregards the notion of wisdom. This can be exposed in the fact that we can identify a new object by just having a description of it, possibly leveraging its similarities with the previously learned concepts, without requiring an image of the new object <cit.>.In the absence of object annotations, zero-shot learning (ZSL) aims at recognizing object classes not seen at the training stage. In other words, ZSL intends to bridge the gap between the seen and unseen classes using semantic (and syntactic) information, which is often derived from textual descriptions such as word embeddings and attributes. Emerging work in ZSL attempt to predict and incorporate semantic embeddings to recognize unseen classes <cit.>. As noted in <cit.>, semantic embedding itself might be noisy. Instead of a direct embedding, some methods <cit.> utilize global compatibility functions, e.g. a single projection in <cit.>, that project image features to the corresponding semantic representations. Intuitively, different seen classes contribute differently to describe each unseen class. Enforcing all seen and unseen classes into a single global projection undermines the subtle yet important differences among the seen classes. It eventually limits ZSL approaches by over-fitting to a specific dataset, visual and semantic features (supervised or unsupervised). Besides, incremental learning with newly added unseen classes using a global projection is also problematic due to its less flexibility. Traditionally, ZSL approaches (e.g., <cit.>) assume that only the unseen classes are present in the test set. This is not a realistic setting for recognition in the wild where both unseen, as well as seen classes, can appear during the test phase. Recently <cit.> tested several ZSL methods in generalized zero-shot learning (GZSL) settings and reported their poor performance in this real world scenario. The main reason of such failure is the strong bias of existing approaches towards seen classes where almost all test unseen instances are categorized as one of the seen classes. Another obvious extension of ZSL is few/one-shot learning (F/OSL) where few labeled instances of each unseen class are revealed during training. The existing ZSL approaches, however, do not scale well to the GZSL and FSL settings <cit.>. To provide a comprehensive and flexible solution to ZSL, GZSL and FSL problem settings, we introduce the concept of principal directions that adapt to classes. In simple terms, CAPD is an embedding of the input image into the semantic space such that, when projected onto CAPDs, the semantic space embedding of the true class gives the highest response. A visualization of the CAPD concept is presented in Fig. <ref>. As illustrated, the CAPDs of a Leopard (Fig. <ref>) and a Persian cat image (Fig. <ref>) point to their true semantic label embedding shown in violet and blue respectively, which gives the highest projection response in each case.Our proposed approach utilizes three main sources of knowledge to generalize learning from seen to unseen classes. First, we model the relationships between the visual features and semantics for seen classes using the proposed `Class Adapting Principal Directions' (CAPDs). CAPDs are computed using class-specific discriminative models which are learned for each seen category in the `visual domain' (Sec. <ref>). Second, our approach effectively models the relationships between the seen and unseen classes in the `semantic space' defined by CAPDs. To this end, we introduce a mixing transformation, which learns the optimal combination of seen semantics which are sufficient to reconstruct the semantic embedding of an unseen class (Sec. <ref>). Third, we learn a distance metric for the seen CAPDs such that samples belonging to the same class are clustered together, while different classes are mapped further apart (Sec. <ref>). This learned metric transfers cross domain knowledge from visual domain to semantic embedding space. Such a mapping is necessary because the class semantics, especially those collected from unsupervised sources (e.g., word2vec), can be noisy and highly confusing. The distance metric is then used to robustly estimate the seen-unseen semantic relationships.While most of the approaches in the literature focus on specific sub-problems and do not generalize well to other related settings, we present a unified solution which can easily adapt to ZSL, GZSL and F/OSL settings. We attribute this strength to two key features in our approach: a)a highly `modular learning' scheme and b) the two-way inter-domain `knowledge sharing'. Specifically for the GZSL,we present a novel method to generalize seen CAPDs that avoids the inherent bias of prediction towards seen classes (Sec. <ref>). The generalized seen CAPD balances the seen-unseen diversity in the semantic space, without any direct supervision from the visual data. In contrast to ZSL and GZSL, the F/OSL setting allows few or a single training instance of the unseen classes. This information is used to update unseen CAPDs based on the learned relationships between visual and semantic domains for unseen classes (Sec. <ref>). The overall pipeline of our learning and prediction process is illustrated in Fig. <ref>. We hypothesize that not all seen classes are instrumental in describing a novel unseen category. To validate this claim, we introduce a new constraint during the reconstruction of semantic embedding of the unseen classes. We show that automatically reducing the number of seen classes in the mixing process to obtain CAPD of each unseen class results in a significant performance boost (Sec. <ref>).We perform extensive experimental evaluations on four benchmark datasets and compare with several state-of-the-art methods. Our results demonstrate that the proposed CAPD based approach provides superior performance in supervised and unsupervised settings of ZSL, GZSL and F/OSL.To summarize, our main contributions are: * We present a unified solution by introducing the notion of class adapting principal directions that enable efficient and discriminative embeddings of unseen class images in the semantic space.* We propose a semantic transformation to link the embeddings for seen and unseen classes based on a learned distance measure. * We provide an automatic solution to select a reduced set of relevant seen classes resulting in a better performance.* Our approach can automatically adapt to generalized zero-shot setting by generalizing seen CAPDs to match seen-unseen diversity.* Our approach is easily scalable to few/one-shot setting by updating the unseen CAPDs with newly available data. § RELATED WORK Class Label Description: It is a common practice to employ class label descriptions to transfer knowledge from seen to unseen class in ZSL. Such descriptions may come from either supervised or unsupervised learning settings. For the supervised case, class attributes can be one source as well <cit.>. These attributes are often generated manually, which is a laborious task. As a workaround, word semantic space embeddings derived from a large corpus of unannotated text (e.g. from Wikipedia) can be used. Among such unsupervised word semantic embeddings, word2vec <cit.> and GloVe <cit.> vectors are frequently employed in ZSL <cit.>. These ZSL methods are sometimes (arguably confusingly) referred as unsupervised zero-shot learning <cit.>. Supervised features tend to provide better performance than the unsupervised ones. Nevertheless, unsupervised features provide more scalability and flexibility since they do not require expert annotation. Recent approaches attempt to advance unsupervised ZSL by mapping textual representations (e.g. word2vec or GloVe) as attribute vectors using heuristic measures <cit.>. In our work, we use both types of features and evaluate on both supervised and unsupervised ZSL to demonstrate the strength of our approach.Embedding Space: ZSL strategies aim to map between two different sources of information and two spaces: image and label embeddings. Based on the mapping scheme, ZSL approaches can be grouped into two categories. The first category is attribute/word vector prediction. Given an image, they attempt to approximate label embedding and then classify an unseen class image based on the similarity of predicted vector with unseen attribute/word vector. For example, in an early seminal work, <cit.> introduced a semantic output code classifier by using a knowledge base of attributes to predict unseen classes. <cit.> proposed direct and indirect attribute prediction methods via a probabilistic realization. <cit.> formulated a discriminative model of category level attributes. <cit.> proposed an approach of transferring semantic knowledge from seen to unseen classes by a linear combination of classifiers. The main problem with such direct attribute prediction is the poor performance when noisy or biased attribute annotations are available. Jayaraman and Grauman <cit.> addressed this issue and proposed a discriminative model for ZSL. Instead of predicting word vectors, the second category of approaches learn a compatibility function between image and label embeddings, which returns a compatibility score. An unseen instance is then assigned to the class that gives the maximum score. For example, <cit.> proposed a label embedding function that ranks correct class of unseen image higher than incorrect classes. In <cit.>, authors use the same principle but an improved loss function and regularizer. Qiao et al. <cit.> further improved the former approach by incorporating a component for noise suppression. In a similar work, Xian et al. <cit.> added latent variables in the compatibility function which can learn a collection of label embeddings and select the correct embedding for prediction. Our method also has similar compatibility function based on inner product of CAPD and corresponding semantic vector. The use of CAPDs provide an effective avenue to recognition.Similarity Matching: This type of approaches build linear or nonlinear classifiers for each seen class, and then relate those classifiers with unseen classes based on class-wise similarity measures <cit.>. Our method finds similar relation but instead of classifiers, we relate CAPDs of seen and unseen classes. Moreover, we compute this relation on a learned metric of semantic embedding space which let us consider subtle discriminative details.Few/One-shot Learning: FSL has a long history of investigation where few instances of some classes are used as labeled during training <cit.>. Although ZSL problem can easily be extended to FSL, established ZSL methods are not evaluated in FSL settings. A recent work <cit.> reports FSL performance of only two ZSL methods e.g. <cit.>. In another work, <cit.> presented FSL results on ImageNet. In this paper, we extend our approach to FSL settings and compare our method with the reported performance in <cit.>.Generalized Zero-shot Learning: GZSL setting significantly increases the complexity of the problem by allowing both seen and unseen classes during testing phase <cit.>. This idea is related to open set recognition problem where methods consider to reject unseen objects in conjunction with recognizing known objects <cit.>. In open set case, methods consider all unseen objects as one outlier class. In contrast, GZSL represents unseen classes as individual separate categories. Very few of the ZSL methods reported results on GZSL setting <cit.>. <cit.> proposed a joint visual-semantic embedding model to facilitate the generalization of ZSL. <cit.> offered a novelty detection mechanism which can detect whether the test image came from seen or unseen category. Chao et al. <cit.> proposed a calibration mechanism to balance seen-unseen prediction score which any ZSL algorithm can adopt at decision making stage and proposed an evaluation method called Area Under Seen-Unseen accuracy Curve (AUSUC). Later, several other works <cit.> adopted this evaluation strategy. In another recent work, Xian et al. <cit.> reported benchmarking results for both ZSL and GZSL performance of several established methods published in the literature. In this paper, we describe extension of our ZSL approach to efficiently adapt with GZSL settings. < g r a p h i c s > 0ex5exOverall pipeline of conventional ZSL, FSL and GZSL method. Conventional ZSL: An image is passed through a deep network to get an image feature . Then,is fed to seen classifiers _s to produce seen CAPDs, _s. Afterwards, unseen CAPDs, _u are computed by linearly combining seen CAPDs using α (or β for reduced case). Finally, prediction is done by computing the maximum projection response of _u and unseen semantic embeddings 𝐞_u. FSL:is fed to unseen classifiers _u to produce another version of unseen CAPDs '_u which are combined with previously computed _u through δ'_u and δ_u to find an updated version of unseen CAPDs ^f_u. Final prediction is done by maximum response of ^f_u and 𝐞_u.GZSL: Seen CAPDs, _s of conventional ZSL setting are generalized using γ to produce generalized seen CAPDs, ^g_s. For prediction, both ^g_s and _u are considered for calculating maximum response of CAPDs and their corresponding semantic embeddings, 𝐞_s and 𝐞_u. § OUR APPROACH Problem Formulation: Suppose, the set of all class labels is = ^∪^ where ^={1, ... , } and ^ = {+1, ... , +} are the sets of seen and unseen class labels respectively, with no overlap i.e., ^∩^ = ∅. Here, anddenote the total number of seen and unseen classes, respectively. For all classes in the seen and unseen class sets, we have associated semantic class embeddings (either attributes or word vectors) denoted by the sets 𝐄^ = {_s : s ∈^} and 𝐄^ = {_u : u ∈^} respectively, where _s, _u ∈ℝ^d. For every seen (s) and unseen (u) class, we have a number of instances denoted by n_s and n_u respectively. The matrices _s = [_s^1, ..., _s^n_s] for s∈^, and _u = [_u^1, ..., _u^n_u] for u∈^ represent the image features for the seen class s and the unseen class u, respectively, such that _s, _u∈ℝ^k. Below, we define the three problem instances addressed in this paper:* Zero Shot Learning (ZSL): The image features of the unseen classes _u are not available during the training stage. The goal is to assign an unseen class label u ∈^ to a given unseen image using its feature vector _u. * Generalized Zero Shot Learning (GZSL): The image features of the unseen classes _u are not available during the training stage similar to ZSL. The goal is to assign a class label l ∈ to a given image using its feature vector . Notice that, the true class ofmay belong to either a seen or an unseen class. * Few/One Shot Learning (FSL): Only a few/one randomly chosen image features from _u are available as labeled examples during the training stage. The goal is same as the ZSL setting above. In Secs. <ref> and <ref>, we first provide a general framework of our approach mainly focused on ZSL. Afterwards, in Secs. <ref> and <ref> we extend our approach to FSL and GZSL settings, respectively. Before describing our extensive experimental evaluations in Sec. <ref>, we also provide an in-depth comparison with the existing literature in Sec. <ref>.§.§ Class Adapting Principal Direction We introduce the concept of `Class Adapting Principal Direction' (CAPD), which is a projection of image features onto the semantic space. The CAPD is computed for bothseen and unseen classes, however the derivation of the CAPD is different for both cases. In the following, we first introduceour approach to learn CAPDs for seen classes and then use the learned principal directions to derive CAPDs for unseen classes.§.§.§ Learning CAPD for Seen ClassesFor a given image feature _s belonging to the seen class s, we define its CAPD _s in terms of a linear mapping parametrized by _s as,_s = _s^T _s.Our goal is to learn theclass-specific weights _s such that the output principal directions are highly discriminative in the semantic space (rather than the image feature space). To this end, we introduce a novel loss function which uses the corresponding semantic space embedding _s ofseen class s to achieve maximum separability.Proposed Objective Function: Given the training samples _s for the seen class s, _s is learned such that the projection of _s on the semantic space embedding _s, defined by the inner product ⟨_s,_s ⟩, generates a strong response. Precisely, the following objective function is minimized: min__s1/κ∑_c=1^∑_m = 1^n_clog( 1 + exp{ L(_c^m;_s) }) + λ_s/2∥_s ∥_2^2where L is the cost for a specific input _c^m, λ_s is the regularization weight set using cross validation and κ = ∑_c=1^n_c. We define the cost L as:L(^m_c;_s)={[ ⟨_s,_c ⟩ - ⟨_s,_s ⟩,c ≠ s; ⟨_s,1/-1∑_t≠ s_t ⟩ - ⟨_s,_s ⟩,c = s ].In the above loss function, two different scenarios are tackled depending on whether the training samples (image features) are from the same (positive) or different (negative) classes. For the negative samples (c≠ s), the projection of _s on the correct semantic embedding _s is maximized while its projection on the incorrect semantic embedding _c is minimized. For the positive samples (c=s), our proposed formulation directs the projection on the correct semantic embedding _s to be higher than the average response of projections on the incorrect semantic embeddings. In both cases, ⟨_s,_s ⟩ is constrained to produce a high response.Our loss formulation is motivated by <cit.>, with notable differences such as the class-wise optimization, explicit handling of positive samples and the extension of their ranking loss for image tagging to the ZSL problem. We optimize Eq. <ref> by Stochastic Gradient Descent (SGD) to obtain _s for each seen class. Note that, _s=_s^T _c^m in the above cost function, thus for any sample _c^m, _s changes when _s is updated at each SGD iteration. Also, the learning process of _s for each seen class is independent of other classes. Therefore, all _s can be learned jointly in a parallel fashion.Once the training process is complete, given an input visual feature ^m_c, we generate one CAPD _sfor each seen class using Eq. <ref>. As a result, 𝐏^ = [_1 ... _] ∈ℝ^d× accumulates the CAPDs of all the seen classes. Each CAPD is the mapped version of the image feature on the class specific semantic space. The CAPD vector and its corresponding semantic space embedding vector point to similar direction if the input feature belongs to the same class.§.§.§ Learning CAPD for Unseen Classes In ZSL settings, the images of the unseen classes are not observed during the training. For this reason, we cannot directly learn a weight matrix to calculate _u using the same approach as _s. Instead, for any unseen sample, we propose to approximate _u using the seen CAPD of the same sample. Here, we consider a bilinear map, in particular, a linear combination of the seen class CAPDs to generate the CAPD of the unseen class u:_u = ∑_s=1^θ_s,u_s = 𝐏^θ_uwhere, θ_u = [θ_1,u ... θ_,u]^T ∈ℝ^ is the coefficient vector that, in a way, aggregates the knowledge of seen classes into the unseen one.The computation of θ_u is subject to the relation between CAPDs and semantic embeddings of classes. We detail our approach to approximate θ_u below. Metric Learning on CAPDs: The CAPDs reside in the semantic embedding space. In this space, we learn a distance metric to better model the similarities and dissimilarities among the CAPDs. To this end, we assemble the sets of similar 𝐀 and dissimilar 𝐀̅ pairs of CAPDs that correspond to the pairs of training samples belonging to the same and different seen classes, respectively. Our goal is to learn a distance metric d_𝐌 such that the similar CAPDs are clustered together and the dissimilar ones are mapped further apart. We minimize the following objective which maximizes the squared distances between theminimally separated dissimilar pairs: max_𝐌min_(i,j)∈𝐀̅ d^2_𝐌 (_i,_j) s.t. ∑_(i,j)∈𝐀 d^2_𝐌 (_i,_j) ≤ 1where d_𝐌 = √((_i-_j)^T𝐌(_i-_j)) is the Mahalanobis distance metric <cit.>.After training, the most confusing dissimilar CAPD pairs are pulled apart while the similar CAPDs are clustered together by learning an optimal distance matrix 𝐌.Our intuition is that, given a learning metric 𝐌 in the semantic embedding space, the relation between the semantic label embeddings of the seen _s and the unseen classes _u is analogous to that of their principal directions. Since the semantic label embedding of unseen classes are available, we can estimate their relation with the seen classes. For simplicity, we consider a linear combination of semantic space embeddings:_u = ∑_s=1^α_s,u_s = 𝐄^α_uwhere, _u is the approximated semantic embedding of _u corresponding to unseen class u. We compute α_u = [α_1,u ... α_,u]^T ∈ℝ^ by solving:min_α_u (_u- _u)^T 𝐌 (_u - _u) + λ_u/2∥α_u∥_2^2where λ_u is a regularization parameter which is set via cross validation.As we mentioned above, using the learned metric 𝐌, the relationship between the seen-unseen semantic embeddings α_u is analogous to the relationship between the seen-unseen CAPDs θ_u, thus θ_u ≈α_u. Accordingly, we approximate the unseen CAPDs with seen CAPDs by rewriting Eq. <ref> as:_u ≈𝐏^α_u.We derive a CAPD, _u for each unseen class using Eq. <ref>. In test stage of ZSL setting, we assign a given image featureto an unseen class using the maximum projection response:ŷ = max_u ⟨_u, _u ⟩§.§ Reduced Set Description of Unseen Classes When describing a novel object, we often resort to relating it with the similar known object categories. It is intuitive that a subset of the known objects is sufficient for describing an unknown one. We incorporate this observation by proposing a modified version ofEq. <ref>. The term α_u contains the contribution of each seen class to describe the unseen class u ∈^ by reconstructing _u using all seen classes semantic label embeddings. We reconstruct _u by only a small number of seen classes (<). Theseseen classes can be selected using any similarity measure (Mahalanobis distance in our case). The reconstruction of _u becomes:_u = ∑_i=1^β_i,u_iHere, β_u∈ℝ^ is the coefficients of selected seen classes. We learn β_u by a similar minimization objective as in the Eq. <ref>. By replacing α_u with β_u in the Eq. <ref>, it is possible to compute the CAPD of unseen class u using a reduced set of seen classes. Such CAPDs are shown in Fig. <ref> in dashed lines.Appropriate Choice of Seen Classes:In Fig. <ref>, we show comparisons when different approaches are used to select a subset of seen classes to describe the unseen ones. The results illustrate that the seen classes having the semantic space embeddings close to that of a particular unseen class are more suitable to describe it. Here, we considered nearest neighbors of the unseen class semantic vector _u using the Mahalanobis distance. Using a less number of seen classes is inspired by the work Norouzi et al. <cit.> where they applied convex combination of selected semantic embedding vector based on outputs of the softmax classifier of corresponding seen classes. The main drawback of their approach is that the softmax classifier output does not take the semantic embeddings into consideration, which can ignore important features when describing the unseen class. Instead, our method performs an independent optimization (Eq. <ref>) that jointly considers image feature, CAPD and semantic embedding relations via the learned metric 𝐌. As a result, the proposed strategy is better able to determine the optimal combination of selected seen semantic embeddings (see Sec. <ref>).AutomaticSelection for Each Unseen Class: While <cit.> proposed a fixed number of selected seen classes to describe an unseen class, we suggest a novel technique to automatically select the number of most informative seen classes (). First, for an unseen class semantic embedding _u, we calculate the Mahalanobis distances (using learned metric 𝐌) from _u to all _s and perform mean normalization. Then, we apply kernel density estimation to obtain a probability density function (pdf) for the normalized distances. Fig. <ref> shows the pdf for each unseen semantic embedding vector of the AwA dataset. For a specific unseen class, the number of seen classes with the highest probability score is assigned as the value of . Unlike <cit.>, this scheme allows choosing a variable number of the seen classes for different unseen classes. In Sec. <ref> of this paper, we have reported an estimation of the average numbers of seen classes selected for the tested unseen classes.Sparsity: Using a reduced number of the seen classes in Eq. <ref> indirectly imposes sparsity in the coefficient vector α_u in the Eq. <ref>. This is similar to Lasso (ℓ_1) regularization (instead of ℓ_2 regularization) in the loss function in Eq. <ref>. We observe that the above selection solution is more efficientand accurate than the Lasso-based regularization. This is because the proposed solution is based on the intuition that the semantic embedding of an unseen class can be described by closely embedded seen classes. In contrast, Lasso is a general approach and do not consider any domain specific semantic knowledge.Having discussed the ZSL setting in Secs. <ref> and <ref> above, we present the extension of CAPDs to the GZSL problem. §.§ Generalized Zero-shot Learning ZSL setting considers only unseen class images during the test phase. This setting is less realistic, because new images can belong to both seen and unseen classes. To address this scenario, generalized ZSL (GZSL) has recently been introduced as a new line of investigation <cit.>. Recent works suggest that most of the existing ZSL approaches fail to cope up with the GZSL setting. When both seen and unseen classes come into consideration for prediction, the prediction score function becomes highly biased towards seen classes because only seen classes were used for training. As a result, majority of the unseen test instances are misclassified as seen examples. In other words, this bias notably decreases the classification accuracy on unseen classes while maintains relatively high accuracy on seen classes. To solve this problem, available techniques attempts to estimate the prior probability of an input belonging to either a seen or an unseen class <cit.>. However, this scheme heavily depends on the original data distribution used for training. Considering the above aspects, a competent GZSL method should possess the following properties: * Equilibrium: It should be able to balance seen-unseen diversity so that the performances of both seen and unseen classes achieve a balance.* Reduced data dependency: It should not receive any supervision signal (obtained from either training or validation set images) determining the likelihood of an input belonging to seen or unseen class.* Consistency: It should retain its performance on the conventional ZSL setting as well.In this work, we propose a novel GZSL algorithm to adequately address these challenges. Generalized CAPD for Seen Class: In Sec. <ref>, we described the CAPD of seen classes for a given input image is 𝐏^ = [_1 ... _]. Each seen CAPDs is obtained using the class-wise learned classifier matrix _s. It is obvious that each _s is biased to seen class `s'. For the same reason, each _s is also biased to class `s'. Since there was no seen instance available during the testing phase in conventional ZSL setting, seen CAPDs were not used for prediction (Eq. <ref>). Therefore, the inherent bias of seen CAPDs was not affecting ZSL performance. In contrast, for GZSL settings, all seen and unseen CAPDs are considered for prediction. Thus, biased seen CAPDs will dominate as expected and significantly affect the unseen class performances. To solve this problem, we propose to develop a generalized version of each seen CAPD as follows:𝐩^g_s= 𝐏^γ_s,where,γ_s denotes a parameter vector for seen class `s'.Proposed Objective Function: Our hypothesis is that the bias towards seen classes that causes high scores during prediction can be resolved using the semantic information of classes. To elaborate, γ_s is computed solely in semantic label embedding domain and later applied to generalize CAPD of seen class instances. We minimize the squared difference of two complementary losses to obtain γ = [γ_1 ... γ_] ∈ℝ^×, as:min_γ∥1/∑_s=1^ (𝐄^γ_s-_s)^2^mean generalized seen loss-1/∑_u=1^(𝐄^α_u-_u)^2^mean unseen reconst. loss∥_2^2 + λ_γ/2∑_s=1^∥γ_s∥_2^2,where λ_γ is the regularization weight set using cross validation.The objective function inEq. <ref> minimizes the squared difference between the mean of two loss components. The first component is the mean generalized seen loss which measures the reconstruction accuracy of seen class embedding _s using the generalization parameters γ_s. The second component measures the reconstruction accuracy of unseen class embedding _u from seen classes.By reducing the squared difference between these two components, we indirectly balance the distribution of seen-unseen diversity which effectively prevents the domination of seen classes in the GZSL setting (the `equilibrium' property). The interesting fact is that our proposed generalization mechanism does not directly use CAPDs, yet it is strong enough to stabilize the CAPD of different classes during the prediction stage (the `less data dependence' property). Furthermore, the formulation does not affect the computation of unseen CAPDs i.e. _u which preserves the conventional ZSL performance (the `consistency' property).Prediction: For a given image feature , we can derive generalized CAPDs of seen classes ^g_s and CAPD of unseen classes _u using the description in Sec. <ref>. In test stage, we consider both the projection responses of seen and unseen classes to predict a class.ŷ = max_l ∈⟨_l, 𝐯_l ⟩where, _l ∈_u ∪_s^g and 𝐯_l ∈_s ∪_u.§.§ Few-shot LearningThe few-shot learning (FSL) is a natural extension of ZSL. While ZSL considers no instance of an unseen class during training, FSL relaxes this restriction by allowing a few instances of an unseen class as labeled during the training process. Another variant of FSL is called one-shot learning (OSL), which allows exactly one instance of an unseen class (instead of few) as labeled during training. An ideal ZSL approach should be able to benefit from the labeled data for unseen classes under F/OSL settings. In this section, we explain how our approach is easily adaptable to FSL.Updated CAPD for Unseen Class. In ZSL setting, for a given input image feature, we can calculate the unseen CAPD, _u for everyunseen class `u'. Now, in the FSL setting, we optimally use the newly available labeled unseen data to update _u. To this end, new classifiers _u are learned for eachunseen class `u' similar to the case of seen classes (Sec. <ref>). For a given image feature, , we can calculate unseen CAPDs by '_u = ^T_u. These CAPDs are fused with _u, which were derived from the linear combination of seen CAPDs (Eq. <ref>). The updated CAPD for unseen class `u' is represented as ^f_u, given by:^f_u = δ_u_u + δ'_u'_u,s.t. δ_u + δ'_u = 1where, δ_u and δ'_u are the contribution of the respective CAPDs to form an updated CAPD of an unseen class. During prediction, we use ^f_u instead of _u in Eq. <ref>.Calculation of δ_u and δ'_u: The weights δ_u and δ'_u are set using training data such that they encode the reliability of _u and '_u respectively. Recall that our prediction is based on the strength of projection of a CAPD on the semantic embedding vector. Therefore, we need to maximize the correspondence between a CAPD and the correct semantic embedding vector i.e., a high ⟨_u, _u ⟩. The unseen CAPD among _u and '_u that provides higher projection response with u^th unseen class semantic vector gets a strong weight during the combination in Eq. <ref>.We derive _u and '_u for each training image feature, ∈^ = {_s : s ∈^}, and the classification matrix of unseen class `u'. Then, we find the summation of maximum projection response of the CAPD (either _u or '_u) with its respective semantic vector. This maximum projection response finds the response of most similar (or confusing) unseen class of any image. The summation of this response across all training images can estimate the overall quality of CAPDs from the two sources. Finally, we normalize the summations to get δ_u and δ'_u as follows:δ_u = ∑_∈^max_u ⟨_u, _u⟩/∑_∈^max_u ⟨_u,_u⟩ + ∑_∈^max_u ⟨'_u,_u⟩, δ'_u = ∑_∈^max_u ⟨'_u,_u⟩/∑_∈^max_u ⟨_u,_u⟩ + ∑_∈^max_u ⟨'_u,_u⟩.§ COMPARISON WITH RELATED WORK §.§ ZSL SettingsOur method has similarities with two streams of previous efforts on ZSL. Here, we discuss the significant differences. In terms of class-specific learning, a number of recent studies <cit.> report competitive performances when they rely on handcrafted attributes (`supervised' source). However, we observe that these methods failwhen they use `unsupervised' source of semantics (e.g. word2vec and GloVe). The underlying reason is that they do not leverage on the semantic information during the training of the classifiers. Moreover, the attribute set is less noisy than unsupervised source of semantics. Although our work follows the same spirit, the main novelty lies in using the semantic embedding vectors explicitly during the learning phase for each individual class. This helps the classifiers to easily adapt themselves to a wide variety of semantic information sources, e.g. attributes, word2vec and GloVe. Another body of work <cit.> considers semantic information during the training process. However, these approaches do not take the benefits of class-specific learning. Using a single classifier, they compute a global projection. Generalizing all classes by one projection is restrictive and it fails to encompass subtle variations among classes. These approaches do not leverage the flexibility of suppressing irrelevant seen classes while describing an unseen class. Besides, the semantic label embeddings are subject to tuning based on the visual image features. As they cannot learn any metric on semantic embedding space, these methods fail to work accurately across different semantic embeddings. In contrast, by taking the benefits of class-specific learning, our approach computes CAPD for each classifier that can significantly enhance the learned discriminative information. In addition, our approach describes the unseen class with automatically selected informative seen classes and learns a metric on the semantic embedding space to further fine-tune the semantic label information.We also like to point out that the existing methods seem to overfit on a specific dataset, specific image features, and specific semantic features (supervised-attributes or unsupervised-GloVe). Our method, on the other hand, consistently provides improved performance across all the different problem settings.§.§ GZSL settingsWe automatically balance the diversity of seen-unseen classes in an unsupervised way, without strongly relying on CAPD or image visual feature. Previous efforts used a supervision mechanism either from training or validation image data to determine if any input image belongs to a seen or an unseen class. Chao et al. <cit.> proposed a calibration based approach to rescale the seen scores and evaluated using Area Under Seen-Unseen accuracy Curve (AUSUC) <cit.>. As prediction scores of GZSL are strongly biased to seen classes, they proposed to calibrate seen scores by adding a constant negative bias termed as a calibration factor. This factor is calculated on a validation set and works as a prior likelihood of a data point being from a seen/unseen class. The drawback of such an approach is that it acts as a post-processing mechanism applied at the decision making stage, not dealing with the generalization at the basic algorithmic level.Another alternative work, CMT method <cit.> incorporates a novelty detection approach which estimates the outlier probability of an input image. Again, the outlier probability is determined using training images which provides an extra image-based supervision to GZSL model. In contrary, our method considers the seen-unseen biasness in the semantic space at the algorithmic level.The overall prediction scores are then balanced to remove the inherent biasness towards the seen classes. We show that such an approach can be useful for both supervised attributes and unsupervised word2vec/GloVe as semantic embedding information. As our approach does not follow the post-processing strategy like <cit.>, we do not evaluate our work with AUSUC. In line with the recommendation in <cit.>, we use harmonic mean based approach for GZSL evaluation. § EXPERIMENTS Benchmark Datasets: We use four standard datasets for our experiments; aPascal & aYahoo (aPY) <cit.>, Animals with Attributes (AwA) <cit.>, SUN attributes (SUN) <cit.>, and Caltech-UCSD Birds (CUB) <cit.>. The statistics of these datasets are given in Table <ref>. We follow the standard protocols (seen/unseen splits of classes) used in the literature. To be specific, we have exactly followed <cit.> for AwA and CUB datasets, <cit.> for aPY and SUN-10 and <cit.> for SUN. To increase the complexity of GZSL task for SUN, we used a different of seen/unseen split introduced in <cit.>. In line with the standard protocol, the test images correspond to only unseen classes in ZSL settings. In Few/One-shot settings, we randomly choose three/one instances per unseen class to use in training as labeled examples. Again, in GZSL settings, we perform a 80-20% split of each seen class instances; 80% portion is used in training and rest 20% for testing in conjunction with all unseen test data. We report the average results of 10 random trails for Few/One shot or GZSL settings. In a recent work, Xian et al. <cit.> proposed a different seen/unseen split for the above mentioned four datasets. We perform GZSL experiments on that setting as well.Image Features: Previous ZSL approaches use both shallow (SIFT, PHOG, Fisher Vector, color histogram) and deep features <cit.>. As reported repeatedly, deep features outperform shallow features by a significant margin <cit.>. For this reason, we consider only deep features from the pretrained GoogLeNet <cit.> and VGG-verydeep-19 <cit.> models for our comparisons. For feature extraction from GoogLeNet and VGG-verydeep-19, we exactly followChangpinyo et al. <cit.> and Zhang et al. <cit.>, respectively. The dimension of visual features extracted from GoogLeNet is 1024, and VGG-verydeep-19 is 4096. While using the recent Xian et al. <cit.> seen/unseen split, we use the same 2048-dim features from top-layer pooling units of the 101-layered ResNet <cit.> for a fair comparison.Semantic Space Embeddings: We analyze both supervised and unsupervised settings of ZSL. For the supervised case, we use 64, 85, 102 and 312 dimensional continuous valued semantic attributes for aPY, AwA, SUN, and CUB datasets, respectively. We dismiss the binary version of these attributes since <cit.> showed that continuous attributes are more useful. For the unsupervised case, we test our approach on AwA and CUB datasets. We consider both word vector embeddings i.e., word2vec (w2v) <cit.> and GloVe (glo) <cit.>. We use ℓ_2 normalized 400-dimensional word vectors, similar to <cit.>. Evaluation Metrics: This line of investigation naturally applies to two different tasks; recognition and retrieval <cit.>. We measure the recognition performance by the top-1 accuracy, and the retrieval performance by the mean average precision (mAP). The top-1 accuracy is the percentage of the estimated labels (the ones with the highest scores) that match the correct labels. The mean average precision is computed over the precision scores of the test classes. In addition, <cit.> proposed to useHarmonic Mean (HM) of the accuracies of seen and unseen classes (acc_s and acc_u respectively) to evaluate GZSL performance, as follows:HM = 2 × acc_s × acc_u/acc_s + acc_u.The main motivation of using HM is its ability to estimate the inherent biasness of any method towards seen classes. If a method is too biased to seen classes then acc_s will be very high compared to acc_u and harmonic mean based GZSL performance drops down significantly <cit.>.Implementation Details:[The code of our method will be released.] We initialize each classifier W_s from a N(0,1/k) distribution where k is the dimension of the image feature <cit.>.We use a constant learning rate over 100 iterations in training of each class: 0.001 for AwA and 0.005 for aPY, SUN and CUB datasets. For each dataset, we select the value of parameters λ_s, λ_u and γ_s using a validation set. We use the same value of λ_s and λ_u across all seen and unseen classes in the optimization task (Eq. <ref> and <ref> respectively). To choose the validation dataset, we divide all seen classes into two groups, and use one group as the unseen set (no test data is used in the validations). Our results on multi-fold cross-validation and single-validation are very similar.§.§ Results for Reduced Set In the reduced set experiment as describe in Sec. <ref>, for each unseen class, we select four subsets of the seen classes having one-third of the total number. First three subsets contain the farthest away, mid-range, and nearest seen classes of each unseen class in the semantic embedding space, and the last one is the random selection. For all subsets, we determine the proximity of the unseen classes by Mahalanobis distance with learned metric 𝐌. In our experiments, a different unseen class will get a different set of seen classes to describe it. We report the top-1 accuracy on test data of those four subsets in Fig. <ref>. We observe that only one-third of seen classes closest to each unseen class perform the best among the four subsets. The farthest away, mid-range and randomly selected subsets fail to describe an unseen class with high accuracy.This experiment suggest that using only some nearest seen classes located in the semantic embedding space can efficiently approximate an unseen class embedding. The nearest case experiment performances are not the best accuracies reported in the paper because we consider an automatic seen class selection process in our final experiments. Top-1 accuracy (in %) of the various versions of CAPD using the attributes. V: VGG-verydeep-19, G: GoogLeNet image features.24emMethod 2|c|aPY 2|c|AwA 2|c|SUN 2|c|CUB2-9 V G V G V G V GOurs [all-seen] 45.84 50.64 73.19 64.74 84.5 87.00 39.86 42.31Ours [reduced-Lasso] 36.54 37.22 74.16 75.76 78.50 84.50 37.47 37.37Ours [reduced-auto] 54.69 55.07 78.53 80.43 85.00 79.00 43.01 45.31From the discussion in Sec. <ref>, we also know that for different unseen classes our method automatically chooses different sets of useful seen classes. The numbers of seen classes in those sets can be different. In Table <ref>, we report the average number of seen classes in the sets. One can observe that the average number of the seen classes required is around 50% across different datasets. This means, in general, only half of the total seen classes are useful to describe one unseen class. Such a reduced set description of the unseen class not only maintains the best performance but also reduces the complexity of the sparse representation of each unseen class.§.§ Benchmark ComparisonsWe discuss benchmark performances of ZSL recognition and retrieval for both supervised (attributes) and unsupervised semantics (word2vec or GloVe).§.§.§ Results for ZSL with Supervised Attributes[For fairness, inductive test performances from DSRL <cit.>, MFMR <cit.> and DMaP <cit.> are reported in the tables.]We present the top-1 ZSL accuracy results of different versions of our method in Table <ref>. In the all-seen case, we have considered all seen classes to describe an unseen class (Eq. <ref>). In Lasso, we report the performance using Lasso regularization in place of ℓ_2 in Eq. <ref>. The results demonstrate that using a reduced number of seen classes to describe an individual unseen class can improve ZSL accuracy. In Table <ref>, we compare the overall top-1 accuracy of our method with many recent ZSL approaches. Our approach outperforms other methods in most of the settings. In Fig. <ref>, we show confusion matrices of a recent approach <cit.> and ours. Similar to recognition, ZSL can also perform retrieval task. ZSL retrieval is to search images of unseen classes using their class label embeddings. We test the attributes set as a query to retrieve test images. In Table <ref>, we compare our ZSL retrieval performance with four recent approaches on four datasets. Our approach performs consistently better or comparable to state-of-the-art methods. §.§.§ Results for ZSL with Unsupervised SemanticsZSL with pretrained word vectors <cit.> as sematnic embedding is the focus of attention nowadays since it is difficult to generate manually annotated attribute sets in real-world applications. Therefore, the ZSL research is pushing forward to eliminate dependency on manually assigned attributes <cit.>. In line with this view, we adapt our method to unsupervised settings by replacing attribute set with word2vec <cit.> and GloVe <cit.> vectors. Our results on two standard datasets, AwA and CUB, are reported in Table <ref>.We compare with very recent approaches keeping same experimental protocol. One can notice that our approach performs consistently in the unsupervised settings as well in a wide variety of feature and semantic embedding combinations. We provide the average precision-recall curves of ours and two very recent approaches using word2vec embeddings in Fig. <ref>. As shown, our method is superior to others by a significant margin. Our observation is that ZSL attains better performance with supervised attributes as semantics than unsupervised ones because the semantic descriptors (word2vec and GloVe) are often noisy and cannot describe a class as good as attributes. To address this performance gap, some worksinvestigate ZSL with transductive learning <cit.>, domain adaptation techniques <cit.>, and class attribute associations <cit.>. In our study, we consider these improvements as future work.§.§.§ Results for GZSLGZSL is a more realistic scenario than conventional ZSL because GZSL setting tests a method with not only the unseen class instances but also seen class instances. In this paper, we extend our method to work on GZSL setting as well. Although GZSL is a more interesting problem than ZSL, usually standard ZSL methods do not report any results on GZSL in the original papers. However, recently a few efforts have been published to establish the standard testing protocol for GZSL <cit.>. In the current work, we test our GZSL method on both testing protocols of <cit.> and <cit.>. Xian et al. <cit.> tested 10 ZSL methods with a new seen-unseen split of datasets ensuring unseen classes are not used during pre-training of deep network (e.g., GoogLeNet, ResNet) which was used to extract image features. They used ResNet as image features and attributes as semantic embedding for SUN, CUB, AwA and aPY dataset. With this exact settings, in Table <ref>, we compare our GZSL results with the reported results of <cit.>. In terms of Harmonic based (HM) measure, our results consistently outperform other methods by a large margin. Moreover, our method balances the seen-unseen diversity in a robust manner which helps to achieve the best unseen class accuracy (acc_u). In contrast, seen accuracy (acc_u) moves down because of the trade-off while balancing the bias towards seen classes. In the last row, we report the ZSL performance of this experiment where only unseen class test instances are classified to only unseen classes (not considering both seen-unseen classes together). This accuracy is actually an oracle case (upper bound) for acc_u of GZSL case of our method. This is because, if an instance is misclassified in the ZSL case, it must be misclassified in the GZSL case too. Another important point to note is that the parameters of our method are tuned for GZSL setting in this experiment. Therefore, ZSL performance in the last row may increase if one tunes parameters for the ZSL setting.Chao et al. <cit.> experimented GZSL with standard seen-unseen split used in ZSL literature. Keeping this split, they kept random 80% seen class images for training and held out the rest of 20% images for testing stage during GZSL. We perform the same harmonic mean based evaluation like previous setting. In Table <ref>, we compare our results with the reported results in <cit.>. Using the same settings, we also compare with two recent methods, MFMR <cit.> (Table <ref>) and DMAP <cit.> (Table <ref>). For the sake of comparison with DMAP <cit.>, we compare mean Top1 accuracy (not standard though) instead of harmonic mean because acc_u and acc_s are not reported separately in the <cit.>. Again, our method performs consistently well across datasets. More results on GZSL for AwA, CUB, SUN and aPY datasets are reported in Tables <ref>, <ref>, <ref> and <ref>.§.§.§ Results for FSLAs stated earlier, our method can easily take the advantage when new unseen class instances become available as labeled data for training. To test this scenario, in FSL settings, we assume three instances of each unseen class (randomly chosen) are available as labeled during training. In Table <ref>, we report our results for FSL on AwA and CUB dataset while using attribute, word2vec and GloVe as semantic information. The compared methods, DeViSE <cit.> and CMT<cit.>, did not report FSL performance in the original paper. But, <cit.> reimplemented the original work to adapt FSL. The exact three instances of each unseen class used in <cit.> are not publically available. However, to make our results comparable with others, we report the average performance of 10 random trails. Our method performs consistently better than comparing methods except one case: mAP of CUB-att (58.0 vs 58.5). Another observation from these results is that the performance gap between unsupervised semantics (like word2vec and GloVe) and supervised attribute semantics is significantly reduced compared to ZSL settings where unsupervised semantics always ill-performed than supervised attributes across all methods. The reason is that the FSL setting alleviates the inherent noise of unsupervised semantics to perform better (and as good as) supervised semantic. We also experiment on the OSL task, where all conditions are same as FSL setting except a single randomly picked labeled instance is available for each unseen class during training. More results of OSL and FSL for AwA, CUB, SUN and aPY datasets are reported in Table <ref>, <ref>, <ref> and <ref>.For any given image, our FSL method described in Sect. <ref> utilizes the contribution of unseen CAPDs coming from two sources: one by combining the CAPDs of seen classes from zero-shot setting and another by using unseen classifier from few-shot setting. In Eq. <ref>, two constants (δ_u and δ_u^') combine the respective CAPDs to compute the updated CAPD of the unseen class. In this experiment, we visualize the contribution of δ_u and δ_u^' for AwA and CUB dataset in Fig. <ref>. Few observations from this figure are below: * In most cases, few-shot contribution from classifier (δ_u^') contributes higher than zero-shot contribution (δ_u). The reason is that few instances of unseen class can make better generalization than no instance during training. * Zero-shot contribution (δ_u) contributes higher on supervised attribute case than word2vec or GloVe across two datasets. The reason is that supervised attributes contain less noise which gives high confidence to zero-shot based CAPD. * While comparing OSL and FSL, few-shot contributionfrom classifier (δ_u^') contributes higher in FSL than OSL case. The reason is that in FSL settings, any unseen classifier becomes more confident than OSL settings as FSL observes more than one instances during training. * While comparing word2vec and GloVe for both OSL and FSL settings, zero-shot contribution (δ_u) contributes higher for word2vec than GloVe semantics. It suggests that word2vec is a better semantic embedding than GloVe for FSL task. * While comparing AwA and CUB, zero-shot contribution (δ_u) contributes lower than few-shot contribution from classifier (δ_u^') for CUB across all semantics used. The reason is that CUB is a more difficult dataset than AwA in zero-shot setting. One can find that the overall performance on CUB is lower than AwA in all cases (i.e., ZSL, F/OSL and GZSL). §.§.§ All results at a glance. With experiment setting of <cit.>, we juxtapose all results of OSL, FSL, ZSL and GZSL for AwA, CUB, SUN and aPY datasets in Table <ref>, <ref>, <ref> and <ref> respectively. Some overall observations from these results are below: * Performance improves from OSL to FSL settings. This is expected because in FSL setting, more than one (three to be exact) instances of unseen class are used as labeled during training. * The performance gap between supervised attributes and unsupervised word2vec or GloVe is greatly reduced in OSL and FSL. It suggests that getting few instances as labeled during training helps to greatly compensate the noise of unsupervised semantics. * O/FSL setting should always outperform ZSL because more information of unseen is revealed in O/FSL settings. However, we got one exception in SUN dataset where OSL perform worse than ZSL. The reason is that the SUN dataset has 717 classes and only one labeled instance of unseen class could not provide discriminative information which eventually confuses our auto unseen CAPD weighting process. * ZSL results are different from Table <ref>, <ref> and <ref> because here our method is tuned for GZSL case not on ZSL. In addition, random selection of 80% training instance of seen classes across 10 different trails affects the result. * Performance of acc_u of GZSL is always lower than ZSL because ZSL accuracy is the oracle case of acc_u. §.§ Discussion Based on our experiments, we draw the following contributions of our work:Benefits of CAPD: A CAPD points out the most likely class. If a semantic space embedding vector of a class and the CAPD of the image lies close to each other, there is a strong confidence for that class. One important contribution of this paper is the derivation of the CAPD for each unseen class. Conventional ZSL approaches in this vein of thought essentially calculate one principal direction <cit.>. Generalizing all seen-unseen classes with only one principal direction cannot capture the differences among classes effectively. In our work, each CAPD is obtained with the help of bilinear mapping (matrix multiplication). One can extend this by incorporating latent variables, in line with the work Xian et al. <cit.> where a collection of bilinear maps along with a selection criterion is used. Benefits of Nearest Seen Classes: Intuitively, when we describe a novel object, rather than giving a dissimilar object as an example, we use a similar known object. This hints that we can reconstruct the CAPD of an unseen class with the CAPDs of the similar seen classes. This idea helps to improve the prediction performance. How Many Seen Classes are Required? Results presented in Fig. <ref> support the idea that all seen classes are not always necessary. We propose a simple yet effective solution for selecting adaptively the number of similar seen classes for each unseen class (see the discussion in Sec. <ref>). This scheme allows different set of useful seen classes required to describe an unseen class.Extension to GZSL Setting: ZSL methods are biased to assign high prediction scores towards seen classes while performing GZSL task. Due to this reason, conventional ZSL methods fail to achieve good performance in GZSL. Our proposed method solves this problem by adapting seen-unseen class diversity in a novel manner. Unlike <cit.>, our adaptation technique does not take any extra supervision from training/validation image data. We show that class semantic information can be used to adapt seen-unseen diversity.Extension to Few/One Shot Settings: In some applications, a few images of a new class may become available for training. To adapt with such situations, our method can train a model for the new class without disturbing the previous training. The CAPD from the new model is combined with its previous CAPD (of unseen settings) to obtain an updated CAPD with few-shot refinement. We propose an automatic way of combining CAPDs from two sources by measuring the quality of prediction responses of training images. Our updated CAPD provides better fitness score for unseen class prediction. § CONCLUSION We propose a novel unified solution to ZSL, GZSL and F/OSL problems utilizing the concept of class adaptive principal direction (CAPD) that enables efficient and discriminative embeddings of unseen class images in semantic space for recognition and retrieval. We introduce an automatic solution to select a reduced set of relevant seen classes. As demonstrated in our extensive experimental analysis, our method works consistently well in both unsupervised and supervised ZSL settings and achieves the superior performance in particular for the unsupervised case. It provides several benefits including reliable generalization and noise suppression in the semantic space. In addition to ZSL, our method also performs very well in GZSL settings. We propose an easy solution to match the seen-unseen diversity of classes at the algorithmic level. Unlike conventional methods, our GZSL strategy can balance seen-unseen performance to achieve overall better recognition rates. We have extended our CAPD based ZSL approach to adapt with FSL settings. Our approach easily takes the advantage of few examples available in FSL task to fine tune unseen CAPDs to improve classification performance. As a future work, we will extend our approach to transductive settings and domain adaptation.ieee | http://arxiv.org/abs/1706.08653v2 | {
"authors": [
"Shafin Rahman",
"Salman H. Khan",
"Fatih Porikli"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170627024255",
"title": "A Unified approach for Conventional Zero-shot, Generalized Zero-shot and Few-shot Learning"
} |
Institute for Fusion Studies, University of Texas at Austin, Austin, Texas, 78712 A combined analytic and computational gyrokinetic approach is developed to address the question of the scaling of pedestal turbulent transport with arbitrary levels of E × B shear.Due to strong gradients and shaping in the pedestal, the instabilities of interest are not curvature-driven like the core instabilities. By extensive numerical (gyrokinetic) simulations, it is demonstrated thatpedestal modesrespond to shear suppression very much like the predictions of a basic analytic decorrelation theory. The quantitative agreement between the two provides us with a new dependable, first principles (physics based) theoretical framework to predict the efficacy of shear suppression in burning plasmas that lie in a low-shear regime not accessed by present experiments.Flow Shear Suppression of Pedestal Turbulence—A First Principles Theoretical Framework S.M. Mahajan Received 26th June 2017 / Accepted 25th January 2018 ======================================================================================Introduction.–The interplay between shear flow and turbulence is a central component in the self-organization of wide-ranging fluid and plasma systems. In neutral fluids, for example, shear flow is a common driver of turbulence (i.e., Kelvin-Helmholtz instabilities).In contrast, the primary source of turbulence in a typical fusion plasmas is the immense free energy contained in extreme temperature and density gradients. How does the shear flow interact with this class of drift-generated turbulence?It is found that, as opposed to its role in hydrodynamic turbulence, the shear flow, in fact, suppresses drift turbulence <cit.>, and is thought to be the main mechanism underlying the formation and sustenance of the edge transport barrier (called the pedestal) characteristic of the tokamak H-mode <cit.>.Since the residual turbulence mediates the structure of the pedestal and, consequently, largely determines plasma confinement, it (and its interplay with shear flow) is of central importance to fusion energy.The earliest theoretical investigations of shear suppression <cit.>, which we will call decorrelation theories, predicted reduced fluctuation amplitudes due to the combined advection by background shear flow and self-consistent turbulent flow. Amongst these, the analytic theory of Zhang and Mahajan <cit.> has compared rather favorably with experimental observations of shear suppression <cit.>, albeit in experimental setups, perhaps, less challenging than a fusion-relevant H-mode pedestal.It turns out, however, that the predictions of these basic theories are in striking agreement with gyrokinetic simulations (using the Gene code <cit.>) of the pedestal, providing a sound basis for a deep fundamental understanding of the reduction of turbulence by shear flow.We can, thus, with greater confidence, apply a combination of analytical and numerical approaches to study the scaling of turbulence with flow shear, even in the extreme environment of the H-mode pedestal.In effect, we will seek a first principles (physics-based) answer to the crucial question: how does turbulence in the pedestal react to a systematic reduction of flow shear rate?Since all proposed burning plasma machines will lie in the low shear regime, a reliable answer to this question is crucial to the future of fusion energy via Tokamaks operating in H-modes.After the decorrelation theories of shear suppression (worked out in simple geometry) were proposed, subsequent work emphasized the importance of toroidal effects <cit.>.Toroidal effects are indeed prominent for the conventional instabilities in the plasma core. Driven by toroidal curvature (i.e., via resonances with the magnetic drift frequencies), such fluctuations peak in the low magnetic field region of the torus. Interestingly, however, the dynamics of shear suppression manifests differently under conditions characteristic of the pedestal where, due to steep gradients and geometric shaping, the relevant modes are typically not curvature-driven, and consequently are insensitive to toroidal effects <cit.>. It is expected, then, that the influence of shear flow on pedestal turbulencemaybe very different from what could be extrapolated from the notions pertinent to the core plasma.In fact, we reach the surprising conclusion that, despite the substantial complexity and computational challenges involved in pedestal turbulence simulations, the early decorelation theories of shear suppression become highly relevant.The importance of this work should be framed as follows.First, it provides a natural extension of the theory of shear suppression to a pedestal context (perhaps its most important application). Second, building on recent numerical work <cit.>, it establishes the theoretical underpinnings necessary to understand and therefore to predict/estimate pedestal transport over the transition to lower shear burning plasma regimes. Decorrelation Theories— The decorrelation theories of shear suppressionbegin with a generic fluid equation of the form∂_t ξ + v̅(x) ∂_y ξ + ṽ(x,y,t) ∂_x ξ = q(x,y,t),where x is a radial coordinate, y is the corresponding binormal coordinate, ṽ is the fluctuating E × B velocity, v̅(x) is the macroscopic steady state shear flow, q is a gradient-driven source term, and ξ is a fluid quantity like density or temperature.Here we analyze the perpendicular temperature fluctuations (i.e. ξ = T̃_⊥—hereafter denoted by T̃) so that Eq. <ref> may be viewed as a simple analog to Eq. 11 from Ref. <cit.>, which is derived from a moment expansion of the gyrokinetic equations.The decorrelation theories are based on the so-called clump theory described in Ref. <cit.>, and apply basic turbulence closures to solve for properties of the two point correlation function (note that Ref. <cit.> derives similar results via an alternative approach to clump theory).For the purpose of this study, we have reproduced a calculation very similar to the original ZM theory <cit.> (due to the close connection, we will refer to our model simply as the ZM theory—details can be found in Appendix A).The calculation arrives at the following relation describing the reduction of turbulence by shear flow,P(P-1/3)(P-1) = 2/3 W^2 P^2 α,where P^-1 represents the reduction in fluctuation amplitudes, P^-1≡Δ_x0^2/Δ_x^2⟨T̃^2 ⟩/⟨T̃^2 ⟩_0,and W is the normalized shear rateW=γ_E × Bτ_c0Θ.In these expressions, 0 subscripts denote shear free quantities, γ_E × B = dv/dx is the shear rate, the brackets represent ensemble averages (in practice, averages over space and time), τ_c0 is the shear-free correlation time, Δ_x,y is the correlation length in the x,y direction (respectively), Θ=Δ_x/Δ_y accounts for anisotropy, and α is a near-unity scaling parameter, which will be described below.The ZM theory has two features that distinguish it from other decorrelation theories.Both are indispensable for the quantitative comparisons that will be described below.First, the theory is non-asymptotic in shear rate, describing shear suppression seamlessly across the weak and strong shear limits.Second, and most importantly, it accounts for the fact that fluctuation levels and nonlinear diffusivity are intimately connected and are both sensitively dependent on shear rate.This is accomplished via an ad hoc expression relating the two: D = D_* ⟨T̃^2 ⟩^α,where D is the nonlinear diffusivity, D_* is a constant proportionality factor, and α is the relevant scaling parameter related to the strength of the turbulence (α∼ 0.5/1.0 for strong/weak turbulence, respectively). In summary, given a shear rate, correlations lengths, and the scaling parameter α, Eq. <ref> predicts the relative suppression of turbulent fluctuation amplitudes. In essence, the theory captures the nonlinear decorrelation of turbulence when subject simultaneously to a background shear flow and a self-consistent turbulent flow.By balancing this decorrelation with a generic gradient drive, an expression is derived for the reduction of turbulence by shear flow.Notably, the theory neglects parallel dynamics (e.g. Landau damping), zonal flows <cit.>, toroidal effects, non-local (i.e., global) effects <cit.>, details of the driving instability, coupling with damped eigenmodes <cit.>, and non-monotonic flow profile variation, all of which are included in our simulations.Thus, to the extent that simulation and theory agree, it can be concluded that the underlying mechanism of shear suppression in the pedestal is described by a few relatively simple ingredients.Presently, we make such comparisons.Comparisons between Theory and Simulation— The gyrokinetic Gene simulations described here are designed to include a wide range of relevant pedestal effects while also facilitating clear comparisons with theory.To this end, we employ an adiabatic electron approximation in order to reduce computational demands and limit the dynamics to the ion temperature gradient (ITG) driven turbulence of interest.We note that this ITG turbulence is not the dominant pedestal transport mechanism in most present day experiments precisely due to its suppression by shear flow.The most important fluctuations are likely electron temperature gradient turbulence <cit.>, microtearing modes <cit.>, and low-n (toroidal mode number) magnetic fluctuations <cit.>—all of which are expected to be much less sensitive to shear flow than ITG.Recent related work (Refs. <cit.>) explores the implications of the expected ρ_* scaling of pedestal flow shear (ρ_* is the ratio of the sound gyroradius to the minor radius ρ_s / a).The pedestal shear rate is effectively determined by the self-organization of the pedestal by means of force balance between the radial electric field and the pressure gradient.This force balance, which is well-described by neoclassical theory and well-founded experimentally <cit.>, results in shear rates that scale linearly with ρ_*: γ_E × Ba/v_Ti∝ρ_* (v_Ti is the ion thermal velocity). Refs. <cit.> identify two classes of pedestal transport from gyrokinetic simulations—one that scales close to the expected gyroBohm ρ_* scaling , and a second (ITG turbulence) that is small throughout most of the experimentally accessible parameter space but has an unfavorable ρ_* scaling due to its sensitivity to shear flow.These studies predict the latter mechanism—shear-sensitive ITG turbulence—to be relevant on JET (which has access to the lowest values of ρ_* among active experiments) and to become increasingly important in the transition to low-ρ_* regimes.To summarize, ITG turbulence in the pedestal is suppressed by the strong shear rates characteristic of present-day experiments.We are addressing the question of the manner in which it re-emerges as shear rates decrease.In order to make comparisons with the ZM theory (Eq. <ref>), the time- and box-averaged squared temperature fluctuation amplitude, ⟨T̃^2 ⟩, and the radial and binormal correlation lengths (Δ_x, and Δ_y) are calculated from simulation data.The correlation lengths are calculated for temperature fluctuations at the top of the torus where the fluctuation levels peak.The heat flux, which is embedded in the corresponding temperature moment of the gyrokinetic nonlinearity, provides an appropriate proxy for the nonlinear diffusivity D.The scaling factor α (recall Eq. <ref>) is extracted from a comparison of Q_i and⟨T̃^2 ⟩ and lies in the range α = [0.81,0.97] for the cases studied here.In order to connect the shear rate W (Eq. <ref>) with its corresponding quantity from the simulations, we use the inverse linear growth rate at k_y ρ_s=0.1 as a proxy for the shear-free correlation time τ_c0, the standard definition of the E × B shear rate used in the Gene code <cit.>, and an anisotropy factor Θ = Δ_x/Δ_y defined by the correlation lengths.One free parameter is used to scale the shear rate and is selected to minimize the discrepancy between simulation and theory.Encouragingly, this free parameter remains of order unity in all cases studied (varying from 0.58 to 2.1). The simulations are based on the low ρ_* pedestal setup described in Ref. <cit.>, which uses JET profile shapes <cit.> in conjunction with ITER geometry and projected ITER pedestal parameters.Profiles for this base case are shown in Fig. <ref> (a).The extensive simulation campaign described below entails scans of E × B shear rate for four different scenarios, which are designed to isolate various effects and gauge variation in shear suppression dynamics.The first case, the local constant shear (LCS) case, is designed to match the assumptions of the ZM theory as closely as possible by employing a local approximation (i.e., taking plasma parameters, gradients, and shear rate at a single radial location and neglecting effects from radial profile variation).A comparison between theory and simulation is shown in Fig. <ref> (a); it exhibits a remarkable quantitative prediction of the simulations by the theory.As demonstrated with the global constant shear (GCS) case, which includes self consistent global profile variation (but retains a radially constant shear rate), the theory is robust to the addition of global effects (see Fig. <ref> (b)).The global full shear (GFS) case additionally includes non-monotonic flow profiles whose shapes are set by the standard neoclassical expression for the radial electric field (we define the pedestal shear rate to be the radially averaged quantity).This is particularly significant since it introduces a region of zero shear in the simulation domain (see Fig. <ref> (b)), raising the possibility of non-trivial interactions between the turbulence and the flow profile.We note in this case anomalous behavior in the low shear limit, as shown in Fig. <ref> (c), where a discontinuity in P^-1 between the low shear and the zero shear cases is observed.Extended simulations targeted at reducing statistical uncertainty produce very minor differences in fluctuation levels, but a persistent (∼ 20 %) decrease in radial correlation length for the low shear case, suggesting that the non-monotonic shear profile acts as a singular perturbation to the length scales.Consequently, we normalize to the low shear (as opposed to zero shear) case and implement a small corresponding offset in the shear rate.With this adjustment the simulations for the GFS case also find very good agreement with the theory, as seen in Fig. <ref> (c).Scan of ρ_*— While valuable for the purpose of theoretical verification, the three cases examined thus far may be characterized as idealized (and somewhat artificial) setups that exploit the flexibility of our simulation capabilities to independently scan E × B shear rates.In an experimental context there is little external control over the shear rate.As described above, the shear rate is set by the self-organization of the pedestal by means of force balance between the radial electric field and the pressure gradient, resulting in direct proportionality between ρ_* and pdestal shear rates.Consequently, the most experimentally relevant simulation scenario (called the global rho star [GRS] case) involves a fully self-consistent scan of ρ_*, which holds the pedestal width (in magnetic flux coordinates) fixed along with all other dimensionless parameters (i.e., safety factor q, ν_*, β).In this scenario, the E × B shear profile is determined self-consistently from the density and temperature profiles using the standard neoclassical expression <cit.>.This scenario involves an additional level of complexity since it conflates the effects of shear suppression with intrinsic ρ_* effects, which are well-known to independently affect turbulence levels (i.e., produce deviations from gyroBohm scaling) as ρ_* is raised above a certain threshold <cit.>.We address this additional complexity with a straightforward modification to the ZM theory.We assume that finite ρ_* effects are limited to two mechanisms—1) E × B flow shear, and 2) ρ_* effects manifest in the linear instability drive.The latter enters the theory while balancing the decorrelation and gradient drive ⟨T̃^2/T_0^2 ⟩/τ_c = γ_lin(ρ_*)(v_Ti/a)D/L^2,where γ_lin is the ρ_*-dependent linear growth rate, τ_c is the decorrelation time, and L is a macrosopic gradient scale length.Note that the decorrelation theories (e.g., <cit.>) use Eq. <ref> without the inclusion of the linear growth rate (see Appendix A for additional details).With this generalization, we find excellent agreement with simulation results, as shown in Fig. <ref> (d).Clearly, the agreement between simulation and theory is substantial in all four cases studied.This agreement strongly supports a quantitative connection between the simulations and the ZM theory, particularly in light of the following observations: 1) There is good agreement in all four cases studied, which represent substantial variation in physical comprehensiveness.2) The agreement has been achieved using only a single free parameter, which remains order unity in all cases. 3) As an additional test, we make comparisons while neglecting various aspects of the theory.The discrepancy increases significantly when the anisotropy facter Θ and/or the the radial correlation lengths are left out of the theory. The results described here suggest that the scaling of pedestal shear suppression of ITG turbulence is determined by the basic ingredients in the decorrelation theory, namely the interplay between turbulent and background advection—both balanced by a gradient drive mechanism.By extension, the complex physics included in the simulations (zonal flows, Landau damping, damped eigenmodes, global profiles effects, non-monotonic shear profiles, etc.)—which are important for determining absolute fluctuation levels—have little influence on the scaling of turbulence with shear flow. High Shear Limit— We now examine the scaling of turbulence in the high shear regime.The appropriate scaling can be readily derived by taking the P ≫ 1 limit of Eq. <ref>P^-1 = (2/3)^1/(2α-3)W^2/(2α-3).Fig. <ref> shows a comparison between this high shear scaling and fits to the asymptotic simulation data points.Note that the asymptotic scaling is strongly dependent on the α factor.The self-consistent values of α produce a much better match than either the weak turbulence (α = 1 translating to W^-2) or strong turbulence (α = 0.5 corresponding to W^-1) limits.This high shear scaling has implications for the ρ_* dependence of pedestal transport.As shown in Fig. <ref>, turbulence reduction scales roughly as shear rate to the -3/2 power in the high shear limit.Translating the theory-based quantities (P^-1 and W) to more-intuitive quantities (gyroBohm-normalized heat flux Q/Q_GB and ρ_*) produces (empirically) the rough scaling Q/Q_GB∝ρ_*^-2, which indicates a transport mechanism that is independent of ρ_*.This scaling is roughly observed for the ρ_* scan described above as well as for similar scans described in Refs. <cit.>, suggesting its robustness.This fundamental theoretical prediction must be considered when extrapolating to low ρ_* regimes.Discussion— The present study demonstrates that the underlying mechanism of pedestal shear suppression involves a relatively small set of transparent physical ingredients, which are insensitive to parameter variations and difficult to modify. We have, thus, the makings of a first-principles, physics-based theory of turbulence suppression (via shear flow) that can be exploited to estimate/predict turbulent transport in regimes that have not been experimentally probed yet. In fact, ITER and all the future burning plasma experiments fall in the low shear regime not accessed in most current experiments.We end the paper by pointing out that the simulations/theory suggest several possible routes for controlling turbulent transport in pedestals.Since the relevant suppression parameter W is the shear rate normalized to the linear growth rate, the most promising route to optimized pedestal performance in low ρ_* regimes is through the minimization of ITG growth rates.As demonstrated in Refs. <cit.>, this can be accomplished by at least two mechanisms—1) ion dilution via impurity seeding, and 2) reduction of η_i (the ratio of the density and temperature gradient scale lengths) through the separatrix boundary condition.The latter route will rely heavily on optimization of divertor performance.§ APPENDIX A—THEORY SUMMARYHere we succinctly outline a derivation of the Zhang-Mahajan (ZM) shear suppression theory <cit.> used in this study.This derivation is intended to be more accessible than the rigorous but very involved calculation described in <cit.>.Our simplified derivation first reproduces the orbit equations from Biglari-Diamond-Terry <cit.> using the standard clump theory <cit.>.Thereafter, the distinctive elements of the ZM theory are applied, namely a non-asymptotic (in shear rate) treatment of the problem and the use of an ansatz relating the nonlinear diffusivity and fluctuation amplitude (described below).The original ZM theory builds on an alternative set of orbit equations, which stem from an independent approach that does not rely on the standard clump theory.The model described below exhibits only minor qualitative differences from the original ZM theory. We use coordinates (x,y), where x represents the radial direction and y varies in some direction (other than that of ) on the magnetic surface.Parallel gradients, along with such geometrical details as magnetic curvature, are neglected.A perturbed fluid quantity, such as temperature or density, is denoted by ξ(x,y,t) and assumed to satisfy the equation_tξ + v̅(x)_yξ + ṽ(x,y,t)_xξ = q(x,y,t)Here v̅(x) is the shear flow, a slowly varying equilibrium flow in the y-direction, while ṽ(x,y,t)≪v̅ is a turbulent flow. We assume that the turbulent flow is incompressible, and that the variation of ξ on flux surfaces is smaller than its radial variation, so the x-component of the turbulent velocity dominates. Rather than trying to solve (<ref>), we derive from it an approximate equation for the correlation functionC_12≡⟨ξ(x_1,y_1,t)ξ(x_2,y_2,t)⟩≡⟨ξ_1ξ_2⟩where the angle brackets denote a statistical average. Standard renormalization methods, based primarily on the random phase approximation, yield the evolution equation(_t +ω_s x_-_y_- - _x_-(k_0i^2x_i-^2)D_x_-)C_12= Qwhere x_- = x_1 - x_2 is the relative coordinate, Q is the source, ω_s≡v̅_c' is the shearing rate, D is a turbulent diffusion coefficient and_0 is a spectral-averaged wave number, related to the width Δ of an eddy: k_0x,y= 1/Δ_x,y. The corresponding Green's function G(_-,t;_-',t') satisfies the homogeneous version of (<ref>) with initial data G(_-,t;_-',t) = δ(_- - _-').We are content to study the moments M^ij(t) ≡∫G(, t; _0, 0) x^ix^jwhere (x^1,x^2) = (x_-,y_-) and _0 is some initial value. Integration by parts yields the dynamical moment equations_tM^11 =2D k_⊥^2(3M^11+ sin^2θ M^22)_tM^12 = ω_sM^11 + 2Dk_⊥^2M^12 _tM^22 =2ω_sM^12Here k_⊥^2≡ k_0x^2 and sinθ≡ k_0y/k_0x. Denoting the characteristic time for change in M_ij by τ_c and introducing the nominal diffusion time τ_D≡ (k_⊥^2D)^-1 = Δ^2/D we obtain the characteristic equationz(z-2)(z-6) = 4(ω_sτ_D)^2sin^2θwhere z ≡τ_D/τ_c.Since the gradients relax through turbulent diffusion, the source for turbulence is measured by D/L^2.This observation leads to the estimate1/τ_c⟨T̃^2/T_0^2⟩ = D/L^2We concentrate on the fastest relaxation rate, z_0 = 6, where the subscript refers to zero shear.The effects of shear are displayed through the ratio z/z_0 = Δ^2/Δ_0^2⟨T̃^2⟩_0/⟨T̃^2⟩≡ PFollowing <cit.>, we adopt the ansatz D = D_*⟨T̃^2⟩^γ, where D_* is independent of the turbulence level and scale.Then (<ref>) becomesP(P - 1/3)(P -1)= 2/3W^2P^2γwhere W = τ_c0(Δ^2/Δ_0^2)^1-γ. 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"authors": [
"D. R. Hatch",
"R. D. Hazeltine",
"M. K. Kotschenreuther",
"S. M. Mahajan"
],
"categories": [
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"primary_category": "physics.plasm-ph",
"published": "20170626143704",
"title": "Flow Shear Suppression of Pedestal Turbulence--A First Principles Theoretical Framework"
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http://arxiv.org/abs/1706.08393v2 | {
"authors": [
"A. Chen",
"M. F. Kling",
"A. Emmanouilidou"
],
"categories": [
"physics.atom-ph"
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"published": "20170626142226",
"title": "Controlling electron-electron correlation in frustrated double ionization of molecules with orthogonally polarized two-color laser fields"
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|
Universidade Federal do Espírito Santo, Departamento de Ciências Naturais, Rodovia BR 101 Norte, km. 60, São Mateus, ES, Brazil. Facultad de Ciencias, Universidad Nacional de Ingeniería, Av Túpac Amaru. 210,Rimac, Lima, Perú. Núcleo Cosmo-ufes & Departamento de Física, CCE, Universidade Federal do Espírito Santo, 29075-910, Vitória-ES, Brazil. Núcleo Cosmo-ufes & Departamento de Física, CCE, Universidade Federal do Espírito Santo, 29075-910, Vitória-ES, Brazil. National Research Nuclear University “MEPhI”, Kashirskoe sh. 31, Moscow 115409, Russia. UniversidadeFederaldeJuizdeFora, DepartamentodeFísica–ICE,JuizdeFora, CEP 36036-330,MG, Brazil. Tomsk State Pedagogical University, 634041, Tomsk, Russia. Tomsk State University, 634050, Tomsk, Russia. International Institute of Physics (IIP), Universidade Federal do Rio Grande do Norte (UFRN) CP 1613, 59078-970 Natal-RN, Brazil. InstituteofTheoreticalAstrophysics, UniversityofOslo,0315Oslo,Norway.Reduced Relativistic Gas (RRG) is a useful approach to describe the warm dark matter (WDM) or the warmness of baryonic matter in the approximation when the interaction between the particles is irrelevant. The use of Maxwell distribution leads to the complicated equation of state of the Jüttner model of relativistic ideal gas. The RRG enables one to reproduce the same physical situation but in a much simpler form. For this reason RRG can be a useful tool for the theories with some sort of a “new Physics”. On the other hand, even without the qualitatively new physical implementations, the RRG can be useful to describe the general features of WDM in a model-independent way. In this sense one can see, in particular, to which extent the cosmological manifestations of WDM may be dependent on its Particle Physics background. In the present work RRG is used as a complementary approach to derive the main observational exponents for the WDM in a model-independent way. The only assumption concerns a non-negligible velocity v for dark matter particles which is parameterized by the warmness parameter b. The relatively high values ofb ( b^2≳ 10^-6)erase the radiation (photons and neutrinos) dominated epoch and cause an early warm matter domination after inflation. Furthermore, RRG approach enables one to quantify thelack of power in linear matter spectrum at small scales and in particular, reproduces the relative transfer function commonly used in context of WDM withaccuracy of ≲ 1%. A warmness with b^2≲ 10^-6 (equivalent to v≲ 300 km/s) does not alter significantly the CMB power spectrum and is in agreement with the background observational tests. On general features ofwarm dark matter with reduced relativistic gas L. Casarini[ E-mail:[email protected]] December 30, 2023 ======================================================================§ INTRODUCTION In the last decades cosmological observations provided numerous evidence for the two dark components nominated dark matter (DM) and dark energy (DE), which are responsible for ∼ 96% of the content of the universe. In particular, the confirmation of existence of these two dark components come from the measurements of the luminosity redshift of type Ia supernova <cit.>, baryon acoustic oscillations <cit.>, anisotropies of the cosmic microwave background (CMB)<cit.> andother observations <cit.>.The standard interpretation suggests that DE is necessary to accelerate the expansion of the universe. On the other hand the DM has non-baryonic nature and is important, in particular, to describe the formation of cosmic structure. The standard cosmology, ΛCDM model, assumes that the DE is a cosmological constant, and regards DM as a non-relativistic matter withnegligible pressure (cold dark matter). ΛCDMprovides an excellent agreement with most of the data (see, e.g., <cit.> for a general review), however this agreement is not perfect due to the tensions with part of theobservational data (see for example <cit.>).In part due to these difficulties, some alternative models have been proposed and studied as possible DE and DM candidates (see for example <cit.>). Let us note that some of these alternative models aim to describe fluids that replace both DM and DE (see for example <cit.>) or describe interaction between DE and DM <cit.>.Some of the mentioned ΛCDM difficulties are related with the choice of the cold dark matter (CDM)paradigm <cit.>.For instance, at small scales the issues such as the missing satellites problem <cit.>, core/cusp problem <cit.>, and the Too big to fail problem <cit.>, can be alleviated by assuming that the DM is not completely cold. In contrast to the CDM,theHot Dark Matter (HDM) scenario implies that the free streaming due to a thermal motion of particles is important to suppress structure formation at small scales. Nevertheless this scenario was ruled out and opens the space for the Warm Darm Matter (WDM) scenario. The main feature of WDM models is that thermal velocities of the DM particles are not so high as in the HDM scenario and, on the other hand, not negligible like in the CDM scenario. Typically, the WDM models assume that it is composed by particles of mass about keVinstead of GeV which is “typical” for CDM and eV which is the standard case for the HDM. The standard approach to explore the possible warmness of DM and its consequences for structure formation arebased on to solution of the Hierarchy Bolztmann equation, taking into account the specific properties of the given WDM candidate <cit.> [One has to remember that equations for DM are always coupled to the Bolztmann equations for other components of the universe.].For example,relation between mass and warmness for each WDM candidate comesfrom the particle physics arguments. This is in fact very good, because the ultimate knowledge of the DM nature may be achieved only within the particle physics and, more concrete, by means of laboratory experiments.Until the moment when the DM will be detected in the laboratory experiments, one can always assume that the properties of DM derived within a particle physics models may be violated by some the qualitatively new scenarios for the DM, which can be never ruled out completely <cit.>. From this perspective, it is useful to develop also model-independent approaches to investigate the cosmological features of a WDM. In the present work we will explore the consequences and impacts of warmness in the process of structure formation and CMB anisotropies, but using a model-independent approach which is based on the RRGapproximation. The RRG is a model of ideal gas of relativistic particles, which has a very simple equation of state. This nice property is due to the main assumption – that the particles of the ideal gas have non-negligible but equal thermal velocities. Regardless of this simplicity, the model has long history which started in a glorious way. The RRG equation of state was first introduced by A.D. Sakharov in order to explore the acoustic features of Cosmic Microwave Background (CMB) in the early universe <cit.>. Using this model Sakharov predicted the existence of oscillations in CMB temperature spectra long before its observational discovery (see <cit.> for the historical review).Recently, RRG was reinvented in <cit.>, where the derivation of its equation of state was first presented explicitly. The simplicity of the equationof state is due to the assumption that all particles of relativistic gas have equal kinetic energies, i.e., equal velocities. Therefore RRGis a reduced version of well-known Jüttner model of relativistic ideal gas <cit.>. A comparison between the equations of state of the relativistic ideal gas and RRG shows that the difference does not exceed 2.5% even in the low-energy region <cit.> and becomes completely negligible at higher energied. Further considerations have shown that RRG model enables one to achieve a simple and reliable description of the matter warmness in cosmology. In Ref. <cit.> RRG was used to decribe WDM and its perturbations were compared with the Large Scale Structure data. Furthermore, the general analytic solutions for the several backgroundcosmological models involving RRG were discussed in Ref. <cit.>.The RRG was successfully used in <cit.> as an interpolation between radiation and dark matter eras in cosmology.An upper bound on the warmness coming from RRG <cit.> is very closeto the one obtained from much more complicated analysis based on a complete WDM treatment, based on the Boltzmann equation. This standard approach requires specifying the nature of the particle physics candidate for the WDM contents <cit.>, while the approach based on RRG requires only one parameter, that is the warmness of DM. In this sense RRG represents a really useful tool for exploring WDM cosmologywithout specifying a particular candidate for the WDM. Such a model may be helpful for better understanding of the model-dependence or independence of the cosmological features of WDM.The main goal of the present work is to take advantage of the RRG and its analytical solutions for the background cosmology and apply it to WDM, instead of considering full set of WDM hierarchical Boltzmann equations. The RRG enables one to make greater part of considerations analytically and hence provide better qualitative understanding of the results. In this way we consider the formation of large-scale structure and the problem of CMB anisotropy in a model-independent manner. Following <cit.>, we shall establish the bounds for the thermal velocities of the WDM particles in a general way. With this objective in mind we consider the model of the spatially flat Friedmann-Roberston-Walker universe filled by radiation [Of course, WDM has a radiation behavior in early epochs but their physical processes are different than photons or neutrinos. For this reason we will diferenciate in all paper WDM in early stages from “standard“radiation (photons and neutrinos).] and RRG, representing WDM. Gravity is described by the general relativity, with the cosmological constant representing DE. We shall refer to this model as to ΛWDM. All the perturbative treatments will be performed at the linear order only, and using the normalization with the scale factor at present a_0=1. With these notations, the WDM space of parameters is reduced by using the most recent data from SNIa, H(z) and BAO.The paper is organized as follows. Sec. <ref> describes the dynamics of the WDM in the framework of RRG, both at the background and perturbative levels. It is shown that high level of warmness may erase ”standard" radiation era from the cosmic history. Starting from this point one can establish an upper bound for the velocity of the RRG particles, which preserves the “standard“ primordial scenario for the universe. This bound is used as a physical prior in the consideration of Sec. <ref>, devoted to the statistical analysis using the background data. In this framework we reduce the space of parameters for WDM and use this reduced space in the consequent analysis. At the next stage the CAMB code is modified and used to quantify the relation between the DM warmness andthe total matter density contrast, linear matter power spectrum and CMB power spectrum. We show that theRRG is capable to reproduce the main feature of the WDM, i.e., the suppression of matter over-densities at small scales. Furthermore, in Sec. <ref> we discussproprierties of thermal relics via RRG. Finally, Sec. <ref> includes discussions and conclusions.§ A DESCRIPTION FOR A WARM DARK MATTER FLUIDLet us start with the background notions. The reader can consult <cit.> or recent <cit.> for further details.In the RRG approach WDM is treatedas an approximation of a Maxwell-distributed ideal gas formed by massive particles. All these particles have equal kinetic energies, or equal velocity β = v/c <cit.> (c is the light speed). This leads to the following relation between WDM pressure p_w and WDM energy density ρ_w, p_w=ρ_w/3[1-(mc^2/ϵ)^2],where m is the WDM particle mass, and ϵ is the kinetic energy of each particle of the system which is given by,ϵ = mc^2/√(1-β^2).Here ρ_c is introduced as a notation for the rest energy density, i.e.,the energy density for the v=0 case. Thus, ρ_c=ρ_c0a^-3=n mc^2, where n is the numberdensity a=a(t) is the scale factor of the metric. Using this relation, Eq. (<ref>) can be cast into the formp_w=ρ_w3[1-(ρ_c/ρ_w)^2],which can be regarded as equation of state (EoS) of the WDM fluid.Using Eq. (<ref>) in the energy conservation relation, the solution for ρ_w has the formρ_w(a) = ρ_w0 a^-3√(1+b^2 a^-2/1+b^2) ,b=β/√(1-β^2) .Thus, b parameter measures velocity and warmness of the WDM particles at present. In the limit v≪ c we have b ≈ v/c. Note also that for b=0 the CDM case is recovered. Combining Eqs. (<ref>) and (<ref>), one can find a posteriori state parameter,w(a)=p_w/ρ_w=1/3-a^2/3(a^2+b^2) .Here we called this term as a state parameter a posteriori because the "natural" EoS for the RRG description, given by Eq. (<ref>), implicitly depends on the scale factor and on the nowadays WDM energy density. However after the integration of continuity equation it is possible to write the state parameter that depends only on the scale factor. This form will prove useful in the perturbative analysis.In what follows we consider themodel with cosmological constant, which does not agglomerate, WDM described by a RRG, baryons and radiation. All of them are assumed being interacting gravitationally and only photons and baryons also interacting via Thomson scattering before recombination. In this situation Hubble rate takes the formH^2=H_0^2(Ω_Λ0+Ω_w0a^3√(1+b^2 a^-2/1+b^2) + Ω_b0a^3 + Ω_r0a^4).In the last equation Ω_x0 (with x=Λ,w,b and r) is the value of the DE, WDM, baryons and radiation density parameters at present, while Ω_Λ0=1-Ω_w0-Ω_b0 -Ω_r0 since we deal with a spatially flat universe. It is easy to see that the expressions (<ref>), (<ref>), (<ref>) and (<ref>) interpolate between the dustat b → 0 and radiation at b →∞ cases. Because of this interpolation feature, RRG can be used to investigate the cosmological consequencesof the transition between epochs of radiation anddust <cit.>.One can note that the WDM with EoS (<ref>) has a remarkable consequence at early times, when RRG becomes very close to radiation. This feature could cause an early warm matter domination and erase the ”standard" radiation dominated epoch. In order to ensure the existence of a “standard" radiation dominated era we must impose that in the very early universe the radiation energy density is bigger than WDM energy density[We must emphasize that even thoughin a primordial universe RRG behaves like radiation atbackground, this is not true at perturbative level. On the other hand, processes involvingWDM radiation limit will be, in general, different that thoseinvolving the standard radiation (photons and neutrinos). The case where WDM dominates even in early times deserves a more carefully study of earlier processes like nucleosynthesis, reionization, reheating, etc and is beyond the scope of this paper.]. This requirement leads to an upper bound on the warmness b-parameter, lim_a→ 0Ω_r(a)/Ω_w(a)>1 ⇒ b^2 < Ω_r0^2/Ω_w0^2-Ω_r0^2.Note that in early times, radiation dominates over baryons which decay as a^-3. For this reason we do not take them into account in Eq. (<ref>). Since the present-day values are Ω_r0∼ 10^-4and Ω_w0∼ 10^-1, we expect that b^2≲ 10^-6, which corresponds to a DM particle velocity approximately equal to 300 km/s. Mathematically WDM dominating over "standard”radiation means the absence of a real value for z_eq,that is the redshift at the point of radiation and matter equilibrium. One can evaluate z_eq from the relationΩ_r(z_eq)=Ω_w(z_eq)+Ω_b(z_eq) .with the following solution,1+z_eq=(Ω_b0Ω_r0/Ω^2_b0-Ω^2_w0-√((1+b^2)Ω^2_w0Ω^2_r0+b^2Ω^2_w0[(1+b^2)Ω^2_b0-Ω^2_w0])/(1+b^2)Ω^2_b0-Ω^2_w0)^-1.The early domination of WDM is shown in Fig. <ref>. The left panel of the Fig. <ref> shows the densities of radiation and WDM for different values of parameter b. We can see that for b^2-values higher than ∼ 10^-6 there is no radiation-dominated era. After inflation the universe is always dominated by WDM. Moreover, for any value of b^2-parameter smaller than ∼ 10^-6, the equality between WDM and radiation happens earlier compared to the CDM case. In the right panel of theFig. <ref> one can see the plot for the scale factor dependence of fractional abundances (i.e. Ω_i(a)/Ω_T(a)) for radiation, baryons and WDM (here Ω_T(a) is the total density parameter). The plot in the top panel corresponds to the case b^2=10^-5 and clearly shows that WDM always dominates, while in the bottom panel, for b^2=10^-6, we still have an epoch dominated by standard radiation. In both cases the baryons contribution is subdominant. Consider now the structure formation process, which is strongly dependent on the behavior ofWDM both at the background and perturbative level. The dynamics of WDM perturbations has been described in <cit.>, so we can just write down the main result for the dynamics of WDM perturbations. The energy and momentum balance equation, in Fourier space for each k-mode in flat universelead to following equations:δ̇_w+(1+w)(θ_w+ḣ/2)+3H(c^2_s-w)δ_w+9H^2(c^2_s-w)(1+w)θ_w/k^2 +3Hẇθ_w/k^2 = 0,θ̇_w+H(1-3c^2_s)θ_w -k^2c^2_s/1+wδ_w=0.For the sake of convenience we used synchronous gauge and hence h is the trace of the scalar metric perturbations, δ_w≡δρ_w/ρ_w is the WDM density contrast and θ_w is the velocity.We have followed conventions for metric signature and Fourier transform of <cit.>, and the dot represents derivative with respect to conformal time η and H=ȧ/a. Note that for w=0, CDM case is reproduced in the above equations. The equations are written in the frame which is co-moving to the WDM fluid and hence here was considered the rest-frame sound speed c^2_s <cit.>. We shall consider WDM as adiabatic fluid. Actually, as far as we are dealing with thermal systems, it is possible that some intrinsic non-adiabaticity traces could be present. However, as a first approximation, we suppose that they are negligible. Then one can use the relationδ p_w = c^2_sδρ_w,where c^2_s=ṗ_w/ρ̇_w. Eqs. (<ref>) and (<ref>) require analytical expression for the rest-frame sound speed and the derivative of the state parameter with respect to the conformal time. Using the background quantities it is straightforward to obtain,ẇ=-H/3a^2/(a^2+b^2) c^2_s = w-ẇ/3H(1+w) .In order to solve the system it is necessary to fix initial conditions. The WDM initial conditions can be implemented in the super-horizon regime and deep into the radiation-dominated epoch, i.e., a∝η.In the fluid description, for the early radiation era, WDM case can be described by the following equations:δ̇_w+4/3θ_w+2/3ḣ=0 θ̇_w -k^2/4δ_w =0.By solving equation for h in the super-horizon limit and in the radiation era we arrive to the well-known solution h∝ (kη)^2. With the last solution we found, for the relevant limits, that δ_w =-2/3C(kη)^2 and θ_w=-1/18Ck(kη)^3 are appropriate initial conditions. Of course, equations (<ref>) and (<ref>) are coupled with DE via background solutions and with baryons and radiation both at the background and perturbative level. One has to solve the complete system in order to analyse the consequences of DM warmness for the observables such as, e.g., CMB power spectrum, linear matter power spectrum and the transfer function.§ CONSEQUENCES OF DM WARMNESS VIA RRGIn addition to Eqs. (<ref>) and (<ref>) we need also the equations describing perturbations for baryons and radiation. These equations can be found for example in <cit.> and we will not repeat them here. To integrate the system including baryons, radiation, WDM and cosmological constant, we modify the Boltzmann CAMB code <cit.>. The initial value Ω_b0=0.0223 h^-2 is chosen to provide the agreement with Big Bang nucleosynthesis <cit.>, while Ω_r0 is taken to agree with CMB measurements <cit.>. The free parameters related to WDMare H_0, Ω_w0 and b, and in principle they have as priors 0<H_0<100, 0<Ω_w0<1 and 0<b^2 ≲ 10^-6 to ensure a radiation dominated era. In this way we consider a reduction of this WDM space of parameters by using backgroundobservational tests. Thus, in what follows we limit our analysis to the valuesofWDM parameters such that they are in 1σ CL (confidence level) region of thejoint analysis based on SNIa, BAO and H(z) data. This shall help us to get more realisticand measurablewarmness effects that do not contradict observations, at least at the background level. §.§ Background testsThe backgroundtests related to SNIa, BAO and H_0 are based on the likelihood computed using the χ^2 function,χ^2(θ)=Δ y (θ)^T𝐂^-1Δ y(θ),where θ =(H_0,Ω_m0,b) andΔ y(θ) = y_i-y(x_i;θ). Here y(x_i;θ)represents the theoretical predictions for a given set of parameters, y_i the data and 𝐂is the covariance matrix. Note that, for convenience,the total matter density parameter Ω_m0=Ω_w0+Ω_b0 was used here as a free parameter instead of Ω_w0.In order to perform the background statistical analysis it was used the numerical code CLASS <cit.> combined with the statistical code MontePython <cit.>. For the data set we have used the complete SNeIa data and correlation matrix from the JLA sample <cit.>, H_0 is considered from <cit.>, and for BAO test we have used data from 6dFGS<cit.>, SDSS <cit.>, BOSS CMASS <cit.> and WiggleZ survey <cit.>.The 6dFGS, SDSS and BOSS CMASS data are mutually uncorrelated and also they are not correlated with WiggleZ data, however we must take into account correlation beetween WiggleZ data pointsgiven in <cit.>. The set offree parameters θ can be divided in two parts: the cosmological free parameters Ω_m0, h and b; and the nuissance parametersα, β, M and Δ_M, related to SNe Ia data. The results of the complete statistical analysis is presented in TABLE <ref> and thecontour curves are presented in FIG. <ref>. These results are in agreement with the previous results <cit.> but here we have updated the results anderror was reduced due to the improved quality of observational data in recent years.§.§ Perturbative analysisThe reducedspace of parameters is in agreement with the SNIa+BAO+H(z) data, which we will use to study consequences of the DM warmness in the two relevant observables, namely the structure formation and CMB anisotopies. Before starting the coprresponding consideration, let us illustrate the consequences of the free-streaming of WDM in the matter perturbations.Concerning the structure formation, a relevan quantity is the total matter density contrast, δ_m≡δρ_m/ρ_m=δρ_w+δρ_b/ρ_w+ρ_b.By recalling that for each component δρ_x=ρ_x δ_x, one can write analytic expression for the total matter density contrast in the RRG-based model,δ_m=Ω̃_w(a) δ_w+Ω_b0 δ_b/Ω̃_w(a)+Ω_b0,Ω̃_w0(a)=Ω_w0√(1+b^2 a^-2/1+b^2) .Let us note that in this case, different from ΛCDM, the total matter density contrast depends on the scale factor. Furthermore, after decoupling, the contribution of warm matter to total matter density varies from ∼ 100% for a≪1 (δ_m ≈δ_w) to ∼ 87% when a=1 (i.e δ_m ≈ 0.87δ_w +0.13 δ_b) while in ΛCDM the contribution is always constant and of the order ∼ 85% (i.e δ_m ≈ 0.85δ_w +0.15δ_b).The left panel of the FIG. <ref> shows the total matter density contrast for different scales and for b^2 = 10^-14. In the top panel it is shown δ_m for scale k=2 h Mpc^-1 and in the bottom panel it is shown δ_m for scale k=5 h Mpc^-1. In the first case the difference with CDM case is minimal and ∼ 5% at maximum. However, in the second case, this difference goes to ∼ 20%. These results indicatea strong suppression of the growth of matter perturbations at the small scales, in contrast with the CDM case. Once again, one can see that the RRG enables one to reproduce known features of WDM in a very economic way.One should expect that the suppression in the total matter density contrast caused by DM warmness also appears in the linear matter power spectrum and in its transfer function. The linearmatter power spectrum is computed as P(k)∝ k^n_sT(k)^2, where n_s is the scalar spectral index and T(k) is the transfer function. The transfer function is defined as T(k)≡δ_m(k,z=0) δ_m(0,z=0)/δ_m(k,z→∞) δ_m(0,z→∞) . At the next stage we use our modified CAMB code to compute the linear matter power spectrum and transfer function. The results for b^2=10^-13,10^-14 and 10^-15 are shown in the right panel of the FIG. <ref>.The top right panel shows the linear matter power spectrum while bottom right panel shows the transfer function for these cases. From these plots one can conclude that at large scales there is no much deviation from the ΛCDM result, while at the small scales there is considerably lack of power proportional to the value of warmness b in relation to the ΛCDM. This situation by itself is not new at all, it is regarded as one of the main features of WDM models. However, it is remarkable that one can reproduce itby using the simple RRG description, in a model-independent way and without any supposition about particle physics models.One can wonder how CMB power spectrum is affected by the suppression on matter overdensitiesin small scales. The FIG. <ref> shows the CMB temperature power spectrum for different b-values . One can observe that even with the strong suppression in P(k), the CMB temperature power spectrum is not considerably affected for b^2≲10^-10. At large scales, when l≲ 30, all curvescoincide. The differences only appear at the scales smaller than l ∼ 30. The most of the difference is at the intermediate scales30 ≲ l ≲ 1300. In order to quantify deviations from ΛCDM we compute difference Δ D_l = D^Λ CDM_l-D^Λ WDM_l,D_l =l(l+1)C_l/2πwhere C_l represents the CMB temperature power spectrum. Bottom panel of Fig. <ref> shows Δ D_l. Notice that the maximum difference is ≲ 0.015% and takes place for b^2=10^-10. However, Δ D_l could be slightly higher for the values of b larger than b^2 = 10^-10. Even thoughlarge scalesl ≲ 30 are not influenced by thermal velocities of dark matter, the rest of the spectrum does. It is possible to show that some velocities ≳ 30 km/s (b^2 ≳ 10^-8) produce strong distortions in the interval 30 ≲ l ≲ 1300, such that it would hardly fit the data.It is interesting to compare the RRG-based results with the oneswhich are based on different approaches. In the context of WDM,the effect of the free-streaming on matter distribution is quantifiedbya relative function transferT̅(k) which is defined as T̅(k)≡[P_ΛWDM(k)/P_ΛCDM(k)]^1/2,where P_ΛWDM and P_ΛCDM are linear matter power spectra for ΛWDM andΛCDM cases, respectively. The function T̅(k) canbe approximated by the following fitting expression <cit.>, T̅(k)=[1+(α k)^2ν]^-5/ν,where α and ν are fitting parameters. For the sake of comparison, let us denote the relative function transfercomputed via RRG by T̅_RRG(k). After computing T̅_RRG(k), we perform a fit for Eq. (<ref>) and findparameters α and ν for different values of b. The resultsare summarized in three first entries of Table II. The results show that RRG reproduces the relative transferfunction which is consideredstandard in the WDM framework with accuracy of ≲ 1%, which can be seen in FIG. <ref>.In the left top panel of FIG. <ref> we showT̅_RRG(k) with b^2 =10^-15 and T̅(k)with α=0.0147 and ν=1.12, and in its bottom panel it is shown the relative error between T̅_RRG(k) and T̅(k). The same plotsare shown in right panel of FIG. <ref> for the case where the T̅_RRG(k)was computed with b^2 =10^-14 and T̅(k) was computed with α=0.0242 and ν=1.12. Note that, for both cases,the relative error is ≲ 1%. In what follows we shall consider a more detailed comparison between RRG approach and the wellestablished particle physics candidate for WDM associated to thermal relics. Our comparison shall includesome non-linear features.§ THERMAL RELICS VIA RRGIn the context of thermal relics it is well-known that there is a lowerbound[This limit comes from high redshift Lyman-alpha forest data <cit.>.]of 3.3 keV for the WDM particle mass. Hence it would be interestingto perform a comparison between thermal relics with such a bound and RRG approach. For this reason, it is necessary first to find a equivalence between thermal relics mass scales and RRG b-parameter. Thus we recall that for relics we have ν=1.12and the parameter α, in units of h^-1Mpc, isrelated to the mass scale m_w via<cit.>,α = 0.049 (m_w/1 keV)^-1.11(Ω_w0/0.25)^0.11(0.01 H_0/0.7)^1.22 . In order to obtain a complete association of the RRG parameter b^2 and the mass of WDM particles in the thermal relics context, it was used the following particular set of values for the b^2,b^2=(1× 10^-15,2× 10^-15, …,9× 10^-12,1× 10^-11) .For each pointit was defined a χ^2 function,χ^2=(T̅(k)^th.-T̅(k)^num.)^2 ,where the T̅(k)^th. is given by the equation (<ref>) and the T̅(k)^num. is obtained with theCAMB code for each value of b^2 in the set (<ref>). Then we minimize the equation (<ref>) in order to find the best-fit α-value for each b^2.Using the equation (<ref>) we found, for each value of b^2, the corresponding mass m_w. This correspondence is shown in FIG. <ref>,where the dots indicate the best-fit value for m_w found through the equation (<ref>) and the solid line is the linear regression in the loglogframe. This linear regression results in the following fit-formula,m_w=4.65· 10^-6 (b^2)^-2/5 keV. By using eq.(<ref>) we found that a mass scale of3.3 keV in thermal relics is equivalent tob^2=2.36 × 10^-15∼10^-15 in RRG approach. This value for b^2 bringsdifficulties in distinguishing between WDM and CDM scenarios both at backgroundand linear level. At background this can be seen in the left panel of the FIG. <ref>, where for b^2 ≲ 10^-6 the expansion dynamics aftermatter-radiation equality is indistinguishable from CDM case and also, the best fit value for Ω_m0 is almost the same as ΛCDM (see FIG. <ref>). On the other hand, at linear regime, CMB signal via RRG for b^2≲10^-10is completly indistinguishable of ΛCDM(see FIG. <ref>) and linear matter power spectrum has the expected little suppreession in small scales (see FIG. <ref>). In order to better observe such tiny differences, we can for example, recompute the matter power spectrum in the Fourierspace in its dimensionless form, denoted by Δ^2(k)=k^3/2π^2P(k) . The left panel of the FIG. <ref> shows Δ^2(k) at z=0 for both cases: standard treatment forthermal relicswith mass 3.3 keV and thermal relics via RRG approach withb^2=10^-15. We can see thatCDM and WDM (thermal relics) are indistinguishable untilΔ^2(k) ∼ 30 where Δ^2(k) falls off too rapidly.In fact, in the context of thermal-like candidates, this feature has been used as justification to claim that albeit such a cut-off is still consistent with constraints based on Lyman-α forest data, for masses larger than 3.3 keV, the WDM appears no better than CDM in solving the small scaleanomalies <cit.>. Of course, this affirmation needs to be better investigated in the context of others WDM candidates and, if possible,in a model-independent way. Thus, we believe that RRG can be very helpful in such direction.On the other hand, we can also compute the time scale where the perturbations reach the non-linear regime. In the right panel of the FIG. <ref> it isshown the time (redshift) scale z_nl in which the matter perturbationscale R reachs the non-linear regime. z_nl is the redshift where σ^2(R)=1, being σ^2(R) the mass variance at the scale R. Againboth cases are considered: standardtreatment forthermal relicswith mass 3.3 keV and thermal relics via RRG approach withb^2=10^-15. We can see that, in WDM thermal-like candidates context, the non-linearity is reached more recently than in ΛCDM for scales R≲1Mpc h^-1.Finally, by using Δ^2(k), z_nl and a top-hat filter normalized with the results from Planck <cit.>it is possible to obtain the density of halos.This quantity gives us the number of collapsed objects above agiven mass M. Here, the mass function was computed usingthe method of <cit.>. In FIG. <ref>, the mass function obtained followingthe RRG approachis compared with the one computedfor thermalrelics in the standard way and with CDM case. Clearly in WDM context, there is less collapsed objetcs than inΛCDM. Furthermore, from FIGS. <ref> and <ref> it is possible to see the huge similarity between the RRGapproach and the WDM thermal-like standard description also in the non-linear regime. In fact, the relative error in both casesis ≲ 1%. Our results in this and in the previous sectionindicate thatRRG approach is good enough to capture important featuresof WDMin linear regime. Specially the suppression on small scales structures and lack of power in matter spectrum at such scales. Also, in theparticular case of thermal relics, RRG reproduces with high precision,the potentialities and weakness of the candidate in both linear andnon linear regime. Thus, RRGapproach could be considered as a complementary alternative approach to investigatewarm matter and specially for understanding general behavior of the WDM scenario in a model-independent way. § DISCUSSION AND CONCLUSIONSWe have shown that the RRG approach enable one to model WDM, which is treated as a gas of particles with non-negligible thermal velocities. The simplifying aspect of RRG is that all such velocities are taken to be equal. The presence of warmness produce consequences on the dynamics of the universe both for the background and perturbations.For the background the most important is that radiation era may be smearing out for greater warmness. In this case the universe is dominated by WDM up to the DE dominated era. This scenario could have deep impact on the primordial nucleosynthesis, reionization, recombination and other effects in the early universe. This new non-standard scenario may be deserving more careful and specific investigation, which we leave for the future work. Our analysis here was limited by the relatively small warmness, with b^2 ≲ 10^-6. In this case the radiation dominated era is still maintained, but the radiation-matter equality takes place before than in the CDM models. After the equality point, there are no serious differences with expansion is dominated by cold matter, as it is shown at the left panel of FIG. <ref>.Instead of dealing with the full space of the WDM parameters, we restrict consideration to a more reduced set. At the backgroundlevel this is achieved by using recent data from SNIa, H_0 andBAO. As a result we arrive at reduced space which does notcontradict more recent background observations at 1σ CL, i.eb∈ [0,2.1× 10^-5] and Ω_m0=[0.27,0.35]. Taking this into account one can expectthat the quantification of imprint of warmness on observables shouldbecome more significant. Our analysis in this reducedspace ofparameters shows that velocities which satisfy ν≲ 3 km/swould agree with the CMB observations. This limit is essentiallysmaller than the bound for HDM, which may have velocities which are just two order of magnitude smaller than the speed of light. Thevelocities bound which were found here agree with the ones foundearlier by other approaches<cit.>. This fact shows that, regardless of that the RRG is technically simple,it is a sufficiently reliable approach to probe new physics within theWDM approach. The consequences of a warmness of DM are is more evident in the dynamics of cosmic perturbations. Since DM is supposed to be themain source of forming gravity potentials and overdensities, theimpact of the DM warmness on structure formation and CMBanisotropies is evident. In the case of WDM thermal velocities causefree-streaming out from overdense regions, delaying and inhibiting thegrowth of fluctuations at certain scales. Another way to interpret thiseffect is by relating velocity to pressure. The non-negligible pressureof WDM, together with the radiation pressure, are resisting thegravitational compression and therefore suppress the power. Thiseffect is stronger in small scales, as can be seen in Fig. <ref>. Furthermore, from Fig. <ref> one can conclude thatthermalvelocitiesdo not affect considerablythe CMB temperature powerspectrum for b≲ 10^-10. Indeed, the situation canbe opposite for sufficiently large values of b.As a next step we reproduced features of thermal relics by using RRG prescription. First it was necessary tofind a relation between the mass scale in relics context and the b^2 parameter of RRG. The b^2 ∼ 10^-15 equivalent to the lower bound known for thermal relics (3.3 keV) brings the necessity to look more carefully matter perturbations. Thus, we computed the matter perturbations in the Fourierspace Δ^2(k), the time scale where the perturbations reach the non-linear regime (z_nl) and the mass function. In all those cases, our results indicate thatRRG approachis good enough to capture important featuresof WDMeven in non-linear regime.Therefore, wehave proved that RRG is a reliable model,and can be considered asa complementary, greatly simplifiedalternative approach to investigate warm matter, in particular for understanding the behavior of theWDMin a totally model-independent way.One can foresee the possibility of detailed investigations of the newParticle Physics candidates to WDM, which includes the comparisonwith model-independent RRG. In our opinion some aspects of thispossibility would be quite interesting. Forexample, let us mentiona relation between T̅_RRG(k) and WDM candidate models,a more complete and comprehensive exploration of the space of parameters via Markov Chain Monte Carlo (MCMC) or verifying how RRG would work inthe nonlinear regime of structure formation throughnumerical simulations and the possibility ifWDMis really capable of solve small scale problems. The work on these aspects of the model is currently in progress.§ ACKNOWLEDGMENTSAuthors are grateful to Winfried Zimdahl for useful discussions. J.F. and R.M. wish to thankCAPES, CNPq and FAPES for partial financial support. I.Sh. acknowledges the partial support from CNPq, FAPEMIG and ICTP.ieeetr | http://arxiv.org/abs/1706.08595v3 | {
"authors": [
"W. S. Hipolito-Ricaldi",
"R. F. vom Marttens",
"J. C. Fabris",
"I. L. Shapiro",
"L. Casarini"
],
"categories": [
"astro-ph.CO",
"gr-qc"
],
"primary_category": "astro-ph.CO",
"published": "20170626210513",
"title": "On general features of warm dark matter with reduced relativistic gas"
} |
zju]Xin Wang [email protected]]Lejun Zouzju]Xiaohua Shen [email protected]]Yupeng Renzju]Yi Qin[zju]Department of Earth Sciences, Zhejiang University Yuquan Campus, 38 Zheda Road, Hangzhou 310027, ChinaConventional manual surveys of rock mass fractures usually require large amounts of time and labor; yet, they provide a relatively small set of data that cannot be considered representative of the study region. Terrestrial laser scanners are increasingly used for fracture surveys because they can efficiently acquire large area, high-resolution, three-dimensional (3D) point clouds from outcrops. However, extracting fractures and other planar surfaces from 3D outcrop point clouds is still a challenging task. No method has been reported that can be used to automatically extract the full extent of every individual fracture from a 3D outcrop point cloud. In this study, we propose a method using a region-growing approach to address this problem; the method also estimates the orientation of each fracture. In this method, criteria based on the local surface normal and curvature of the point cloud are used to initiate and control the growth of the fracture region. In tests using outcrop point cloud data, the proposed method identified and extracted the full extent of individual fractures with high accuracy. Compared with manually acquired field survey data, our method obtained better-quality fracture data, thereby demonstrating the high potential utility of the proposed method.Outcrop fracture surveys; Terrestrial laser scanner; LiDAR; Point cloud; Automatic extraction; Region-growing-based algorithm § INTRODUCTION The manual surveying of fractures and other planar rock mass surfaces is one of the most fundamental but time-consuming activities performed by field geologists. The surveyed fracture data usually comprise the fracture location, orientation, and surface roughness, which can support models and/or hypotheses in various applications (e.g., structural and geomechanical analysis, flow modeling, reservoir characterization, and engineering rock mass classification). These surveys are conventionally performed in situ with standard fieldwork instruments, such as a handheld compass, clinometer, and possibly a digital camera to record the fracture locations. However, the development of remote sensors (e.g., LiDAR-based scanners) and their availability as research equipment have prompted geoscientists to develop new methods that improve the analysis, avoid access problems, reduce time and labor, and result in a more representative dataset. The terrestrial laser scanner (TLS) is one of the most widely used instruments in Earth science applications, and it is very useful for acquiring high-quality, high-resolution, three-dimensional (3D) point clouds from outcrops <cit.>. In addition, the GPS receiver module in a typical TLS allows the point cloud, a set of points in a 3D coordinate system, to be transformed into different geographic coordinate systems, so the data can be processed and used for different purposes, such as topographic feature extraction and orientation estimation for planar surfaces.During the last few years, because of the widespread use of TLS in Earth science applications, there has been a growing need for point cloud processing methods to perform analyses and interpretation. The extraction of fractures and other planar surfaces from 3D outcrop point clouds has been the focus of much research by the geological community because fracture data have a wide range of applications. Many semi-automatic and automatic methods have been developed in the last 10 years, and certain algorithms can be used to extract (or segment) the points of fracture surfaces from the point cloud. <cit.> and <cit.> derived triangulated irregular networks from the point cloud and then grouped neighboring polygons with a similar orientation to obtain planar features. <cit.>, <cit.>, and <cit.> used random sample consensus (RANSAC) algorithm-based methods to segment point clouds into subsets, each of which comprise points that belong to the same discontinuity surface. Recently, other methods have also been proposed based on k-means clustering <cit.>, moving sampling cube <cit.>, point attributes <cit.>, neighboring points coplanarity testing <cit.>, and principal component analysis (PCA) <cit.>. However, using these methods, either the full extent of the individual fracture surface is not extracted <cit.> or human supervision is required <cit.>. The automatic method proposed by <cit.> splits the point cloud into four subsets (quadrants) iteratively to detect planar structures, but the full extent of the individual fracture surface was not fully extracted and was detected as several planar structures in most cases.In this study, we propose an algorithm using a region-growing approach for the automatic extraction of the full extent of individual outcrop fractures from point clouds and for estimating their orientation. The main novel feature of this algorithm is the application of region growing to the extraction of outcrop fractures from point clouds. Instead of growing the region locally without a global view of the fracture surface, we use a seed point selection criterion to consider the overall fracture occurrence, as well as criteria for determining the initial seed point and controlling the growth of the region. The region-growing concept is simple, and by using carefully designed criteria, our algorithm can extract the full extent of every individual fracture in an automatic and robust manner.§ STUDY AREA AND DATABASEThe study site is a road-cut rock slope located along a country road in Nanbaoxiang, Chengdu, Sichuan Province, China (N 30^∘ 24' 25.35”, E 103^∘ 11' 8.34”) (fig:figure_1). The rock slope mainly comprises thin to thick layered sandstone, and the area of the study outcrop is about 30 m^2. A RIEGL VZ-1000 terrestrial laser scanning system (mainly comprising a 3D laser scanner, digital camera, and GPS receiver) was used to perform a high-resolution LiDAR scan of the rock slope. This TLS system uses the time-of-flight technique, which utilizes the emission and return time of highly collimated electromagnetic radiation to calculate the distance from the instrument's optical center to a reflecting target surface <cit.>. An outcrop 3D point cloud was acquired with an average point spacing of < 1 cm; there were about 21 million points. To test the proposed region-growing-based algorithm, we selected the central part of the point cloud (delineated by the white rectangle in fig:figure_1), where less vegetation and fallen stone were present. Conventional measurements of fracture surface orientations using a handheld compass were also performed on the rock slope to compare with the results obtained by the proposed algorithm, and 65 orientation measurements (dip direction and dip angle) were acquired from fracture faces distributed over the outcrop.§ METHODOLOGY Region growing is an element-based segmentation method, which has many advantages compared with other methods. The concept of region growing is simple; only a small number of seed points and a few criteria are required to grow the region. Region growing can correctly segment regions that share the same defined properties. The seed points and the criteria can be selected freely to suit different applications. A drawback of region growing is that it lacks a global view of the problem; however, this drawback can be addressed by selecting criteria that consider the global view of the problem during the region growing, and this we do in our proposed algorithm.The proposed region-growing-based algorithm works directly with the LiDAR point cloud instead of using an interpolated 2.5D mesh surface. Many detailed geometrical features extracted from the point cloud of the outcrop surface can be used for the segmentation, but the proposed algorithm uses mainly the local surface normal and curvature.The proposed method comprises three main steps, as follows. First step: Local surface normal and curvature estimation, which involves a nearest-neighbor search as well as the estimation of the least-squares fitting plane and the curvature of the neighboring points. This task is described in sec:normal_and_curvature_estimation. Second step: Region growing, which extracts the fracture face by using criteria based on the local surface normal and curvature to select the seed points and to control the growth. This step is explained in sec:region_growing. Third step: Fracture orientation estimation, which employs a patch of the point cloud after its growth is complete. This part is described in sec:orientation_estimation. §.§ Local surface normal and curvature estimationOne or more properties of each point in the point cloud are required for region-growing segmentation, i.e., segmenting the regions that comprise points with similar defined properties. The local surface normal and the local surface curvature are two basic properties that can be used to define planar surfaces such as fractures.To estimate the local surface normal and curvature for each point, its neighboring points, which together form the local topography, are needed. We refer to a point cloud as P, a collection of 3D points p_i = {x_i, y_i, z_i}∈ P. Let p_q be the query point in the problem of estimating the local surface normal and curvature, and let P^k be the K-nearest neighbors of p_q, in which k is chosen by the user to find the k nearest neighbors according to their Euclidean distance to p_q.The method we use for estimating the local surface normal is based on least-squares plane fitting with P^k, as proposed by <cit.>. The least-squares plane fitting method is based on PCA. The local surface normal n_q of point p_q is obtained by analyzing the eigenvalues and eigenvectors of P^k's covariance matrix C = 1/k∑_i=1^k (p_i - p̅) · (p_i - p̅)^T, where p_i ∈ P^k and p̅ = 1/k·∑_i=1^k p_i. If we let λ_0, λ_1, and λ_2 be the eigenvalues of C that satisfy 0 ≤λ_0 ≤λ_1 ≤λ_2 and if v_0 is the corresponding eigenvector of λ_0, then n_q = v_0if v_0 · (v_p - p_q) > 0-v_0if v_0 · (v_p - p_q) < 0 , where v_p is the viewpoint from which the point cloud is acquired.The method used to estimate the local surface curvature was proposed by <cit.>. This method can estimate the curvature directly from the eigenvalues of P^k's covariance matrix C without needing to first create a surface from the point cloud. The local surface curvature σ_q of p_q is estimated as follows. σ_q = λ_0/λ_0 + λ_1 + λ_2 §.§ Region growingLet P_r⊂ P be the set of points that have not yet been assigned to any fracture regions. For each fracture region, the initial seed point that starts this region's growth is selected from P_r, and the point p_min∈ P_r with the minimum curvature is selected as a reasonable initial seed point for planar surfaces such as fractures.Next, the criterion that controls the growth from the seed points to their neighboring points is defined as the local surface normal deviation threshold θ_th given by the user. For the neighboring point p_i, the local surface normal is n_i, and the seed point's local surface normal is n_s, so p_i is added to the current region if cos^-1(n_s·n_i/‖n_s‖‖n_i ‖) < θ_th, i.e., if the angle between n_s and n_i is less than θ_th.The new seed points are then selected from the newly added points. The criterion defined as the transmission error threshold t_th, which is also given by the user, determines whether the newly added p_i is selected as a new seed point. For the newly added p_i, the local surface normal is n_i and the initial seed point p_min's local surface normal is n_min, so the newly added p_i is selected as a new seed point if cos^-1(n_min·n_i/‖n_min‖‖n_i ‖) < t_th, i.e., if the angle between n_min and n_i is less than t_th. Therefore, the overall occurrence of the fracture, which is represented by n_min, serves to control the growth of the region instead of its being allowed to grow blindly.The region's growth from the newly selected seed points and the selection of new seed points are then performed iteratively until no new seed point can be selected and the region's growth is complete. The growth of the other regions is completed for those remaining in the point cloud until all of the points in the point cloud have been processed.The local surface normal deviation threshold θ_th and the transmission error threshold t_th that yield the best segmentation result are related to the fracture surface's geometrical nature and the weathering condition of the outcrop. According to their definitions and the functions described above, θ_th is related to the local roughness, whereas t_th is related to the overall flatness of the fracture. For example, if a flat fracture has a rough local surface, then θ_th should be sufficiently large to allow small protrusions and dents in the fracture region. If the uneven fracture has a smooth local surface, such as weathered fracture surfaces, then t_th should be sufficiently large to allow the uneven fracture to grow into one region. The flow chart for the region-growing step of the proposed method is shown in fig:flowchart. §.§ Fracture orientation estimationFor a patch of a point cloud that has completed its growth, many features of the extracted fracture can be estimated, such as the fracture orientation. The fracture orientation can be estimated as the normal ⟨ n_x, n_y, n_z ⟩ of the least-squares fitting plane in the point cloud patch that represents the fracture. The normal can also be transformed into the ⟨ Dip direction, Dip ⟩, and the transformation may vary with the geographic coordinate systems employed for the point cloud. For example, if the y-axis of the coordinate system points north, the x-axis points east, and the z-axis points vertically up, then the Dip direction and Dip will be as follows. Dip direction =0^∘ ifn_x = 0 & n_y ≥ 0180^∘ ifn_x = 0 & n_y < 090^∘ - tan^-1(n_y/n_x)ifn_x > 0270^∘ - tan^-1(n_y/n_x)ifn_x < 0Dip =0^∘ ifn_x^2 + n_y^2 = 090^∘ - tan^-1(| n_z |/√(n_x^2 + n_y^2))ifn_x^2 + n_y^2 ≠ 0 § RESULTS AND DISCUSSION The point cloud used to test our algorithm and the conventional measurements of the fracture surface orientations for the same outcrop were described in sec:study_area_and_point_cloud_acquisition. The application of the proposed algorithm to the entire point cloud will provide a great number of planar regions with various dimensions, and this makes it difficult to show the detailed results obtained by the proposed algorithm. Therefore, before applying the proposed algorithm to the entire point cloud, we tested the algorithm with a portion of the point cloud, as described in sec:details_of_the_result. The results obtained for the entire point cloud and comparisons with manual field survey results are discussed in sec:overall_result_and_its_comparison. The performance of the proposed algorithm in terms of time consumption was tested using a set of outcrop point cloud data, and the number of planes detected from the same point cloud data with different configurations of θ_th and t_th was also investigated. These are discussed in sec:performance_and_parameter_configuration. §.§ Details of results for a portion of the point cloudA portion of the outcrop (fig:local_point_cloud_segmentation_resulta and its point cloud fig:local_point_cloud_segmentation_resultb) was processed using our method. The local surface normal deviation threshold θ_th was set to 6^∘, and the transmission error threshold t_th was set to 20^∘, which were tuned to obtain the best results.These settings for θ_th and t_th can be applied to similar outcrop conditions. fig:local_point_cloud_segmentation_resultc is the result of the proposed algorithm; it shows the fracture regions having more than 800 points so that they can be conveniently compared with the manually identifiable fractures in fig:local_point_cloud_segmentation_resulta. The threshold value of 800 can be modified if smaller or larger fracture regions are required. Different fracture regions are indicated by different colors; the non-fracture regions are shown in red.The results show that most of the fractures extracted by the proposed algorithm could be identified as real fractures in fig:local_point_cloud_segmentation_resulta and also that most of the major fractures identified in fig:local_point_cloud_segmentation_resulta were extracted by the proposed algorithm. In addition, the results show that unlike other existing methods, the proposed region-growing-based algorithm can extract the full extent of every individual fracture automatically.The estimated fracture planes and the least-squares fitting planes for the extracted fracture regions are shown in fig:local_point_cloud_segmentation_resultd. The fracture orientations were estimated using these planes according to the methods described in sec:orientation_estimation. §.§ Results for entire point cloud and comparison with manual field survey resultsWe applied the proposed method, with the local surface normal deviation threshold θ_th set to 6^∘ and the transmission error threshold t_th set to 20^∘, to the entire point cloud (fig:overall_point_cloud_segmentation_resulta). The resulting fracture regions having more than 100 points are shown by different colors in fig:overall_point_cloud_segmentation_resultb, and the non-fracture regions are shown in red.The results demonstrate that our proposed algorithm can extract many small fracture faces, cases for which conventional measurements cannot be obtained. Thus, besides accurately extracting fractures in the same manner as a conventional manual survey as demonstrated by the detailed results shown in sec:details_of_the_result, our method may also provide additional information about fractures (particularly small fractures) that cannot be acquired from conventional manual surveys.We extracted 157 fracture regions having more than 100 points and estimated their orientations. To compare the performance of the manual field survey and our proposed algorithm, we stereographically projected 65 orientations from the manual field survey and 157 orientations from the results obtained by the proposed algorithm and plotted the density of their poles (fig:field_contour_auto_contoura and fig:field_contour_auto_contourb, respectively).A comparison of the pole density plots shows that the major clusters of poles representing different sets of fractures from the field manual survey, such as f_1,...,f_8 in fig:field_contour_auto_contoura, can also be found in the results obtained using our algorithm (a_1,...,a_8 in fig:field_contour_auto_contourb). The comparison also shows that the proposed algorithm has the advantage of locating clusters of fracture sets more accurately. For example, if many fractures are perpendicular to the bedding (as was found at the study site considered), then the stereographic plot poles of those fractures should be near the arc of the bedding, and obviously the proposed algorithm has better descriptions than the manual field survey. Thus, the red arc of bedding in fig:field_contour_auto_contourb fits the distribution of poles better than the red arc of bedding in fig:field_contour_auto_contoura, which indicates that our algorithm was better at locating the cluster of bedding and the other clusters of fracture sets. Therefore, in general, our algorithm was able to obtain fracture data whose quality was as good as or better than that from the manual field survey.Furthermore, our algorithm provides additional information about fractures, which may be useful for the analysis. For example, clusters a_9, a_10 and a_11 in fig:field_contour_auto_contourb, as well as the symmetries between a_6 and a_7, a_9 and a_10, a_6 and a_10, and a_7 and a_9, may be interesting information that merits further discussion and study.The only disadvantage of the proposed algorithm may be the presence of some possible outlier clusters in the fracture sets (examples can be seen in fig:field_contour_auto_contourb). The nature and removal of these outliers should be studied in future work. §.§ Performance and parameter configuration of the proposed algorithmA set of point cloud data (tbl:datasets_for_performance_testing) was used to test the performance of our algorithm using a desktop computer with a CPU of 3.60 GHz and 4 GB RAM. The average point spacing in all the point cloud datasets used was 0.01 m, the dataset size ranged from 134,067 points to 1,096,948 points, and the outcrop areas ranged from 5.5 m^2 to 67.3 m^2. The same processing parameters were used for all the point cloud datasets to highlight the variation in the time consumption.The performance testing result is shown in fig:performance. The figure shows that there was a steep increase in time consumption as the point cloud size reached 1 million points, but the time consumption is still acceptable. Therefore, we conclude that our algorithm is suitable for point cloud datasets whose outcrop area ≤ 70 m^2. For datasets larger than that, the computing power should be increased or the algorithm should be modified.Using the same point cloud as the one shown in fig:overall_point_cloud_segmentation_resulta, we investigated the number of planes detected using different configurations of θ_th and t_th. The results are shown in fig:p2theta_p2t, which shows that an increase in θ_th or t_th resulted in a decrease in the number of planes detected. It is interesting to note that θ_th = 6^∘ in fig:p2theta_p2ta and t_th = 20^∘ in fig:p2theta_p2tb, the configuration we judged to yield the best results, are turning points: before these points, the number of planes detected decreases quickly; after these points, the number of planes detected decreases much more slowly. As we know, small changes in the configuration of θ_th and t_th should not greatly influence the number of planes detected, so the point cloud may be over-segmented before these turning points. Thus, an analysis of the number of planes detected under different configurations of θ_th and t_th may help find the configuration that yields the best results.§ CONCLUSION In this paper, we have proposed an innovative region-growing-based method for automatically extracting outcrop fractures from 3D point clouds. Two local topographic features of the point cloud, i.e., the local surface normal and curvature, are used to define planar surfaces such as fractures. By their definitions, the local surface normal deviation threshold θ_th and the transmission error threshold t_th are designed to control the growth of the fracture regions; t_th considers the overall occurrence of the fracture while controlling the region growth so that it is not allowed to grow blindly. The orientations are estimated for each extracted fracture.We tested the proposed method using a 3D point cloud acquired for a real outcrop at the study site, and the results obtained were compared with data collected by a manual field survey for the same outcrop. The test results showed that unlike the existing automatic or semi-automatic methods, the new algorithm can extract the full extent of every individual fracture automatically and accurately. The comparison between our method and the manual field survey shows that the proposed region-growing-based algorithm can obtain fracture data whose quality is as good as or better than that of the manual field survey, thereby demonstrating the potential utility of our method. The performance test using a set of point cloud data showed that the proposed algorithm is suitable for point cloud datasets whose outcrop area ≤ 70 m^2. The analysis of the number of planes detected under different configurations of θ_th and t_th helped explain the configuration we had judged to yield the best results; such analysis may provide a way to find the configuration that yields the best results.Further research should focus on improving the proposed method by removing possible non-fracture regions (outlier clusters in the fracture sets) and analyzing the results obtained by the region-growing-based algorithm, such as the relationship between the roughness of the fracture and the weathering condition, fracture type, and orientation, as well as assessing the performance of the proposed method with different rock types and weathering conditions.§ ACKNOWLEDGMENTS This study was funded by the Chinese National Science and Technology Major Project (2011ZX05009001). We are grateful to Prof. Changjiang Li for valuable comments on earlier drafts.We would like to thank the editor and two anonymous reviewers for their valuable comments and suggestions, which have improved the paper. | http://arxiv.org/abs/1707.03266v1 | {
"authors": [
"Xin Wang",
"Lejun Zou",
"Xiaohua Shen",
"Yupeng Ren",
"Yi Qin"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170627025143",
"title": "A region-growing approach for automatic outcrop fracture extraction from a three-dimensional point cloud"
} |
[email protected] Institute of Science and Technology, Federal University of São PauloAvenida Cesare G. M. Lattes 1201, 12247-014 São José dos Campos, SP, BrazilIf Ψ is a quaternionic wave function, then iΨ≠Ψ i. Thus, there are two versions of the quaternionic Schrödinger equation (QSE). In this article, we present the second possibility for solving the QSE, following on from a previous article. After developing the general methodology, we present the quaternionic free particle solution and the scattering of the quaternionic particle through a scalar barrier. QuaternionicQuantumParticles: New SolutionsSERGIO GIARDINO December 30, 2023 ===============================================- 0.3mm§ INTRODUCTION Quaternions <cit.>, represented by ℍ, are hyper-complex numbers with three anti-commuting complex units. In general, ifq∈ℍ, thenq=x_0+x_1 i+x_2 j+ x_3 k,where x_0, x_1, x_2 and x_3 are real and the complex units i, j and k satisfyij=-ji=k ijk=-1.Using a notation called symplectic, we express (<ref>) asq=z +ζ j z, ζ∈ℂ.Quantum mechanics is a physical theory based on complex numbers. A generalization that replaces complexes with quaternions has been attempted, and Stephen Adler's textbook <cit.> surveys this endeavor. In this formulation, quaternionic quantum mechanics (QQM) deploys anti-hermitian operators (AHO), instead of the usual hermitian operators of complex quantum mechanics (CQM).An anti-hermitian operator 𝒜 satisfies 𝒜^†=-𝒜, where 𝒜^† is the adjoint operator of𝒜. QQM has been formulated in terms of AHO, among other reasons, to preserve the conservation of the probability current.In spite of this advantage, the adoption of AHO has several drawbacks. The most visible is the breakdown of Ehrenfest's theorem <cit.>. The lack of simple solutions in anti-hermitian QQM is another drawback. Of course, there are anti-hermitian quaternionic solutions, like <cit.>, but they are difficult to grasp and do not have the simplicity that allows them to be compared to either CQMor classical solutions.Experimental tests have been performed as well <cit.>, but no quaternionic effect has been observed at the present timeMore recently, novel alternatives have been attemptei to build a consistent QQM. One may replace usual QQM wave functions with regularquaternionic functions <cit.>, and maintain anti-hermitian formalism. A more radical approach emerged after the discovery of quaternionicsolutions obtained throughout non-anti-hermitian (NAH) Hamiltonians in the study of the quaternionic Aharonov-Bohm (AB) effect <cit.>. This discovery has enabled a formal expression of an NAH-QQM, where the probability current and the expectation value are redefined <cit.>. Following these results, a solution for the QSE was developed, and the first quaternionic particle solution was obtained <cit.>. Because of the non commutativity between quaternionic functions and the complex unit i, there are twopossibilities for the QSE, according to the position of i. The first possibility has already been considered in <cit.>, and in this article we entertain the second possibility. This article is organized as follows: Section <ref> presents a general time-dependent solution for QSE, whereas the time-independent QSE is solved in Section <ref>. In Section <ref>, the solution method is used in several simple situations and, in Section <ref>, the quaternionic free particle is obtained. Section <ref> describes the scattering of the quaternionic free particlethrough a scalar step potential, while section <ref> rounds off the article with our conclusions and future perspectives.§ TIME-DEPENDENT EQUATION The time-dependent quaternionic Schrödinger is i ħ ∂Ψ/∂ t=ℋΨ,where Ψ is a quaternionic wave function and ℋ is a quaternionic Hamiltonian. We comment on the position of the complex unit i on the left side of the time-derivative of Ψ in (<ref>). We separate the time variable of the wave function asΨ( x, t)=Φ( x)Λ(t),where Φ( x) and Λ(t) are both quaternionic. In a complex wave function, a complex exponential, which is a unitary complex, carries the time dependence. By analogy, the time-dependent function Λ of the quaternionic wave function will be chosen to be a unitary quaternionic function. Accordingly, in symplectic notation (<ref>), we getΛ=cosΞe^iX+sinΞe^i Υj,ΛΛ^*=1.Ξ, X and Υ are time-dependent real functions and Λ^* is the quaternionic conjugate of Λ. In order to eliminate the dependence on time from the wave equation, we multiply the right hand side of (<ref>) by Λ^* andimposeΛ̇Λ^*=κ/ħ,where the dot denotes a time derivative and κ∈ℍ is the separation constantκ=κ_0+κ_1 j,κ_0, κ_1∈ℂ. Consequently, from (<ref>) we geti( Ẋcos^2Ξ+Υ̇sin^2Ξ)+[Ξ̇+isinΞcosΞ(Υ̇-Ẋ)]e^i(X+Y)j=κ/ħ.We can obtain several solutions for (<ref>). For Ξ̇=0, Ẋ=Υ̇=-iκ_0 and κ_1=0, we obtainΛ=exp[-iℰ/ħt]Λ_0,Λ̇Λ^*=-iℰ/ħ,where ℰ is the energy and Λ_0 is a unitary quaternionic constant that can multiply both sides of the complex exponential indifferently. The time solution (<ref>) is very similar to the complex case; conversely, we achieved something more interesting by imposing Ξ̇=0 and X+ Υ=τ_0, where τ_0 is a real constant. Thus, we getΛ={cosΞexp[-iℰ/ħt]+sinΞ exp[i(ℰ/ħt+τ_0)]j}Λ_0,Λ̇Λ^*=iℰ/ħ(-cos2Ξ+sin2Ξ e^iτ_0j).We remark that (<ref>) may be written schematically asΛ=Λ_1exp[-iℰ/ħt]Λ_0where Λ_1 is an arbitrary constant quatenion. Solution (<ref>) is absolutely new, it recovers (<ref>) when sinΞ=0, but the value of the constant quaternion contributes to the eigenvalue. This kind of influence of a constant over the eigenvalue of an eigenfucntion is unknown in CQM. We expect that novel quantum solutions, unattainable through CQM mayemerge. Nevertheless, there is another simple possibility for time-dependent solutions. If Ẋ=Υ̇=0, we achieveΛ=[cos(ℰ/ħt)e^-iX+ sin(ℰ/ħt)e^i(X+τ_0)j]Λ_0Λ̇Λ^*=ℰ/ħ e^iτ_0 j.In the same fashion as (<ref>), the solution given by (<ref>) is absolutely new, it does not have a complex counterpart and a complex limit is meaningless. If something physical may be described with it, it is probably unknown to CQM.§ GENERAL TIME-INDEPENDENT EQUATION Using (<ref>-<ref>), the time-independent Schrödinger equation readsℋΦ=i Φ κ.In order to solve (<ref>), we need several assumptions. First of all, we propose the spatial wave functionΦ=ϕ λ,where λ is a time-independent quaternionic function given byλ=ρ K, |λ|=ρ, K=cosΘe^iΓ+sinΘe^i Ωj ,KK^*=1;and ϕ is a time-independent complex solution of Schrödinger equation with energy E, so that ℋϕ=Eϕ, where the energy E is real. Consequently, the Hamiltonian ℋ is the hermitian operatorℋ=- ħ^2/2m∇^2+V,where V is a real scalar potential. More general hamiltonian operators, with complex potentials and complex energies,are interesting directions for research and they may be examined in a separate article. Using (<ref>-<ref>), (<ref>), we get∇^2λ+2/ϕ∇ϕ·∇λ=2m/ħ^2(Eλ -iλκ).We also adopt∇ K= pe^iΓ+ qe^iΩj∇^2 K=ue^iΓ+ ve^iΩj,wherep=-sinΘ ∇Θ+icosΘ ∇Γ, q= cosΘ ∇Θ+i sinΘ ∇Ω, u= -cosΘ ( |∇Γ|^2+|∇Θ|^2 )-sinΘ ∇^2Θ +i(cosΘ ∇^2Γ-2sinΘ ∇Γ·∇Θ) v=-sinΘ ( |∇Ω|^2+|∇Θ|^2 )+cosΘ ∇^2Θ +i(sinΘ ∇^2Ω+2cosΘ ∇Ω·∇Θ).The complex and quaternionic parts of (<ref>) give1/ρ(∇+2/ϕ∇ϕ)·∇ρ+ 2/ρϕ∇(ρϕ)· p/cosΘ+u/cosΘ= 2m/ħ^2[E-iκ_0+iκ^*_1tanΘ e^i(Ω-Γ)] 1/ρ(∇+2/ϕ∇ϕ)·∇ρ+ 2/ρϕ∇(ρϕ)· q/sinΘ+v/sinΘ= 2m/ħ^2[E-iκ_0^*-iκ_1Θ e^i(Γ-Ω)].Specific values chosen for κ furnish the three time-dependent solutions (<ref>-<ref>). Four real equations are obtained from the two complex equations (<ref>-<ref>). First of all, we note thatiκ_0∈ℝκ_1=|κ_1|e^iτ_0.Thence, we define1/ρ(∇+2/ϕ∇ϕ)·∇ρ=𝒵_0,2/ρϕ∇(ρϕ)· p/cosΘ=𝒵_12/ρϕ∇(ρϕ)· q/sinΘ=𝒵_2,where 𝒵_0, 𝒵_1 and 𝒵_2 are complex functions. We finally separate (<ref>-<ref>) into real components, so that(𝒵_0+𝒵_1)-|∇Γ|^2-|∇Θ|^2-tanΘ∇^2Θ = 2m/ħ^2(E-iκ_0+|κ_1|tanΘsin W) (𝒵_0+𝒵_2)-|∇Ω|^2-|∇Θ|^2+ Θ∇^2Θ= 2m/ħ^2(E+iκ_0+|κ_1|Θsin W) (𝒵_0+𝒵_1)+(∇-2tanΘ∇Θ)·∇Γ= 2m/ħ^2|κ_1|tanΘcos W(𝒵_0+𝒵_2)+(∇+2Θ∇Θ)·∇Ω=-2m/ħ^2|κ_1|Θcos Wwhere W=Γ-Ω+τ_0,and (𝒵) and (𝒵) are the real and the imaginary components of a complex 𝒵, respectively.Equations (<ref>-<ref>) are the most general time-independent solutions obtained from the three time-dependent cases.In the following section, we examine several simple solutions, and leave more complicated cases for future research.§ WAVE FUNCTIONSAssuming an arbitrary one-dimensional complex wave function ϕ and also ∇Θ= 0, we impose theconstraints ∇ϕ·∇ρ=0,∇ϕ·∇Γ=0,∇ϕ·∇Ω=0,∇ρ·∇Γ=0,∇ρ·∇Ω=0,remembering that ∇Γ·∇Ω≠ 0. Thus, (<ref>-<ref>) turn into two equations1/ρ∇^2ρ-|∇Γ|^2=2m/ħ^2(E-iκ_0+|κ_1|tanΘsin W)1/ρ∇^2ρ-|∇Ω|^2=2m/ħ^2(E+iκ_0+|κ_1|Θsin W),∇^2Γ= 2m/ħ^2|κ_1|tanΘcos W∇^2Ω=-2m/ħ^2|κ_1|Θcos WFrom (<ref>), we suppose that there is no common variable between either ρ and Γ or between ρ and Ω; therefore, ∇^2ρ/ρ has to be constant. After the action of the gradient operator over (<ref>-<ref>),we recover (<ref>-<ref>) with changed signs. This result imposes∇^2Γ=∇^2Ω=0.Hence, we gain the constraint|κ_1|cos W=0,and consequently two cases to consider.Let us examine the first one. §.§ |κ_1|=0 We suppose a three-dimensional space, and thus ∇Γ and ∇Ω are necessarily collinear. The other two directions of the space given by ∇ϕ and ∇ρ. From (<ref>) and (<ref>), we obtain1/ρ∇^2ρ=2mE/ħ^2+|∇Γ|^2+|∇Ω|^2/2,2mℰ/ħ^2=|∇Γ|^2-|∇Ω|^2/2,where ℰ=iκ_0 is the quaternionic energy. Therefore, we reach the following solutionΓ=γ· x+Γ^(0),Ω=ω· x+Ω^(0),ρ=Ae^ α· x+B e^-α· x,where Γ^(0), Ω^(0), A and B are real scalar constants, and α is a constant real vector. There is no solutionfor the constant ρ, and we must have |α|^2=2mE/ħ^2+|γ|^2+|ω|^2/2|γ|^2-|ω|^2≥ 0.The most general wave function is thusΦ= ϕ( x) ρ( x)[(cosΘe^ iΓ+sinΘe^ iΩj)C_1+ (cosΘ e^ iΓ+sinΘ e^-iΩj)C_2+ +(cosΘe^-iΓ+sinΘe^ iΩj)C_3+ (cosΘ e^-iΓ+sinΘ e^-iΩj)C_4 ],where C_1, C_2, C_3 and C_4 are arbitrary complex constants. (<ref>) is not satisfied for quaternionic integration constants.We can make ϕ constant, so that E=0, and thus obtain the simplest solution of the case, a truly quaternionic free particle.However, we stress that every one-dimensional complex wave function ϕ generates the same kind of quaternionic solution, where the quaternionic solution may be understood as a geometric phase. A general study concerning quaternionic phases is an interesting direction for research. §.§ cos W= 0 and κ_0≠ 0The solutions of this case obeyW= Γ-Ω+τ_0=(n+1/2)π∇Γ=∇Ω,with n∈ℤ. Using (<ref>-<ref>), (<ref>) and (<ref>), we getℰ=-|κ_1| 2Θsin W,so that 1/ρ∇^2ρ=2m/ħ^2(E-ℰ2Θ)+|∇Γ|^2.Inasmuch as there is no defined sign on the right hand side of (<ref>), two kinds of solutions for ρ are admitted,either real exponentials or a linear combination of sines and cosines. Even a ∇^2ρ=0 is admitted, and hence there are more possibilities for ρ in this situation than has been found in the previous |κ_1|=0 case. The general solution is thusΦ= ϕ( x) ρ( x)( C_1e^iΓ+C_2e^-iΓ)(cosΘ-isin W sinΘ e^iτ_0 j)with C_1 and C_2 arbitrary complex constants and Γ given by (<ref>). The solution (<ref>) comprises a complex solution and a quaternionic unitary constant that multiplies its right-hand side.The quaternionic energy accomplished through (<ref>) isℰ=cos2Θ[E+ħ^2/2m(|γ|^2±|α|^2)],where the sign of |α|^2 is defined by ρ. It is totally unexpected that the quaternionic constant influences the energy of the system, and there is no counterpart to this phenomenon in CQM. A physical system described with this solution would be remarkable. We notice that low quaternionic energies, where ℰ<E, are admitted, depending on ρ.This curious and new effect is probably due to the quaternionic character of the eigenvalue κ, which is connected to the energy through (<ref>).§.§ cos W= 0 and κ_0 = 0 This solution is related to the time-dependent solution (<ref>). The solutions of this case follow (<ref>), but (<ref>-<ref>) additionally imposesΘ=(ñ+1/2)π/2,ñ∈ℤ.Consequently, the quaternionic wave function is obtained from (<ref>), and the quaternionic energy isℰ=|κ_1|=-Θ/sin W[E+ħ^2/2m(|γ|^2±|α|^2)]. §.§ The ∇Γ=∇Ω= 0 case Assuming an arbitrary one-dimensional complex wave function ϕ, we impose theconstraints ∇ϕ·∇ρ=0,∇ϕ·∇Θ=0,∇ρ·∇Θ=0.Thus, (<ref>-<ref>) turns into two equations1/ρ∇^2ρ-|∇Θ|^2=2m/ħ^2(E-iκ_0cos2Θ+|κ_1|sin W sin2Θ)∇^2Θ=2m/ħ^2(iκ_0sin2Θ+|κ_1|sin Wcos2Θ),and the constraint (<ref>) that has already been found for the ∇Θ= 0 cases. However, the orthogonality of ∇ρ and ∇Θ implies that ∇^2ρ/ρ is a constant. Applying the gradient operator over (<ref>), we recover (<ref>) with a changed sign, and thus ∇^2Θ=0, which forces a constant Θ because of (<ref>). With exception for an exotic zero energy solution, where κ=0 and ∇Θ is constant.Therefore, we do not have a simple solution for non-constant Θ. Maybe this kind of solution exists when (<ref>) includes a non-orthogonal ∇Θ. The research of such solutions is potentially an interesting subject for future work.§ THE FREE QUATERNIONIC PARTICLE Let us take the complex free particle as a reference. Its expression isϕ( x)=A_1 e^ ik· x+A_2 e^-ik· x, | k|^2=2mE/ħ^2,and A and B are complex integration constants. In order to understand the physics of the previously calculated solutions, we willentertain the probability current defined in <cit.>, namely j=1/2m{(p̂Ψ)Ψ^*+[(p̂Ψ)Ψ^*]^* }p̂Ψ=-i ħ∇Ψ.Using the general quaternionic time independent wave function (<ref>), we getj=ρ^2[j_0+ħ/m|ϕ|^2(cos^2Θ∇Γ+sin^2Θ∇Ω)] |C|^2where C is an arbitrary quaternionic constant and j_0 is the probability current due to the complex free particle (<ref>).There is no probability flux along the ∇Θ and ∇ρ directions. This fact does not mean that there is no motionalong these directions. We remember the complex square well, where a particle oscillates along a confined region without generating a non-zero probability flux. Thus, we interpret that the quaternionic particle freely propagates along directions∇Γ and ∇Ω only, while directions ∇Θ and ∇ρ allow oscillatory motions only. A simple quaternionic free particle obtained from (<ref>) is Φ=ρ( x)(cosΘe^iγ· x+sinΘe^iω· x j)C⇒ j= ħ/mρ^2|C|^2(cos^2Θ γ+sin^2Θ ω),where C is a quaternionic constant and∇Θ= 0. From (<ref>), we get|α|^2=|γ|^2+|ω|^2/2,2mℰ/ħ^2=|γ|^2-|ω|^2/2.An exotic zero-energy wave function is possible for|α|=|γ|=|ω|.The situation is different from the complex case, where null energies imply null momenta.Another particularity is that we cannot recover a complex solution by simply imposing Θ=|ω|=|α|=0. We remember that |α|=0 is prohibited by (<ref>), and that a simple quaternionic constant may change the energy of the particle (<ref>-<ref>). Thus, QQM solutions cannot be understood as encompassing a complex solution plus an independent pure quaternionic part. In fact, we have a different theory, which may recover CQM for several situations,but not for all cases.§ THE STEP POTENTIALLet us consider the scalar step potentialV={[ 0x<0,; V_0 x≥ 0, , ].where V_0 concerns a real positive constantand the potential V divides the three-dimensional space into two parts bordered by the Oyz plane. We propose the wave functionΦ_I =ρ_k[cosΘ_k e^i( k+γ_k^⊥)· x+sinΘ_k e^i(- k+ω_k^⊥)· x j ] + R ρ_q[cosΘ_q e^i(- q+γ_q^⊥)· x+sinΘ_q e^i( q+ω_q^⊥)· x j ] Φ_II = Tρ_p[cosΘ_p e^i( p+γ_p^⊥)· x+sinΘ_p e^i(- p+ω_p^⊥)· x j ]with R and T complex constants and k,q,p, γ^⊥_a and ω^⊥_a real vectors for a=k, q, p.We also adopt real constants for Θ_a and real functions for ρ_a. An arbitrary vector v is decomposed asv= v^∥ + v^⊥,where the component of v parallel to k isv^∥,and the component of v normal to k is v^⊥. We expect that the three terms of the wave function (<ref>) have identical properties, or describe particles of the same type. We choose the time-dependent solution (<ref>), where E=0. From (<ref>), we obtain several energy relations2mℰ/ħ^2=|γ_k|^2-|ω_k|^2/2=|γ_q|^2-|ω_q|^2/2,2m(ℰ-V_0)/ħ^2=|γ_p|^2-|ω_p|^2/2and also that|α_a|^2=| a|^2+|γ_a|^2+|ω_a|^2/2 a=k, q, p.At the point of incidence x_0=(0, y_0, z_0), which we set to x_0=(0, 0, 0) without loss of generality, we consider the continuity of the wave function, Φ_I ( x_0)=Φ_II( x_0) ⇒{[ρ_k(0)cosΘ_k^(0)+Rρ_q(0)cosΘ_q^(0)=Tρ_p(0)cosΘ_p^(0); ρ_k(0)sinΘ_k^(0) +Rρ_q(0)sinΘ_q^(0)=Tρ_p(0)sinΘ_p^(0) ].∇Φ_I^∥ ( x_0)=∇Φ_II^∥( x_0) ⇒{[ kρ_k(0)cosΘ_k^(0)- q Rρ_q(0)cosΘ_q^(0)= p Tρ_p(0)cosΘ_p^(0);-kρ_k(0)sinΘ_k^(0)+ q Rρ_q(0)sinΘ_q^(0)=- p Tρ_p(0)sinΘ_p^(0). ].,and thus|T|^2=| k+ q|^2/| p+ q|^2(ρ_k(0)/ρ_p(0))^2|R|^2=| k- p|^2/| p+ q|^2(ρ_k(0)/ρ_q(0))^2.The normal directions place further boundary conditions∇Φ_I^⊥ ( 0)=∇Φ_II^⊥( 0)⇒{[ ∇ρ_k(0) K_k+∇ρ_q(0) R K_q=∇ρ_p(0) T K_p; γ_k^⊥ ρ_k(0) cosΘ_k^(0)+Rγ_q^⊥ ρ_q(0) cosΘ_q^(0)=Tγ_p^⊥ ρ_p(0) cosΘ_p^(0);ω_k^⊥ ρ_k(0) sinΘ_k^(0)+Rω_q^⊥ ρ_q(0) sinΘ_q^(0)=Tω_p^⊥ ρ_p(0) sinΘ_p^(0), ].whereK_a=cosΘ_a^(0)+sinΘ_a^(0) j,∇ρ_a(0)=α_a(A_a-B_a) a=k, q, p.We propose a solution using the constraintsin^2Θ_k^(0)=sin^2Θ_q^(0)=sin^2Θ_p^(0).If | k|=| q|, we benefit from the identity| k|-| q|R ρ_q(0)/ρ_k(0)=| p|T ρ_p(0)/ρ_k(0),which is valid for the scattering of quantum particles in CQM. Thus, (<ref>), (<ref>) and (<ref>) give rise to|γ_q|/|γ_k|=|ω_q|/|ω_k|=|∇ρ_q(0)|/|∇ρ_k(0)|=1|γ_p|/|γ_k|=|ω_p|/|ω_k|=|∇ρ_p(0)|/|∇ρ_k(0)|=| p|/| k|.From (<ref>-<ref>) and (<ref>) we finally obtain|α_p|^2/|α_k|^2=| p|^2/| k|^2| p|^2/| k|^2=1-V_0/ℰ.Consequently, every parameter of the reflected and transmitted particles may be written in terms of the incident particle parameters. Theresults are similar to the CQM case, but the transmission and reflection coefficients have multiplying factors that are characteristicof quaternionic particles. These factors are related to the oscillation along the ∇ρ direction; as discussed in section <ref>, the quaternionic particle propagates along ∇Γ and ∇Ω, whereasalong ∇ρ and ∇Θ it only oscillates.§ CONCLUSIONIn this article, we have presented a general method for solving the QSE (<ref>). The results are complementaryto the solution of the QSE with a right multiplying complex unit i <cit.>. We have proven that there aretime-dependent solutions for Schrödinger equation (<ref>), which are fundamentally different from the time-function of CQM. We have also developed a general method for solving the time-independent solution, and we have obtained free particle solutions without a CQM counterpart. These solutions present oscillations that are normal to the propagation direction. This result is unknown in CQM, and now we need to find more physical examples where this kind of situation is found. Another interesting new feature is the influence of quaternionic constants on the energies of solutions.This is another quaternionic feature unknown in CQM, and other physical systems where this kink of behavior is found are also of interest for further research.In summary, we expect that these two articles will foster new developments in QQM. First of all, because the presented solutions are simple,and do not assume the anti-hermitian constraint that has been supposed for the QQM. Secondly, we have a method for finding new solutions, and then every result of CQM is at risk of being studied using the quaternionic formalism deployed here. We expect that more physical solutions may be found in the future, which may inspire either mathematical or physics investigations, including experimental ones. § ACKNOWLEDGEMENTSSergio Giardino is grateful for the hospitality at the Institute of Science and Technology of Unifesp in São José dos Campos.unsrt | http://arxiv.org/abs/1706.08370v1 | {
"authors": [
"Sergio Giardino"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170626134627",
"title": "Quaternionic quantum particles: new solutions"
} |
IPMU16-0192 Department of Physics and Astronomy, University of California, Los AngelesLos Angeles, CA 90095-1547, USA Department of Physics and Astronomy, University of California, Los AngelesLos Angeles, CA 90095-1547, USA Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIASThe University of Tokyo, Kashiwa, Chiba 277-8583, Japan Scalar condensates with large expectation values can form in the early universe, for example, in theories with supersymmetry. Thecondensate can undergo fragmentation into Q-balls before decaying.If the Q-balls dominate the energy density for some period of time, statistical fluctuations in their number density can lead to formation of primordial black holes (PBH).In the case of supersymmetry the mass range is limited from above by 10^23g.For a general charged scalar field, this robust mechanism can generate black holes over a much broader mass range, including the black holes with masses of 1–100 solar masses, which is relevant for LIGO observations of gravitational waves. Topological defects can lead to formation of PBH in a similar fashion.Primordial black holes from scalar field evolution in the early universe Alexander Kusenko December 30, 2023 ========================================================================§ INTRODUCTION It is well established that stellar core collapse can lead to formation of black holes. However, it remains an open question whether some processes in the early universe could produce primordial black holes (PBH) <cit.>. PBHs can account for all or part of dark matter <cit.>.Furthermore, they could be responsible for some of the gravitational wave signals observed by LIGO <cit.>, In addition, PBHs can invade and destroy neutron stars, ejecting neutron rich material in the process, which can account for for all or part of ther-process nucleosynthesis, as well as the 511-keV line in the galactic center <cit.>. Finally, PBHs could provide seeds for supermassive black holes <cit.>. A number of scenarios for black hole formation have been considered <cit.>, and many of them rely on a spectrum of primordial density perturbations that has some additional power on certain length scales, which can be accomplished by means of tuning an inflaton potential.It was recently pointed out that PBHs can form in a very generic scenario, which does not require any particular spectrum of density perturbations from inflation <cit.>. Scalar fields with slowly growing potentials form a coherent condensate at the end of inflation <cit.>. In general, the condensate is not stable, and it breaks up in lumps, which evolve into Q-balls <cit.>. The gas of Q-balls contains a relatively low number of lumps per horizon, and the mass contained in these lumps fluctuates significantly from place to place. This creates relatively large fluctuations of mass density in Q-balls across both subhorizon and superhorizon distances. Since the energy density of a gas of Q-balls redshifts as mass, it can come to dominate the energy density temporarily, until the Q-balls decay, returning the universe to a radiation dominated era. The growth of structure during the Q-ball dominated phase can lead to copious production of primordial black holes.In this paper we will investigate this scenario in further detail.Formation of Q-balls requires nothing more than some scalar field with a relatively flat potential at the end of inflation. For example, supersymmetric theories predict the existence of scalar fields with flat potentials. PBH formation in supersymmetric theories is, therefore, likely, even if the scale of supersymmetry breaking exceeds the reach of existing colliders. A similar process can occur with topological defects, which can also lead to relatively large inhomogeneities. The discussion of topological defects is complicated by their non-trivial evolution. We will focus primarily on Q-balls, and will briefly comment on topological defects.The format of this paper is as follows: in Section <ref>, we describe the fragmentation of the condensate and the production of Q-balls, then in Section <ref> we derive the formalism for calculating the statistical moments of collections of Q-balls. In Section <ref> we use the results of the previous Section to calculate the expected PBH density and mass spectrum, and in Section <ref> we account for the effects on cosmological thermal history and evolve the PBH distribution to the present day. In Section <ref>, we then compare our results with current observational constraints, and in Section <ref> explore the available parameter space. In Section <ref>, we comment on the applicability of this mechanism to topological defects.§ FORMATION OF Q-BALLSFormation of a scalar condensate after inflation and its fragmentation <cit.> is a fairly generic phenomenon.While supersymmetry is a well-motivated theory for scalar fields carrying global charges and having flat potentials <cit.>, our discussion can be easily generalized to an arbitrary scalar field with a global U(1) symmetry in the potential. Supersymmetric potentials generically contain flat directions that are lifted only by supersymmetry breaking terms.Some of the scalar fields that parameterize the flat directions carry a conserved U(1) quantum number, such as the baryon or lepton number.During inflation, these field develop a large vacuum expectation value (VEV) <cit.>, leading to a large, nonzero global charge density. When inflation is over, the scalar condensate ϕ(t) = ϕ_0(t) exp{iθ (t)} relaxes to the minimum of the potential by a coherent classical motion with θ̇≠ 0 due to the initial conditions and possible CP violation at a high scale.The initially homogeneous condensate is unstable with respect to fragmentation into non-topological solitons, Q-balls <cit.>. Q-balls exist in the spectrum of every supersymmetric generalization of the Standard Model <cit.>, and they can be stable or long-lived along a flat direction <cit.>. In the case of a relatively large charge density (which is necessary for Affleck-Dine baryogenesis <cit.>), the stability of Q-balls can be analyzed analytically <cit.>; these results agree well with numerical simulations <cit.>. One finds that the almost homogeneous condensate develops an instability with wavenumbers in the range 0<k<k_max, where k_max=√(ω^2 - V”(ϕ_0)), and ω=θ̇. The fastest growing modes of instability have a wavelength ∼ 10^-2± 1 of the horizon size at the time of fragmentation, and they create isolated lumps of condensate which evolve into Q-balls.Numerical simulations <cit.> indicate that most of the condensate ends up in lumps. However, since the mass of Q-balls is a non-linear function of the Q-ball size, Q-ball formation, in general, leads to a non-uniform distribution of energy density in the matter component represented by the scalar condensate.Q-balls can also form when the charge density is small or zero, in which case both positively and negatively charged Q-balls are produced <cit.>; here we do not consider this possibility. Depending on the potential, the Q-balls with a global charge Q have the following properties <cit.>:ω∼Λ Q^α-1,R ∼ |Q|^β/Λ,ϕ_0 ∼Λ |Q|^1-α,M ∼Λ |Q|^α, where Λ is the energy scale associated with the scalar potential, Q is the global U(1) charge and 0<α<1, 0<β<1 denotes which type of Q-ball is under consideration (and also depends on the form of the scalar potential). For “flat direction" (FD) Q-balls, α = 3/4 (β = 1/4), and for “curved direction" (CD) Q-balls, α = 1 (β = 1/3) <cit.>. § Q-BALL CHARGE/MASS DISTRIBUTIONSNumerical simulations of condensate fragmentation and Q-ball formation have been performed in the past, from which we are able to determine the resulting charge and mass distributions <cit.>. These distributions appear to be very sensitive to initial conditions in the condensate, such as the ratio of energy to charge density (x=ρ/mq), and to the details of the scalar potential. In addition, the resultant charge distribution can be very non-Gaussian due to the high degree of nonlinearity and chaos in the fragmentation process.It should be understood that the results of these simulations are statistical in nature: a large number of Q-balls are created within the simulation volume so that the charge distribution tends towards a statistical average. In reality, if one were to perform a large simulation and look at the charge distributions in a number of small sub-volumes, you will find a large degree of variation, with more variation on smaller scales due to small sample sizes (this is not to say that the variance will be larger, just that the differences between distributions are large), as can be seen in Fig. <ref>.It is these large fluctuations relative to the mean that will be the source of density perturbations.Once the resultant charge distribution of Q-balls f_Q(Q)dQ has been calculated from these numerical simulations, we can use this to calculate the mass distribution for single Q-balls using M=Λ|Q|^α (we will absorb all numerical factors into the definition of Λ without loss of generality):f_M(M) = M^1-α/α/αΛ^1/α[f_Q((M/Λ)^1/α) + f_Q(-(M/Λ)^1/α)].It is important to note that a distribution well-localized in charge is also well-localized in mass. We can also use probability theory to calculate the mass of a collection of Q-balls. Under the assumption that a charge Q_tot is distributed amongst N Q-balls whose distribution is described by f_Q(Q), the probability distribution function (PDF) for the total mass of this collection of Q-balls is given byf_M(M,Q_tot,N) = ψ(M,Q_tot,N)/∫ dM ψ(M,Q_tot,N), ψ(M,Q_tot,N)= ∫(∏_i=1^N dQ_if_Q(Q_i)) ×δ(M - Λ∑_i=1^N |Q_i|^α) δ(Q_tot - ∑_i=1^N Q_i),where ψ admits a simple-looking Fourier transform in M and Q_tot:ψ̃(ξ_M,ξ_Q_tot,N) = [ ∫ dQe^i(ξ_M |Q|^α + ξ_Q_tot Q) f_Q(Q) ]^N.The power of α prevents analytic calculation of this PDF for all but the simplest charge distributions. Specifically, if we take the charge distribution to be a delta function: f_Q(Q) = δ(Q - Q_0), then the mass distribution is also a delta function: f_M(M,Q_tot,N) = δ(M - N M_0), where M_0 = Λ Q_0^α, and Q_0 = Q_tot/N to satisfy charge conservation (this constraint comes from a mathematical issue that arises due to the canceling of a delta function of the form δ(Q_tot-NQ_0)/δ(Q_tot-NQ_0); we can see that if we consider δ(x) as the limit of a smooth function that approaches this distribution, then this ratio is unity provided Q_0 = Q_tot/N).For ease of computation, we will assume the delta function charge/mass distribution for the rest of this paper. This also has good theoretical motivation, as the Affleck-Dine baryogenesis scenario requires a large nonzero charge density, which tends to result in a highly-localized charge distribution. §.§ Single length scaleOne should notice that the mass distribution function calculated earlier is also a function of both the total charge Q_tot and the number of Q-balls N. During the chaotic fragmentation procedure, the number of Q-balls will fluctuate between horizons. So in order to get a full description of the fluctuations, we must supplement the mass distribution with a number distribution p(N). This can be calculated from a simulation by simply counting the number of Q-balls within the simulation volume. Here, we will assume that the number of Q-balls per horizon N is described by a Poisson distribution, as is typical for a random process such as fragmentation:p(N) = e^-N_fN_f^N/N!,where N_f is the average number of Q-balls per horizon at fragmentation. We then combine Equations <ref> and <ref> to create a joint PDF which describes the distribution of mass M within a horizon composed of N Q-balls (we also set Q_tot = Q_f, the total charge on the horizon at t_f): F_Q(M,N) = f_M(M,Q_f,N) p(N). This is manifestly normalized since ∑_N ∫ dMF_Q = ∑_N (∫ dMf_M) p = ∑_N p = 1. We can then use this to calculate statistical moments such as the average horizon mass, average horizon Q-ball number, RMS fluctuations, etc:⟨M|=⟩∑_N=1^∞∫_0^∞ dMM F_Q(M,Q_f,N) ≈ M_f,⟨N|=⟩∑_N=1^∞∫_0^∞ dMN F_Q(M,Q_f,N) = N_f, N_RMS = [∑_N=1^∞∫_0^∞ dMN^2 F_Q(M,Q_f,N)]^1/2 = N_f,where M_f = Λ Q_f^α N_f^1-α is the horizon mass of Q-balls at t_f (the first relation is only approximate because ⟨N^1-α|≈⟩N_f^1-α, though the relative error scales as |⟨N^1-α|-⟩ N_f^1-α|/⟨N^1-α|≈⟩1/10 N_f, so totally negligible for large N_f). §.§ Multiple length scalesThe previous treatment has the shortcoming that it can only describe Q-ball distributions with spatial extent the size of the horizon at the time of fragmentation. We now generalize this to handle distributions on an arbitrary scale. First, when considering a physical volume V at the time of fragmentation, the charge contained within this volume (assuming initial uniformity of the condensate) is given by Q_tot = Q_V = Q_f (V/V_f), where V_f = 4π/3 t_f^3 is the horizon volume at t_f. Second, the number distribution is altered so that the number of Q-balls within volume V (assuming the same average number density n_f = N_f/V_f across all scales) is described byp(N,V) = e^-N_f V/V_f(N_f V/V_f)^N/N!.The joint PDF for a mass M composed of N Q-balls contained within a volume V at the time of fragmentation t_f is then given byF_Q(M,V,N) = δ(M - M_f (N/N_f)^1-α(V/V_f)^α) p(N,V).Note that if V=V_f, this reduces to the single-scale, horizon-size treatment.In addition to being able to calculate quantities on each scale V individually, we will also want to sum the contributions from each scale in some cases (such as contributions to the PBH density from both subhorizon and superhorizon modes). To do so, we will sum over all volume scales from V_min to V_max using a coarse-graining method. We consider an arbitrary function of volume g(V). The sum of the contributions from each scale V_i = V_max/χ^i-1 is then given by∑_{V} g(V)= ∑_i=1^⌊ 1 + log_χV_max/V_min⌋ g(V_i) ≈∫_1^ 1 + log_χV_max/V_min dig(V_max/χ^i-1) = 1/lnχ∫_V_min^V_maxdV/Vg(V),where we have used Euler-Maclaurin to approximate the sum, and χ∼few is a parameter of the spacing between intervals of the coarse-graining procedure. We will take χ = e from now on for simplicity; another choice does not significantly affect the outcome provided it is not too close to unity. V_min is the smallest volume under consideration; there will be a natural cutoff due to the fact that Q-balls have a finite size, and so this scale is generally defined as the volume which contains some number N_min∼ 10 Q-balls on average: V_min/N_min = N_f/V_f.§ Q-BALL AND PBH DENSITIESUsing the framework of Section <ref>, we are now in a position to begin calculating the energy densities associated with the Q-balls, fluctuations in that energy density, and the resulting density of black holes. §.§ Q-ball density at fragmentationUsing the formalism of Section <ref>, we can calculate the background energy density (over the largest scales) of Q-balls at t_f: ⟨ρ_Q(t_f)|=⟩lim_V→∞⟨M|⟩/V = M_f/V_f,which will be important in the discussion of density perturbations in Section <ref>. Since Q-balls are formed at rest, the evolution of the Q-ball density after fragmentation is simply that of decaying nonrelativistic matter ⟨ρ_Q(t)|=⟩⟨ρ_Q(t_f)|(⟩a_f/a)^3 e^(t_f-t)/τ_Q, where τ_Q = 1/Γ_Q is the lifetime of the Q-balls. Q-balls are generally considered stable with respect to decay into the quanta of the scalar field, but it is possible to decay through other processes. For example, if a coupling of the scalar field to a light fermion with mass m<ω exists, Q-balls can decay to these fermions through an evaporation process <cit.>. Q-balls can also decay if the U(1) symmetry is broken by some higher-dimension operators <cit.>. We define Γ_Q to include all such decay channels. §.§ Q-ball density perturbations due to fluctuationsDue to the stochastic nature of the fragmentation process, volumes of space can arise within which the number density of Q-balls exceeds the average number density. Due to the nonlinear relationship between Q-ball mass and charge M=Λ|Q|^α, this also gives rise to fluctuations in the energy density within that volume. The density contrast in Q-balls at fragmentation δ(t_f) for a volume V containing mass M is defined asδ(t_f) = δρ_Q/⟨ρ_Q|⟩ = M/V/⟨ρ_Q|⟩ - 1 = (N/N_f/M/M_f)^1-α/α - 1where in the last line we have used the argument of the delta function in Equation <ref> to eliminate V (this will be justified by an integral over V later). Note that if the Q-ball mass-charge relationship were linear (α=1), the perturbations would vanish identically.The subhorizon density perturbations (V<V_f) are frozen during the initial radiation dominated era, but they grow linearly in the scale factor during the Q-ball dominated epoch: δ(t) = δ(t_f) (a/a_Q) = δ(t_f) = (t/t_Q)^2/3, where t_Q is the beginning of the era of Q-ball domination. The structure growth generally goes nonlinear and decouples from the expansion around δ > δ_c ∼ 1.7, at which point the overdense regions collapse and become gravitationally bound. However, some structures with δ < δ_c can still collapse, and not all structures with δ > δ_c are guaranteed to collapse into black holes. Due to nonsphericity of the gravitationally-bound structures, only a fraction β = γδ^13/2(t_R)(M/M_Q)^13/3 (where γ≈ 0.02 is a factor due to the nonsphericity, M_Q = M_f(t_Q/t_f)^3/2 is the horizon mass at the beginning of the Q-ball dominated era, and t_R is the end of the Q-ball dominated era, when the radiation comes to dominate again) will actually collapse to black holes <cit.> by the end of the Q-ball dominated era. We assume that structures with δ≥δ_c do not continue to grow past the point of nonlinearity, as they have already collapsed and had their chance to form a PBH; for these perturbations we set β = γδ_c^13/2 (M/M_Q)^13/3 for δ(t_R) > δ_c. This refinement may not be necessary, as the average density perturbations are generally so small they never reach δ_c, and indeed, changing the value of δ_c does not seem to significantly alter the outcome.Additional care must be taken to extend this to scales which enter the horizon at later times, and thus may not subject to the same amount of growth as subhorizon modes. Those that enter the horizon between t_f < t < t_Q can be treated as effectively subhorizon since they enter the horizon before the Q-ball dominated epoch begins, and thus fluctuations are subject to the same amount of amplification as initially subhorizon modes. This includes all volumes V < V_Q, where V_Q = 4π/3 t_Q^3 (t_f/t_Q)^3/2 is defined as the (initially superhorizon) physical volume at t_f which enters the horizon at t_Q: (a_Q/a_f)^3 V_Q = 4π/3 t_Q^3. Fluctuations which enter the horizon during the Q-ball dominated epoch are treated slightly differently, as they are only subjected to amplification from the time they enter the horizon t_h until the radiation comes to dominate again at t_R. We can account for this by calculating t_h for a given scale V via (a(t_h)/a_f)^3 V = 4π/3 t_h^3 (which gives us t_h = (3/4π)(V/t_f^3/2 t_Q^1/2)), and then replacing the scale factor a_R/a_Q with a_R/a(t_h) in the definition of β above. This treatment is valid for all scales between V_Q < V < V_R, where V_R = (4π/3) t_R^3 (t_Q/t_R)^2 (t_f/t_Q)^3/2 is the physical volume at t_f which enters the horizon at t_R.In addition to these details, we also enforce the constraint β≤ 1 in order to prevent PBH production probabilities over unity, though this does not become relevant unless the Q-ball dominated era is extremely long. §.§ Primordial black hole densityWe are now in a position to calculate the average energy density in PBH created during the Q-ball dominated era. We do this by calculating the energy density of Q-balls at t_f that will eventually form black holes by t_R by weighting the Q-ball energy density M/V by the collapse fraction/probability β evaluated at t_R, summing over all scales V, and then redshifting this value appropriately. The expression for this procedure is given by⟨ρ_BH(t_R)|=⟩(a_f/a_R)^3 ∑_N=1^∞∫_V_min^V_RdV/V∫_0^∞ dM (βM/V) F_Q,where it should be understood that the integral over V is broken up into two separate domains, [V_min,V_Q] and [V_Q,V_R], where separate definitions of β apply, as described in Section <ref>. Due to the complicated piecewise nature of the function β, the authors are unaware of any analytic solution, and further progress must be made numerically.We find that Equation <ref> can be rewritten in such a way that it only depends on the dimensionless numbers N_f, r_f = t_Q/t_f, and r = t_R/t_Q. r_f and r can be interpreted as measures of the duration of the era between the fragmentation and the beginning of Q-ball domination, and the length of the Q-ball dominated era, respectively. The effect of these parameters on the black hole density can be seen in Figure <ref>.Larger r_f will reduce the fraction of Q-ball energy that goes into making black holes due to the dilution of the number density and increased horizon mass at t_Q due to the delay of the Q-ball dominated era. Larger r leads to an increased fraction of Q-ball energy that goes into black holes due to more amplification of the density perturbations, leading to a higher probability of PBH formation. Larger N_f reduces the fraction because of higher suppression of fluctuations due to the Poisson statistics. The form of the contours in this plot suggest that this ratio roughly scales as ⟨ρ_BH|/⟩⟨ρ_Q|∼⟩(4.6× 10^-4) r_f^-5.9 r^4.4 N_f^-3.7. §.§ Black hole mass spectrumOne can derive the mass spectrum of the black holes by not integrating over M in Equation <ref>:d⟨ρ_BH|⟩/dM = (a_f/a_R)^3 ∑_N=1^∞∫_V_min^V_RdV/V(βM/V) F_Q.This yields the differential black hole energy density d⟨ρ_BH|/⟩dM. We find that the spectrum can be rewritten in terms of the parameter η = M/M_f (fraction of horizon mass at t_f), along with the previously mentioned parameters r_f=t_Q/t_f, r=t_R/t_Q, and N_f. Calculation of this function can be done by evaluating Equation <ref> at multiple values of η and then interpolating. An example is given in Figure <ref>.First, it's obvious from the normalization of each curve that the lower the number of Q-balls per horizon, the more black holes that are created. This is expected, as the Poisson statistics suppress the density fluctuations for large Q-ball number. The normalization also increases with r, as explained in Figure <ref>. Second, there is a hard lower cutoff in the PBH mass, which occurs at η = N_min/N_f, which is due to the lower cutoff in the volume mentioned earlier. Above that, the BH number sharply increases with a power law ∝η^2.85± 0.15; the extent of this region depends on the magnitude of r, with larger values leading to a larger range. We suspect that this is due to the fact that the small-scale density fluctuations have already reached their critical value δ_c and can no longer continue growing, whereas the large-scale fluctuations (which started out smaller) still have room to do so. Above that, the spectrum becomes approximately flat (∝η^-0.15), meaning that the number of black holes in each decade of mass are comparable. Of course, the upper end of this range dominates the energy density of the distribution. Then, at around M = M_Q, there is a sharp transition and the slope becomes strongly negative (∝η^-4.5) due to the reduced growth the superhorizon modes are subject to. Then, there is an upper exponential cutoff at η∼ 10^8/N_f due once again to the Poisson statistics (the cutoff appears to take precedence over previously mentioned transitions).§ COSMOLOGICAL HISTORYWe now give a detailed account of how the Q-balls, radiation, and black holes evolve throughout the history of the universe up until the present day, as seen in Figure <ref>.In summary: we assume an initial period of inflation and reheating in order to create a radiation dominated era with a uniform charged scalar field as a subdominant component of the energy density. The scalar field fragments into Q-balls at t_f, which then come to dominate the energy density at t_Q. During the Q-ball dominated epoch, primordial black holes are produced, and at t_R, their density is frozen in and evolves as nonrelativistic matter. After this initial matter dominated epoch, the Standard Model of cosmology resumes, and evolves through all the eras we are familiar with (BBN, matter-radiation equality, etc.) up to the present day.The functions used to model the energy density evolution for each species is summarized in Appendix <ref>. §.§ Initial radiation dominated eraAfter the end of inflation, the Universe enters a brief matter dominated era due to the coherent oscillations of the inflaton field. The decay of the quanta of this field at time t_RH = Γ_I^-1 reheats the Universe, which enters a radiation dominated epoch with temperature T_RH = 0.55 g_*^-1/4 (Γ_I M_p)^1/2 and radiation energy densityρ_R(t_RH) = π^2/30 g_*(T_RH) T_RH^4 ≈π^2/327Γ_I^2 M_p^2,where Γ_I ∼ 1/t_RH is the decay width of the inflaton oscillations. The radiation density redshifts as ρ_R(t) = ρ_R(t_RH) (a_RH/a)^4 = ρ_R(t_RH) (t_RH/t)^2 during this epoch, which ends up canceling the factor of Γ_I to give usρ_R(t) = π^2 M_p^2/327 t^2; t_RH < t < t_QAt some point t_f, the scalar condensate fragments into Q-balls, resulting in an energy density given by Equation <ref>. The Q-balls then redshift as decaying nonrelativistic matter:⟨ρ_Q(t)| ⟩= ⟨ρ_Q(t_f)|⟩(a_f/a)^3 e^-(t-t_f)/τ_Q= 3 Λ Q_f^α N_f^1-α/4π t_f^3/2 t^3/2 e^-(t-t_f)/τ_Q ; t_f < t < t_Q§.§ Q-ball dominated eraAt some point t_Q, the Q-balls come to dominate the energy density. This time is defined by ρ_R(t_Q) = ⟨ρ_Q(t_Q)|$⟩, using the equations of the previous section. During this era, Q-ball decays begin to affect the radiation density, causing the radiation temperature to decrease less slowly than it normally would due to the expansion. Following the analysis of Scherrer and Turner <cit.>, the radiation density in this epoch due to the decay of the Q-balls can be modeled asρ_R(t) = [ ρ_R(t_Q) + ⟨ρ_Q(t_Q)|∫⟩_x_0^x dx'z(x') e^-x'] z^-4,wherex ≡Γ_Q t,x_0 = Γ_Q t_Q, andz=(x/x_0)^2/3. The Q-balls continue to redshift and decay, leading to⟨ρ_Q(t)|=⟩3 Λ Q_f^α N_f^1-α t_Q^1/2/4π t_f^3/2 t^2 e^-(t-t_f)/τ_Q ; t_Q < t < t_RAs the Q-balls decay, eventually the radiation comes to dominate again att_R, defined byρ_R(t_R) = ⟨ρ_Q(t_R)|$⟩. Using Equations <ref>, <ref>, <ref> and <ref>, this gives us the relation1 + (t_R/τ_Q)^-2/3Γ(5/3, t_Q/τ_Q, t_R/τ_Q) = (t_R/t_Q)^2/3 e^(t_Q - t_R)/τ_Q,where Γ is the generalized incomplete gamma function. This allows us to solve (numerically) for r_Q ≡τ_Q/t_Q as a function of r=t_R/t_Q. At this point, if we specify t_f, r_f and r, we can calculate the other parameters t_Q, t_R, τ_Q, and Λ Q_f^α viat_Q = t_f r_f,t_R = t_f r_f r,τ_Q = t_f r_f r_Q(r), Λ Q_f^α = 4π M_p^2 t_f/3· 327 r_f^1/2 N_f^1-α e^(1-1/r_f)/r_Q(r),from which we can calculate all other quantities of interest (M_f = Λ Q_f^α N_f^1-α, M_Q = M_f r_f^3/2, etc.). §.§ Standard cosmological eraAfter the Q-balls have decayed sufficiently, the universe returns to a radiation dominated era, and the standard cosmology begins. In order to evolve the radiation, Q-ball, and black hole densities to the present day, one would naïvely use a_1/a_2 = (t_1/t_2)^n, where n=1/2 (2/3) in a radiation (matter) dominated era, keeping in mind that the universe transitions between the two at z_eq≈ 3360, or t_eq≈ 4.7× 10^4 yr. However, due to the extended era of matter domination, the time at which cosmological events (such as BBN, matter-radiation equality, or recombination) occur are not the same as in the standard cosmology. Instead, one must evolve according to the universe's thermal history, where cosmological events occur at specific temperatures. In this case, one must use a_1/a_2 = g_*S^1/3(T_2) T_2/g_*S^1/3(T_1) T_1 and evolve from T_R (defined by ρ_R(t_R) = (π^2/30) g_*(T_R) T_R^4) to T_0 = 2.7K = 2.3meV. This has the advantage of accurately accounting for any deviation from cosmological history. We can then find the time at which some event X occurs by solving ρ_R(t_X) = (π^2/30) g_*(T_X) T_X^4 = ρ_R(T_R) (a(t_R)/a(t))^4. In order to ensure that this early matter dominated era does not spoil the canonical cosmological thermal history, we enforce an additional constraint T_R > T_BBN∼MeV, so that the entropy injection from Q-ball decays does not interfere with nucleosynthesis.§ OBSERVATIONAL CONSTRAINTSWe now examine the observational constraints on primordial black holes and where our results fit in. The constraints come from a wide variety of sources, such as extragalactic gamma rays from evaporation <cit.>, femtolensing of gamma ray bursts (GRB) <cit.>, capture by white dwarfs <cit.>, microlensing observations from HSC <cit.>, Kepler <cit.>, and EROS/MACHO/OGLE <cit.>, measurements of distortion of the CMB <cit.>, and bounds on the number density of compact X-ray objects <cit.> (constraints summarized in <cit.>). The constraints are typically expressed in a form that assumes a monochromatic distribution of PBH masses. However, in the case of an extended mass distribution (such as we have in this scenario), care must be taken to apply the limits correctly. To do this, we follow the procedure outlined in <cit.>, which amounts to dividing the mass spectrum into a number of bins (labeled by the index i), then integrating the dark matter fraction over the interval contained in the bin:f_i = 1/ρ_DM∫_M_i^M_i+1 dM d⟨ρ_BH(t_0)|⟩/dM,which is then compared with the constraints on a bin-by-bin basis. We find that for sufficient choices of the parameters N_f, t_f, r_f, and r, our model can produce black holes over practically the entire parameter space allowed by the constraints (see Fig. <ref>).Notably, this Figure illustrates two interesting points: 1) that this mechanism is capable of generating black holes which can account for both 100% of the dark matter in the region M ∼ 10^20 g and production or r-process elements <cit.>, and 2) it is also capable of generating black holes with sufficient mass to explain the recent LIGO observation GW150914 <cit.>. Some studies have even argued that PBH can account for 100% of the DM in this range by contesting the CMB constraints <cit.>. The three contours in Figure <ref> are, however, simply chosen by hand for illustrative purposes, and are therefore not representative of the entire parameter space allowed to this mechanism, which is much wider than suggested by the given parameters.§ PARAMETER SPACEWe now explore the parameter space available to this mechanism in which it is possible to account for a considerable fraction of the dark matter while avoiding observational constraints. To do so, we develop an algorithm to accomplish this task in the following manner: since the function ((a_f/a_R)^3/⟨ρ_Q(t_f)|)⟩(d⟨ρ_BH(t_R)|/⟩dη) is determined solely by the parameters r_f and r, we generate a list of such functions by sampling the r - r_f plane at various points. Then, for each (r,r_f) pair, we vary t_f using a weighed bisection method until max(|1-f_i/f_con(M_i)|) < ϵ = 10^-1, with f_i given by Equation <ref>. This determines the value of t_f which gives the maximum dark matter fraction allowed by the constraints for the given values of r and r_f. Once that has been determined, we can calculate all other relevant quantities of interest, such as M_BH,peak, T_R, and the dark matter fraction f=Ω_PBH/Ω_DM. The results for N_f = 10^6 are shown in Figure <ref>. We can see from this Figure that the contours of constant f are highly correlated with the values of M_BH,peak. This is due to the fact that the observational constraints are solely a function of M, and the calculated spectrum f(M) is quite sharply peaked at M_BH,peak. The region of f ≥ 1 roughly follows the contour of M_BH,peak=10^20 g, where the constraints are weakest, due to this fact. The general tendency appears to be that for increasing r, the spectrum favors heavier black holes, while for increasing r_f, it favors lighter black holes. This is because the longer the period of structure formation, the more the large mass perturbations (which initially have small δ) grow, increasing the probability that they collapse to black holes, whereas the longer the period between fragmentation and Q-ball domination, the more the energy density of Q-balls is diluted, so that the value of t_f has to be lower in order to achieve the same value of f that one would with smaller r_f. The smaller t_f, the smaller the horizon mass, and the smaller the PBH masses. This can be quantified, as the contours of constant f very roughly follow r^0.63/r_f ∼const.However, at around r ≳ 3.2× 10^5, the character of the plot changes so that the spectrum appears to no longer depend on r, only on r_f. This is likely due to the definition of β discussed in Section <ref>, in which density perturbations are not allowed to grow past δ_c, which makes it so that further increases in r have no effect on the spectrum. Further increases in r_f, however, serve to further dilute the Q-ball density before structure formation can occur, lowering the necessary t_f as mentioned above.The pink region where T_R > 1MeV is ruled out since in this region Q-ball decays begin to interfere with nucleosynthesis. We can see that it is correlated with high mass, as the later the fragmentation (meaning larger horizon mass), the less time there is for the Q-balls to dominate before the universe cools sufficiently enough that nucleosynthesis begins. One might notice that the mass of the black holes seen at LIGO in the GW150914 event lies beyond the range indicated in this Figure, but in Figure <ref> we have plotted a contour that has M_BH,peak∼ 30M_⊙. This is because the contour in Fig. <ref> is for N_f = 10^5, whereas Fig. <ref> has N_f fixed at 10^6. We can see in Fig. <ref> that the density spectrum has a peak at higher masses for lower values of N_f, so that the equivalent plot to Fig. <ref> for N_f < 10^6 would see the T_R < 1MeV constraint pushed to higher values of M_BH,peak; enough so that they could explain the 30M_⊙ black holes while avoiding the nucleosynthesis constraint.§ TOPOLOGICAL DEFECTSTopological defect formation can also lead to the production of PBHs if the topological defects come to dominate the energy density. The analysis is sufficiently different from that of Q-balls, primarily because typically only one defect per horizon is produced at the time of formation due to the Kibble mechanism <cit.>. However, the general mechanism remains the same: small number densities of defects lead to large fluctuations relative to the background density, these fluctuations become gravitationally bound and collapse to form black holes once the relic density has come to dominate, and the relics decay due to some instability (such as gravitational waves or decay to Nambu-Goldstone bosons in the case of cosmic strings). In order to accurately model production of PBHs from these defects, one should calculate the expected density perturbations on initially superhorizon scales, which only begin to grow once these scales pass back within the horizon and the defects come to dominate the universe's energy density.Cosmic strings are probably the most likely candidate for primordial relics due to the fact that they are typically cosmologically safe, as the energy density in string loops is diluted during expansion at the same rate as radiation, a^-4 <cit.>.In contrast, the string “network" (i.e. infinitely long strings) energy density redshifts as a^-2 so that they quickly come to dominate the universe's energy density. However, once long strings start intercommuting to produce loops, and these loops subsequently self-intersect to fragment into smaller loops, the string network approaches a scaling solution which leads to the a^-4 dilution of string loops <cit.>.However, this scaling solution critically relies on the probability of string intercommutation being very close to unity so that the long strings can efficiently break into small loops. If this probability was sufficiently low, then the string density could redshift as a^-2 or a^-3, survive until a matter/string dominated era, initiate structure formation, collapse to form PBHs, and then subsequently decay. As long as these conditions are satisfied, cosmic strings could act as a source of PBH. In addition, there exists a large class of solutions to the string equations of motion which never self-intersect <cit.>, making this scenario plausible.§ DISCUSSIONThe mechanism we have discussed has a number of advantages over some other models. It is extremely effective in creating primordial black holes across a broad range of masses, and it does not require the tuning the inflaton potential <cit.>. We did not have to make any ad-hoc assumptions regarding density fluctuations; the fluctuations are calculable from first principles.The mechanism is also generally applicable to practically any complex scalar field with a conserved global charge and flat potential, so that the formation of PBH is now a general prediction of any theory containing such charged scalars. In particular, supersymmetric extensions to the Standard Model typically have such fields, making the production of primordial black holes a general prediction of such theories. For the case of supersymmetric Q-balls with the SUSY-breaking scale Λ_SUSY > 10 TeV, the fragmentation time cannot be much longer than the Hubble time H^-1∼ M_p/g_*^1/2Λ_SUSY≲ 8 × 10^-15 s, which corresponds to peak PBH masses of about 10^23g (assuming N_f ∼ 10^6). The solid black curve illustrated in Figure <ref> satisfies this bound, thus primordial black holes from supersymmetric Q-balls can account for 100% of the dark matter.Supersymmetric Q-balls themselves have been suggested as the source of dark matter in models where they are entirely stable <cit.>. However, the stability is model-dependent, and it only applies to Q-balls carrying baryonic charge (so-called B-balls), since those carrying leptonic charge (L-balls) would quickly evaporate to neutrinos <cit.>. In our scenario, a short evaporation timescale is precisely what is needed to end the early era of Q-ball domination before nucleosynthesis begins. The L-balls would then be composed of slepton fields which subsequently decay to neutrinos at an early time. Since neutrinos do not decouple from the plasma until just before nucleosynthesis (T ≳few MeV), they thermalize quickly. If they decay early enough, it may even be possible to generate the baryon asymmetry through conversion of lepton number to baryon number via sphaleron processes during the electroweak phase transition <cit.>.There are some remaining open questions, such as how well the assumption that all Q-balls within the volume V are the same charge models this scenario. Simulations show that for scalar condensates with a high ratio of charge density to energy density, this is a good prediction, as all the Q-balls formed from this initial condition typically have similar sizes. This is also theoretically understood for a scalar condensate with a sufficiently large charge density <cit.>. However, condensates with a large energy density and small charge density generally produce broad charge distributions, nearly symmetric about Q=0, since the excess energy cannot be contained in Q-balls with the same sign of Q while also conserving charge. We suspect that in this scenario the production of PBH will be reduced, since the charge conservation does not play as significant a role. Loss of energy due to scalar radiation in the fragmentation process may still be able to produce energy density inhomogeneities, but this will require further study. In the same manner, the production of oscillons from the fragmentation of a real scalar field may be able to produce significant numbers of PBH as well.One may also wonder what sort of mechanism is needed in order to ensure the Q-balls decay at the correct time. As an example, following the work of <cit.>, the lifetime of a Q-ball with initial charge Q_0 decaying to pseudo-Goldstone bosons through the effects of a charge-violating operator of the form V_Q(ϕ) = g ϕ^n (ϕ^*)^m/Λ_*^n+m-4 + c.c. is given byτ≈1-Q_0^1-a/(a-1)Γ_0,wherea = 1/4(7+2(n+m-2)), Γ_0 = 112.7 |g|^2 e^-0.236(n+m) (n-m)^2 J_nmΛ(Λ/Λ_*)^2(n+m)-8,and J_nm∼ O(10^-7 - 10^-6). For g ∼ 0.1, Λ_* ∼ 10^16 GeV, and Λ∼ 10^9GeV, the lifetime of a Q-ball decaying through an operator with (n,m) = (2,3) is about τ∼ 10^-13 s, which is sufficient to explain the curve of Figure <ref> (and satisfies the SUSY bound). Decay through these higher-dimension operators isn't the only way to induce the decay of Q-balls though; many other scenarios have been explored in the literature <cit.>.This work also begs the question of what possible observables exist that could show the Q-ball clusters collapse to black holes. We assume that the collapse will produce a stochastic gravitational wave background <cit.>, which could be detected by future observatories (or put constraints on the model). Further evolution of the PBH population could see successive mergers, which in addition to creating another stochastic GW background <cit.>, could also alter the distribution of black hole masses (in addition to evaporation/accretion effects <cit.>). We propose to calculate the gravitational wave spectrum in a future publication.§ CONCLUSIONIn summary, we have shown that the number density fluctuations of a Q-ball population in the early universe can lead to production of primordial black holes with sufficient abundance to explain the dark matter. Scalar fields and Q-ball formation are general features of supersymmetric extensions to the Standard Model, which provides a good motivation for this mechanism. A similar mechanism using solitons, topological defects, or other compact objects associated with scalar fields in the early universe can also lead to a copious production of primordial black holes.§ ACKNOWLEDGEMENTSThis work was supported by the U.S. Department of Energy Grant No. DE-SC0009937.A.K. was also supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan.§ EVOLUTION OF ENERGY DENSITIESHere we tabulate the functional form of the energy density for each species (radiation, Q-ball, black hole) up until the present day. The values for t_f, t_Q, t_R, and τ_Q are taken as input parameters (subject to some self-consistency conditions), while the values of t_eq and t_0 are calculated from the procedure described in Section <ref>. §.§ RadiationThe radiation density begins after reheating and is given by Equation <ref>. From this point we evolve it through time to the present day, taking into account the contribution due to Q-ball decays during the period t_Q < t < t_R. ρ_R(t) = π^2 M_p^2/327 t^2t_RH < t < t_Qπ^2 M_p^2/327 t_Q^2[ 1 + (τ_Q/t)^2/3Γ(5/3, t_Q/τ_Q, t/τ_Q) ] (t_Q/t)^8/3t_Q < t < t_Rπ^2 M_p^2/327 t_Q^2[ 1 + (τ_Q/t_R)^2/3Γ(5/3, t_Q/τ_Q, t_R/τ_Q) ] (t_Q/t_R)^8/3(t_R/t)^2 t_R < t < t_eq π^2 M_p^2/327 t_Q^2[ 1 + (τ_Q/t_R)^2/3Γ(5/3, t_Q/τ_Q, t_R/τ_Q) ] (t_Q/t_R)^8/3(t_R/t_eq)^2 (t_eq/t)^8/3t_eq < t < t_0§.§ Q-ballsThe Q-balls are created at the time of fragmentation t_f, and evolve as decaying nonrelativistic matter. The magnitude of the energy density becomes insignificant shortly after t_R. M_f = Λ |Q_f|^α N_f^1-α can be determined from specifying t_f, r_f, r, and N_f, as given in Section <ref>.⟨ρ_Q(t)|=⟩3 M_f/4π t_f^3(t_f/t)^3/2 e^-(t-t_f)/τ_Qt_f < t < t_Q3 M_f/4π t_f^3(t_f/t_Q)^3/2(t_Q/t)^2 e^-(t-t_f)/τ_Qt_Q < t < t_R3 M_f/4π t_f^3(t_f/t_Q)^3/2(t_Q/t_R)^2(t_R/t)^3/2 e^-(t-t_f)/τ_Qt_R < t < t_eq 3 M_f/4π t_f^3(t_f/t_Q)^3/2(t_Q/t_R)^2(t_R/t_eq)^3/2(t_eq/t)^2 e^-(t-t_f)/τ_Qt_eq < t < t_0§.§ Black holesThe black holes are created towards the end of the initial Q-ball dominated era, and their density at t_R is given by Equation <ref>:⟨ρ_BH(t_R)| ⟩= (a_f/a_R)^3 ∑_N=1^∞∫_V_min^V_RdV/V∫_0^∞ dM (βM/V) F_Q= (t_f^3/2 t_Q^1/2/t_R^2) M_f /V_f∑_N=1^∞∫_x_min^x_R dx β(x,N) x^N+α-2 e^-xwhereβ(x,N) = ((N/x)^1-α - 1) (t_R/t_*)^2/3≥δ_c : γδ_c^13/2(N/N_fx^α/N^α)^13/3 r_f^-13/2 β≤ 1 1β > 1((N/x)^1-α - 1) (t_R/t_*)^2/3 < δ_c : γ((N/x)^1-α - 1)^13/2(t_R/t_*N/N_fx^α/N^α)^13/3 r_f^-13/2 β≤ 1 1β > 1andt_* =t_Q x_min < x < x_Q3 V_f x/4π N_f t_f^3/2 t_Q^1/2x_Q < x < x_Rwhere x_min = N_f V_min/V_f = N_min, x_Q = N_f V_Q/V_f, and x_R = N_f V_R/V_f. After this has been evaluated, the evolution of the black hole density is fairly straightforward:⟨ρ_BH(t)|=⟩⟨ρ_BH(t_R)|⟩(t_R/t)^3/2t_R < t < t_eq ⟨ρ_BH(t_R)|⟩(t_R/t_eq)^3/2(t_eq/t)^2t_eq < t < t_0 | http://arxiv.org/abs/1706.09003v1 | {
"authors": [
"Eric Cotner",
"Alexander Kusenko"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170627183608",
"title": "Primordial black holes from scalar field evolution in the early universe"
} |
This paper initiates the study by analytic methods of the generalized principal series Maass forms on GL(3). These forms occur as an infinite sequence of one-parameter families in the two-parameter spectrum of GL(3) Maass forms, analogous to the relationship between the holomorphic modular forms and the spherical Maass cusp forms on GL(2). We develop a Kuznetsov trace formula attached to these forms at each weight and use it to prove an arithmetically-weighted Weyl law, demonstrating the existence of forms which are not self-dual. Previously, the only such forms that were known to exist were the self-dual forms arising from symmetric-squares of GL(2) forms. The Kuznetsov formula developed here should take the place of the GL(2) Petersson trace formula for theorems “in the weight aspect”. As before, the construction involves evaluating the Archimedian local zeta integral for the Rankin-Selberg convolution and proving a form of Kontorovich-Lebedev inversion. [2010]Primary 11F72; Secondary 11F55Evolution of quantum entanglement with disorder in fractional quantum Hall liquids R. N. Bhatt December 30, 2023 ==================================================================================§ INTRODUCTION The non-principal, generalized principal series forms for GL(3) are forms of minimal K-type attached to the (2d+1)-dimensional Wigner- matrix d (see section <ref>) with d ≥ 2 and spectral parametersμ = μ(r) := d-12+r,-d-12+r,-2r.These generate strict subrepresentations of principal series representations which are induced from representations on the 2,1 Levi subgroup. As a one-parameter family of Maass forms, one might compare them to the spherical Maass cusp forms on GL(2), but as a lower-dimensional subspace of the full GL(3) spectrum, one might also compare them to the point spectrum, i.e. holomorphic modular forms, on GL(2). The purpose of this paper is to initiate the analytic aspects of their study. This is the last of the three spectral Kuznetsov formulae on GL(3) (for full level over ), the others were constructed in <cit.> and <cit.>. From the adelic perspective, we are handling the case of ramification at the place at infinity and this would be largely unaffected by considering quotients by congruence subgroups, aka. ramification at the finite places.The structure of this paper is similar to the previous one, with the chain of constructionsStade's formula ⇒ Kontorovich-Lebedev inversion ⇒ Kuznetsov's formula ⇒ Weyl law,but the technical details of producing an infinite sequence of formulae are much more involved. In particular, the proof of Stade's formula (Theorem <ref> below) becomes difficult, and this is the piece we were unable to complete in the preceeding paper; the reader will notice the construction of section <ref> is quite intricate. On the other hand, the proof of the Weyl law becomes even easier since the Mellin-Barnes integrals for the Kuznetsov kernel functions are much simplified over the principal series case.Let 𝒮^d*_3 be a basis of vector-valued minimal-weight cusp forms attached to the (2d+1)-dimensional representation of SO(3,). Denote the spectral parameter of such a Maass form φ by r_φ∈ and notice the Ramanujan-Selberg conjecture r_φ∈ i is known for these forms <cit.>. Let 0r'∈ i and d ≥ 3, T > M> 1 such that d+T→∞. Then if we assume (as we may) that 𝒮^d*_3 consists of Hecke eigenforms, ∑_r_φ-T r' < M1/L(1,φ) = 3/2π∫_r-T r' < M^d(r) dr+Od(d+T)^2, where ^d(r) is the spectral weight ^d(r) = 116π^4 i(d-1)d-12-3rd-12+3r. The main term in the Weyl law is ≍ dM(d+T)^2, and we fail to give an asymtotic exactly when M ≪ 1, but a more careful analysis in that range would tighten up the error term to save a small power of log(d+T) over the main term (so the asymptotic holds for M down to a slightly negative power of log(d+T)), as the argument is actually using a Gaussian of width (log(d+T))^-1/2; we leave this to future explorers. One can see from (<ref>) that this is the limit of the method (up to loglog factors, etc.), and this (roughly) conforms to the rule of thumb that the Kuznetsov formula cannot resolve a ball of radius less than one in the spectrum.From the Weyl law, we see that the generalized principal series forms are fewer in number than the principal series forms of the previous papers: If we identify the weight d=0 and weight d=1 tempered spectra with μ∈^3(μ)=0,μ_1+μ_2+μ_3=0, then those Weyl laws (see <cit.>; the d=0 case follows almost verbatim using <cit.> as it appears in <cit.>) may be written∑_φ∈𝒮^d*_3 μ_φ∈ TΩ1/L(1,φ) = 3/2π∫_TΩ^d(μ) dμ+OT^4+ϵ,where Ω is some nice subset of the spectrum, dμ=dμ_1 dμ_2, and the spectral weights are^0(μ) = 1/384π^4∏_i<j (μ_i-μ_j)tanπ/2(μ_i-μ_j),^1(μ) = 1/64π^4∏_i<j (μ_i-μ_j)π/2(μ_1-μ_3) π/2(μ_2-μ_3) tanπ/2(μ_1-μ_2).The main terms of the two principal series Weyl laws are ≍ T^5, while an equivalent statement for fixed d ≥ 3 would have main term ≍ T^3. However, in every case, the size of the spectral measure is (generically) ≍∏_i<jμ_i-μ_j, and a sum of generalized principal series forms over d ≍ T restores equality between the two types. This should not be too surprising, as the number of holomorphic modular forms of weight k ≍ T is also of similar size to the number of spherical GL(2) Maass cusp forms of spectral parameter μ≍ T.Prior to the above theorem, existence theorems for Maass forms on GL(3) were limited to two types: First, Müller's Weyl law <cit.> indicates the existence of GL(3) Maass forms by counting all forms having a given K-type (including lifts of lower-weight forms). Second, the symmetric-square lift construction of Gelbart and Jacquet <cit.> directly produces a GL(3) Maass form from a given GL(2) Maass form (usually in the adelic viewpoint, so computing the effect of ramification in the classical sense can be trying). The group of forms studied here is too small to be detected by Müller's theorem (they are drowned out by the lifts of the d=0,1 forms; see previous paragraph), and the symmetric square lifts of GL(2) holomorphic modular forms of weight k are known to occur at d=2k-1 with r=0, so for φ∈𝒮^d*_3 when the weight d ≥ 3 is even or when the spectral parameter r_φ is non-zero, the existence of such forms is new.The unweighted Weyl laws (i.e. without 1/L(1,φ)) can be obtained from the Kuznetsov formulae given below by a weight-removal technique, see <cit.>. One then expects the spectral measures ^d(μ) dμ to correspond to the measure in Harish-Chandra's Plancherel theorem for SL(3,) (see <cit.>), up to some predictable (but difficult) constant, and this is easy to see for the spherical case, as the measure is nicely written out in <cit.>; meanwhile, the remaining cases d ≥ 1 are somewhat difficult to extract from the literature. The theorem above also does not handle the case d=2 for technical reasons, and we will discuss this in section <ref>, where we provide a rough upper bound on the Weyl law for those forms (which is needed in <cit.>). § SOME NOTATION FOR GL(3) Let G=PSL(3,) = GL(3,)/^× and Γ=SL(3,). The Iwasawa decomposition of G is G=U() Y^+ K using the groups K=SO(3,),U(R) = 1 x_2 x_31 x_1 1 x_i∈ R,R ∈,,, Y^+ = y_1 y_2,y_1,1 y_1,y_2 > 0.The measure on the space U() is simply dx := dx_1 dx_2 dx_3, and the measure on Y^+ isdy := dy_1 dy_2/(y_1 y_2)^3,so that the measure on G is dg :=dx dy dk, where dk is the Haar probability measure on K (see <cit.>). We generally identify elements of quotient spaces with their coset representatives, and in particular, we view U(), Y^+, K and Γ as subsets of G.Characters of U() are given byψ_m(x) = ψ_m_1,m_2(x) = m_1 x_1+m_2 x_2, t = e^2π i t,where m∈^2; we say ψ=ψ_m is non-degenerate when m_1 m_20. Characters of Y^+ are given by the power function on 3× 3 diagonal matrices, defined byp_μa_1,a_2,a_3 = a_1^μ_1a_2^μ_2a_3^μ_3,where μ∈^3. We assume μ_1+μ_2+μ_3=0 so this is defined modulo ^×, renormalize by ρ=(1,0,-1), and extend by the Iwasawa decompositionp_ρ+μx y k = y_1^1-μ_3 y_2^1+μ_1,x∈ U(),y∈ Y^+,k∈ K.Integrals in μ use the permutation-invariant measure dμ=dμ_1 dμ_2.The Weyl group W of G contains the six matrices[I=11 1,w_2= - 1 1 1,w_3= -1 11,;w_4= 1 1 1,w_5=1 11,w_l= -11 1, ]with the relations w_3 w_2=w_4, w_2 w_3=w_5 and w_2 w_3 w_2=w_3 w_2 w_3=w_l. The Weyl group induces an action on the coordinates of μ by p_μ^w(a):=p_μ(waw^-1), and we denote the coordinates of the permuted parameters by μ^w_i := (μ^w)_i, i=1,2,3. Explicity, the action isμ^I = μ_1,μ_2,μ_3,μ^w_2 = μ_2,μ_1,μ_3,μ^w_3 = μ_1,μ_3,μ_2,μ^w_4 = μ_3,μ_1,μ_2,μ^w_5 = μ_2,μ_3,μ_1,μ^w_l = μ_3,μ_2,μ_1.The group of diagonal, orthogonal matrices V ⊂ G contains the four matrices v_ε_1,ε_2=ε_1,ε_1 ε_2,ε_2, ε∈± 1^2, which we abbreviate V = ++, +-, -+, –. We write Y=Y^+ V for the diagonal matrices in G so that Y ∩ K = V. We tend not to distinguish between elements of Y and pairs in (^×)^2, as the multiplication is the same.We will also require the Bruhat decomposition G=U() Y W U(); the decomposition becomes unique if we replace, for each element w∈ W, the right-hand copy of U() with U_w() where U_w = (w^-1U w) ∩ U. When taking the Bruhat decomposition of an element γ∈Γ, we have γ=bcvwb' with b,b'∈ U(), v∈ V, w∈ W and c of the form1/c_2c_2/c_1 c_1,c_1, c_2∈.For such a matrix and a pair of characters ψ_m,ψ_n m,n∈^2, we define the Kloosterman sum attached to the Weyl element w∈ W byS_w(ψ_m,ψ_n,c) = ∑_b cw b' ∈ U()\Γ/(V U_w())ψ_m(b) ψ_n(b'),provided it is well-defined, and zero otherwise; these are given explicitly in <cit.>.The unitary, irreducible representations of K, up to isomorphism, are given by the Wigner -matrices d:K→ GL(2d+1,) for 0 ≤ d ∈. We treat d primarily as a matrix-valued function with the usual propertiesd(kk')=d(k)d(k'), d(k) = d(k)^-1 = d(k^-1).The entries of the matrix-valued function d are indexed from the center:d = d-d-d … d-dd ⋮ ⋱ ⋮ dd-d … ddd.The entries, rows, and columns of of the derived matrix- and vector-valued functions (e.g. the matrix-valued Whittaker function, see section <ref>) will be indexed similarly. As the Wigner -matrices exhaust the equivalence classes of unitary, irreducible representations of the compact group K, they give a basis of L^2(K), as in <cit.>, by the Peter-Weyl theorem. We tend to refer to the index d as the “weight” of the Wigner -matrix and any associated objects (e.g. the Whittaker function, Maass forms, etc.). The Wigner -matrices on the v-matrices occur frequently, and these are given by (see <cit.>)d(v_ε,+1) = ε^d,…,ε^-d,dm'm(v_ε,-1) =(-1)^dε^m'δ_m'=-m. A Maass form (cuspidal or Eisenstein) of weight d and spectral parameters μ for Γ is a row vector-valued (or matrix-valued) smooth function f:Γ\ G→^2d+1 that transforms under K as f(gk)=f(g)d(k), satisfies a moderate growth condition and is an eigenfunction of both Casimir operators with eigenvalues matching p_ρ+μ (see <cit.>). We denote the quadratic Casimir operator by Δ_1 and the cubic Casimir operator by Δ_2; precise expressions for these operators will not be necessary, here. The action of the Lie algebra of G on Maass forms gives rise to five operators Y^a, a≤ 2 on vector-valued Maass forms (see <cit.>) that change the weight d ↦ d+a, and a Maass form is said to have minimal weight if it is sent to zero by the lowering operators Y^-1, Y^-2 and an eigenfunction of the Y^0 operator.Throughout the paper, we take the term “smooth”, in reference to some function, to mean infinitely differentiable on the domain. The letters x,y,k,g, v and w will generally refer to elements of U(), Y^+, K, G, V, and W, respectively. The letter ψ will generally refer to a character of U(), and μ will always refer to an element of ^3 satisfying μ_1+μ_2+μ_3=0. Vectors (resp. matrices) not directly associated with the Wigner -matrices, e.g. elements n ∈^2 are indexed in the traditional manner from the left-most entry (resp. the top-left entry), e.g.n=(n_1,n_2). We do not use the primed notation F' for derivatives, but rather to distinguish functions and variables with similar purpose.§ RESULTSThroughout we assume d ≥ 2 is an integer. Suppose r∈ i and y =(y_1 y_2,y_1,1) in Y^+ ≅ (^+)^2, the space of positive diagonal matrices as in <cit.>. Then for m'≤ d, we write m'=ε m with ε = ± 1 and 0 ≤ m ≤ d, and defineW^d*_ε m(y, r) = 1/2^d+1π√(2dd+m)∑_ℓ=0^mε^ℓmℓ∫_(s)=𝔰 (2π y_1)^1-s_1 (2π y_2)^1-s_2×Γd-12+s_1-rΓd-12 + s_2+r Bd-m+s_1+2r2,ℓ+s_2-2r2ds/(2π i)^2,for any 𝔰∈ (^+)^2. The completed Whittaker function W^d*(y,r) attached to the minimal K-type d is the row vector with coordinates W^d*_-d,…,W^d*_d.We start in section <ref> by generalizing Stade's formula to the Rankin-Selberg convolution of two generalized principal series forms of the same minimal weight (see <cit.>). Define Ψ^d=Ψ^d(r,r',t) = ∫_Y^+ W^d*(y,r) W^d*(y,r') (y_1^2 y_2)^t dy, then Ψ^d=2^4-d-4t-r-r'π^2-3tΓt+r+r'Γd-12+t+r-2r'Γd-12+t+r'-2r×Γ(d-1+t+r+r') Γt2-r-r' / Γ3t2. In the preceeding paper, we essentially followed Stade's original proof <cit.>, but that fails here for technical reasons; the proof of this result is dramatically more complicated, and this is the central, new step that we were unable to accomplish previously.Define the spectral weights1/^d(r) := 2π i/3lim_t→ 0^+ tΨ^d(r,-r,t) = 2^5-dπ^3 i Γ(d-1) Γd-12-3rΓd-12+3r,1/^d(r) := Ψ^d(r,-r,1)= 2^1-d/πΓ(d) Γd+12+3rΓd+12-3r,^d(r) := ^d(r)/^d(r) = 116π^4 i(d-1)d-12-3rd-12+3r. In section <ref>, we apply Stade's formula to generalize Kontorovich-Lebedev inversion to our choice of Whittaker functions, using the method of Goldfeld and Kontorovich <cit.>. For f:Y^+→^2d+1 define f^♯(r) = ∫_Y^+ f(y) W^d*(y,r) dy, and for F:i→, define F^♭(y) = ∫_(r)=0 F(r) W^d*(y,r) ^d(r) dr. If F(r) is holomorphic and Schwartz-class on a vertical strip r(r)<δ for some δ > 0, then (F^♭)^♯(r) = F(r). Once again, we are making no claim as to the image of F ↦ F^♭ beyond the necessary convergence.In section <ref>, the Kuznetsov formula will follow from Stade's formula and Kontorovich-Lebedev inversion. The Kuznetsov kernel functions are defined from the power-series solutions (<ref>),(<ref>) as the linear combinationsK_I^d(y;r) =1, 4πcosπd2+3r K_w_4^d(y;r) = (-ε_1 i)^d J_w_4(y,μ(r)^w_4) exp-ε iπ2d-12-3r -(-ε_1 i)^d J_w_4(y,μ(r)), K_w_5^d(y;r) =K_w_4^d(-+ y^ι;-r), -4πcosπd2+3r K_w_l^d(y,r) = δ_ε_1=-1 (ε_2)^d J_w_l(y,μ(r)^w_4) + δ_ε_2=-1 (-ε_1)^d J_w_l(y,μ(r))-δ_ε_1 ε_2=-1 (-ε_1)^d J_w_l(y,μ(r)^w_3).These functions are the kernels for the integral transformsH_w(F; y) = 1/y_1 y_2∫_(r)=0 F(r) K^d_w(y, r) ^d(r) dr. Let 𝒮_3^d* be an orthonormal basis of SL(3,) cusp forms of minimal weight d and 𝒮_2^d* an orthonormal basis of SL(2,) cusp forms of minimal weight d (i.e. those coming from holomorphic modular forms of weight d). Note that 𝒮_2^d* is empty for odd d and the symmetric-squares of holomorphic modular forms only occur for odd d, but we will see there are forms in 𝒮_3^d* with r0 for all d. The particular normalization of the Fourier-Whittaker coefficients ρ_φ^*(m) of φ∈𝒮_3^d* and ρ_ϕ^*(m;r) of the maximal parabolic Eisenstein series attached to ϕ∈𝒮_2^d* is given in (<ref>).We are now ready to state the Kuznetsov formula for this class of forms. Let d ≥ 3. Let F(r) be Schwartz-class and holomorphic on (r)<1/4+δ for some δ > 0. Then for m,n∈^2 with m_1 m_2 n_1 n_20, 𝒞+ℰ=𝒦_I+𝒦_4+𝒦_5+𝒦_l, where 𝒞 = ∑_φ∈𝒮_3^d* F(r_φ) ρ_φ^*(m)ρ_φ^*(n)/^d(r_φ),ℰ = 2/2π i∑_ϕ∈𝒮_2^d*∫_(r)=0 F(r) ρ_ϕ^*(m;r)ρ_ϕ^*(n;r)/^d(r) dr, 𝒦_I = δ_m_1=n_1 m_2=n_2 H_I(F;(1, 1))𝒦_4 = ∑_ε∈±1^2∑_c_1,c_2∈ ε_1 m_2 c_1=n_1 c_2^2S_w_4(ψ_m,ψ_ε n,c)/c_1 c_2 H_w_4F; ε_1 ε_2 m_1 m_2^2n_2c_2^3n_1,1,𝒦_5 = ∑_ε∈±1^2∑_c_1,c_2∈ ε_2 m_1 c_2= n_2 c_1^2S_w_5(ψ_m,ψ_ε n,c)/c_1 c_2 H_w_5F; 1,ε_1 ε_2 m_1^2 m_2 n_1c_1^3 n_2,𝒦_l = ∑_ε∈±1^2∑_c_1,c_2∈S_w_l(ψ_m,ψ_ε n,c)/c_1 c_2 H_w_lF; ε_2 m_1 n_2 c_2c_1^2, ε_1 m_2 n_1 c_1c_2^2. The absolute convergence of the Kuznetsov formula can be shown through contour shifting and relies on the holomorphy condition; we will demonstrate this explicitly in the proof of the Weyl law.It will be useful to compare the current kernel functions to those for the other minimal K-types. Using the K^±±_w_l functions of <cit.> and <cit.>, and setting ε=(y), we have-4πcosπd2+3r K_w_l^d(y,r) = 0ε=(1,1), K^-+_w_l(y,μ(r)^w_4)ε=(-1,1), (-1)^d K^+-_w_l(y,μ(r))ε=(1,-1), K^–_w_l(y,μ(r))ε=(-1,-1).On the other hand, we can provide a much more compact statement of the Mellin-Barnes integrals at μ=μ(r); define[B^–_w_l(s,r) := (-1)^d Bs_1+3r,s_2-3r,; B^-+_w_l(s,r) := Bs_2-3r,1-s_1-s_2,; B^+-_w_l(s,r) := Bs_1+3r,1-s_1-s_2,; B^++_w_l(s,r) := 0, ] Q(d,s) := Γd-1/2+s/Γd+1/2-s,thenK_w_l^d(y,r) = 1/4π^2y_2/y_1^r ∫_-i∞^+i∞∫_-i∞^+i∞4π^2 y_1^1-s_14π^2 y_2^1-s_2 B^ε_w_ls,r Q(d,s_1) Q(d,s_2) ds_2/2π ids_1/2π i.And of course, the K_w_4^d integral is new once again:K_w_4^d(y;μ(r)) = 4^r-1(ε_1 i)^d/π^3/2∫_-i∞^+i∞4π^3 y_1^1-r-s Q(d,s) Γs+3r/2/Γ1-s-3r/2+ε_1 i Γ1+s+3r/2/Γ2-s-3r/2ds/2π i = (ε_1 i)^d /4π^2∫_-i∞^+i∞8π^3 y_1^1-r-s Q(d,s) Γs+3rexpε_1 iπ2s+3rds/2π i.The author would like to point out that this is essentially the only possibility for a Mellin-Barnes integral (on a vertical contour) for any linear combination of the J_w_4 (which is a 02 hypergeometric function), which lends some credence to the hope that we have not missed any v-matrices along the way.In section <ref>, we prove our first application of the Kuznetsov formula, which is a technical version of the Weyl law with both analytic and arithmetic weights: Let F(r) and δ be as in Theorem <ref>, then we have ∑_φ∈𝒮^d*_3 F(r_φ) ρ_φ^*(1)^2/^d(r_φ) = ∫_(r)=0 F(r) ^d(r) dr+OE_1+E_2, where E_1 := ∫_(r)=0F(r) (d+r)^1+ϵdr, E_2 := ∫_(r)=-1/4-ηF(r)+F(-r) d (d+r)^-1/2+ϵdr, and we assume 0 < η < δ satisfies η = O(ϵ) and ϵ>0 arbitrarily small.In section <ref>, by a careful choice of test function, we will remove the analytic weights: For Ω,r' and T and M as in Theorem <ref>, ∑_r_φ-T r' < Mρ_φ^*(1)^2/^d(r_φ) = ∫_r-T r' < M^d(r) dr+Od(d+T)^2. We have made no attempt to optimize the error terms in Theorem <ref>, as this will be sufficient for what is likely the best-possible error term in the corollary due to the sharp cut-off.A Rankin-Selberg argument using Stade's formula gives the Kuznetsov formula on Hecke eigenvalues, and this is discussed in section <ref>. When the bases of Theorem <ref> are taken to be Hecke eigenfunctions, the left-hand side may be written as 𝒞 = 2π/3∑_φ∈𝒮_3^d* F(r_φ) λ_φ(m)λ_φ(n)/L(1,φ),ℰ = 2/i∑_ϕ∈𝒮_2^d*∫_(r)=0 F(r) λ_ϕ(m,r)λ_ϕ(n,r)/L(ϕ,1+3r) L(ϕ,1-3r) L(1,ϕ) dr. where λ_φ(m) and λ_ϕ(m,r) as in <cit.> are the Hecke eigenvalues of the associated forms. The main theorem now follows. § BACKGROUND §.§ The Whittaker functions The Whittaker function W^d*(y, r) naturally extends to a function on G by the Iwasawa decomposition (see <cit.>),W^d*(xyk, r) = ψ_1,1(x) W^d*(y, r) d(k).In <cit.>, this Mellin-Barnes integral derives from the matrix-valued Whittaker functionW^d(g,μ,ψ) := ∫_U() I^d(w_l u g, μ) ψ(u) du,I^d(xyk,μ) := p_ρ+μ(y) d(k),in the formΛ^*(r) W^d_-d(y, μ(r), ψ_1,1) =W^d*(y, r),W^d_d,m = 0,whereΛ^*(r) =Λ^*(μ(r)) = (-1)^d/π (2π)^-d-12-3rΓ(d) Γd+12+3r. The functional equations in μ of the matrix-valued Whittaker function are given by <cit.>,W^d(g,μ,ψ_1,1) = T^d(w,μ) W^d(g,μ^w,ψ_1,1),w ∈ W,and these are generated by the matricesT^d(w_2,μ) := π^μ_1-μ_2Γ^d_𝒲(μ_2-μ_1,+1), T^d(w_3,μ) := π^μ_2-μ_3d(–w_l) Γ^d_𝒲(μ_3-μ_2,+1) d(w_l–),where Γ^d_𝒲(u,ε) is a diagonal matrix coming from the functional equation of the classical Whittaker function <cit.>: If 𝒲^d(y,u) is the diagonal matrix-valued function with entries (see <cit.>)𝒲^d_m,m(y,u) = ∫_-∞^∞1+x^2^-1+u/21+ix/√(1+x^2)^-m-yx dx = (πy)^1+u/2/yΓ1-ε m+u/2 W_-ε m/2, u/2(4πy) y0,2^1-uπ Γ(u)/Γ1+u+m/2Γ1+u-m/2y=0,(where W_α,β(y) is the classical Whittaker function), then for y 0, we have the functional equations𝒲^d(y,-u) =(πy)^-uΓ^d_𝒲(u,(y)) 𝒲^d(y,u)Γ_𝒲,m,m^d(u,ε) = Γ1-ε m+u/2/Γ1-ε m-u/2. The matrices T^d(w,μ) commute with the action of the v-matrices according to <cit.>d(v) T^d(w,μ) = T^d(w,μ) d(w^-1 v w). We can express Γ_𝒲^d(u,+1) in terms of 𝒲^d(0,-u) and visa-versa by <cit.> and its inverseΓ_𝒲^d(u,+1) = i Γ(1+u)/2^1+uπexpiπ u2d(–)-exp-iπ u2d(+-)d(w_2)𝒲^d(0,-u),𝒲^d(0,u) =2^-uΓ(u) expiπ u2d(–)+exp-iπ u2d(+-)d(w_2) Γ_𝒲^d(-u,+1).§.§ The Spectral ExpansionThere exists a symmetric differential operator Λ_d-1/2 of <cit.>, whose kernel in the weight d forms is exactly the span of the forms with spectral parameters of the form μ(r), r∈. Then for a Schwartz-class function f:Γ\ G→^2d+1 satisfyingf(gk) =f(g)d(k),Λ_d-1/2 f = 0,the spectral expansion of <cit.> takes the formf(g) = ∑_φ∈𝒮_3^d*φ(g)∫_Γ\ G f(g') φ(g') dg'+2/2π i∑_ϕ∈𝒮_2^d*∫_(r)=0 E^d_d(g, ϕ, r)∫_Γ\ G f(g') E^d_d(g', ϕ, r) dg' dr,where 𝒮_3^d* is a basis of SL(3,) cusp forms of minimal weight d and 𝒮_2^d* is a basis of SL(2,) cusp forms of minimal weight d. Note that there are no SL(2,) cusp forms and hence no Eisenstein series when d is odd. We have dropped all terms whose spectral parameters are not of the form μ(r) by the usual orthogonality argument:0=Λ_d-1/2 f, g = f, Λ_d-1/2 g = λ_gΛ_d-1/2f,g,whenever g is an eigenfunction of the Casimir operators. The extra 2 in the coefficient of the maximal parabolic Eisenstein series comes from the equalityE^d_d(g', ϕ, r) = E^d_-d(g', ϕ, r).The cusp forms φ∈𝒮_3^d* are normalized by <φ,φ>=1, in place of <cit.>.§.§ Integrals of gamma functions and hypergeometric identities We require a number of identities of Mellin-Barnes integrals, and we collect them here for ease of disposition. Let 𝒞 denote any contour from -i∞ to i∞ which obeys the Barnes integral convention (no gamma function in the numerator should have its argument pass through the negative real axis) and allow 𝒞 to vary from line to line. The parameters a,b,c,d∈ and arguments x,y∈, z∈ may be any values such that corresponding integrals and hypergeometric series converge absolutely and avoid the relevant branch cuts. We will not need to worry about the branch cuts, but more precise statements can be found in the references.The hypergeometric functions in general have the integral description <cit.>∏_i=1^p Γ(a_i)/∏_i=1^q Γ(b_i)pqa_1,…,a_pb_1,…,b_q-z = ∫_𝒞Γ(s)∏_i=1^p Γ(a_i-s)/∏_i=1^q Γ(b_i-s) z^-sds/2π i,for q ≤ p+1.By Mellin inversion and the definition of the beta function,∫_𝒞Γ(s) Γ(a+1)/Γ(a+1+s) x^-sds/2π i = (1-x)^a 0<x<1, 0 x ≥ 1. From (<ref>) and Thomae's theorem <cit.>∫_𝒞Γ(s) Γa-sΓa+b-d-s/Γ2-c+sΓa+b-sΓa+c-d-sds/2π i= Γ(a)/Γ(2-c) Γ(d) Γc-b∫_𝒞Γ(s) Γ(1-s) Γ(d-s) Γc-b-s/Γc-sΓ1+a-sds/2π i. The Euler integral representationsB(u,v) = ∫_0^1 x^u-1 (1-x)^v-1 dx,and <cit.>∫_0^1 (1-x)^a x^b (1+y x)^c dx = Γ(a+1)Γ(b+1)/Γ(a+b+2)21b+1,-ca+b+2-y. The Pfaff transformation <cit.> at z=1/221a,bc12 = 2^-b21a,c-bc-1. Barnes' first lemma <cit.>∫_𝒞Γ(a+s)Γ(b+s)Γ(c-s)Γ(d-s) ds/2π i = Γ(a+c)Γ(a+d)Γ(b+c)Γ(b+d)/Γ(a+b+c+d). Gauss' theorem <cit.>21a,bc1=Γ(c)Γ(c-a-b)/Γ(c-a)Γ(c-b). And several more, apparently unnamed, identities:∫_𝒞 x^-s/2Γa+sΓb-sds/2π i = 1+√(x)^-a-b x^a/2 Γ(a+b), <cit.>, ∫_𝒞Γ(s-a) Γb-s/Γb-ads/2π i = 10b-a-1 = 2^a-b, <cit.>, 1/a21a+b,aa+1-1+1/b21a+b,bb+1-1 =B(a,b), <cit.>. § STADE'S FORMULA The proof of Theorem <ref> is quite complicated and involves a number of seemingly random manipulations for which the author has no intuition beyond their simple effectiveness. To assist the reader, we give a brief summary: The proof proceeds by the usual application of Parseval's formula for the Mellin transform in (<ref>) followed by expanding the beta functions using Euler's integral (<ref>), at which point we can evaluate all of the sums and inverse Mellin transforms to produce a single, elementary, two-dimensional integral (<ref>) for Ψ^d. The elementary transformation (<ref>) splits the integral into two, simpler integrals and Mellin-expanding the resulting hypergeometric integrals gives Ψ^d as a sum of two, three-dimensional Mellin-Barnes integrals at 1 in (<ref>). Applying Thomae's theorem allows us to evaluate one of the three one-dimensional integrals, at which point the hypergeometric identities (<ref>) and (<ref>) recombine the sum of two, two-dimensional integrals into one, one-dimensional integral, which can be evaluated by Barnes' first lemma.It seems that the hypergeometric manipulations would be better realized as substitutions on the elementary integral; in particular, splitting the integral into two pieces only to later recombine them suggests we missed an elementary substitution which kept the two pieces together in the first place. Of course, even knowing such a thing should exist doesn't necessarily make it easy to find.We denote the contour for an n-dimensional Mellin-Barnes integral as 𝒞^n, continuing on from section <ref> for simplicity of notation.Now to the proof: Starting from the definition of Ψ^d, we apply the definition of the completed Whittaker function (<ref>), and Parseval's formula for the Mellin transform so thatΨ^d= (2π)^4-3t/2^2d+2π^2∫_𝒞^2 F(s) Γd-12+s_1-rΓd-12 + s_2+r×Γd-12+2t-s_1-r'Γd-12 + t-s_2+r'ds/(2π i)^2, F(s) := ∑_m'=-d^d 2dd+m'∑_ℓ_1=0^mε^ℓ_1mℓ_1∑_ℓ_2=0^mε^ℓ_2mℓ_2 Bd-m+s_1+2r2,ℓ_1+s_2-2r2× Bd-m+2t-s_1+2r'2, ℓ_2+t-s_2-2r'2,after collecting the sums of beta functions. Now we apply the Euler integral (<ref>) for each beta function, and evaluate the sums, using the binomial theorem in the forms∑_ℓ=0^mε^ℓmℓ (1-x)^ℓ/2 = 1+ε√(1-x)^m,∑_m'=-d^d 2dd+m' u^m' = 2+u^-1+u^d.And so F is given by the two-dimensional integralF(s) =2^d ∫_0^1 ∫_0^1 x_1^s_1+2r/2-1 (1-x_1)^s_2-2r/2-1 x_2^2t-s_1+2r'/2-1(1-x_2)^t-s_2-2r'/2-1×1+√(x_1x_2)+√(1-x_1)√(1-x_2)^d dx_1 dx_2,where we have used several times the fact that√(x)/1+√(1-x)=1-√(1-x)/√(x). Returning to Ψ^d, we haveΨ^d= (2π)^4-3t/2^d+2π^2∫_[0,1]^2 x_1^r-1 (1-x_1)^-r-1 x_2^t+r'-1 (1-x_2)^t-2r'/2-11+√(x_1x_2)+√(1-x_1)√(1-x_2)^d ×∫_𝒞x_2/x_1^-s_1/2Γd-12+s_1-rΓd-12+2t-s_1-r'ds_1/2π i×∫_𝒞1-x_2/1-x_1^-s_2/2Γd-12 + s_2+rΓd-12 + t-s_2+r'ds_2/2π i dx. Next we apply (<ref>) to achieve the elementary integral description,Ψ^d= (2π)^4-3t/2^d+2π^2Γ(d-1+2t-r-r') Γ(d-1+t+r+r') ×∫_[0,1]^2 x_1^d-1/2+2t+2r-r'/2-1 (1-x_1)^d-1/2 +t+r'-2r/2-1 x_2^d-1/2+2t+2r'-r/2-1 (1-x_2)^d-1/2 +t+r-2r'/2-1×1+√(x_1x_2)+√(1-x_1)√(1-x_2)^d ×√(x_1)+√(x_2)^-(d-1+2t-r-r')√(1-x_1)+√(1-x_2)^-(d-1+t+r+r') dx. Now split the integral at x_1 = x_2, and perform the substitutions{[x_1 ↦ x_1^2 x_2,then 1-x_21-x_1^2 x_2↦ x_2^2onx_1<x_2,;x_2 ↦ x_1 x_2^2,then 1-x_11-x_1 x_2^2↦ x_1^2onx_2<x_1. ].We haveΨ^d= (2π)^4-3t/2^d π^2Γ(d-1+2t-r-r') Γ(d-1+t+r+r') ×(∫_[0,1]^2 x_1^d-3/2+2t+2r-r' (1-x_1)^t/2-r-r'-1 (1+x_1)^-3t/2 x_2^d-3/2+t+r-2r'× (1-x_2)^t+r+r'-1 (1-x_1 x_2)^1-3t/2 (1+x_1 x_2)^1-d-3t/2 dx+ ∫_[0,1]^2 x_1^d-3/2+t+r'-2r (1-x_1)^t+r+r'-1 x_2^d-3/2+2t+2r'-r× (1-x_2)^t/2-r-r'-1 (1+x_2)^-3t/2 (1-x_1 x_2)^1-3t/2 (1+x_1 x_2)^1-d-3t/2 dx).Notice the three-summand d-th power in (<ref>) has factored due to the substitutions.Temporarily assuming 0<t<2/3, we apply (<ref>) to the factor (1-x_1 x_2)^1-3t/2, so that we may apply (<ref>) and (<ref>) twice to produceΨ^d= (2π)^4-3t/2^d π^2Γ(d-1+2t-r-r') Γ(d-1+t+r+r') Γ2-3t/2Γt/2-r-r'/Γd-1+3t/2Γ3t/2×Γt+r+r'∫_𝒞^3Γ(s_1) Γ(s_2) Γ(s_3) Γd-1+3t/2-s_2Γ3t/2-s_3/Γ2-3t/2+s_1×(Γd-1/2+t+r-2r'-s_1-s_2Γd-1/2+2t+2r-r'-s_1-s_2-s_3/Γd-1/2+2t+2r-r'-s_1-s_2Γd-1+5t/2+r-2r'-s_1-s_2-s_3 + Γd-1/2+t+r'-2r-s_1-s_2Γd-1/2+2t+2r'-r-s_1-s_2-s_3/Γd-1/2+2t+2r'-r-s_1-s_2Γd-1+5t/2+r'-2r-s_1-s_2-s_3) ds/(2π i)^3 Now applying Thomae's theorem in the form (<ref>) to the s_1 integral usinga = d-12+t+r-2r'-s_2,b=t+r+r',c=3t2,d=s_3,we may evaluate the s_3 integral using (<ref>). The s_1 integral becomes a 21 at 1/2 by (<ref>), and after a Pfaff transformation (<ref>), we haveΨ^d=2^4-d-4t-r-r'π^2-3tΓ(d-1+2t-r-r') Γ(d-1+t+r+r') Γt+r+r'Γ(t2-r-r')/Γd-1+3t/2Γ3t/2×∫_𝒞Γ(s_2) Γd-1+3t2-s_2×(Γ(d-1/2+t+r-2r'-s_2)/Γ(d+1/2+t+r-2r'-s_2)21d-1/2+t+r-2r'-s_2,t2-r-r'd-1/2+t+r-2r'-s_2+1-1 + Γ(d-1/2+t+r-2r'-s_2)/Γ(d+1/2+t+r'-2r-s_2)21d-1/2+t+r'-2r-s_2,t2-r-r'd-1/2+t+r'-2r-s_2+1-1) ds_2/2π i. In the second term, we send s_2 ↦ d-1+3t2-s_2, which allows us to use (<ref>), so thatΨ^d=2^4-d-4t-r-r'π^2-3tΓ(d-1+2t-r-r') Γ(d-1+t+r+r') Γt+r+r'Γt2-r-r'/Γd-1+3t/2Γ3t/2×∫_𝒞Γ(s_1)Γd-1+3t2-s_1 Bd-12+t+r-2r'-s_1,1-d2-t2+r'-2r+s_1ds_1/2π i.The theorem now follows from Barnes' first lemma (<ref>).§ KONTOROVICH-LEBEDEV INVERSION If F(r) is holomorphic in a neighborhood (r) < δ < 1/10, then we can argue that the Y^+ integral of (F^♭)^♯ converges absolutely (via contour shifting in r' and the Mellin-Barnes integral) and defineF(r,ϵ) := ∫_Y^+ F^♭(y) W^d*(y,μ(r)) (y_1^2 y_2)^ϵ dy = ∫_(r')=0 F(r') Ψ^d(r',-r,ϵ) ^d(r') dr',where we assume η := δ/2 > ϵ > 0, (r)=0 and r 0.Shift the r' integral to (r')=η, picking up a residue at r'=ϵ/2+r. The shifted integral is zero in the limit ϵ→ 0 (by the Γ(3ϵ/2) in the denominator), and the residue of Ψ^d islim_ϵ→ 0^+_r'=ϵ/2+rΨ^d(r',-r,ϵ) =2^4-dπ^2 Γ(d-1) Γ(d-12-3r) Γ(d-12+3r) = 1/2π i ^d(r).§ ASYMPTOTICS AND FUNCTIONAL EQUATIONS OF THE WHITTAKER FUNCTIONSAs in the previous papers, we will require certain first-term asymptotics of the Whittaker function. Define (2d+1)-dimensional row vectors ^d_j with entries ^d_j,m' = δ_m'=j. Then the asymptotics of the completed Whittaker function are given by the following lemmas. Assume r0, then as y → 0, we have W^d*(y,r) ∼ (-1)^d/π (2π)^d+3/2-3r p_ρ+μ^w_4(y) Γd-12 + 3r^d_d d(– w_l) +1/π (2π)^d+3/2+3r p_ρ+μ(y) Γd-12 - 3r^d_d -(-1)^d (2π)^3d+1/2+3r p_ρ+μ^w_3(y)/Γd+1/2+3rsinπd-1/2+3r^d_d d(– w_l) T^d(w_2,μ^w_4), for μ=μ(r), in the sense that W^d*(y,r) is a sum of (vector multiples of) three power series with the given leading terms. Assume r0, then as y_1 → 0, we have W^d*(y,r) ∼y_1^d+1/2-r y_2^1-2rΛ^*(r) Γ-d-12+3r(2π)^3d-1/2-3r/(d-1)!×i^d exp-iπ2d-12-3r^d_-d+i^-dexpiπ2d-12-3r^d_d× T^d(w_3,μ) d(w_2) 𝒲^d-y_2,d-12+3r+(-i)^d (2π)^1+6r y_1^1+2r y_2^3-d/2+rΛ^*(-r) Γd-1/2-3r/Γd+1/2+3r^d_-dd(–w_l) × T^d(w_5,μ^w_4) d(w_2) 𝒲^d(-y_2,d-1), for μ=μ(r), in the sense that W^d*(y,r) is a sum of (vector multiples of) two power series with the given leading terms. The functional equation takes the form W^d*(g, r) = (-1)^d W^d*(– g^ι w_l,-r), where g^ι = w_l (g^-1) w_l.We will use Lemma <ref> in the proof of Lemma <ref>, which in turn is used in the proof of Lemma <ref>. §.§ Double asymptotics of the Whittaker Function We now prove Lemma <ref>. Assume r0. We know that W^d*_m'(y,r) is a linear combination of power series with leading terms p_ρ+μ^w(y) and it is clear that the terms with w∈w_2,w_5,w_l do not occur since W^d*_m'(y,r) ≪y_1 y_2 for (r)=0. We need to find the coefficients of the remaining first terms; these occur as poles of the integrand in the Mellin-Barnes integral. As in the definition of the completed Whittaker function, we write m'=ε m with ε=±1 and m ≥ 0.The residue at s_1=-d-1/2+r, s_2=2r isR_1 := 2/2^d+1π√(2dd+m) (2π y_1)^d+1/2-r (2π y_2)^1-2rΓd-12 + 3r.By <cit.>, we haveddm'(– w_l) = (-1)^m'ddm'(0) = (-1)^d 2^-d√(2dd+m'),soR_1 = (-1)^d/πddm'(– w_l) (2π y_1)^d+1/2-r (2π y_2)^1-2rΓd-12 + 3r. The residue at s_1=-2r, s_2=-d-1/2-r isR_2 := δ_m=d2/2^d+1π (2π y_1)^1+2r (2π y_2)^d+1/2+rΓd-12 - 3r∑_ℓ=0^d ε^ℓdℓ = δ_m'=d1/π (2π y_1)^1+2r (2π y_2)^d+1/2+rΓd-12 - 3r. The residue at s_1=-d-1/2+r, s_2=-d-1/2-r isR_3 := 1/2^d+1π√(2dd+m) (2π y_1)^d+1/2-r (2π y_2)^d+1/2+rΓd+1/2-m+3r2∑_ℓ=0^mε^ℓmℓΓℓ-d-1/2-3r/2/Γ1-m+ℓ/2.The terms with ℓ≢m2 are zero, and the sum of the remaining terms may be evaluated by converting to a 21 at 1 and applying (<ref>):∑_ℓ=0^mε^ℓmℓΓℓ-d-1/2-3r/2/Γ1-m+ℓ/2 =-ε^m 2^d-1/2+3r/Γd+1/2+3rΓd+1/2+3r+m/2/sinπ/2d-1/2+3r-m.Using (<ref>)-(<ref>), we may write this asR_3 =-(-1)^d (2π)^d-1/2+3r(2π y_1)^d+1/2-r (2π y_2)^d+1/2+r/Γd+1/2+3rsinπd-1/2+3rd(– w_l) T^d(w_2,μ(r)^w_4)_d,m'. Then in a formal sense, we haveW^d*_m'(y,r) ∼ R_1+R_2+R_3as y → 0, or in other words, W^d*_m' is given by a sum of three power series with those leading terms. This will be sufficient to identify the particular linear combination of power series occuring in the K^d_w_l functions, as those power series are also distinguished by their leading terms.§.§ The dual Whittaker function We must briefly resort to the differential operators Y^a. As in <cit.>, if we take the dual f(g) := f(– g^ι w_l) of the function f(g) :=W^d*(g, -r), then the action of the lowering operators isY^-1f(g) = -Y^-1f(g) = 0,Y^-2f(g) = -Y^-2f(g) = 0,by the minimality of f.Assume (r)=0, then from <cit.> and (<ref>), we havef(g) = Λ^*(-r) ^d_-dd(– w_l) T^d(w_2,μ(r)^w_4) W^d(g,μ(r)^w_3,ψ_1,1),so f(g) lies in the rowspace of W^d(g,μ(r)^w_3,ψ_1,1). If we also assume r0, then <cit.>, <cit.>, and (<ref>) (and the w_3 functional equation (<ref>)) imply f(g) = C W^d*(g, r) for some scalar C=C(d,r). But then Lemma <ref> implies C=(-1)^d by comparing asymptotics, and this extends to an equality of meromorphic functions.§.§ Single asymptotics of the Whittaker Function We now prove Lemma <ref>. If (r) is large, then as in <cit.>, we haveW^d*(y,r) ∼p_ρ+μ^w_l(y) Λ^*(r) (-2π)^d y_1^d-1/(d-1)!^d_-d∫_^2 (1+u_3^2)^-1+d-1/2-3r/2 (1+u_2^2)^-1-d-1/2-3r/2×d(w_3) d-u_3-i/√(1+u_3^2)d(w_3) d1-iu_2/√(1+u_2^2)-y_2 u_2 du,as y_1 → 0. The integrals may then be expressed in terms of the 𝒲^d function as in (<ref>):W^d*(y,r) ∼y_1^d+1/2-r y_2^1-2rΛ^*(r) (2π)^d/(d-1)!^d_-dd(w_3) 𝒲^d0,-d-12+3r×d(w_5) 𝒲^d-y_2,d-12+3r. From the functional equation of Lemma <ref>, for (r) highly negative, we haveW^d*(y,r) ∼(-1)^d y_1^1+2r y_2^3-d/2+rΛ^*(-r) ^d_-d W^d(I,-μ^w_2,ψ_y_2,0) d(–w_l),as y_1 → 0. We then insert the computed value <cit.>:W^d*(y,r) ∼(-1)^d y_1^1+2r y_2^3-d/2+rΛ^*(-r) ^d_-dd(–w_l) 𝒲^d0,-d-12-3r×d(w_3) 𝒲^d0,d-12-3rd(w_5) 𝒲^d(-y_2,d-1). As with the y → 0 asymptotic, we know that the term y_1^3-d/2-r does not occur. Applying (<ref>), (<ref>) and^d_± dd(v_ε_1,ε_2) = (ε_1 ε_2)^d ^d_±ε_2 d, ^d_± dd(w_2) = (-1)^d i^± d^d_∓ dcompletes the lemma, keeping in mind that d(+-w_2)=di commutes with diagonal matrices such as T^d(w_2,μ).§ KUZNETSOV'S FORMULA We consider a Fourier coefficient of a Poincaré series of the formP_m(g,F) = ∑_γ∈ U()\Γ∫_(r)=0 F(r) W^d*(mγ g,r) ^d(r) dr, m=(m_1 m_2,m_1,1).There are some technical issues with the convergence of this series, especially for d=3 and d=2, and we will discuss them in section <ref>.Define the Fourier-Whittaker coefficients of a Maass form ξ with Langlands parameters μ(r_ξ) by∫_U()\ U()ξ(xyk) ψ_m(x) dx = ρ_ξ^*(n)/m_1 m_2 W^d*(m yk, r_ξ),and define the integral transformH_w(F; y, g) = 1/y_1 y_2∫_U_w()∫_(r)=0 F(r) W^d*(ywxg,r) ^d(r) drψ_1,1(x) dx,for w∈ W, y∈ Y, g∈ G.The spectral expansion and Bruhat decomposition give the pre-Kuznetsov formula∫_ℬ F(μ_ξ) ρ_ξ^*(m)ρ_ξ^*(n) W^d*(n yk, μ_ξ) dξ= n_1 n_2/m_1 m_2∫_U()\ U() P_m(xyk,F) ψ_n(x) dx = ∑_w∈ W∑_v∈ V∑_c_1,c_2≥1S_w(ψ_m,ψ_n^v,c)/c_1 c_2H_wF; m c w v n^-1 w^-1,n yk,where ∫_ℬ… dξ serves as an abbreviation for the sums and integrals occuring in the spectral expansion (<ref>). The details of the Bruhat decomposition can be found in <cit.>.Note: To be precise, in the development of the Kuznetsov formula, we must initially require F to be holomorphic on (r) < d/6+δ and have sufficient exponential decay to overcome the growth of the Fourier-Whittaker coefficients for absolute convergence of the Poincaré series. We may relax to (r) < 1/4+δ once we reach the pre-Kuznetsov formula.In section <ref> below, we show Let F be holomorphic and Schwartz-class on a neigborhood of (μ)=0, then for w=I,w_4,w_5,w_l, we have H_w(F; y, g) = 1/y_1 y_2∫_(r)=0 F(r) K_w^d(y,r) W^d*(g,r) ^d(r) dr, with K_w^d(y,r) as in (<ref>)-(<ref>). Then replacing F(r) withF(r) W^d*(n yk,r) (y_1^2 y_2)(in a suitable manner), and integrating in y with Stade's formula gives the theorem. There is a small technical point that the pre-Kuznetsov formula (<ref>) is an equality of vectors and we wish to apply Stade's formula, which involves a dot product, inside the r-integral; this may be accomplished, e.g., by taking the central entry of (<ref>), replacing F(r) with the central entry of the previous display, and integrating over k. §.§ Power series for the Kuznetsov kernel functionsThe functions K_w^d(y,r) are solutions to the differential equationsΔ_i K_w(g,r) = λ_i(μ(r)) K_w(g,r),K_w(u g (wu'w^-1),r) = ψ_1,1(uu') K_w(g,r),where g∈ G, w∈ W, u∈ U(), u'∈ U_w() andλ_1(μ) = 1-μ_1^2+μ_2^2+μ_3^22, λ_2(μ) = μ_1 μ_2 μ_3.These were solved in the paper <cit.>, under the assumption that μ_i-μ_j ∉, i j, but of course that fails for μ=μ(r).When w=w_l, the original power-series solutions areJ_w_l(y,μ) = 4π^2 y_1^1-μ_34π^2 y_2^1+μ_1∑_n_1,n_2≥ 0Γn_1+n_2+μ_1-μ_3+1(4π^2 y_1)^n_1 (4π^2 y_2)^n_2/∏_i=1^3 Γn_1+μ_i-μ_3+1Γn_2+μ_1-μ_i+1.By comparing asymptotics, we can see that J_w_l(y,μ), J_w_l(y,μ^w_3) and J_w_l(y,μ^w_4) are distinct solutions, but we have the relationsJ_w_l(y,μ(r)^w_2) = (y_2)^d-1 J_w_l(y,μ(r)), J_w_l(y,μ(r)^w_l) = (y_1)^d-1 J_w_l(y,μ(r)^w_4), J_w_l(y,μ(r)^w_5) = (y_1 y_2)^d-1 J_w_l(y,μ(r)^w_3).For w=w_4,w_l and w',w̃∈ W, we define the linear combinationY^d_w(y,μ,w',w̃,α) = J_w(y,μ^w̃)-α^d-1 J_w(y,μ^w')/sinπ(μ_1-μ_2).Then the remaining three long-element solutions and their first-term asymptotics as y → 0 areY^d_w_l(y,μ(r), I, w_2,(y_2)) ∼ 4π^2 y_1^1+2r4π^2 y_2^3-d/2+r(d-2)!/πΓ3-d/2+3rΓd+1/2+3r, Y^d_w_l(y,μ(r),w_3, w_5,(y_1 y_2)) ∼ 4π^2 y_1^3-d/2-r4π^2 y_2^3-d/2+r(d-2)!/πΓ3-d/2+3rΓ3-d/2-3r, Y^d_w_l(y,μ(r),w_4, w_l,(y_1)) ∼ 4π^2 y_1^3-d/2-r4π^2 y_2^1-2r(d-2)!/πΓ3-d/2-3rΓd+1/2-3r.In each case, either y_1 or y_2 has an exponent less than one, and will not appear in the Kuznetsov kernel functions. (For d ≤ 3, the main asymptotics as y → 0 are actually given by some logarithmic factors; again, such terms cannot appear in the kernel functions.)For w=w_4, the power-series solutions are (see <cit.>),J_w_4(y,μ) = 8π^3 y_1^1-μ_3∑_n=0^∞(-8π^3 i y_1)^n/n!Γn+1+μ_1-μ_3 Γn+1+μ_2-μ_3,and J_w_4(y,μ(r)^w_4) is distinct from J_w_4(y,μ(r)), butJ_w_4(y,μ(r)^w_5) =(i (y_1))^d-1 J_w_4(y,μ(r)^w_4).The requisite third solution is given byY^d_w_4(y,μ(r), w_4,w_5,i(y_1)) ∼ 8π^3 y_1^3-d/2-r(d-2)!/πΓ3-d/2-3r,and again, this will not appear in the Kuznetsov kernel functions. §.§ The weight functions Having done the relevant technical work related to the analytic continuation in previous papers, we regard the functions K_w^d(y,r) as being defined by the Riemann integralK_w^d(y,r) W^d*(g,r) = ∫_U_w() W^d*(ywxg,r)ψ_1,1(x) dx, Let x^* y^* k^* = wxg and replace y ↦ v_ε_1,ε_2 y with y∈ Y^+, then formallyK_w^d(y,r) W^d*(g,r) = ∫_U_w() W^d*(y y^*,r) d(v_ε_1,ε_2 k^*)ψ_v_ε_1,ε_2 y(x^*) ψ_1,1(x) dx. The process for obtaining K_w^d(y,r) is the same as for the previous two cases, but we must shift the s-integrals to (s)=(-d-1/2-ϵ,-d-1/2-ϵ) to see the term y_1^d+1/2-r y_2^d+1/2+r, and the test function needs holomorphy out to (r) < d/6+ϵ for absolute convergence of the integrals and sums of Kloosterman sums for the terms y_1^1+2r y_2^d+1/2+r and y_1^d+1/2-r y_2^1-2r. Lastly, the latter two terms require many more applications of the integration by parts proceedure on the function X_3' in <cit.> to reach (r)=0, but as explained in the preceeding paper, this is always possible.§.§.§ The long element functionAs y → 0,K_w_l^d(v_ε_1,ε_2 y,r) W^d*(g,r) ∼ 2 (-1)^d p_ρ+μ^w_4(y) (2π)^d+1/2-3rΓd-12 + 3r^d_d d(– w_l v_ε_1,ε_2) ×∫_U() p_ρ+μ^w_4(y^*) d(k^*) ψ_1,1(x) dx+ 2 p_ρ+μ(y) (2π)^d+1/2+3rΓd-12 - 3r^d_d d(v_ε_1,ε_2) ∫_U() p_ρ+μ(y^*) d(k^*) ψ_1,1(x) dx -(-1)^d (2π)^3d+1/2+3r p_ρ+μ^w_3(y)/Γd+1/2+3rsinπd-1/2+3r^d_d d(– w_l) T^d(w_2,μ^w_4) d(v_ε_1,ε_2) ×∫_U() p_ρ+μ^w_3(y^*) d(k^*) ψ_1,1(x) dx,in the sense of Lemma <ref>.The x integrals give incomplete Whittaker functions by comparing to the definition (<ref>), and the functional equations (<ref>) and Lemma <ref> (keeping in mind (<ref>) and (<ref>)) implyK_w_l^d(v_ε_1,ε_2 y,r) ∼2 (-ε_2)^d p_ρ+μ^w_4(y) δ_ε_1=-1(2π)^d+1/2-3rΓd-12 + 3r/Λ^*(-r) + 2 (-ε_1)^d p_ρ+μ(y) δ_ε_2=-1(2π)^d+1/2+3rΓd-12 - 3r/Λ^*(r)-(ε_1)^d p_ρ+μ^w_3(y) δ_ε_1 ε_2=-1(2π)^3d+1/2+3r/Γd+1/2+3rsinπd-1/2+3rΛ^*(-r).Then comparing asymptotics with J_w_l(y,μ(r)^w_4), J_w_l(y,μ(r)), and J_w_l(y,μ(r)^w_3) gives (<ref>).§.§.§ The w_4 functionAs y_1 → 0 on ε_2=y_2=1, with μ=μ(r),K_w_4^d(ε_1,1y,r) W^d*(g,r) ∼ y_1^d+1/2-rΛ^*(r) Γ-d-12+3r(2π)^3d-1/2-3r/(d-1)!×i^d exp-iπ2d-12-3r^d_-d+i^-dexpiπ2d-12-3r^d_d T^d(w_3,μ) d(w_2 v_ε_1,1) ×∫_U_w_4() (y_1^*)^1-μ_2 (y_2^*)^1+μ_3𝒲^d-y_2^*,μ_1-μ_3d(k^*)x_2^*-x_2 dx+(-i)^d (2π)^1+6r y_1^1+2rΛ^*(-r) Γd-1/2-3r/Γd+1/2+3r^d_-dd(–w_l) T^d(w_5,μ^w_4) d(w_2 v_ε_1,1) ×∫_U_w_4() (y_1^*)^1-μ_3 (y_2^*)^1+μ_2𝒲^d(-y_2^*,μ_1-μ_2) d(k^*)x_2^*-x_2 dx,in the sense of Lemma <ref>.The x integrals give incomplete Whittaker functions by comparing to <cit.>, so thatK_w_4^d(ε_1,1y,r) W^d*(g,r) ∼ y_1^d+1/2-rΛ^*(r) Γ-d-12+3r(2π)^3d-1/2-3r/(d-1)!×i^d exp-iπ2d-12-3r^d_-d+i^-dexpiπ2d-12-3r^d_d×d(v_ε_1,ε_1) T^d(w_3,μ) W^d(g,μ^w_3,ψ_1,1)+(-i)^d (2π)^1+6r y_1^1+2rΛ^*(-r) Γd-1/2-3r/Γd+1/2+3r^d_-dd(v_ε_1,1) d(–w_l) × T^d(w_5,μ^w_4) W^d(g,μ,ψ_1,1). Then applying the functional equations (<ref>) and Lemma <ref>, this becomesK_w_4^d(ε_1,1y,r) ∼y_1^d+1/2-rΓ-d-12+3r(2π)^3d-1/2-3r/(d-1)! (ε_1 i)^d exp-ε_1 iπ2d-12-3r+(ε_1 i)^d (2π)^1+6r y_1^1+2rΓd-1/2-3r/Γd+1/2+3r. The expression (<ref>) follows by comparing asymptotics with J_w_4(y,μ(r)) and J_w_4(y,μ(r)^w_4). §.§.§ The w_5 functionThe definition of the involution ι and two applications of Lemma <ref> giveK_w_5^d(y,r) W^d*(g,r) =(-1)^d ∫_U_w_4() W^d*(– y^ι w_4 –x –g^ι w_l,-r)ψ_1,1(x) dx =K_w_4^d(y^ι-+;-r) W^d*(g,r),and this implies (<ref>). § THE TECHNICAL WEYL LAW We now prove Theorem <ref>. By either Stirling's formula or the Phragmén-Lindelöf principle, when (s) > d is in some fixed compact set, we haveQ(d,s)≪ d+s^2(s)-1.(Note that, eg. Q(d,s)=d-1/2+s^-1 on (s)=0.) Stirling's formula also impliesB^ε_w_l(s,r)≪ 1+s_1+3r^-1-ϵ1+s_2-3r^-1-ϵ1+s_1+s_2^3/2+2ϵ,on (r)=0, (s)=-1/2-ϵ,-1/2-ϵ.On (r)=0, as in <cit.>, we can show∫_(s)=-1/2-ϵ,-1/2-ϵ1+s_1+s_2^3/2+2ϵ/1+s_1+r^1+ϵ1+s_2-r^1+ϵd+s_1^-2d+s_2^-2ds_1 ds_2≪d,r^1/2+ϵ/d(d+r)^2by following the methods of <cit.>.Starting from (<ref>) and the Mellin-Barnes integral (<ref>), we shift the s-contours back to (s)=-1/2-η,-1/2-η, picking up poles at s_1=-3r or s_2=3r (but not both simultaneously), where we shift the r contour to ±(1/4+η) and place the remaining s-contour at -1/4-η, givingy_1 y_2^-1/2-ϵH_w_l(F;y)≪∫_(r)=0F(r)∫_(s)=-1/2-η,-1/2-η1+s_1+3r^-1-ϵ1+s_2-3r^-1-ϵ×1+s_1+s_2^3/2+2ϵd+s_1^-2d+s_2^-2ds_1 ds_2 d(d+r)^2 dr+ ∫_(r)=-1/4-ηF(r)∫_(s_1)=-1/4-ηd+s_1^-3/2ds_1Q(d,3r) d(d+r)^2 dr+ ∫_(r)=1/4+ηF(r)∫_(s_2)=-1/4-ηd+s_2^-3/2ds_2Q(d,-3r) d(d+r)^2 dr≪ E_1 +E_2.Note: The residues in s are actually given by J-Bessel functions, so we could apply known bounds for those, but the bounds above are sufficient for our purposes.For the w_4 term, we start with the second form of (<ref>), and shift to (s) = -1/2. (We only need (s)=-ϵ, but shifting farther would give a better bound; this choice gives the more concise statement). The residue at s=-3r we shift up to (r)=1/4. The residue is trivial to handle, and for the shifted contour, we use∫_(s)=01+s+r^-1d+s^-2ds≪ 1/d(d+r).The w_5 term is handled by symmetry.For the Eisenstein series term, we assume 𝒮^d*_2 contains Hecke eigenforms so that we may (skip ahead a little and) use ℰ in the form from Theorem <ref>. It is well-known that 𝒮^d*_2≪ d and the quotient by L-functions is bounded by d^ϵ (1+r)^ϵ <cit.> (see the second remark on page 164 of <cit.>). § THE WEYL LAW In this section, we prove Corollary <ref>. Taking a test function F(r) = exp (r-Tr')^2, it follows from Theorem <ref> that∑_r_φ-T r' < 100ρ_φ^*(1)^2/^d(r_φ)≪ d(d+T)^2for r'∈ i.Let χ_r_φ-T r' < M be the characteristic function of the set r_φ-T r' < M, then we define our test function by convolution with an approximation to the identity:F(r) = -i√(log (d+T)/π)∫_(r')=0χ_r_φ-T r' < M(r-r') (d+T)^(r'^2) dr',Substituting r'↦ r-r', this extends to an entire function of r.As in the previous paper,0 < F(r) < 1on r∈ i χ_r_φ-T r' < M(r)-F(r)≪ (d+T)^-100 on r∈ i, r-T r'± iM≥ 10,and in general F satisfies the boundF(r) ≪ (d+T)^(r)^2+ϵχ_r_φ-T r' < M+10(i(r))+(r+d+T)^-97 on(r)<1.Applying (<ref>) in Theorem <ref>, we see that the integrals E_1 and E_2 are small compared to the error d(d+T)^2 resulting from the sharp cut-off. That error is again obtained by covering the inflated boundary r-Tr'± iM < 10 by 2 balls of radius 11, and applying (<ref>) and (<ref>). § RANKIN-SELBERG The computation on φ∈𝒮^d*_3 in Theorem <ref> follows precisely as in <cit.>, but some more work is required for the maximal parabolic Eisenstein series: From <cit.>, (<ref>), andΛ^*(r)Λ^*(r) = (d-1)!/π^d ^d(r)we haveρ_ϕ^*(m;r)ρ_ϕ^*(n;r)/^d(r) =4π^d/(d-1)!λ_ϕ(m,r)λ_ϕ(n,r)/L(ϕ,1+3r) L(ϕ,1-3r),when ϕ is Hecke-normalized (as in the Φ^d_H normalization of <cit.>). When ϕ is L^2-normalized, this becomesρ_ϕ^*(m;r)ρ_ϕ^*(n;r)/^d(r) =2πλ_ϕ(m,r)λ_ϕ(n,r)/L(ϕ,1+3r) L(ϕ,1-3r) L(1,ϕ).§ ABSOLUTE CONVERGENCE AND WEIGHT D=2 In this section, we give an upper bound for the Weyl law at d=2, as the naive Kuznetsov formula just fails to converge absolutely. Before we begin, we take a moment to discuss the convergence of the Kuznetsov formula for all d ≥ 3.In our development of the spectral Kuznetsov formulae above, we are using implicitly the polynomial dependence on T in the Weyl law. Ideally, this would follow from Müller's Weyl law <cit.> and existing bounds for the sup. norm of the cusp forms, but the author is unaware of any results of sufficient generality to cover non-spherical forms on a non-compact manifold. On the other hand, this may be proved quite easily by the method given for d=2 below.For d=3, the Poincaré series (<ref>) just fails to converge absolutely; the contour shifting in the Whittaker function encounters a pole at (s_1,s_2)=(-1+r,-1-r) and there is no way to shift the r contour so that the powers on both y coordinates simultaneously exceed 2. On the other hand, if we modify the kernel of the Poincaré series in (<ref>) to include an extra factor (y_1 y_2)^u, the spectral expansion will now also have the lifts of the d ≤ 2 forms, but will converge rapidly and uniformly on compact sets in (u) ≥ 0 by the usual Fourier-type analysis on the Whittaker functions of the spectral basis. The Bruhat decomposition also converges to a holomorphic function of u by the usual contour shifting as in section <ref>, and we see that (<ref>) holds for d=3 in spite of the conditional convergence of the Poincaré series.The true difficulty arises for d=2; in this case, even the sum of Kloosterman sums in the Fourier coefficients (just) fails to converge absolutely, due to a pole at (s_1,s_2)=(-1/2+r,-1/2-r) in the contour shifting. It will follow from the arithmetic Kuznetsov formula <cit.> that the naive d=2 Kuznetsov formula still holds, with the sum of Kloosterman sums converging in the conditional sense, but first one needs a reasonable upper bound on the Weyl law, so we prove:For T > 2,∑_φ∈𝒮^2*_3 r_φ± iT≤ 1ρ^*_φ(1,1)^2/^2(r_φ)≪ T^3.We consider the L^2-norm of the vector-valued Poincaré seriesP_F(g) = ∑_γ∈ U()\Γ F(γ g), F(xyk) :=f(xy) 0 0 1 0 02(k), f(xyk) := ψ_1,1(x) (2π y_1)^1+s_1 (2π y_2)^1+s_2exp(-2π(y_1+y_2)),where s_1=s_2=σ± iT with σ > 1 to be chosen later.Then if φ∈𝒮_3^2* has spectral parameters μ(r), r∈ i, we haveP_F,φ =2π^3 √(6)Λ^*(r) ρ^*_φ(1,1) F(r),F(r) := 1/Λ^*(r)∫_(s')=0Γ(s_1-s'_1) Γ(s_2-s'_2) Γ12+s'_1-rΓ12+s'_2+r× B2+s'_1+2r2,s'_2-2r2ds'/(2π i)^2.Now shift the s' contours back to -1+ϵ and suppose r± iT≤ 1, so F = F_1+F_2+F_3 with F_1 having the shifted contours, F_2 having the residue at s_2'=2r (and the s_1' contour at, say, (s_1')=0), and F_3 having the double residue at (s_1',s_2')=(-1/2+r,-1/2-r). Then F_1(r) ≪ T^-2+ϵ, F_2(r) decays exponentially in T, and F(r) ≍F_1(r) ≍ T^-3/2.By general L^2 theory, we may complete the orthonormal set of cusp forms 𝒮_3^2* to some basis (which won't be a spectral basis) of the whole L^2 space and apply positivity to Plancherel's theorem:∑_r_φ± iT≤ 1ρ^*_φ(1,1)^2/^2(r_φ)≪ T^3 P_F,P_F.On the other hand,P_F,P_F≤4 ∑_w∈ W∑_c_1,c_2≥ 1 S_w(1,1,c) F_w(c), F_w(c) := ∫_Y^+∫_U_w()f(cwxt) dx (2π t_1)^1+σ (2π t_2)^1+σexp(-2π(t_1+t_2)) dt.We conjugate xt=tu and for each w∈ W, we choose some β=β(w)∈^3 so exp(-y_1-y_2) ≪ p_-β(w)(y) andf(cwtu) p_ρ+σρ(t) dx/du≪ p_ρ+σρ^w+σρ(t) p_ρ+σρ(cwu) p_-β(cwtu) ≪ p_3ρ(cwu) t_1^h_1 t_2^h_2with h_1,h_2 > 2 also depending on w. This is sufficient for convergence of the x and t integrals and the c sum so that P_F,P_F≪ 1.Taking σ=4, a valid choice of β(w) isβ(I) = β(w_2) = β(w_3) = 0, β(w_4)=β(w_5)=β(w_l)=2ρ.Note that the above method only relied on the evaluation of the Mellin transform of a single entry of the vector-valued Whittaker function; in fact, this particular entry was not only the simplest in form, but also the easiest to evaluate <cit.>, and it is reasonable to expect this type of behavior to exist on a wide variety of groups. So we expect that the above method would generalize nicely to give (weak, but simple) bounds on arithmetically-weighted Weyl laws in those situations, as well. This is also interesting from the L-function standpoint as it gives a lower bound on the adjoint-square L-function at 1 (aka. the residue of the Rankin-Selberg L-function).amsplain | http://arxiv.org/abs/1706.08816v3 | {
"authors": [
"Jack Buttcane"
],
"categories": [
"math.NT",
"11F72 (Primary), 11F55 (Secondary)"
],
"primary_category": "math.NT",
"published": "20170627123409",
"title": "Kuznetsov, Petersson and Weyl on $GL(3)$, II: The generalized principal series forms"
} |
Combinatorial approach to detection offixed points]Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamicsYeshiva University, Department of Mathematical Sciences, New York, NY 10016, [email protected] University, Department of Mathematical Sciences, New York, NY 10016, [email protected]^†, ^♭ Research of M.G. and Y.S. was partially supported by NSF grantDMS-0635607,NSF grant DMS-1700154,and bytheAlfred P. Sloan Foundation grant G-2016-7320.We present a combinatorialapproach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well asto find numerical approximations of such objects. Our approach relies on the methodof `correctly aligned windows'. We subdivide the `windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way we encode the dynamics information into a combinatorial structure. We use a version of the Sperner Lemma saying that if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. Our arguments are elementary. [ Yitzchak Shmalo ^♭====================== § INTRODUCTIONNumerical investigations ofdiscrete-time dynamical systems often require the approximation of the phase space and of the underlying map via a fine grid. Henceforth, the dynamics information is encoded into a combinatorial structure. From the computational point of view it is important that such combinatorial structure should be as simple as possible. While simplicial structures appear to be more elegant, cubicalstructures present many practical advantages, includingthe possibility of using cartesian coordinates, simple numericalrepresentation of maps asmultivalued maps, and lower computational costs ofhigherdimension homologies. See. e.g., <cit.>.In this paper we develop a combinatorial topology-based approachto detect fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems. The approach relieson the method of `correctly aligned windows', also known as `covering relations'. This method goes back to the geometric ideas of Conley, Easton and McGehee <cit.>, while more recent, topological versions of the method have been developed in <cit.>. The method can be described concisely as follows. A `window' (also known as an `h-set') is a multi-dimensional rectangle, whose boundary consists of an `exit set' and an `entry set'. One window is correctly aligned with (or `covers') another window under the map if the image of the first window is going acrossthesecond window, with the exit set of the first window coming out through the exit set of the second window, and without the image of the first window intersecting theentry set of the second window. There is an additional condition that the crossing of the windows should be topologically nontrivial, which can be expressed in terms of the Brouwer degree. The main results about correctly aligned windows can be summarized as follow: (i) If a window is correctly aligned with itself, then there is a fixed point inside that window;(ii) For a finite sequenceof windows (with a circular ordering), if each window is correctly aligned with the next window in the sequence, then there exists a periodic orbit inside those windows;(iii) For an infinite sequence of windows, if each window is correctly aligned with the next window, then there exists an orbit inside those windows; (iv)For a finite sequence of pairwise disjoint windows, with the correct alignment of pairs of windows described by a 0-1 transition matrix,there exists an invariant set inside those windows on which the dynamics is semi-conjugate to a topological Markov chain associated to that matrix.In principle,this method only yieldsexistential type of results on fixed points/perodic orbits/orbits with prescribed itineraries.In this paper, we provide an algorithmic approach to verify the that the crossing of the windows is topologically non-trivial, andto detect numerically, up to a desired level of precision, fixed points/perodic orbits/orbits with prescribed itineraries. Our method is constructive, can be implemented numericallyquite easily, and does not requirethe computation of algebraic topology-type invariants(e.g., Conley index, homology, Brouwer degree). The approach proposed in this paper is based on combinatorial topology, particularly on the classical SpernerLemma <cit.>. We regard each window as a multi-dimensional cube, and we construct a cubical decomposition of it. Then we assign labels to all vertices of the cubical decomposition. Thelabel of a vertex xis given by the hyperoctant where the vector f_χ(x)-x lands, where f_χ is the map that defines the dynamical system expressed in certain coordinates. A cubeis called completely labeled if the intersection of the hyperoctants corresponding to its labels is the zero vector. We can also assign an index to the labeling of a cube. This index turns out to be related to the Brouwer degree (see Proposition <ref>). Thus the index can be computed via a simple recursive formula (see (<ref>)). A non-zero index is a sufficient condition, but not necessary,for thelabeling to be complete.To check that one window is correctly aligned with another, it is sufficient to check that the labels of the vertices of the cubical decomposition that lie on the boundary of one of thewindows satisfy certain explicit conditions, and that the above mentioned index is non-zero. If a window is correctly aligned with itself, a version of the Sperner Lemmashows the existence of at least one small cube in the cubical decomposition that has non-zero index, hencecompletely labeled. There may also exists small cubes that are completely labeled that have zero index. In this setting, a small cubewith non-zero index yields a true fixed point, while a small cube that is completely labeled yields a numerical approximation of a fixed point. Similarly, in the case of sequences of windows, small cubes inside those windows that have non-zero index yield true periodic orbits/orbits with prescribed itineraries, while small cubes that are completely labeled yieldnumerical approximations of periodic orbits/orbits with prescribed itineraries.Checking that a small cube has non-zero index can be done via a finite computation. Completely labeled small cubes can be searched using a Nerve Graph algorithm similar to <cit.>.In Section <ref>, we recall the classical Sperner Lemma and some generalizations. In Section <ref> we provide a newversion of the Sperner Lemma, for cubical complexes, which will be used in the subsequent sections. In Section <ref> we provide sufficient conditions for correct alignment of windows, in terms of the labeling of vertices of a cubical decomposition. We also prove several results on detection– in terms of the non-zero index/completely labeled cubes – of true/approximate fixed points, periodic orbits, and orbits with prescribed itineraries. In Section <ref> we illustrate the above procedure by an example, in which we find periodic orbits for the Hénon map.§ PRELIMINARIES §.§ Sperner's Lemma for simplicesConsider an n-simplex T, and T=⋃_i∈ I T_i, with I finite, a simplicial decomposition of T. A labeling of T is a map ϕ:T→{1,2,…,n+1}. In particular, each vertex v of T and of any of the T_i's is assigned a unique label ϕ(v)∈{1,2,…,n+1}. A simplex T is said to be completely labeled if its vertices are assigned all labels from {1,2,…,n+1}.The labeling ϕ is called a Sperner labeling if everypoint p that lies on some face S of Tis assigned one of the labels of the vertices of S.In the case of a completely labeled simplex, the Sperner condition is equivalent to anon-degenerate labeling condition,that no (n-1)-dimensional face of T contains points of(n+1) or more different labels.In the most basic form the Sperner Lemma states the following:Given an n-simplex T with a complete Sperner labeling,a simplicial decomposition T=⋃_i T_i. Then the number of completely labeled simplices T_i is odd. In particular,there exists at least one simplex T_i that is completely labeled. See Figure <ref>. The Sperner Lemma can be used to derive an elementary proof of the Brouwer Fixed Point Theorem: any continuous map f:B→ B from a(homeomorphic copy of an) n-dimensional ball to itself has a fixed point. See, e.g., <cit.>.The Sperner Lemma can be used to numerically find fixed points, and it has been used extensively in numerical works,see, e.g., <cit.>. §.§ Sperner's Lemma for polytopesWe now describe a more general Sperner Lemma-type of result for polytopes, following <cit.>. A relatedresult can be found in <cit.>.Let P be a convex n-dimensionalpolytope. We consider a labeling ϕ:P→{1,2,…, n+1} of P. As in the simplex case, a polytope is said to be completely labeled ifits vertices are assigned all labels from {1, 2, …, n+1}. Also, a labeling ϕ:P→{1,…,n+1} is said to be non-degenerate if no (n-1)-dimensional face of P contains points which take (n+1) or more different values. In the general case of a polytope, the non-degeneracy condition on the labeling is not equivalent to the Sperner condition. In the sequel, we will assume that all labelings of the polytopes under consideration are non-degenerate. Inapplications, such as below, we often consider a polytope (cube) P divided into a finite number of smaller polytopes (cubes) P_i, i∈ I. In suchcontexts, we only need to verify that the labels of the vertices of P and the P_i's lying on the faces of P satisfy the non-degeneracy condition. That is, we only need to verify such conditionon a finite number of points. Weintroduce some tools.Consider the standard n-dimensional simplex T=conv(0, e_1,e_2,… ,e_n)⊂ℝ^n, where we denote by (e_1,e_2,… ,e_n) the standard basis of ℝ^n, and by conv(·) the convex hull of a set. Let us declare the standard labeling of the vertices of T, given byϕ(0)=1 and ϕ(e_i)=i+1, i=1,…,n, as positively oriented.We also declare any labeling obtained by an even number of permutations of the labels of the vertices in the standard labeling also positively oriented, andany labeling obtained by an odd number of permutationsas negatively oriented. We define the oriented index of a labeling ϕ of any oriented, convex polytopeP, recursively:ind_P(ϕ)={[1, if (P)=0 and ϕ(V(P))={1};;± 1, if (P)=1 and ϕ(V(P))={1,2},; according to its orientation;;0, if (P)=k and ϕ(V(P))≠{1,2,…,k+1};; ∑_S∈ℱ_k-1(P)ind_S(ϕ), if (P)=k and ϕ(V(P))={1,2,…,k+1},; where ind_S(ϕ) is counted with orientation.; ].Above, V(P) denotes the vertices of P, andℱ_k-1( P) denotes the set of all (k-1)-faces ofP. The orientation is taken into account in the following way. The orientation of a k-dimensional polytope P induces an orientation of every (k-1)-dimensional face S of P, and on all the lower dimensional faces of S.In the definition of the index, only those (k-1)-dimensional faces that carry all labels{1,2,…,k} count in the summation. The definition of the index yields a choice of sign ± 1 to eachlower dimensional face carrying all labels which appears in the recursive definition.By taking into account the orientation of the polytope, we get here a definition of the index slightly different from the one in <cit.>, which uses mod 2 summation in the above recursive definition. We note that the definition of the index in <cit.> is inconsistent with some of the results later in that paper, e.g., with Proposition <ref> quoted below. Notice that the index of a labeling ϕ of a polytopedefined above only depends on the values of ϕ at the vertices V(P) of P.The following result is immediate:If T is a simplex and ϕ:T→{1,…,n+1} is a non-degenerate labeling, then_T(ϕ)={[ ± 1 , if T is completely labeled;;0,otherwise. ]. Given an n-dimensional, oriented, convex polytopeP, a labelingϕ:P→{1,…,n+1}, and the standard n-simplex T of vertices a_1,…,a_n+1, a realization of ϕ is a continuous map Φ: P → T, satisfying the following condition: (i) If v is a vertex of P then Φ(v)=a_ϕ(v), i.e., Φ(v) is the vertex a_i ofT with the index i equal to the label of v;(ii) If S is face of Pwith vertices v_1,…,v_k,then Φ(S)⊂conv(a_ϕ(v_1),…, a_ϕ(v_k)).Informally, a realization of P is a continuous mapping of P onto T that `wraps' ∂ P around ∂ T, such that the labels of the vertices of P match with the indices i of the vertices of T. Such Φ is in general non-injective.In the sequel we denote by (·) the oriented Brouwer degree of a continuous function (see, e.g., <cit.>). Recall that for a smooth, boundary preserving map Φ:(M,∂ M)→ (N,∂ N) between two oriented n-dimensional manifolds with boundary, (Φ)=(∫_ MΦ^*η)/(∫ _ Nη) where η is a volume form on N; the definition is independent of the volume form. Let ω be such thatdω=η. By Stokes' theorem and the fact that dΦ^*ω=Φ^*dω=Φ^*η, we have (Φ)=(∫_∂ MΦ^*ω)/(∫ _∂ Nω). This implies the following property of the degree: (Φ)=(∂Φ), where∂Φ:∂M→∂ N is the map induced by Φ on the boundaries. Equivalently, the degree of the map Φ can be defined as the signed number of preimages Φ^-1(p)={q_1,…,q_k} of a regular valuep of the map Φ, where each point q_i is counted with a sign ± 1 depending on whetherdΦ_q_i:T_q_iM→ T_pN is orientation preserving or orientation reversing. That is, (Φ)=∑_q∈Φ^-1(p)sign ( (dΦ_q)), where p∈N∖∂ N is a regular value of Φ. The definition of the Brouwer degree extends via homotopy to continuous maps. Let Pbe a convexn-dimensional polytope. (i) Any non-degeneratelabeling ϕ of P admits a realization Φ;(ii) Any two realizations of the same labeling are homotopic as maps of pairs (P,∂ P)↦ (T,∂ T);(iii) The index _P(ϕ) of the labeling ϕ is equal to the degree (Φ) of any realization Φ of ϕ, up to a sign_P(ϕ)=±(Φ). (iv) If _P(ϕ)≠ 0 then P is completely labeled. Let us considera convex, n-dimensional polytope P that is subdivided into finitely manyn-dimensional polytopes {P_i}_i∈ I, with I finite, such that P=⋃_i∈ IP_i, and for i≠ j, (P_i)∩ (P_j)=∅ and P_i∩ P_j is either empty or a face of both P_i and P_j. In particular, each k-dimensional face of P is the union of finitely many k-dimensional faces of P_i's. For the following result from<cit.> we provide an alternative proof.If the labeling ϕ is non-degenerate, then_P(ϕ)=∑_i _P_i (ϕ). The proof follows by induction on the dimension n of the polytope. When n=1 the identity is immediate. For the induction step,we use(<ref>). Note that each (n-1)-dimensional face S of P is the union of (n-1)-dimensional faces of P_i's. Each (n-1)-dimensional face of a P_i that is not lying on a (n-1)-dimensional face of P is shared by two polytopes P_i and P_j, and so it is counted twice withopposite orientations. Thus, the sum of the indices of the (n-1)-dimensional faces of the P_i's reduces to the sum of the the indices of the (n-1)-dimensional faces of the P_i's that lie on (n-1)-dimensional faces of P. The fact that the index of each (n-1)-dimensional face S^j of P is the sum of the indices of the (n-1)-dimensional faces S^j_i of the P_i's that lie on S^j follows from the induction hypothesis. In summary, we have:∑_i_P_i(ϕ) =∑_i∑_S'∈ℱ_n-1(P_i)ind_S'(ϕ)=∑_i∑_S^j_i∈ℱ_n-1(P_i)∩ℱ_n-1(P)ind_S^j_i(ϕ)=∑_S^j∈ℱ_n-1(P)ind_S^j(ϕ)=_P(ϕ). The following is a generalization of the Sperner Lemma from<cit.>.Assume that P is an n-dimensional polytope, P=⋃_i∈ IP_i is a decomposition of P into polytopes as above, and ϕ:P→{1,…,n+1} is a non-degenerate labeling. If _P(ϕ)≠ 0, then there exists a polytope P_i such that _P_i(ϕ)≠ 0; in particular, P_i is completely labeled. Follows immediately from Lemma <ref> and Proposition <ref>.Assume that P is an n-dimensional polytope, and P=⋃_i∈ IT_i is a simplicial decomposition of P. If anon-degenerate labeling ϕ satisfies _P(ϕ)≠ 0, then there exists a simplex T_i that is completely labeled.Follows from Lemma <ref>.Note that in Corollary <ref> the assumption that P is completely labeled alone is not sufficient to ensure that there exists a completely labeled simplex in the decomposition; the condition that _P(ϕ)≠ 0is necessary. See Example <ref>, (ii), (iii). (i) Consider the polygon P, the simplicial decomposition, and the labeling shown in Fig. <ref>-(a). We have_P(ϕ)=2; there exists a completely labeled triangle.(ii) Consider the polygon P', the simplicial decomposition,and the labeling shown in Fig. <ref>-(b). We have _P'(ϕ)=0; there is no completely labeled simplex.(iii) Consider the polyhedron P”, the simplicial decomposition,and the labeling shown in Fig. <ref>-(c). We have_P”(ϕ)=0;there is no completely labeled simplex.§ SPERNER'S LEMMA FOR CUBICAL COMPLEXES In this section we present a new version of the SpernerLemma for cubical decompositions. The main difference from the previous sections will be the labeling. For an n-dimensional cube and a corresponding cubical decomposition, it will be convenient in Section <ref> to use 2^n labels that are n-dimensional vectors with coefficients ± 1; whereas, in Section <ref>, for an n-dimensional polytope we have used only (n+1) labels. Hence,we will have to re-define what a complete labeling means in terms of the new labeling convention, and relate with the old labeling convention.Considervector-labels ℓ∈{(± 1,…, ± 1)}=𝒵^n, where we denote 𝒵={± 1}. Each label ℓ=(± 1,…, ± 1) corresponds toa hyperoctant𝒪_ℓ={(x_1,x_2,…, x_n)∈^n | ∀ i, ℓ_i x_i≥ 0 }.Note that ⋃_ℓ∈𝒵^n𝒪_ℓ=^n.We call a collection of labels {ℓ_1,ℓ_2,…,ℓ_n+1}complete if 𝒪_ℓ_1∩𝒪_ℓ_2∩…∩𝒪_ℓ_n+1={ 0}. Equivalently, {ℓ_1,ℓ_2,…,ℓ_n+1} is complete if for each coordinate index i∈{1,…,n}, there exists a pair of labels ℓ_j, ℓ_k such that the i-th coordinates of ℓ_j and ℓ_k have opposite signs, that is, π_i(ℓ_j)=-π_i(ℓ_k).A labeling ϕ is said to be non-degenerate if no face of Pcarries a complete set of labels. A convexpolyhedral cone is a convex conein ^n bounded by a finite collection of hyperplanes of the form x_k=0; alternatively, it can be characterizedas an intersection of finitely many half-spaces (e.g., spaces of the form {x∈^n | x_k≥ 0} for some k); see <cit.>.A specialconvexpolyhedral cone partition of ^n is a collection of (n+1) convexpolyhedral cones 𝒩_j, j=1,…,n+1, satisfying the following properties: (a) ⋃ _j 𝒩_j=^n;(b) (𝒩_j)∩(𝒩_l)=∅ for j≠ l;(c) ⋂_j 𝒩_j={0}. (i) Given a specialconvexpolyhedral cone partition 𝒩_1, …, 𝒩_n+1 of ^n. Any set of (n+1) hyperoctants 𝒪_ℓ_1,…,𝒪_ℓ_n+1, with the property that each 𝒪_ℓ is contained in exactly one 𝒩_j, and no two 𝒪_ℓ's are contained in the same 𝒩_i, satisfies 𝒪_ℓ_1∩𝒪_ℓ_2∩…∩𝒪_ℓ_n+1={ 0}.(ii) Givena set of (n+1) hyperoctants 𝒪_ℓ_1,…,𝒪_ℓ_n+1 with 𝒪_ℓ_1∩𝒪_ℓ_2∩…∩𝒪_ℓ_n+1={ 0}. There exists a specialconvexpolyhedral cone partition 𝒩_1, …, 𝒩_n+1 of ^n,such that each 𝒪_ℓ_i is contained exactly in one 𝒩_j. (i) We have𝒪_ℓ_1∩𝒪_ℓ_2∩…∩𝒪_ℓ_n+1⊆𝒩_1∩…∩𝒩_n+1={ 0}.(ii) Consider aset ofhyperoctants 𝒪_ℓ_1,…,𝒪_ℓ_n+1 with 𝒪_ℓ_1∩𝒪_ℓ_2∩…∩𝒪_ℓ_n+1={ 0}. Each hyperplane will separate the set of hyperoctants into two non-empty collections on each side of the hyperplane (since having all hyperoctants on the same side of a hyperplane would imply that their intersections is more than the zero vector).Thus, cutting ^n by the n hyper-planes will imply that each pair of hyperoctants is on opposite sides of some hyperplane. First,select 𝒩_1 as the largestintersection of half-spaces in ^n (i.e., an intersection by the minimum number of half-spaces) that contains only one hyperoctant 𝒪_ℓ_i_1. Then select 𝒩_2 as the largestintersection set of half-spaces in ^n∖𝒩_1that contains only one hyperoctant 𝒪_ℓ_i_2, with i_1≠ i_1. Continue this procedure up to the last hyperoctant 𝒪_ℓ_i_n+1 in the given collection, which will provide 𝒩_n+1.The following example shows some simple changes of labels:Consider the following special partition into convex polyhedral cones:𝒩_1 ={x∈^n | x_1≥ 0}, 𝒩_2 ={x∈^n | x_1≤ 0, x_2≥ 0},⋯ 𝒩_n ={x∈^n | x_1≤ 0, …,x_n-1≤ 0,x_n≥ 0}, 𝒩_n+1 ={x∈^n | x_1≤ 0, …,x_n-1≤ 0,x_n≤ 0}. The correspondingchange of labels ψfrom vector-labels ℓ∈𝒵^n to labels j ∈{1,2,…, n + 1} is given byψ(1,± 1, …, ± 1)= 1, ψ(-1,1, ± 1, …, ± 1)= 2,⋯ ψ(-1, …, -1, 1,± 1, …,± 1)= min{j| ℓ_j = 1},⋯ ψ(-1,-1, …, -1)= n+1.Consider a labeling ϕ:P→𝒵^n of an n-dimensional polytope P. Let ψ:𝒵^n→{1,…,n+1} be a change of labels. We can define a re-labeling ψ∘ϕ:P→{1,…,n+1}. This re-labeling is asin Section <ref>, hence all the results from that section can be applied in this context. In particular, any re-labeling ψ∘ϕ: P→{1,…,n+1} admits a realization. Let P be a polytope,ϕ:P→𝒵^na labeling,ψ_1,ψ_2:𝒵^n→{1,…,n+1} two changes of labels, andψ_1∘ϕ,ψ_2∘ϕ:P→{1,…,n+1}the corresponding re-labelings.Then_P (ψ_1∘ϕ)=±_P (ψ_2∘ϕ). LetΦ_i:P→ T be a realization of ψ_i∘ϕ, i=1,2. ByProposition <ref>, _P(ψ_i∘ϕ)=±(Φ_i), for i=1,2. Each change of labels ψ_i corresponds to a special partition into convex polyhedral cones {𝒩^i_j}_j. Either partition can be obtained from the other via a composition of a rotation, a reflection, and a homotopy. These transformation preserve the Brouwer degree up to a sign. Hence (Φ_1)=±(Φ_2),thus _P (ψ_1∘ϕ)=±_P (ψ_2∘ϕ).By Lemma <ref>, we can define the index of a labeling ϕ:P→𝒵^n, up to a sign, by _P(ϕ)=±_P (ψ∘ϕ), where ψ is a change of labels.Now, let us consider Cand a labelingϕ:C→𝒵^n of C. Let {C_i}_i be a cubical decomposition of C. Then C can be regarded as a polytopeby appending to the vertices of C all the vertices of the the C_i's lying on the faces of C, as well as appending to the faces of C all the faces of the C_i's lying on the faces of C. We denote this polytope by C̃. For C̃ we have a polytope (cubical) decomposition C̃=⋃_i C_i. The labeling ϕ:C→𝒵^n of C can be viewed as a labeling ϕ:C̃→𝒵^n of C̃. The following is a version of Sperner's Lemma for cubical decompositions.Let C=⋃_i C_i bea cube together with a cubical decomposition, C̃ the corresponding polytope, and ϕ:C̃→𝒵^n a non-degenerate labeling of C̃. If _C̃(ϕ)≠ 0 then there exists at least one cube C_i such that _C_i(ϕ)≠ 0, hence completely labeled relative to ϕ.Consider a re-labeling ψ∘ϕ of C̃. By definition, we have _C̃(ϕ)=±_C̃(ψ∘ϕ)≠ 0. By Theorem <ref>, there exists acube C_i such that _C_i(ψ∘ϕ)≠ 0, and so C_i is completely labeled relative to ψ∘ϕ. Hence _C_i(ϕ)≠ 0. Lemma <ref> says that a labeling iscomplete relative to ψ∘ϕ if and only ifit is completely labeled relative to ϕ. Let C=⋃_i C_i bea cube together with a cubical decomposition, and ϕ:C→𝒵^n a non-degenerate labeling of C. If _C(ϕ)≠ 0 then there exists at least one cube C_i such that _C_i(ϕ)≠ 0, hence completely labeled relative to ϕ. If the labeling ϕ of C isnon-degenerate, and C̃ is the polytope obtained from the cubical decomposition C=⋃_iC_i, it follows that _C̃(ϕ)=_C(ϕ)≠ 0. Thus Theorem <ref> applies. These last two results are the most important for Section <ref>. In anutshell, they say that given a cubical complex with a non-degenerate, non-zero index labeling, any finer decomposition of the complex into smaller cubes will always have a small cube with non-zero index.§ CORRECTLY ALIGNED WINDOWS AND DETECTION OF FIXED POINTS/PERIODIC ORBITS/ORBITS WITH PRESCRIBED ITINERARIESIn this section we present the definitions of windows and of correct alignment following <cit.>, with a few minor modifications. Then we associate some labeling to the windows, and characterize correct alignment in terms of that labeling. We also use the labeling to find numerical approximations of fixed points, periodic orbits, and orbits with prescribed itineraries. §.§ Approximate fixed points and periodic orbitsGiven δ>0 and a map g:^n→^n, we call a point z=(x_1,…,x_n)∈^n a δ-approximate fixed pointof g if g(z)-z_∞<δ,where z_∞=max_i=1,…,n|x_i|.We remark here that there may be no `true' fixed point near aδ-approximate fixed point, that is,δ-approximate fixed points can be`fake' fixed points. Obviously, any `true' fixed point is a δ-approximate fixed point for any δ>0.Similarly, a finite collection of points z_1,…, z_k is a δ-approximate periodic orbitof g if g(z_j)-z_j+1_∞<δ, for j=1,…,k, and g(z_k)-z_1_∞<δ.As in the case of fixed points, there may be no `true' periodic orbits near aδ-approximate periodic orbit. §.§ WindowsWe considera discrete dynamical system given by a homeomorphism f:^n→^n.We define an equivalence relation on the set of homeomorphisms χ:^n→^n by setting χ_1∼χ_2 if there exists an open neighborhood U of [0,1]^n in ^n such that (χ_1)_| U=(χ_2)_| U.We will use the same notation for anequivalence class as for a representative of that class. A window in ^n consists of:(i) a homeomorphic copy D of a multidimensional rectangle [0,1]^n,(ii) an equivalence class ofhomeomorphisms χ_D:^n→^n with χ_D([0,1]^n)=D,(iii) a choice of stable- and unstable-like dimensions,n_s,n_u≥ 0, respectively,with n_s+n_u=n,(iv) a choice of an `exit set' D^-⊂ (D) and of an `entry set' D^+⊂(D), given byD^- =χ_D(∂[0,1]^n_u×[0,1]^n_s),D^+ =χ_D([0,1]^n_u×∂[0,1]^n_s). We will write [0,1]^n=[0,1]^n_u× [0,1]^n_s. Given D_1,D_2two windows in ^n, and χ_D_1,χ_D_2 two representatives of the corresponding equivalence classes of homeomorphisms, we denote byf_χ_D_1,χ_D_2:^n→^n the homeomorphism f_χ_D_1,χ_D_2:=χ_D_2^-1∘ f ∘χ_D_1. When there is no risk of ambiguity, we use the simplified notation f_χ:= f_χ_D_1,χ_D_2.Denote by Υ :={(x,y)∈^n_u×^n_s | x∉[0,1]^n_u}. Given two windows D_1 and D_2 such that f_χ ([0,1]^n)⊆Υ∪( [0,1]^n_u× (0,1)^n_s),there exists another homeomorphism χ'_D_2 from the equivalence class of homeomorphisms associated toD_2, such that f_χ' ([0,1]^n )⊂^n_u× (0,1)^n_s, where f_χ':=χ'_D_2^-1∘ f ∘χ_D_1. Note that changing χ_D_2 with χ'_D_2 has no effect on the windows D_1 and D_2. §.§ Correctly aligned windows in two-dimensions We first give a definition of correct alignment of windows in the2-dimensional case, that is,n=2. We say that the window D_1 is correctly aligned with D_2 under f if there existcorresponding homeomorphisms χ_D_1,χ_D_2 with the following properties: (i) Case n_u=1, n_s=1:(i.a) f_χ ([0,1]^2 )⊂×(0,1);(i.b)f_χ ({0}× [0,1]) ⊂ (-∞,0)× and f_χ ({1}× [0,1])⊂ (1, +∞) ×, or, f_χ ({0}× [0,1])⊂ (1, +∞) × and f_χ ({1}× [0,1])⊂ (-∞, 0) ×;(ii) Casen_u=0, n_s=2: f_χ( [0,1]^2 )⊂ (0,1)^2;(iii) Casen_u=2, n_s=0:f^-1_χ ([0,1]^2) ⊂ (0,1)^2. As noted above, instead of condition (i.a) we can require the more general condition that f_χ ([0,1]^2 )⊂Υ∪( [0,1]^n_u× (0,1)^n_s). Let g:^2→^2 be a homeomorphism. Let (x_1,x_2) denote the coordinates of a point z∈ [0,1]^2 and (x'_1,x'_2) the coordinatesof g(z). Let Δ z:=(Δ x_1,Δ x_2)=(x_1'-x_1,x'_2-x_2). Denote the quadrants𝒪_(1,1) ={(x_1,x_2) | x_1≥ 0 andx_2≥ 0}, 𝒪_(-1,1) ={(x_1,x_2) |x_1≤ 0 andx_2≥ 0}, 𝒪_(-1,-1) ={(x_1,x_2) | x_1≤ 0 andx_2≤ 0},𝒪_(1,-1) ={(x_1,x_2) | x_1≥ 0 andx_2≤ 0}.These are closed sets which cover ^2, and 𝒪_ℓ_1∩𝒪_ℓ_2∩𝒪_ℓ_3={(0,0)} whenever ℓ_1,ℓ_2,ℓ_3 are all different. With respect to the mapping g, to each point z∈^2 we assign a vector label (± 1,± 1)∈𝒵^2 according to the following:Condition O * if Δ z∈ (𝒪_ℓ) then we assign to z the labelℓ;* ifΔ z∈(𝒪_ℓ_1∩𝒪_ℓ_2)then weassign to z either label ℓ_1 or ℓ_2;* if Δ z=0,we assign to z either label ℓ.A square C⊂^2 labeled according to Condition O is said to be completely labeled if its vertices contain three different labels ℓ_1,ℓ_2,ℓ_3,in which case 𝒪_ℓ_1∩𝒪_ℓ_2∩𝒪_ℓ_3=0.Note that if Δ z∈𝒪_ℓ_1∩𝒪_ℓ_2∩𝒪_ℓ_3 withℓ_1,ℓ_2,ℓ_3mutually distinct, then Δ z=0 and so (x'_1,x'_2)=(x_1,x_2). Suppose that D_1 is correctly aligned with D_2 under f. For the mapping f_χ:[0,1]^2→^2 we assign a labeling ϕ:[0,1]^2→𝒵^2as per Condition O. Denote the vertices of [0,1]^2 as follows: A=(1,0), B=(0,0), C=(0,1), D=(1,1). From the definition of correct alignment, we infer the followinglabeling of the vertices and the edgesof [0,1]^2: (i) Case n_u=1, n_s=1. Using the labeling associated to f_χ yields: (i.a) A→ (1,1), B→ (-1,1), C→ (-1,-1), D→ (1,-1), orB→ (1,1), A→ (-1,1), D→ (-1,-1), C→ (1,-1),(i.b) AB→ (1,1)or (-1,1), CD→(-1,-1) or (1,-1);(i.c) BC→(-1,1) or (-1,-1),AD→(1,1) or (1,-1), orBC→ (1,1) or (1,-1),AD→(-1,1) or (-1,-1);(ii) Casen_u=0, n_s=2. Using the labeling associated to f_χ yields:(ii.a) A→ (-1,1), B→ (1,1), C→ (1,-1), D→ (-1,-1),(ii.b)AB→ (1,1) or (-1,1), AD→ (-1,1) or (-1,-1), CD→ (-1,-1) or (1,-1), BC→ (1,1) or (1,-1);(iii) Casen_u=2, n_s=0. Using the labeling associated to f^-1_χ yields the same labeling rules as in case (ii).See Fig. <ref>.Apossible re-labeling of the points is (1,1)→ 1,(-1,1)→ 2,(1,-1), (-1,-1)→ 3.If D_1 is correctly aligned with D_2 under f, then the labeling of χ^-1_D_1(D_1)=[0,1]^2described aboveis non-degenerate andhasnon-zero index.Conversely, in the case when n_u=1, n_s=1, if D_1, D_2 satisfy:(a) f_χ ([0,1]^2 )⊂×(0,1);(b)f_χ(∂ [0,1]× [0,1])∩ [0,1]^2=∅; and if the labelingas per Condition Ois non-degenerateand ofnon-zero index, then D_1 is correctly aligned with D_2 under f. For the direct statement, the pointson the boundary of [0,1]^2 inherit the labeling explicitly described above, which isnon-degenerateandhasnon-zero index.For the converse statement, we proceed by contradiction. If condition (i.b) from Definition <ref> is not satisfied, it means that the two components of ∂[0,1]× [0,1] are mapped by f_χ on the same side of [0,1]^2 within the strip ×(0,1), which leads to a labeling that fails to benon-degenerate. We note that inProposition <ref>, the converse statement does not include the cases that n_u=0, n_s=2, or n_u=2, n_s=0, since assumingcondition(ii), or (iii) from Definition <ref>, respectively, automatically yields both correct alignment and non-degenerate,non-zero index labeling. Assume that f_χ is abi-Lipschitz function with Lipschitz constant L, relative to the norm ·_∞ on ^2. Consider a subdivision of [0,1]^2 into N^2 squares {C_i}_i=1,…,N^2 of side 1/N. We assign alabeling to all points of [0,1]^2 according to Condition O. Let δ>0 be small, and N>0 be large so that (L+1)/N<δ.If C_i is a completely labeled square, then any point z∈ C_i is a δ-approximate fixed pointof f_χ. Indeed, the complete labeling implies, via the Intermediate Value Theorem,that there exist points ẑ=(x̂_1, x̂_2)∈ C_i and ž=(x̌_1, x̌_2)∈ C_isuch that x̂'_1 =x̂_1 andx̌'_2=x̌_2, respectively. Then, for each z=(x_1,x_2)∈ D_1 we havef_χ(z)-z_∞ =max{|x'_1-x_1|,|x'_2-x_2|}≤max{|x'_1-x̂'_1|+|x̂'_1 -x̂_1|+|x̂_1-x_1|,|x'_2-x̌'_2|+|x̌'_2 -x̌_2|+|x̌_2-x_2|} ≤ (L+1)/N < δ. The next statement is a fixed point theorem in the case of a window correctly aligned to itself. We will distinguish between the cases (i), (ii) of Definition <ref>, and the case (iii), for which the corresponding statement is indicated in parentheses.In the former the labeling as per Condition O is done with respect to f_χ, while in the latter the labeling is done with respect to f^-1_χ.Let D be a window and ϕ:[0,1]^2=χ_D^-1(D)→𝒵^2 be a labeling associated to f_χ as per Condition O (resp., associated to f^-1_χ).(i) If a window D is correctly aligned with itself under f, then f has afixed point in D. (ii) If {C_i} is a subdivision of [0,1]^2=χ^-1_D(D) then there existsa square C_* in the decompositionwith _C_*(ϕ)≠ 0; if C_* further satisfies the non-degeneracy condition on its faces, then f has afixed point in χ_D(C_*) (resp. f has afixed point in χ_D(f_χ^-1(C_*))).(ii) Assume that χ_D is Lipschitz with Lipschitz constant K>1, and thatf_χ is bi-Lipichitz with Lipschitz constant L>1. Then, given δ>0 and a sufficiently fine subdivision of [0,1]^2 into squares {C_i}_i=1,…,N^2 of side 1/N, so thatK(L+1)/N<δ,for every completely labeled square C_*, each point z̃∈χ_D(C_*) is aδ-approximate fixed pointof f (resp. each point z̃∈χ_D(f_χ^-1(C_*)) is aδ-approximate fixed pointof f).(i)Let {C^N_i}be a subdivision of C,with diam(C^N_i)→ 0 as N→∞. For each N, by applying Corollary <ref>, there existsasquare C^N_i^*_N⊂ C with _C^N_i^*_N(ϕ)≠ 0, hence completely labeled; we choose and fix such a C^N_i^*_N.As before, there exist some points ẑ_i^*_N=((x̂_1)_i^*_N, (x̂_2)_i^*_N)∈ C^N_i^*_N and ž_i^*_N=((x̌_1)_i^*_N, (x̌_2)_i^*_N)∈ C^N_i^*_N, such that (x̂_1)_i^*_N=(x̂'_1)_i^*_N and (x̌_2)_i^*_N=(x̌'_2)_i^*_N. By compactness, the sequencesẑ_i^*_N and ž_i^*_N, contain convergent subsequencesẑ_i^*_k_N and ž_i^*_k_N, respectively. Since the diameters of the corresponding completely labeled squares tend to zero, these subsequences approach the same limit z=(x_1,x_2), for which we have x_1=x'_1 and x_2=x'_2. Hence z is a fixed point for f_χ, so z̃=χ_D(z) is a fixed point for f.For an alternativeproof (non-constructive) see <cit.>.(ii) Let {C_i} be a subdivision as in the statement.By Corollary <ref>, there exists a square C_* with _C_*(ϕ)≠ 0. If C_* satisfies the non-degeneracy condition, we further subdivide C_* into smaller squares{C^N_i} as above. The proof for (i) implies that there exists a fixed point z for f_χ in C_*, hence p=χ_D(z) is a fixed point for fin χ_D(C_*).(iii) Suppose that C_i^*^N is a completely labeled square of the subdivision.By (<ref>), for each z∈ C_i^*^N we have f_χ(z)-z_∞<(L+1)/N. Hence, for each z̃=χ_D(z)∈χ_D(C_i^*^N), we havef(z̃)-z̃_∞= χ_D∘ f_χ∘χ^-1_D(χ_D(z))-χ_D(z)_∞≤ Kf_χ(z)-z_∞<K(L+1)/N<δ. In the case of labeling associated to f^-1_χ, if C_i^*^N is completely labeled, applying (<ref>) to f_χ^-1, and using that f_χ is bi-Lipschitz of Lipschitz constant L yields f^-1_χ(z)-z_∞<(L+1)/N for every z∈ C_i^*^N. Thus f_χ(f^-1_χ(z))- f^-1_χ(z)_∞< L(L+1)/N, and so, for z̃=χ_D(f^-1_χ(z))∈χ_D(f^-1_χ(C_i^*^N)) we have f(z̃)-z̃_∞<KL(L+1)/N<δ.§.§ Correctly aligned windows in higher-dimensions As before, let𝒪_ℓ={(x_1,…,x_n)∈ℝ^n | ∀ i,x_i l_i≥ 0},where ℓ= (l_1,…,l_n)∈𝒵^n, with l_i=± 1 for i=1,…,n. Let g:^n→^n be a homeomorphism. For a point z=(x_1,…, x_n) in ^n, letting g(z)=z'=(x'_1,…, x'_n), and Δ z=(x'_1-x_1,…, x'_n-x_n), we assign a label ℓ∈𝒵^n as follows:Condition O * if Δ z∈ (𝒪_ℓ) then we assign to z the labelℓ;* ifΔ z∈(𝒪_ℓ_1∩…∩𝒪_ℓ_k) for some labels ℓ_1,…,ℓ_k that are mutually distinct, then we assign to z either one of the labels ℓ_1,…,ℓ_k.* if Δ z=0 then we assign to z either label ℓ.Apossible re-labeling of the points from labels in 𝒵^n to labels in {1,…,n+1} can be done as in Example <ref>. Let D_1, D_2 be n-dimensional windows. We will assume that the homeomorphism f:ℝ^n→ℝ^n satisfies a bi-Lipschitz condition with Lipschitz constant L. We say that the window D_1 is correctly aligned with D_2 under f if there existcorresponding homeomorphisms χ_D_1,χ_D_2 with the following properties: (i) Case n_u≠ 0, n_s≠ 0:(i.a) f_χ ([0,1]^n )⊂^n_u×(0,1)^n_s;(i.b)f_χ (∂ [0,1]^n_u× [0,1]^s) ⊂(^n_u×(0,1)^n_s)∖ [0,1]^n;(i.c)There exists x^*_s∈(0,1)^n_s such thatthe map L:[0,1]^n_u→^n_u defined by L(x_u)=π_u∘ f_χ(x_u,x^*_s) satisfies (L_| (0,1)^n_u)≠ 0;(ii) Casen_u=0, n_s=n: f_χ( [0,1]^n )⊂ (0,1)^n;(iii) Casen_u=n, n_s=0:f^-1_χ ([0,1]^n) ⊂ (0,1)^n.Suppose that D_1 is correctly aligned with D_2 under f, and let n_u be the unstable-like dimension. We denote by {𝒪_ℓ_u} the octants of ℝ^n_u, where ℓ_u∈𝒵^n_u. In thecases (i) and (ii) of Definition <ref>, we will label all pointsof C=χ^-1_D_1(D_1)according to Condition O applied to f_χ, and in case (iii) of Definition <ref>we will label all pointsof C=χ^-1_D_2(D_2) with labelsaccording to Condition O applied to f_χ. We describe the labeling below,for each of the cases (i), (ii), (iii) of Definition <ref>.Case (i). First consider the case when n_u>0 and n_s>0. Labeling C=[0,1]^n as per Condition O, as we did in the 2-dimensional casein Section <ref>, does not necessarily yield a non-degenerate labeling. We will transformC=[0,1]^n into an n-dimensional polytope C, construct a subdivisionC=⋃_i C_i into smaller cubes,in order to obtain anon-degeneratelabeling.Moreover, we will constructCso that the resulting labeling of its vertices in complete.(As we pointed out earlier, completeness is a necessary, but not sufficient condition for the index to be non-zero.) The faces of the resulting polytope C̃ will consist of the faces of the C_i's in the subdivision that are contained in ∂([0,1]^n). We perform this construction below. Estimates on Δ z. Letρ_s =min{d(p,p') | p∈[0,1]^n_u×∂[0,1]^n_s, p'∈ f_χ([0,1]^n×∂[0,1]^n_s)}>0, ρ_u =min{d(p,p') | p∈∂[0,1]^n_u×[0,1]^n_s, p'∈ f_χ(∂[0,1]^n×[0,1]^n_s)}>0, ρ =min{ρ_s,ρ_u}>0,where d is the distance corresponding to ·_∞. The fact that ρ_u, ρ_s, and hence ρ are positive follows from Definition <ref> (i.a) and (i.b), respectively.This fact implies that for each z∈∂[0,1]^n we haveΔ z=z'-z_∞=f_χ(z)-z_∞>ρ.In particular, for any point z∈∂[0,1]^n we have that Δ z is not contained within a ball of radius ρ around the origin in ^n. Coarse cubical decomposition of C. We divide [0,1]^n into M^n identical cubes {C_i}_i=1,…,M^n, of side 1/M. The quantity M from above is required to satisfy the following condition: Condition P(P1) For each ℓ_u∈𝒵^n_u, there exists a vertex v of a cube C_i lying on ∂[0,1]^n_u×[0,1]^n_s, such that thelabel of z isℓ=(ℓ_u,ℓ_s)∈𝒵^n, for some ℓ_s∈𝒵^n_s. (P2) (L+1)/M<ρ/2.Condition (P1) implies that the cubical decomposition{C_i}_i=1,…,M^nis fine enough so that for the vertices z of the cubes with faces lying on ∂[0,1]^n_u×[0,1]^n_s, the vectors Δ z=(Δ z_u,Δ z_s) have the Δ z_u component taking values in eachof the hyperoctants 𝒪_ℓ_u of ^n_u. The argument for this claim is below.First we note Definition <ref>-(i.a)implies that thecorresponding π_s(Δ z) take values in eachof the sectors 𝒪_ℓ_s of ^n_s. Definition (<ref>)-(i.b) and -(i.c) imply that, for some x^*_s∈(0,1)^n_sthe projection π_u onto [0,1]^n_uof the image of [0,1]^n_u×{x ^*_s} under f_χ contains the rectangle [0,1]^n_u inside its interior, and that the boundary of π_u(f_χ([0,1]^n_u×{x^*_s})) wraps around the boundary of [0,1]^n_u, in the sense that for z∈∂[ π_u(f_χ([0,1]^n_u×{x^*_s}))], the corresponding π_u(Δz) visits all sectors 𝒪_ℓ_u of ^n_u. It follows that Δ z=(Δz_u,Δz_s) take values in a complete set of hyperoctants 𝒪_ℓ of ^n. (This does not mean that Δ z takes values inall hyperoctants 𝒪_ℓ of ^n.) Thus, the corresponding labeling of the vertices of the C_i's is complete.We append the vertices and faces S_i of the C_i's that lie on ∂[0,1]^n to C, thus transforming C into a polytope C (like a Rubik's cube, see Fig. <ref>). Now we discussCondition (P2). Note first that for z_1,z_2∈∂[0,1]^n, we haveΔ z_1- Δ z_2_∞ =(z'_1-z_1)-(z'_2-z_2)_∞≤z'_1-z'_2_∞+z_1-z_2_∞≤ (L+1)z_1-z_2_∞.This implies that, the image of any cube C_i under the map z↦Δ z has diameter less than ρ/2. Hence,the image under z↦Δ z of every face S_i of a cube C_i that lies on∂[0,1]^n, is disjoint from a ρ-ball around the origin. Hence no such a face S_i can carry a complete set of labels. That is, the labeling is non-degenerate.When the windows are correctly aligned, as assumed above, it also follows that the index of the labeling,is non-zero. Condition (i.a) of correct alignment implies that the index relativeto the labels in 𝒵^n_s is non-zero. Also, conditions (i.b) and (i.c), together with (P1), imply that the the index relative to the labels in 𝒵^n_u is non-zero. Proposition <ref>-(iii), saying that the index of a labeling equals the Brouwer degree of a realization, andthe product property of the Brouwer degree, imply that the overall labeling is non-degenerate. Case (ii). Consider the case when n_u=0.We label all points of C=χ^-1_D_1(D_1) according to the quadrant 𝒪_ℓ where f_χ(z)-z lands, as per Condition O. The resulting labeling is non-degenerate and of non-zero index. Case (iii). Consider the case when n_s=0. We label all points of C=χ^-1_D_2(D_2) according to the quadrant 𝒪_ℓ where f^-1_χ(z)-z lands, as per Condition O. The resulting labeling is non-degenerate and of non-zero index. In Fig. <ref> we illustrate a 3Dwindow that is correctly aligned to itself under some map; the points A,B,…, are mapped to the points A',B',…, respectively. The unstable-like dimension is n_u=2 and the stable-like dimension is n_s=1.If we label the vertices A,B,… according to Condition O, we obtain A→ℓ_1:=(-1,1,1), B→ℓ_2:=(-1,1,1), C→ℓ_3:=(-1,-1,1), D→ℓ_4:=(-1,-1,1), E→ℓ_5:=(-1,1,-1), F→ℓ_6:=(-1,1,-1), G→ℓ_7:=(-1,-1,-1), H→ℓ_8:=(-1,-1,-1). We notice that the corresponding set of labels is not complete; indeed, for the corresponding octants we have that 𝒪_ℓ_1∩…∩𝒪_ℓ_8={x∈^n |x_1≤ 0, x_2=0,x_3=0}.In particular, the index is zero. The labels ℓ_u corresponding to the unstable directions take only the values (-1,1) and (-1,-1), so Condition P is not satisfied. However, by taking a coarse cubical decomposition of [0,1]^n satisfying Condition P, as illustrated in Fig. <ref>, we can obtain that the labeling is non-degenerate andhas non-zero index. To summarize, ifD_1 is correctly aligned with D_2 under f, in case (i) and (ii) we start with C=χ_D_1^-1(D_1) and assign a labeling associatedto Δ z=f_χ(z)-z, and in case (iii) we start with C=χ_D_2^-1(D_2) and assign a labeling associatedto Δ z=f^-1_χ(z)-z. In each case, we perform a cubical decomposition of C into smaller cubes {C_i}_i, and transform C into a polytope C with a cubical decomposition {C_i}_i such that the labeling is non-degenerate andhas non-zero index. AssumeD_1 is correctly aligned with D_2 under f, andC={C_i} is a subdivision of [0,1]^n=χ^-1_D(D) satisfying Condition P. Then the labeling described above, in each of the cases (i), (ii), (iii) of Definition <ref>,satisfies_C(ϕ)≠ 0. Conversely, in the case when n_u>0, n_s>0, if D_1, D_2 satisfy:(a) f_χ ([0,1]^n )⊂^n_u×(0,1)^n_s;(b)f_χ (∂ [0,1]^n_u× [0,1]^n_s) ⊂(^n_u×(0,1)^n_s)∖ [0,1]^n; and the corresponding decomposition C=⋃ _i C_i satisfies _C(ϕ)≠ 0, then D_1 is correctly aligned with D_2 under f. The direct statement was shown above.For the converse statement, pick any x^*_s∈(0,1)^n_s and consider the labeling of the n_s-dimensional polytope[0,1]^n_u×{x^*_s} as per Condition O. As before, it follows that the index of the labeling, relative to the labels in 𝒵^n_u, is non-zero. By Proposition <ref>, this index equals to the Brouwer degree of a realization Φ:[0,1]^n_u×{x^*_s}→ T^n_u, where T^n_u is the (n_u)-dimensional simplex. This degree is non-zero and equals, up to a sign, the degree of L(·)=π_u∘ f_χ(·,x^*_s), which concludes the proof. In the statement below, we distinguish between the cases (i), (ii) of Definition <ref>, and the case (iii), for which the corresponding statement is indicated in parentheses. Let D be a window and ϕ:χ_D^-1(D)→𝒵^n be a labeling associated to f_χ as per Condition O (resp., associated to f^-1_χ). Assume thatC={C_i} is a subdivision of [0,1]^n=χ^-1_D(D) satisfying Condition P and Condition O. (i) If D is correctly aligned with itself under f, then f has a fixed point in D.(ii) Let{C'_j}_j=1,…,N^n be afine subdivision of [0,1]^n into cubes of side 1/N, where N is a multiple ofM (so that the family of cubes C'_j subdivide each of the cubes C_i). Then there existsacube C'_* with _C'_*(ϕ)≠ 0 in the decomposition; if C'_* further satisfies the non-degeneracy condition, then f has afixed point p in χ_D(C'_*) (resp., f has a fixed point in χ_D(f_χ^-1(C'_*))).(iii) Assume that χ_D is Lipschitz with Lipschitz constant K>1, and thatf_χ is bi-Lipichitz with Lipschitz constant L>1. Then, given δ>0 and a subdivision {C'_j}_j=1,…,N^n of [0,1]^ninto cubesof side 1/N as above, so thatK(L+1)/N<δ, then for every completely labeled cube C'_j, each point z̃∈χ_D(C'_j) is aδ-approximate fixed pointof f (resp., each point z̃∈χ_D(f_χ^-1(C'_j))).The proof is similar to the proof of Proposition <ref>, and the details are left to the reader. §.§ Detection of periodic points and symbolic dynamicsAssume that p_1 is a periodic point of period k for f; the orbit of p_1 is {p_1,…,p_k}, with f(p_k)=p_1. Let F:(^n)^k→ (^n)^k be given byF(z_1,…, z_k)=(f(z_k), f(z_1),…,f(z_k-1)).Note that {p_1,…,p_k} is a period-k orbit if and only ifF(p_1,…,p_k)=(p_1,…, p_k), that is, (p_1,…,p_k)∈ (^n)^k is a fixed point for F. Now consider a finite sequence of windows D_1,…, D_k in ^n. We are interested in periodic orbits {p_1,…,p_k} withp_j∈ D_j, j=1,…, k. Assume that for j=1,…, k-1, D_j iscorrectly aligned with D_j+1 under f, and also D_k iscorrectly aligned with D_1 under f.Here we only consider correct alignment as in Definition <ref>-(i). See Fig. <ref>. Denote by χ_D_j, the equivalence class of homeomorphisms corresponding to D_j, for j=1,…,k. Let χ_D:(^n)^k→ (^n)^k be given by χ_D(z_1,…, z_k)=(χ_D_1(z_1),…,χ_D_k(z_k)). (i) Let D=χ_D(Π_i=1^k [0,1]^n)⊆(^n)^k, andD^- = χ_D(Π_j=1^k∂[0,1]^n_u×[0,1]^n_s),D^+ = χ_D(Π_j=1^k [0,1]^n_u×∂[0,1]^n_s).Then D is an (nk)-dimensional window, with (n_uk)-unstable-like, and (n_sk)-stable like dimensions.(ii) If for j=1,…, k-1, D_j iscorrectly aligned with D_j+1 under f, and D_k iscorrectly aligned with D_1 under f, then D is correctly aligned with D under F. (i) Follows from elementary set theory. (ii) Follows fromelementary set theory and from the product property of the Brouwer degree. See <cit.>.We associate to each rectangle D_j, j=1,…,k,ann-dimensional polytope P_j, obtained by dividing each underlying cube χ^-1_D_j(D_j)=[0,1]^n into (N_j)^n cubes of side 1/N_j, where N_j is chosen large enough so that Condition P is satisfied. The cubical decomposition of each window D_j determines acoarse rectangular decomposition of χ^-1_D(D)=([0,1]^n)^k into multi-dimensional rectangles of the formC_α= (C_1)_α_1× (C_2)_α_2×⋯× (C_k)_α_k, where α=(α_1,…, α_k)∈𝒜:={1,…,(M_1)^n}×⋯×{1,…,(M_k)^n}. Further, we divide each D_j into small cubes {(C_j)_β}_β=1,…,(N_j)^n, of side 1/N_j, where N_j is a multiple of M_j, obtaining afine rectangular decomposition. For each vertex z_j of a cube (C_j)_β in the cubical decomposition of D_j, we assign a label ℓ_j=(± 1, …, ± 1)∈𝒵^n, based on the sector of 𝒪_ℓ⊆^n where the displacement vector Δ z_j=f_χ_D_j,D_j+1(z_j)-z_j lands.Relative to the product window D, this can also be regarded as an (nk)-dimensional polytope. The cubical decomposition of each window D_j determines arectangular decomposition of D of the formC_β= (C_1)_β_1× (C_2)_β_2×⋯× (C_k)_β_k, where β=(β_1,…, β_k)∈ℬ:={1,…,(N_1)^n}×⋯×{1,…,(N_k)^n}.Note that for each α∈𝒜, (C_α∩ C_β)_β∈ℬ forms arectangular decomposition ofC_α.Each vertex z=(z_1,…,z_k) of a cube C_β is assigned a label ℓ=(ℓ_1,…,ℓ_k)∈(𝒵^n)^k whose component ℓ_j∈𝒵^n is the label corresponding to the vertex z_j, k=1,…, k, according to Condition O.(i) Given a sequence of windows D_1,…, D_k as above, with D_jcorrectly aligned with D_j+1 under f, for j=1,…, k, and D_k correctly aligned with D_1 under f. Then there exists a periodic orbit p_1,…,p_k of f of period k with p_j∈int(D_j), for j=1,…,k.(ii) If {(C_j)_α} is a coarse subdivision of [0,1]^n=χ^-1_D_j(D_j) then for eachj there exists(C_j)_α^*_jwith _(C_j)_α^*_j(ϕ)≠ 0; if each (C_j)_α^*_j further satisfies the non-degeneracy condition on its faces, then f hasa periodic orbit p_1,…,p_k with p_j∈χ_D_j((C_j)_α^*_j),for j=1,…,k. (iii) Assume that χ_D_j is Lipschitz with Lipschitz constant K_j>1, and thatf_χ_D_j,D_j+1 is bi-Lipschitz with Lipschitz constant L_j>1. Let δ>0 and consider a sufficiently fine subdivision of each χ_D_j^-1(D_j)=[0,1]^n into cubes {(C_j)_β_j}_β_j=1,…,N_j^n as above, so thatmax_j{K_j+1(L_j+1)/N}<δ. Then for every sequence of cubes (C_j)_β^*_j⊆χ^-1_D_j(D_j) that are completely labeled,every sequence of points z̃_j=χ_D_j(z_j)∈χ_D_j( (C_j)_β^*_j), j=1,…,k, is a δ-approximate periodic orbit of period k.(i) By Lemma <ref>, the correct alignment of D_j with D_j+1 under f implies that D is correctly aligned with itself under F. The cubical decompositions{(C_j)_α}of [0,1]^n=χ^-1_D_j(D_j), j=1,…,kdetermine a coarse decomposition C_α of ([0,1]^n)^k as in (<ref>). By Theorem <ref>, there exists acube C_α_*=(C_1)_α^*_1×(C_2)_α^*_2×…× (C_k)_α^k_* in this decomposition with _C_α_*(ϕ)≠ 0. The labeling of the vertices of C_α^* with respect to the map F_χ_D, induce a labelingof eachcube(C_j)_α^*_j with respect to the corresponding map f_χ_D_j,χ_D_j+1 such that _(C_j)_α^*_j(ϕ)≠ 0.Take now a sequence of fine cubical decomposition (C_j)_β_j^N of χ^-1_D_j(D_j),as in (<ref>), with diam((C_j)_β_j^N)→ 0 as N→∞. By the above argument, within each subdivision one can find acube (C_j)_β_j^*N with _(C_j)_β_j^*N(ϕ)≠ 0. Since such labeling is also complete, it impliesthat for each i∈{1,…,n} there exists an n-tuple of pointsz^i,N_j∈ (C_j)_β_j^*N, such that for each i we have that π_i(f_χ( z^i,N_j)- z^i,N_j+1)=0. By successively extracting convergent subsequence in each i-coordinate, for each j we obtain n subsequences in (C_j)_α^*_j of the z^i,N_j'sthat are convergent to the same limit z_j∈ (C_j)_α^*_j, and such that π_i(f_χ( z_j)- z_j+1)=0 for all i=1,…,n, that is, f_χ( z_j)= z_j+1. Hence p_j=χ_D_j( z_j), j=1,…,k, is a periodic sequence of period k for f.(ii) Using the non-degeneracy condition on the labeling and that_(C_j)_α^*_j(ϕ)≠ 0 for j=1,…,k,we apply the previous argument to the collection of cubes (C_j)_α^*_j, obtaining a periodic orbit p_j∈χ_D_j((C_j)_α^*_j) for f, where j=1,…,k.(iii) Consider the points z^i,N_j∈ (C_j)_β^*_j as before, with N large enough as in the statement.Let z_j be an arbitrary point in (C_j)_β^*_j, and ẑ_j=χ_D_j(z_j), for j=1,…,k. We havef(ẑ_j)-ẑ_j+1_∞ =χ_D_j+1∘ f_χ_D_j, χ_D_j+1∘χ^-1_D_j(χ_D_j(z_j))-χ_D_j+1 (z_j+1)_∞≤ K_j+1f_χ_D_j, χ_D_j+1(z_j)-z_j+1_∞= K_j+1max_i=1,…, n|π_i(f_χ_D_j, χ_D_j+1(z_j))-π_i(z_j+1)|≤ K_j+1max_i=1,…,n ( |π_i(f_χ_D_j, χ_D_j+1(z_j))-π_i(f_χ_D_j, χ_D_j+1(z^i,N_j))| .+. |π_i(f_χ_D_j, χ_D_j+1(z^i,N_j) -π_i(z^i,N_j+1)| . +. |π_i( z^i,N_j+1)-π_i(z_j+1)| )≤K_j+1(L_j+1)/N<δ.Thus, ẑ_1,…, ẑ_k is a δ-approximate periodic orbit of period k for f. (i) Assume that D_1,…, D_k is a sequence of windows as above. Let Γ=(γ_ij)_i,j=1,…,k be a transition matrix, where γ_ij=0 or 1; assume that for any i,j with γ_ij=1, D_i is correctly aligned with D_j under f. Consider the topological Markov chain associated to the transition matrix Γ defined byΩ_Γ={ω:=(ω_t)_t∈ℤ | ω_t∈{1,…,k} and γ_ω_tω_t+1=1 for allt},and the shift map σ:Ω_Γ→Ω_Γ given by(σ(ω))_t=ω_t+1,t∈ℤ. Then, for every sequence ω∈Ω_Γ, there exists an orbit (p_t)_t∈ℤ of f, withp_t:=f^t(p_0)∈int(D_ω_t), for all t.(ii) Assume that χ_D_j is Lipschitz with Lipschitz constant K>1, and thatf_χ_D_j,D_l is bi-Lipschitz with Lipschitz constant L>1, for all j,l∈{1,…,k}. Let δ>0, T∈ℤ^+, and ω∈Ω_Γ. Consider a sufficiently fine subdivision of each χ_D_j^-1(D_j)=[0,1]^n into cubes {(C_j)_β_j}_β_j=1,…,N^n as above, so thatmax_j{K (L+1)/N}<δ, j=1,…,k. Then for every sequence of cubes (C_ω_t)_β^*_t⊆χ^-1_D_ω_t(D_ω_t) that are completely labeled,every sequence of points z̃_t=χ_D_ω_t(z_ω_t)∈χ_D_ω_t( (C_ω_t)_β^*_t), t=1,…,T, is a δ-approximate orbitof length T, in the following sensed(f(z̃_t), z̃_t+1)<δ,fort=1,…,T.(i) It is enoughto prove that for each ω∈Ω_Γ, for the infinite ofwindows D_ω_t, t∈ℤ, there exists a point p_0 in D_ω_0 such that f^t(p_0)∈ D_ω_t. This follows from the following:Claim 1.If{D_t}_t=1,…, k, is a sequence of windows such that for every t=1,…, k-1, D_t is correctly aligned with D_t+1 under f, then there exists an orbit (p_t)_t=1,…,k such that p_t+1=f(p_t), and p_t∈ D_t for all t.Proof of Claim 1. We can always define a continuous map f such that D_n is correctly aligned with D_0 under f. Then, similarly to (<ref>) we define the mapF(z_1,…, z_k)=(f(z_k), f(z_1),…,f(z_k-1)).Denoting χ_D(z_1,…, z_k)=(χ_D_1(z_1),…,χ_D_k(z_k)), as in Lemma <ref> and Proposition <ref>, we obtain that χ_D(∏_t=1^k [0,1]^n) is correctly aligned to itself under F, hence there is a fixed point for F. This yields an orbit of f as in the claim; the fact that p_1=f(p_k) is irrelevant for the dynamics.Claim 2.IfD_t, t∈ℤ, is a sequence of windows such that for every t, D_t is correctly aligned with D_t+1 under f, then there exists an orbit (p_t)_t∈ℤ such that p_t+1=f(p_t), and p_t∈ D_t for all t.Proof of Claim 2. By Claim 1 for eachfinite sequence of windowsD_-N, …,D_0,…, D_N,there is a point p_0^N∈ D_0 such that f^t(p^N_0)∈ D_t for all t∈{-N,…, N}. Taking a convergent subsequencep^k_N_0 of p^N_0 withp^k_N_0→ p_0 as N→∞, we obtain that the orbit of p_0 is as claimed.(ii) The proof follows in the same way as for Proposition <ref>-(ii). For a related statement to Proposition <ref> see <cit.>.§ APPLICATIONIn this section we illustrate the methodology developed in this paper on a simple example. Namely, we consider the Hénon Map, defined as f(x,y)=(a-x^2+by,x) for a=1.25and b=0.3.We will use the Sperner lemma-based approach to show the existence ofa period-7 orbit.We build a window D around the point (-0.12,-1.36), which is a `first guess' of a period seven point, and compute its seventh iterate f^7(D). See Fig. <ref>. We define a fine grid on D and we label the points of the grid according to Condition O. We further reduce the labeling to only three labels (1,1)→ 1, (-1,1)→ 2, and (1,-1), (-1,-1)→ 3, as in Section <ref>.Thislabeling is shown color coded in Fig. <ref>. It is easy to see that the boundary of D has a non-degenerate labeling and _D(ϕ)=1, thus D is correctly aligned to itself under f^7.A completely labeled square in the grid decomposition occurs where the three different labels `meet'; this square has vertices at x=-0.124198, y=-1.36279 (red), x=-0.124197, y=-1.36279 (red), x=-0.124198, y=-1.36279 (blue), x=-0.124197, y= -1.36279 (green); see Fig. <ref>. The corresponding approximate period-7 orbit is shown in Fig. <ref>.It is easy to see that the above square has non-zero index. Thus, there exists a true period-7 orbit with an initial point near the square.§ ACKNOWLEDGEMENTBoth authors aregrateful to Meir Retter who helped with the computer code for the example in Section <ref>. The first author is gratefulto Zhonggang (Zeke)Zeng, who made us aware of thenumerical analysis literature related to the Sperner Lemma, and to Kathleen Dexter-Mitchell, who read an early version of this work. alpha | http://arxiv.org/abs/1706.08960v1 | {
"authors": [
"Marian Gidea",
"Yitzchak Shmalo"
],
"categories": [
"math.DS",
"math.AT",
"math.CO"
],
"primary_category": "math.DS",
"published": "20170627174545",
"title": "Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics"
} |
Domain reduction techniques for global NLP and MINLP optimization [=================================================================Deep learning has the potential to revolutionize quantum chemistry as it is ideally suited to learn representations for structured data and speed up the exploration of chemical space. While convolutional neural networks have proven to be the first choice for images, audio and video data, the atoms in molecules are not restricted to a grid. Instead, their precise locations contain essential physical information, that would get lost if discretized. Thus, we propose to use continuous-filter convolutional layers to be able to model local correlations without requiring the data to lie on a grid. We apply those layers in SchNet: a novel deep learning architecture modeling quantum interactions in molecules. We obtain a joint model for the total energy and interatomic forces that follows fundamental quantum-chemical principles. Our architecture achieves state-of-the-art performance for benchmarks of equilibrium molecules and molecular dynamics trajectories. Finally, we introduce a more challenging benchmark with chemical and structural variations that suggests the path for further work. § INTRODUCTION The discovery of novel molecules and materials with desired properties is crucial for applications such as batteries, catalysis and drug design. However, the vastness of chemical compound space and the computational cost of accurate quantum-chemical calculations prevent an exhaustive exploration. In recent years, there have been increased efforts to use machine learning for the accelerated discovery of molecules and materials with desired properties <cit.>. However, these methods are only applied to stable systems in so-called equilibrium, i.e., local minima of the potential energy surface E(_1, …, _n) where _i is the position of atom i. Data sets such as the established QM9 benchmark <cit.> contain only equilibrium molecules. Predicting stable atom arrangements is in itself an important challenge in quantum chemistry and material science.In general, it is not clear how to obtain equilibrium conformations without optimizing the atom positions. Therefore, we need to compute both the total energy E(_1, …, _n) and the forces acting on the atomsF_i (_1, …, _n) = - ∂ E/∂𝐫_i (_1, …, _n).One possibility is to use a less computationally costly, however, also less accurate quantum-chemical approximation. Instead, we choose to extend the domain of our machine learning model to both compositional (chemical) and configurational (structural) degrees of freedom.In this work, we aim to learn a representation for molecules using equilibrium and non-equilibrium conformations. Such a general representation for atomistic systems should follow fundamental quantum-mechanical principles. Most importantly, the predicted force field has to be curl-free. Otherwise, it would be possible to follow a circular trajectory of atom positions such that the energy keeps increasing, i.e., breaking the law of energy conservation. Furthermore, the potential energy surface as well as its partial derivatives have to be smooth, e.g., in order to be able to perform geometry optimization. Beyond that, it is beneficial that the model incorporates the invariance of the molecular energy with respect to rotation, translation and atom indexing. Being able to model both chemical and conformational variations constitutes an important step towards ML-driven quantum-chemical exploration.This work provides the following key contributions: * We propose continuous-filter convolutional (cfconv) layers as a means to move beyond grid-bound data such as images or audio towards modeling objects with arbitrary positions such as astronomical observations or atoms in molecules and materials.* We propose SchNet: a neural network specifically designed to respect essential quantum-chemical constraints.In particular, we use the proposed cfconv layers in ℝ^3 to model interactions of atoms at arbitrary positions in the molecule. SchNet delivers both rotationally invariant energy prediction and rotationally equivariant force predictions.We obtain a smooth potential energy surface and the resulting force-field is guaranteed to be energy-conserving. * We present a new, challenging benchmark – ISO17 – including both chemical and conformational changes[ISO17 is publicly available at <www.quantum-machine.org>.]. We show that training with forces improves generalization in this setting as well.§ RELATED WORK Previous work has used neural networks and Gaussian processes applied to hand-crafted features to fit potential energy surfaces <cit.>. Graph convolutional networks for circular fingerprint <cit.> and molecular graph convolutions <cit.> learn representations for molecules of arbitrary size. They encode the molecular structure using neighborhood relationships as well as bond features, e.g., one-hot encodings of single, double and triple bonds. In the following, we briefly review the related work that will be used in our empirical evaluation: gradient domain machine learning (GDML), deep tensor neural networks (DTNN) and enn-s2s. *Gradient-domain machine learning (GDML) <cit.> proposed GDML as a method to construct force fields that explicitly obey the law of energy conservation. GDML captures the relationship between energy and interatomic forces (see Eq. <ref>) by training the gradient of the energy estimator. The functional relationship between atomic coordinates and interatomic forces is thus learned directly and energy predictions are obtained by re-integration. However, GDML does not scale well due to its kernel matrix growing quadratically with the number of atoms as well as the number of examples. Beyond that, it is not designed to represent different compositions of atom types unlike SchNet, DTNN and enn-s2s. *Deep tensor neural networks (DTNN) <cit.> proposed the DTNN for molecules that are inspired by the many-body Hamiltonian applied to the interactions of atoms. They have been shown to reach chemical accuracy on a small set of molecular dynamics trajectories as well as QM9.Even though the DTNN shares the invariances with our proposed architecture, its interaction layers lack the continuous-filter convolution interpretation. It falls behind in accuracy compared to SchNet and enn-s2s. *enn-s2s <cit.> proposed the enn-s2s as a variant of message-passing neural networks that uses bond type features in addition to interatomic distances. It achieves state-of-the-art performance on all properties of the QM9 benchmark <cit.>. Unfortunately, it cannot be used for molecular dynamics predictions (MD-17). This is caused by discontinuities in their potential energy surface due to the discreteness of the one-hot encodings in their input.In contrast, SchNet does not use such features and yields a continuous potential energy surface by using continuous-filter convolutional layers.§ CONTINUOUS-FILTER CONVOLUTIONS In deep learning, convolutional layers operate on discretized signals such as image pixels <cit.>, video frames <cit.> or digital audio data <cit.>. While it is sufficient to define the filter on the same grid in these cases, this is not possible for unevenly spaced inputs such as the atom positions of a molecule (see Fig. <ref>). Other examples include astronomical observations <cit.>, climate data <cit.> and the financial market <cit.>. Commonly, this can be solved by a re-sampling approach defining a representation on a grid <cit.>. However, choosing an appropriate interpolation scheme is a challenge on its own and, possibly, requires a large number of grid points. Therefore, various extensions of convolutional layers even beyond the Euclidean space exist, e.g., for graphs <cit.> and 3d shapes<cit.>. Analogously, we propose to use continuous filters that are able to handle unevenly spaced data, in particular, atoms at arbitrary positions.Given the feature representations of n objects X^l = (^l_1,…,^l_n) with ^l_i ∈ℝ^F at locations R =(_1,…,_n) with _i ∈ℝ^D, the continuous-filter convolutional layer l requires a filter-generating functionW^l: ℝ^D →ℝ^F,that maps from a position to the corresponding filter values. This constitutes a generalization of a filter tensor in discrete convolutional layers. As in dynamic filter networks <cit.>, this filter-generating function is modeled with a neural network.While dynamic filter networks generate weights restricted to a grid structure, our approach generalizes this to arbitrary position and number of objects. The output _i^l+1 for the convolutional layer at position _i is then given by_i^l+1 = (X^l * W^l)_i = ∑_j^l_j ∘ W^l(_i - _j),where "∘" represents the element-wise multiplication.We apply these convolutions feature-wise for computational efficiency <cit.>. The interactions between feature maps are handled by separate object-wise or, specifically, atom-wise layers in SchNet. § SCHNET SchNet is designed to learn a representation for the prediction of molecular energies and atomic forces. It reflects fundamental physical laws including invariance to atom indexing and translation, a smooth energy prediction w.r.t. atom positions as well as energy-conservation of the predicted force fields. The energy and force predictions are rotationally invariant and equivariant, respectively. §.§ ArchitectureFig. <ref> shows an overview of the SchNet architecture.At each layer, the molecule is represented atom-wise analogous to pixels in an image. Interactions between atoms are modeled by the three interaction blocks. The final prediction is obtained after atom-wise updates of the feature representation and pooling of the resulting atom-wise energy. In the following, we discuss the different components of the network. Molecular representation A molecule in a certain conformation can be described uniquely by a set of n atoms with nuclear charges Z=(Z_1, …, Z_n) and atomic positions R=(_1, …_n). Through the layers of the neural network, we represent the atoms using a tuple of features X^l= (_1^l, …_n^l), with ^l_i ∈ℝ^F with the number of feature maps F, the number of atoms n and the current layer l. The representation of atom iis initialized using an embedding dependent on the atom type Z_i:^0_i = _Z_i.The atom type embeddings _Z are initialized randomly and optimized during training.Atom-wise layers A recurring building block in our architecture are atom-wise layers. These are dense layers that are applied separately to the representation ^l_i of atom i:^l+1_i = W^l ^l_i + 𝐛^lThese layers is responsible for the recombination of feature maps. Since weights are shared across atoms, our architecture remains scalable with respect to the size of the molecule. Interaction The interaction blocks, as shown in Fig. <ref> (middle), are responsible for updating the atomic representations based on the molecular geometry R=(_1, …_n).We keep the number of feature maps constant at F=64 throughout the interaction part of the network. In contrast to MPNN and DTNN, we do not use weight sharing across multiple interaction blocks.The blocks use a residual connection inspired by ResNet <cit.>:_i^l+1 = _i^l + _i^l.As shown in the interaction block in Fig. <ref>, the residual _i^l is computed through an atom-wise layer, an interatomic continuous-filter convolution (cfconv) followed by two more atom-wise layers with a softplus non-linearity in between. This allows for a flexible residual that incorporates interactions between atoms and feature maps. Filter-generating networksThe cfconv layer including its filter-generating network are depicted at the right panel of Fig. <ref>. In order to satisfy the requirements for modeling molecular energies, we restrict our filters for the cfconv layers to be rotationally invariant. The rotational invariance is obtained by using interatomic distancesd_ij = _i-_jas input for the filter network. Without further processing, the filters would be highly correlated since a neural network after initialization is close to linear. This leads to a plateau at the beginning of training that is hard to overcome. We avoid this by expanding the distance with radial basis functions e_k(_i-_j) = exp ( -γd_ij - μ_k ^2 )located at centers 0Å≤μ_k ≤ 30Å every 0.1Å with γ=10Å. This is chosen such that all distances occurring in the data sets are covered by the filters. Due to this additional non-linearity, the initial filters are less correlated leading to a faster training procedure. Choosing fewer centers corresponds to reducing the resolution of the filter, while restricting the range of the centers corresponds to the filter size in a usual convolutional layer. An extensive evaluation of the impact of these variables is left for future work. We feed the expanded distances into two dense layers with softplus activations to compute the filter weight W(_i - _j) as shown in Fig. <ref> (right).Fig <ref> shows 2d-cuts through generated filters for all three interaction blocks of SchNet trained on an ethanol molecular dynamics trajectory. We observe how each filter emphasizes certain ranges of interatomic distances. This enables its interaction block to update the representations according to the radial environment of each atom. The sequential updates from three interaction blocks allow SchNet to construct highly complex many-body representations in the spirit of DTNNs <cit.> while keeping rotational invariance due to the radial filters. §.§ Training with energies and forcesAs described above, the interatomic forces are related to the molecular energy, so that we can obtain an energy-conserving force model by differentiating the energy model w.r.t. the atom positionsF̂_i(Z_1, …, Z_n, _1, …, _n) = -∂Ê/∂_i(Z_1, …, Z_n, _1, …, _n).<cit.> pointed out that this leads to an energy-conserving force-field by construction. As SchNet yields rotationally invariant energy predictions, the force predictions are rotationally equivariant by construction. The model has to be at least twice differentiable to allow for gradient descent of the force loss. We chose a shifted softplus ssp(x) = ln(0.5e^x + 0.5) as non-linearity throughout the network in order to obtain a smooth potential energy surface. The shifting ensures that ssp(0) = 0 and improves the convergence of the network. This activation function shows similarity to ELUs <cit.>, while having infinite order of continuity.We include the total energy E as well as forces 𝐅_i in the training loss to train a neural network that performs well on both properties:ℓ(Ê, (E, 𝐅_1, …, 𝐅_n)) = ρE - Ê^2 + 1/n∑_i=0^n 𝐅_i -(-∂Ê/∂_i ) ^2.This kind of loss has been used before for fitting a restricted potential energy surfaces with MLPs <cit.>. In our experiments, we use ρ=0.01 for combined energy and force training. The value of ρ was optimized empirically to account for different scales of energy and forces.Due to the relation of energies and forces reflected in the model, we expect to see improved generalization, however, at a computational cost. As we need to perform a full forward and backward pass on the energy model to obtain the forces, the resulting force model is twice as deep and, hence, requires about twice the amount of computation time.Even though the GDML model captures this relationship between energies and forces, it is explicitly optimized to predict the force field while the energy prediction is a by-product. Models such as circular fingerprints <cit.>, molecular graph convolutions or message-passing neural networks<cit.> for property prediction across chemical compound space are only concerned with equilibrium molecules, i.e., the special case where the forces are vanishing. They can not be trained with forces in a similar manner, as they include discontinuities in their predicted potential energy surface caused by discrete binning or the use of one-hot encoded bond type information.§ EXPERIMENTS AND RESULTSIn this section, we apply the SchNet to three different quantum chemistry datasets: QM9, MD17 and ISO17. We designed the experiments such that each adds another aspect towards modeling chemical space. While QM9 only contains equilibrium molecules, for MD17 we predict conformational changes of molecular dynamics of single molecules. Finally, we present ISO17 combining both chemical and structural changes.For all datasets, we report mean absolute errors in kcal/mol for the energies and in kcal/mol/Å for the forces. The architecture of the network was fixed after an evaluation on the MD17 data sets for benzene and ethanol (see supplement). In each experiment, we split the data into a training set of given size N and use a validation set of 1,000 examples for early stopping. The remaining data is used as test set. All models are trained with SGD using the ADAM optimizer <cit.> with 32 molecules per mini-batch. We use an initial learning rate of 10^-3 and an exponential learning rate decay with ratio 0.96 every 100,000 steps. The model used for testing is obtained using an exponential moving average over weights with decay rate 0.99. §.§ QM9 – chemical degrees of freedom QM9 is a widely used benchmark for the prediction of various molecular properties in equilibrium <cit.>. Therefore, the forces are zero by definition and do not need to be predicted. In this setting, we train a single model that generalizes across different compositions and sizes.QM9 consists of ≈130k organic molecules with up to 9 heavy atoms of the types {C, O, N, F}.As the size of the training set varies across previous work, we trained our models each of these experimental settings. Table <ref> shows the performance of various competing methods for predicting the total energy (property U_0 in QM9). We provide comparisons to the DTNN <cit.> and the best performing MPNN configuration denoted enn-s2s and an ensemble of MPNNs (enn-s2s-ens5) <cit.>. SchNet consistently obtains state-of-the-art performance with an MAE of 0.31 kcal/mol at 110k training examples. §.§ MD17 – conformational degrees of freedomMD17 is a collection of eight molecular dynamics simulations for small organic molecules. These data sets were introduced by <cit.> for prediction of energy-conserving force fields using GDML. Each of these consists of a trajectory of a single molecule covering a large variety of conformations. Here, the task is to predict energies and forces using a separate model for each trajectory. This molecule-wise training is motivated by the need for highly-accurate force predictions when doing molecular dynamics.Table <ref> shows the performance of SchNet using 1,000 and 50,000 training examples in comparison with GDML and DTNN. Using the smaller data set, GDML achieves remarkably accurate energy and force predictions despite being only trained on forces. The energies are only used to fit the integration constant. As mentioned before, GDML does not scale well with the number of atoms and training examples. Therefore, it cannot be trained on 50,000 examples. The DTNN was evaluated only on four of these MD trajectories using the larger training set <cit.>. Note that the enn-s2s cannot be used on this dataset due to discontinuities in its inferred potential energy surface.We trained SchNet using just energies and using both energies and forces. While the energy-only model shows high errors for the small training set, the model including forces achieves energy predictions comparable to GDML. In particular, we observe that SchNet outperforms GDML on the more flexible molecules malonaldehyde and ethanol, while GDML reaches much lower force errors on the remaining MD trajectories that all include aromatic rings.The real strength of SchNet is its scalability, as it outperforms the DTNN in three of four data sets using 50,000 training examples using only energies in training. Including force information, SchNet consistently obtains accurate energies and forces with errors below 0.12 kcal/mol and 0.33 kcal/mol/Å, respectively.Remarkably, when training on energies and forces using 1,000 training examples, SchNet performs better than training the same model on energies alone for 50,000 examples. §.§ ISO17 – chemical and conformational degrees of freedomAs the next step towards quantum-chemical exploration, we demonstrate the capability of SchNet to represent a complex potential energy surface including conformational and chemical changes. We present a new dataset – ISO17 – where we consider short MD trajectories of 129 isomers, i.e., chemically different molecules with the same number and types of atoms. In contrast to MD17, we train a joint model across different molecules. We calculate energies and interatomic forces from short MD trajectories of 129 molecules drawn randomly from the largest set of isomers in QM9.While the composition of all included molecules is C_7O_2H_10, the chemical structures are fundamentally different. With each trajectory consisting of 5,000 conformations, the data set consists of 645,000 labeled examples.We consider two scenarios with this dataset: In the first variant, the molecular graph structures present in training are also present in the test data. This demonstrates how well our model is able to represent a complex potential energy surface with chemical and conformational changes. In the more challenging scenario, the test data contains a different subset of molecules.Here we evaluate the generalization of our model to previously unseen chemical structures. We predict forces and energies in both cases and compare to the mean predictor as a baseline. We draw a subset of 4,000 steps from 80% of the MD trajectories for training and validation. This leaves us with a separate test set for each scenario:(1) the unseen 1,000 conformations of molecule trajectories included in the training set and (2) all 5,000 conformations of the remaining 20% of molecules not included in training.Table <ref> shows the performance of the SchNet on both test sets. Our proposed model reaches chemical accuracy for the prediction of energies and forces for the test set of known molecules. Including forces in the training improves the performance here as well as on the set of unseen molecules. This shows that using force information does not only help to accurately predict nearby conformations of a single molecule, but indeed helps to generalize across chemical compound space.§ CONCLUSIONSWe have proposed continuous-filter convolutional layers as a novel building block for deep neural networks. In contrast to the usual convolutional layers, these can model unevenly spaced data as occurring in astronomy, climate reasearch and, in particular, quantum chemistry. We have developed SchNet to demonstrate the capabilities of continuous-filter convolutional layers in the context of modeling quantum interactions in molecules. Our architecture respects quantum-chemical constraints such as rotationally invariant energy predictions as well as rotationally equivariant, energy-conserving force predictions.We have evaluated our model in three increasingly challenging experimental settings. Each brings us one step closer to practical chemical exploration driven by machine learning. SchNet improves the state-of-the-art in predicting energies for molecules in equilibrium of the QM9 benchmark. Beyond that, it achieves accurate predictions for energies and forces for all molecular dynamics trajectories in MD17. Finally, we have introduced ISO17 consisting of 645,000 conformations of various C_7O_2H_10 isomers. While we achieve promising results on this new benchmark, modeling chemical and conformational variations remains difficult and needs further improvement. For this reason, we expect that ISO17 will become a new standard benchmark for modeling quantum interactions with machine learning.§.§.§ AcknowledgmentsThis work was supported by the Federal Ministry of Education and Research (BMBF) for the Berlin Big Data Center BBDC (01IS14013A). Additional support was provided by the DFG (MU 987/20-1) and from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement NO 657679. K.R.M. gratefully acknowledges the BK21 program funded by Korean National Research Foundation grant (No. 2012-005741) and the Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (no. 2017-0-00451).unsrtnat | http://arxiv.org/abs/1706.08566v5 | {
"authors": [
"Kristof T. Schütt",
"Pieter-Jan Kindermans",
"Huziel E. Sauceda",
"Stefan Chmiela",
"Alexandre Tkatchenko",
"Klaus-Robert Müller"
],
"categories": [
"stat.ML",
"physics.chem-ph"
],
"primary_category": "stat.ML",
"published": "20170626191237",
"title": "SchNet: A continuous-filter convolutional neural network for modeling quantum interactions"
} |
]Numerical modelling of surface water wave interaction with a moving wallG. Khakimzyanov]Gayaz Khakimzyanov G. Khakimzyanov: Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia [email protected] http://www.ict.nsc.ru/ru/structure/Persons/ict-KhakimzyanovGSD. Dutykh]Denys Dutykh^* D. Dutykh: LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, F-73376 Le Bourget-du-Lac Cedex, France [email protected] http://www.denys-dutykh.com/ ^* Corresponding author emptyGayaz KhakimzyanovInstitute of Computational Technologies, Novosibirsk, Russia Denys DutykhCNRS, Université Savoie Mont Blanc, France[t]1.0Numerical modelling of surface water wave interaction with a moving wallemptyLast modified: December 30, 2023[ [ December 30, 2023 =====================emptyIn the present manuscript we consider the practical problem of wave interaction with a vertical wall. However, the novelty here consists in the fact that the wall can move horizontally due to a system of springs. The water wave evolution is described with the free surface potential flow model. Then, a semi-analytical numerical method is presented. It is based on a mapping technique and a finite difference scheme in the transformed domain. The idea is to pose the equations on a fixed domain. This method is thoroughly tested and validated in our study. By choosing specific values of spring parameters, this system can be used to damp (or in other words to extract the energy of) incident water waves.: wave/body interaction; full Euler equations; movable wall; wave run-up; wave damping; wave energy extractionMSC: [2010] 76B15 (primary), 76B07, 76M20 (secondary) PACS: [2010] 47.35.Bb (primary), 47.35.Fg (secondary)empty empty§ INTRODUCTION The mathematical and numerical high fidelity modeling of water waves is a central topic in coastal and naval engineering. The incident waves come and interact with various coastal features. Nowadays the interaction of water waves with bathymetric features is relatively well understood (at least with some special features <cit.>). A more challenging situation is the interaction of waves with various coastal structures <cit.>. At this level the most studied situation is the wave/wall interaction and the wall is assumed to be fixed and impermeable. Violent wave impacts have to be modelled in general using more CFD-like methods in the Eulerian mesh-based <cit.> or Lagrangian particle-based <cit.> approaches. In the present study we apply the free surface approximation by neglecting all the processes happening in the air above the free surface <cit.>. In this line of thinking, the interaction of periodic waves with a fixed vertical wall was recently studied in <cit.> in the framework of a fully nonlinear weakly dispersive wave model <cit.>. The conditions under which an extreme wave run-up on a vertical wall may happen were describe in <cit.> as well.In this study, we focus on wave interactions with a movable vertical wall. The wall motion can be prescribed. In this case we model the wave generation process with a piston-type wave maker <cit.>. For instance, this problem was considered in the framework of Boussinesq-type equations in <cit.>. Otherwise, the wall can move under the action of incident waves. We can even assume that a system of horizontal tension/extension springs (with tunable rigidities) is attached to the wall. Thus, the wall may present a certain resistance to the action of waves. This problem can be regarded also as a piston's free motion under the forcing of water waves. The piston mechanical energy conversion and recuperation is a different technological problem which is out of scope of the present study. The extraction of ocean wave energy on industrial scales is not yet a very common practice <cit.>. However, very active researches in this field are conducted since at least forty years <cit.>. Consequently, the numerical methods developed below can be used to design and to optimize such Wave Energy Conversion (WEC) devices. Movable walls have been used, for example, in a triplate system proposed back in 1977 by Dr. Francis Farley. The rigidity of strings is chosen to minimize the reflected wave amplitude so that most of the wave energy be converted into the mechanical energy of the device. Mathematical and numerical modeling of this type of WEC devices is considered below. The main point is that (ordinary and partial) differential equations posed in time-varying domains are known to pose notorious theoretical and numerical difficulties <cit.>.In the present work we consider a two-dimensional non-stationary problem of surface wave motion in a domain with one moving wall. We assume that the wall remains vertical during its motion. Moreover, at least in a vicinity of the moving wall the bathymetry has to be flat to allow its free motion under the action of waves or to follow a prescribed trajectory. The fluid flow is assumed to be potential and we address the complete (fully nonlinear and fully dispersive) water wave problem <cit.>. We propose first a reformulation of this problem on a fixed and domain (a square) by introducing a new curvilinear coordinate system. Then, we propose a robust finite difference discretization of this problem with mathematically proven good qualities. The performance of this algorithm is illustrated on several examples of practical interest. In particular, we study the influence of springs rigidity on the oscillatory motion of the wave/wall system.The results presented in this study can be viewed under a different angle and, thus, they can be applied to another problem of coastal structures protection from wave loads and impacts. In particular, the catastrophic consequences of 2011 Tohoku earthquake and tsunami are widely known nowadays <cit.>. The protecting structures are made very solid by implementing various security coefficients. However, more economic protections can be designed if we allow them to move under the action of waves. The moving parts can be related to fixed ones by a system of flexible strings and by tuning their rigidity one can reduce significantly the wave run-up on moving parts as well as wave loads on fixed solid structural elements.We would like to mention also related works. The generation of water waves by an accelerating moving vertical plate in a channel of constant depth was studied in <cit.>. The Authors measured the free surface elevation and pressure distributions on the wall for three different values of plate's accelerations. In another recent work <cit.> a nonlinear model for incompressible free surface flows was proposed. Then, this model was used to study wave run-up on a sloping beach and on a moving vertical wall. The two-dimensional unsteady problem of wave interaction with a moving vertical wall was studied analytically in <cit.>. Moreover, the comparisons with numerical solutions obtained with the complex boundary element method was presented as well. Another numerical model was presented in <cit.> as well. In last two works the waves were generated by an initial disturbance located close to the moving wall.The present manuscript is organized as follows. The physical and mathematical problems are formulated in Section <ref> and then it is reformulated on a fixed domain in Section <ref>. The proposed finite difference approximation is described in Section <ref> and studied mathematically in Section <ref>. The construction of the mapping in the fixed domain is discussed in Section <ref> and the numerical algorithm in general is described in Section <ref>. Some numerical results are presented in Section <ref>. Finally, the main conclusions and perspectives are outlined in Section <ref>.§ PROBLEM FORMULATIONConsider a two-dimensional motion of an incompressible, homogeneous and ideal fluid with free surface. The sketch of the fluid domain is depicted in Figure <ref>. The Cartesian coordinate system O x y, = (x, y) is chosen such that the horizontal axis O x is directed along the undisturbed free surface and the axis O y points vertically upwards. The fluid domain Ω (t) is assumed to be simply connected and bounded from below by a horizontal bottom Γ_b{ ∈ Ω (t) | y = -h_0} and from above by the free surface — Γ_s{ ∈ Ω (t) | y = η(x, t)}. On the sides the domain Ω (t) is bounded by two vertical walls. The right wall Γ_r{ ∈ Ω (t) | x = ℓ} is assumed to be fixed[Thus, the length of the undisturbed domain is equal to ℓ > 0.], while the left one Γ_ℓ{ ∈ Ω (t) | x = s (t)} is connected to a system of springs and can move horizontally under the wave action. By s (t) we denote the displacement of the moving wall[We assume additionally that the wall is non-deformable and remains vertical during the interaction with waves.] with respect to its undisturbed position x = 0. The closed domain Ω (t) together with its boundaries is denoted by Ω̅ (t). In order to be able to simulate water waves with a shape, which is not necessarily a graph, we can adopt a parametric representation of the free surface Γ_s:x = ξ (q, t) ,y = η (q, t) ,where q ∈ is a real parameter and t is the time evolution variable. Functions ξ (·, ·) : ×^ + ↦ and η (·, ·) : ×^ + ↦ are assumed to be sufficiently smooth. Without any restriction we can assume that q ∈ [ 0, 1 ]. Below in the text (see Section <ref>) this parametrization will appear more naturally when we map the unknown fluid domain Ω (t) to a fixed reference domain. The fluid flow is described by Euler equations <cit.>:÷ = 0 ,_̆ t + () + 1/ρp=,where = (u, v) is the velocity vector, ρ => 0 is the fluid density, p is the total pressure and = (0,- g) is the gravity acceleration. The subscripts such as _̆ t denote partial derivatives, _̆ tt. These equations have to be supplemented by appropriate boundary conditions in order to have a well-posed problem. The impermeability condition on fixed solid surfaces (the right vertical wall Γ_r and the bottom Γ_b) reads= 0 ,∈ Γ_b ∪ Γ_r ,t ≥ 0 ,whereis the vector of unitary exterior normal to the corresponding boundary of the domain Ω (t). The left wall is impermeable as well and the boundary condition reads= ṡ (t) ,∈ Γ_ℓ ,t ≥ 0 .Taking into account the horizontal character of the wall motion, this condition can be further simplified to giveu = ṡ (t) ,∈ Γ_ℓ ,t ≥ 0 . Traditionally on the free surface we prescribe the kinematicη_ t + u·η_ x = v ,∈ Γ_s ,t ≥ 0and dynamic p = p_a =,∈ Γ_s ,t ≥ 0boundary conditions <cit.>. Here p_a is the constant atmospheric pressure. The boundary conditions mentioned above are enough to obtain a closed system of equations which describe the motion of an ideal fluid. One has to prescribe also the initial conditions, but their form is highly problem-dependent. Consequently, it will not be discussed at the current level of generality. Let us obtain also kinematic free surface boundary conditions when the free surface Γ_s is given in a parametric form (<ref>). The free surface Γ_s is characterized by the fact that its velocity is determined by the velocity of fluid particles constituting this boundary. In other words, the point ∈ Γ_s moves with the velocity of the fluid particle located in this point,t = ((t), t) .By taking the material derivative of both sides of the parametric form (<ref>), we havext≡ ξt + ξq·qt ,yt≡ ηt + ηq·qt .Thus, by taking into account the physical information (<ref>) we obtain the following kinematic boundary conditions for ∀∈ Γ_s and for ∀t ≥ 0:ξt + ξq·qt - u= 0 ,ηt + ηq·qt - v= 0 .In the conditions above the quantity qt is the rate of change of the parameter q for fluid particles located at the free surface. If one uses the Lagrangian description of a flow, fluid particles keep their labels and, thus, qt ≡ 0. However, we use an arbitrary parametrization and in our case qt ≠ 0 in general.§.§ Potential flow model The formulation presented above is still too general. We shall simplify it further by assuming the flow to be irrotational,= -uy + vx = 0 .It implies the existence of a function ϕ : ^2×^+ ↦ which is called the velocity potential such that the velocity field is given by=ϕ .Substituting this form into the incompressibility condition (<ref>) yields that the velocity potential has to be a harmonic functionΔ ϕ = 0 ,(, t) ∈ Ω (t)×^+ .The kinematic boundary condition is obtained straightforwardly by substituting into (<ref>) the velocity components by their representations in terms of the velocity potential:η_ t + ϕ_ x·η_ x = ϕ_ y ,y = η (x, t) .We note also that velocity components u and v in the kinematic boundary conditions (<ref>), (<ref>) are expressed in terms of the velocity potential ϕ according to (<ref>).The momentum equation (<ref>) can be integrated and combined with the dynamic boundary condition (<ref>) to give the so-called Euler–Lagrange integral[The Euler–Lagrange integral becomes the Bernoulli integral for steady flows. We implicitly chose the gauge where the Bernoulli `constant' is identically zero.] evaluated at the free surface:ϕ_ t +ϕ^ 2 + g η = 0 ,y = η (x, t) .On solid stationary walls the velocity potential satisfies the homogeneous Neumann conditionϕ = 0 ,∈ Γ_b ∪Γ_r ,(withdefined above) while on the moving wall Γ_ℓ it satisfies the following non-homogeneous Neumann conditionϕ ≡ ϕx = ṡ (t) ,∈ Γ_ℓ . So, instead of seeking for two components of the velocity field (̆, t) = (u(,t),v(,t)) and the pressure function p(, t), the formulation was simplified to one unknown function, the velocity potential ϕ(, t), thanks to the irrotationality assumption. The potential flow formulation with free surface is known as the full water wave problem and it was shown in numerous studies to be an excellent model for water waves <cit.> (see <cit.> for water wave problem history review).§.§.§ Initial conditionsIn order to obtain a well-posed problem, we have to specify also the appropriate initial conditions. The free surface elevation in the parametric form is initially given by two functions:ξ (q, 0) = ξ_ 0 (q) , η (q, 0) = η_ 0 (q) ,q ∈ [ 0, 1 ] .Let us assume that the initial distribution of fluid particles is known as well:(, 0) = _̆ 0 () ,∈ Ω (0) .Then, we can construct an initial condition for the velocity potential ϕ in the following way:ϕ (, 0) = ϕ_ 0 () ≡ ϕ_ 0 (_ 0) + ∫___ 0u_ 0x + v_ 0y ,where ϕ_ 0 (_ 0) is the value of the velocity potential in an arbitrary point _ 0 ∈ Ω̅ (0) and __ 0⊆ Ω̅ (0) is an arbitrary piecewise smooth curve connecting points _ 0 with . We assume that the initial distribution of the velocities _̆ 0 () is potential. Thus, the curvilinear integral above does not depend on the path __ 0. §.§ Piston motion In order to describe the horizontal motion of the piston we adopt the following very simple model based on the second law of Newton <cit.>:m s̈ + k s = - [(t) -(0) ] ,where m is the wall mass and k is the stiffness coefficient of springs. Finally, (t) is the force acting on the left moving wall. It can be computed by integrating the contributions of the whole water column(t) = ∫_ -h_0^ η_s (t)p (s(t),y,t)y , η_ s (t)η (s (t),t) .If initially the fluid was at rest, the force (0) consists only of the hydrostatic loading on the wall,(0) = g h_ 0^ 2/2 .The pressure p (, t) ≡ p (x, y, t) can be computed at any point inside the fluid thanks to the Euler–Lagrange integral:p (x, y, t) = - ϕ_ t -ϕ^ 2 - g y .A more detailed derivation of equation (<ref>) can be found in Appendix <ref>.The second order nonlinear and non-autonomous ode (<ref>) has to be completed with two initial conditions. If nothing is indicated, we start the integration from the rest state,s (0) = ṡ (0) = 0 .§.§ Dimensionless problemEquations given in previous Section can be further simplified by choosing appropriate scaled variables. Namely, we introduce the following scaled independentx^* = x/h_0 ,y^* = y/h_0 ,t^* = t √(g/h_0)and dependents^* = s/h_ 0 , η^* = η/h_ 0 , ϕ^* = ϕ/h_ 0√(g h_ 0) ,p^* = p/ρ g h_ 0 , ^* = /ρ g h^ 3_ 0variables. The dimensional coefficients and parameters appearing in governing equations scale as follows:ℓ^* = ℓ/h_ 0 ,m^* = m/ρh^ 3_ 0 , k^* = k/ρgh^ 2_ 0 .Finally, dimensionless governing equations read (where for simplicity we drop out all asterisk * symbol from superscripts):^2 ϕ= 0 ,(, t) ∈ Ω (t)×^+ , η_ t + ϕ_ x·η_ x= ϕ_ y ,y = η (x, t) ,ϕ_ t + ϕ^ 2 + η= 0 ,y = η (x, t) ,ϕ_ y= 0 ,y = -1 ,ϕ_ x= ṡ ,x = s (t) ,-1 ≤ y ≤ η (s(t),t) , ϕ_ x= 0,x = ℓ ,-1 ≤ y ≤ η (ℓ, t) ,m s̈ + k s= - [(t) -(0) ] , (t)= ∫_ -1^ η (s(t),t)p (s (t),y,t)y ,p (, t)= - ϕ_t - ϕ^ 2 - y .In numerical simulations below we shall solve this dimensionless system of equations. It is equivalent to setting simply h_0 = 1 and g = 1 in the computer code.§ EQUATIONS ON A FIXED DOMAINThe main difficulty of the problem described above is that the computational domain Ω (t) is time dependent. First of all, it is due to the motion of the free surface, but also due to the motion of the left wall. Consequently, the strategy adopted in this study consists in transforming the problem to a fixed domain ^ 0. Obviously, this transformation will be time dependent due to the unsteady character of the problem. In the past; the idea of using conformal mappings from Complex Analysis has become popular in 2D Hydrodynamics <cit.>. For the 2D water wave problem it was proposed by L. Ovsyannikov (1974) <cit.> and implemented much later numerically by A. Dyachenko (1996) <cit.>. Strictly speaking, in our developments we do not need the conformal property of the underlying mapping. Consequently we shall relax this assumption. Moreover, it will allow us to have a fixed and finite transformed domain ^ 0. For instance, we can choose ^ 0 as a unit square^ 0{(q^ 1,q^ 2) | 0 ≤ q^ 1,q^ 2 ≤ 1} ≡ [0, 1]^2and we consider a smooth bijective mapping ( a diffeomorphism) : ^ 0 ↦ Ω (t):=(,t) ⟺{[ x = x (q^ 1,q^ 2,t) ,; y = y (q^ 1,q^ 2,t) . ]. [font=, column sep = 6em] Ω (t) [bend left = 25]r[font=]^-1 (, t) ^ 0[bend left = 25]l[font=] (, t)Additionally we assume that the Jacobian matrix (, t) of transformation (<ref>) is non-singular,(, t) x_ q^ 1x_ q^ 2y_ q^ 1y_ q^ 2 ≡x_ q^ 1 y_ q^ 2 - x_ q^ 2 y_ q^ 1 ≠ 0 .This situation is schematically depicted in diagram (<ref>) and in Figure <ref>. For more details on the computation of partial derivatives we refer to Appendix <ref>.It is assumed that left Γ_ℓ and right Γ_r walls are images of left and right sides of the square ^ 0, q^ 1 = 0 and q^ 1 = 1 correspondingly (0 ≤ q^ 2 ≤ 1). Similarly, the upper (q^ 2 = 1) and lower (q^ 2 = 0) sides of the square are respectively mapped on the free surface Γ_s and bottom Γ_b. The boundary of the square ^ 0 will be denoted by γ,γ∂ ^ 0 = γ_ s ∪ γ_ b ∪ γ_ r ∪ γ_ ℓ .The practical construction of the mapping (, t) is described in Section <ref>. At the present stage of the exposition we shall assume that mapping (, t) is simply known.In this Section we shall skip the intermediate computations, which are rather standard in the Differential Geometry, for example. We detail them in Appendices <ref> and <ref>. The system of governing equations (<ref>) – (<ref>) posed on the fixed domain ^ 0 reads:q^ α (_ α β ϕq^ β)= 0 ,∈ ^ 0 ,α, β = 1, 2,η_ t + v^ 1·η_ q^ 1= v ,q^ 2 = 1 ,ϕ_ t - (u· x_t + v· y_t) + |_ ϕ|^2 + η= 0 ,q^ 2 = 1 ,_ 2 1 ϕq^ 1 + _ 2 2 ϕq^ 2= 0 ,q^ 2 = 0 ,_ 1 1 ϕq^ 1 + _ 1 2 ϕq^ 2= y_q^ 2 ṡ ,q^ 1 = 0 ,0 ≤ q^ 2 ≤ 1 ,_ 1 1 ϕq^ 1 + _ 1 2 ϕq^ 2= 0,q^ 1 = 1,0 ≤ q^ 2 ≤ 1 , (t)= ∫_ 0^ 1p (0,q^ 2,t)q^ 2 , p=- ϕ_ t+(u· x_t+v· y_t)-|_ϕ|^2-y ,where the implicit convention about the summation over repeated indices is adopted. The transformation of the Laplace equation is detailed in Appendix <ref>. If the free surface is given in the parametric form (<ref>), then we have to add one more kinematic free surface boundary condition:ξ_ t + v^ 1·ξ_ q^ 1 = u ,∈ γ_ s .See also Appendix <ref> for some hints on the derivation of boundary conditions in the transformed domain. Notice that equation (<ref>) is not affected by transformation (<ref>). That is why we do not repeat it here. Above we introduced the following notations. The coefficients in transformed Laplacian are defined as_ 1 1g_ 2 2/ , _ 1 2 ≡ _ 2 1-g_ 1 2/ , _ 2 2g_ 1 1/ ,where = (g_αβ)_α,β=1, 2 is the familiar metric tensorg_ 1 1(x_ q^ 1)^2 + (y_ q^ 1)^2 ,g_ 1 2 ≡ g_ 2 1x_ q^ 1 x_ q^ 2 + y_ q^ 1 y_ q^ 2 , g_ 2 2(x_ q^ 2)^2 + (y_ q^ 2)^2 .In fact, tensor = (_ αβ)_α, β=1,2 can be expressed in a compact matrix form through the metric tensoras=^-1 .The Cartesian (u,v) and the contravariant v^ 1 components of the velocity field are defined asu 1/ [ϕ_q^ 1· y_q^ 2 - ϕ_q^ 2· y_q^ 1] , v 1/ [-ϕ_q^ 1· x_q^ 2 + ϕ_q^ 2· x_q^ 1] ,v^ 1 q^ 1t + u·q^ 1x + v·q^ 1y .In order to compute the derivatives of the inverse mapping, we use the following expressions:q^ 1t = 1/ [ yt xq^ 2 - xt yq^ 2 ] , q^ 1x = 1/ yq^ 2 , q^ 1y = -1/ xq^ 2 . q^ 2t = 1/ [ xt yq^ 1 - yt xq^ 1 ] , q^ 2x = -1/ yq^ 1 , q^ 2y = 1/ xq^ 1 . The mathematical problem of a potential flow formulation with free surface in curvilinear coordinates consists in determining the velocity potential ϕ (q^ 1, q^ 2, t) and the free surface profile η (q^ 1, t). The velocity potential satisfies in the domain ^ 0 an elliptic equation (<ref>) with non-constant coefficients along with boundary conditions (<ref>) – (<ref>) on ∂^ 0. In order to obtain a well-posed problem we have to complete the formulation with two initial conditions at t = 0 directly in the fixed domain:ϕ (q^ 1, 1, 0)= ϕ_ 0(q^ 1) ,= (q^ 1, 1) ∈ Γ_s ,η (q^ 1, 0)= η_ 0 (q^ 1) ,= (q^ 1, 1) ∈ Γ_s .Earlier the initial conditions were discussed in Section <ref>. They can be transposed on the fixed domain ^ 0 using the inverse mapping ^ -1 (, t). §.§ On some properties of elliptic operators in curvilinear coordinatesIt can be noticed that the Laplace equation in curvilinear coordinates (<ref>) has a more complex form comparing to the initial Cartesian coordinates O x y. In particular, constant coefficients become variable in space and time. Moreover, the mixed derivatives appear as well. However, some properties are important to construct qualitatively correct numerical discretizations. For example, below we shall construct a finite difference scheme for equation (<ref>) with a positive definite difference operator. The proof of this fact relies heavily on the uniform ellipticity property of operator (<ref>). Partial differential equation (<ref>) is uniformly elliptic.By our assumptions the mapping (<ref>) is differentiable with bounded derivatives in ^ 0 and non-degenerate with a positive Jacobian > 0, ∀ ∈ ^ 0, ∀ t > 0. Thanks to the identityg_ 1 1· g_ 2 2 ≡ ^ 2 + g_ 1 2^ 2 ,the metric components g_ 1 1 and g_ 2 2 take strictly positive values. Thanks to definitions (<ref>), it is straightforward to conclude that _ 1 1, _ 2 2 are also positive. The matrix= [ _ 1 1 _ 1 2; _ 2 1 _ 2 2 ]is symmetric and its determinant is equal to = 1 due to (<ref>). Consequently, it possesses two orthonormal eigenvectors corresponding to eigenvalues0 ≤ λ_^(-)g_ 1 1 + g_ 2 2 - √()/2≤ 1 , 1 ≤ λ_^(+)g_ 1 1 + g_ 2 2 + √()/2,where the discriminant is defined as(g_ 1 1 + g_ 2 2)^2 - 4 ^ 2 = (g_ 1 1 - g_ 2 2)^2 + 4 g_ 1 2^ 2 ≥ 0 .Then, for any real numbers ∀ξ, ζ ∈ and for any point ∀ ∈ ^ 0 we have the following inequalities:c_1 (ξ^2 + ζ^2) ≤(ξ, ζ) ≤ c_2 (ξ^2 + ζ^2) ,whereis the quadratic form defined as(ξ, ζ)_ 1 1 ξ^ 2 + 2 _ 1 2 ξ ζ + _ 2 2 ζ^ 2 .The constants c_ 1, 2 are defined asc_1 inf_ ∈ ^ 0 g_ 1 1 + g_ 2 2 - √()/2> 0 , c_2 sup_ ∈ ^ 0 g_ 1 1 + g_ 2 2 + √()/2< ∞ .The positivity of c_ 1 > 0 implies the uniform ellipticity property. The constants c_ 1, 2 defined in the proof above can be used to estimate the convergence speed of iterative methods, which depends directly on the conditioning number <cit.> of the matrix corresponding to the difference operator <cit.>. The conditioning number depends on the ratio c_2c_1. High ratio implies a poorly conditioned difference operator and, thus, more iterations are needed to converge to the solution within prescribed accuracy. We underline also that constants c_ 1, 2 depend only on the mapping (<ref>). Below we provide two examples which illustrate situations where the ratio c_2c_1 ≫ 1.§.§.§ Example 1 Let us take a physical domain in the form of a rectangle:Ω_ ▪{ = (x, y) |0 ≤ x ≤ ℓ_ 1 , 0 ≤ y ≤ ℓ_ 2 }with substantially different sizes of the walls, we can assume 0 < ℓ_ 1 ≪ ℓ_ 2. The mapping (<ref>) from ^ 0 ↦ Ω_ ▪ is given explicitly by formulasx = q^ 1·ℓ_ 1 , y = q^ 2·ℓ_ 2 .Then, we can compute explicitly the metric coefficientsg_ 1 1 = ℓ_ 1^ 2 ,g_ 1 2 = 0 ,g_ 2 2 = ℓ_ 2^ 2 ,the Jacobian = ℓ_ 1·ℓ_ 2 ,and eigenvalues of the matrix :c_ 1 = ℓ_ 1/ℓ_ 2 ,c_ 2 = ℓ_ 2/ℓ_ 1 .Thus, the conditioning of the finite difference operator will scale withc_ 2/c_ 1 ≡ (ℓ_ 2/ℓ_ 1)^ 2 ≫ 1 .Thus, for highly distorted domains the ratio c_ 2c_ 1 becomes large. If on the reference domain ^ 0 we use a uniform square grid, than mapping (<ref>) generates a uniform grid with distorted cells. The convergence of iterative methods on such grids will be slowed down. Thus, in real computations we should avoid highly distorted cells since they will penalize the convergence of linear solvers.§.§.§ Example 2 Let us take another physical domain Ω_ □ having the shape of a parallelogram. It can be obtained as an image of the reference domain ^ 0 under the following mappingx = q^ 1 + q^ 2·cosψ ,y = q^ 2·sinψ ,where 0 < ψ ≪ π2 is a small angle. Then, the metric coefficients of this mapping areg_ 1 1 = 1 ,g_ 1 2 = cosψ ,g_ 2 2 = 1 .The Jacobian = sinψ and eigenvalues of the matrixarec_ 1 = tanψ/2 ,c_ 2 = ψ/2 ,and henceforthc_ 2/c_ 1 ≡ ^ 2ψ/2 ≫ 1 .So, we showed that the ratio c_ 2c_ 1 becomes large for domains Ω_ □ featuring a small angle. The problem is that the grid generated by mapping (<ref>) is substantially non-orthogonal, since it consists of parallelograms with an acute angle ψ. On such grids we should expect some reduction of iterative methods convergence speed. Thus, in our computations we should avoid grids featuring small angles.§ FINITE DIFFERENCE SCHEME IN CURVILINEAR COORDINATESIn the numerical simulation of free surface potential flows of an ideal fluid, an elliptic equation to determine the velocity potential ϕ has to be solved at every time step. According to our numerical algorithm (see Section <ref> for more details), we determine first the velocity potential value on the free surface Γ_s, then using this value we reconstruct the velocity potential ϕ in the whole fluid domain (by taking into account other boundary conditions on Γ_b, Γ_r and Γ_ℓ). So, in this Section we assume that we know the values of the velocity potential in nodes which constitute the pre-image of the free surface Γ_s. The goal is to determine the values of the velocity potential in all remaining grid nodes. These values will be determined by solving a system of difference equations constructed below.Moreover, we assume that the curvilinear grid Ω_ h^ n on the nth time layer is already constructed. The nodes _ ȷ^ n are images under the mapping (<ref>) of fixed nodes _ ȷ, which constitute the uniformly distributed grid ^ 0_ h ⊆ ^ 0. Here ȷ(j_ 1, j_ 2) is a multi-index and _ ȷ(q_ j_ 1^ 1, q_ j_ 2^ 2). The uniform grid is traditionally defined asq_ j_ α^ αj_ α· h_ α ,j_ α = 0, …, N_ α ,h_ α1/N_ α , α = 1, 2 .We make an additional geometrical (and not very restrictive) assumption on the mapping (<ref>): boundary components γ_ ℓ, γ_ b, γ_ r and γ_ s are mapped on corresponding boundary components Γ_ℓ, Γ_b, Γ_r and Γ_s of the fluid domain Ω(t) (see Figure <ref> for an illustration). The boundary of the reference domain ^ 0 after the discretization is denoted asγ^ h∂ ^ 0_ h ≡ γ_ ℓ^ h ∪ γ_ b^ h ∪ γ_ r^ h ∪ γ_ s^ h .The sketch of the discretized reference domain ^ 0_ h is depicted in Figure <ref>.In order to construct a finite difference approximation we employ the integro-interpolation method using the terminology of Tikhonov & Samarskii <cit.>. In the western literature this method is closer to the finite volume/conservative finite difference methods <cit.>. The choice for finite differences seems to be natural since we would like to solve an elliptic equation on a simple Cartesian domain <cit.>. For this purpose we replace equation (<ref>) by the following integral relation_∂ϝ^ 1q^ 2 - ϝ^ 2q^ 1 = 0 ,where ⊆ ^ 0 is an arbitrary sub-domain with a piece-wise smooth boundary. We introduced above the notationϝ^ 1_ 1 1·ϕq^ 1 + _ 1 2·ϕq^ 2 , ϝ^ 2_ 2 1·ϕq^ 1 + _ 2 2·ϕq^ 2 .Below we construct discrete approximations based on this integral formulation. Please, notice that the transformed Laplace equation (<ref>) can be compactly rewritten using the new notation:ϝ^ 1q^ 1 + ϝ^ 2q^ 2 = 0 ,= (q^ 1, q^ 2) ∈ ^ 0 . §.§ Approximation in interior nodes Let us consider an interior grid ^ 0_ h node marked with symbol 0 in Figure <ref>(a) along with the integration contour A B C D. For this contour the integral equation (<ref>) takes the form-_ B Cϝ^ 1q^ 2 - -_ A Dϝ^ 1q^ 2 + -_ D Cϝ^ 2q^ 1 - -_ A Bϝ^ 2q^ 1 = 0 .Now, by applying a quadrature rule to all integrals above, one can obtain a discrete analogue of this integral relation. In the present study we employ the quadrature formula of rectangles. For the integral over the segment B C we have-_ B Cϝ^ 1q^ 2 ≈ {_ 1 1 (0) + _ 1 1 (3)/2·ϕ_ 3 - ϕ_ 0/h_ 1 + 1/2[ _ 1 2 (3) ϕ_ 7 - ϕ_ 6/2 h_ 2 + _ 1 2 (0) ϕ_ 4 - ϕ_ 2/2 h_ 2 ]} h_ 2 = 1/2 {(_ 1 1 ϕ_ q^ 1^ ♮)_j_ 1, j_ 2 + (_ 1 1 ϕ_ q^ 1^ ♭)_j_ 1+1, j_ 2 + 1/2 [ (_ 1 2 ϕ_ q^ 2^ ♮)_j_ 1+1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♭)_j_ 1+1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♮)_j_ 1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♭)_j_ 1, j_ 2 ]} h_ 2 ,where we used the grid numbering shown in Figure <ref>(a) and we introduced the following notationϕ_ q^ 1, ȷ^ ♮ ϕ_j_ 1+1, j_ 2 - ϕ_j_ 1, j_ 2/h_ 1 , ϕ_ q^ 1, ȷ^ ♭ϕ_j_ 1, j_ 2 - ϕ_j_ 1-1, j_ 2/h_ 1 ,ϕ_ q^ 2, ȷ^ ♮ ϕ_j_ 1, j_ 2+1 - ϕ_j_ 1, j_ 2/h_ 2 , ϕ_ q^ 2, ȷ^ ♭ϕ_j_ 1, j_ 2 - ϕ_j_ 1, j_ 2-1/h_ 2 .for left- and right-sided finite difference operators. By using similar approximations to discretize other integrals over sides A D, D C and A B we obtain a fully discrete analogue of the integral equation (<ref>):{1/2 [ (_ 1 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2 + (_ 1 1 ϕ_ q^ 1^ ♭)_ j_ 1+1, j_ 2 ] + 1/4 [ (_ 1 2 ϕ_ q^ 2^ ♮)_ j_ 1+1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♭)_ j_ 1+1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♮)_ j_ 1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♭)_ j_ 1, j_ 2 ]} h_ 2- {1/2 [ (_ 1 1 ϕ_ q^ 1^ ♮)_ j_ 1-1, j_ 2 + (_ 1 1 ϕ_ q^ 1^ ♭)_ j_ 1, j_ 2 ] + 1/4 [ (_ 1 2 ϕ_ q^ 2^ ♮)_ j_ 1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♭)_ j_ 1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♮)_ j_ 1-1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♭)_ j_ 1-1, j_ 2 ]} h_ 2+ {1/2 [ (_ 2 2 ϕ_ q^ 2^ ♮)_ j_ 1, j_ 2 + (_ 2 2 ϕ_ q^ 2^ ♭)_ j_ 1, j_ 2+1 ] + 1/4 [ (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2+1 + (_ 2 1 ϕ_ q^ 1^ ♭)_ j_ 1, j_ 2+1 + (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2 + (_ 2 1 ϕ_ q^ 1^ ♭)_ j_ 1, j_ 2 ]} h_ 1- {1/2 [ (_ 2 2 ϕ_ q^ 2^ ♮)_ j_ 1, j_ 2-1 + (_ 2 2 ϕ_ q^ 2^ ♭)_ j_ 1, j_ 2 ] + 1/4 [ (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2 + (_ 2 1 ϕ_ q^ 1^ ♭)_ j_ 1, j_ 2 + (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2-1 + (_ 2 1 ϕ_ q^ 1^ ♭)_ j_ 1, j_ 2-1 ]} h_ 1 = 0 .By dividing the both parts of the last equation by the area h_ 1· h_ 2 of the rectangle A B C D, we obtain more compact difference equations in any interior node _ ȷ ∈ _ h^ 0:1/2 {(_ 1 1 ϕ_ q^ 1^ ♮)_ q^ 1^ ♭ + (_ 1 1 ϕ_ q^ 1^ ♭)_ q^ 1^ ♮}_ ȷ + 1/4 {(_ 1 2 ϕ_ q^ 2^ ♮ + _ 1 2 ϕ_ q^ 2^ ♭)_ q^ 1^ ♮ + (_ 1 2 ϕ_ q^ 2^ ♮ + _ 1 2 ϕ_ q^ 2^ ♭)_ q^ 1^ ♭}_ ȷ +1/2 {(_ 2 2 ϕ_ q^ 2^ ♮)_ q^ 2^ ♭ + (_ 2 2 ϕ_ q^ 2^ ♭)_ q^ 2^ ♮}_ ȷ + 1/4 {(_ 2 1 ϕ_ q^ 1^ ♮ + _ 2 1 ϕ_ q^ 1^ ♭)_ q^ 2^ ♮ + (_ 2 1 ϕ_ q^ 1^ ♮ + _ 2 1 ϕ_ q^ 1^ ♭)_ q^ 2^ ♭}_ ȷ = 0 .These equations approximate the original differential equation (<ref>) to the order Ø(h_ 1^ 2 + h_ 2^ 2) provided that solutions are sufficiently smooth. This is achieved by using the 9-point stencil shown in Figure <ref>(a). §.§ Approximation of boundary conditionsFor the sake of manuscript conciseness we explain the boundary conditions treatment on the example of condition (<ref>) imposed on the fixed[The left wall might be moving in the physical space. However, it becomes fixed in the transformed domain.] boundary γ_ ℓ. Let γ_ ℓ^ h{_ 0, j_ 2 ∈ γ_ h | j_ 2 = 1, 2, …, N_2-1} be the set of `interior' grid nodes belonging to the left boundary γ_ ℓ except two angular nodes _ 0, N_2 and _ 0, 0, which deserve a special consideration. For a boundary node _ ȷ ∈ γ_ ℓ^ h we choose the integration contour as it is shown in Figure <ref>(b). The quadrature rule for the integral over side B C is given in equation (<ref>) (one has just to substitute j_ 1 ↩ 0). On the segment A D we have to use the boundary condition (<ref>) that we can rewrite in a compact form:ϝ^ 1 |_ ∈ γ_ ℓ = y_ q^ 2ṡμ^ 0 .Using this information, we have directly the following approximation:-_ADϝ^ 1q^ 2 ≊ μ_ j_ 2^ 0 h_ 2 .For the side D C the formula of rectangles yield the following approximation of the integral:-_DCϝ^ 2q^ 1 ≊ {_ 2 2 (0) + _ 2 2 (4)/2·ϕ_ 4 - ϕ_ 0/h_ 2 + 1/2 [ _ 2 1 (4)·ϕ_ 7 - ϕ_ 4/h_ 1 + _ 2 1 (0)·ϕ_ 3 - ϕ_ 0/h_ 1 ]} h_ 1/2 ={1/2 [ (_ 2 2 ϕ_ q^ 2^ ♮)_ j_ 1, j_ 2 + (_ 2 2 ϕ_ q^ 2^ ♭)_ j_ 1, j_ 2+1 ] + 1/2 [ (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2+1 + (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2 ]} h_ 1/2 .For the segment A B we give only the final result without intermediate calculations:-_ABϝ^ 2q^ 1 ≊ {1/2 [ (_ 2 2 ϕ_ q^ 2^ ♮)_ j_ 1, j_ 2-1 + (_ 2 2 ϕ_ q^ 2^ ♭)_ j_ 1, j_ 2 ] + 1/2 [ (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2 + (_ 2 1 ϕ_ q^ 1^ ♮)_ j_ 1, j_ 2-1 ]} h_ 1/2 .Now we substitute all these approximations into the integral equation (<ref>) and after dividing its both parts by h_ 2, we obtain the following difference relation in any node γ_ ℓ^ h:(_ 1 1)_ j_ 1+1/2, j_ 2 (ϕ_ q^ 1^ ♮)_ ȷ + 1/4 [ (_ 1 2 ϕ_ q^ 2^ ♮ + _ 1 2 ϕ_ q^ 2^ ♭)_ j_ 1+1, j_ 2 + (_ 1 2 ϕ_ q^ 2^ ♮ + _ 1 2 ϕ_ q^ 2^ ♭)_ ȷ ] +{1/2 [ (_ 2 2 ϕ_ q^ 2^ ♮)_ q^ 2^ ♭ + (_ 2 2 ϕ_ q^ 2^ ♭)_ q^ 2^ ♮ ] + 1/2 [ (_ 2 1 ϕ_ q^ 1^ ♮)_ q^ 2^ ♮ + (_ 2 1 ϕ_ q^ 1^ ♮)_ q^ 2^ ♭ ]}_ ȷ h_ 1/2 = μ_ j_ 2^ 0 .Here we introduced the following notation:( _ 1 1)_ j_ 1+1/2, j_ 2( _ 1 1)_ j_ 1+1, j_ 2 + ( _ 1 1)_ j_ 1, j_ 2/2 .This difference equation is inhomogeneous because of the wall motion. If the (left) wall is fixed, then μ^ 0 ≡ 0 and this relation becomes homogeneous. Please, notice that in all formulas in this Section j_ 1 ≡ 0. §.§.§ Consistency In this Section we show that equation (<ref>) approximates the boundary condition (<ref>) to the order Ø(h_ 1^ 2 + h_ 2^ 2). Indeed, let us substitute a smooth solution ϕ (q^ 1, q^ 2, t) into equation (<ref>) and perform local Taylor expansions. Then, the consistency error ψ can be computed:ψ_ 0, j_ 2 = (_ 1 1 ϕq^ 1)(E) + Ø(h_ 1^ 2) + 1/2 {(_ 1 2 ϕq^ 2)(3) + (_ 1 2 ϕq^ 2)(0) + Ø(h_ 2^ 2)} + {q^ 2(_ 2 2 ϕq^ 2)(0) + Ø(h_ 2^ 2)} h_ 1/2 + {_ 2 1 [ ϕq^ 1 + h_ 1 ^ 2ϕ(q^ 1)^ 2 + Ø(h_ 1^ 2) ] (4) - _ 2 1 [ ϕq^ 1 + h_ 1 ^ 2ϕ(q^ 1)^ 2 + Ø(h_ 1^ 2) ] (2)} h_ 1/4 h_ 2 - μ_ j_ 2^ 0= _ 1 1 ϕq^ 1 + h_ 1/2 q^ 1(_ 1 1 ϕq^ 1) + _ 1 2 ϕq^ 2 + h_ 1/2 q^ 1(_ 1 2 ϕq^ 2) +h_ 1/2 q^ 2(_ 2 2 ϕq^ 2) + h_ 1/2 q^ 2(_ 2 1 ϕq^ 1) - μ_ j_ 2^ 0 = ϝ^ 1 - μ^ 0 + {ϝ^ 1q^ 1 + ϝ^ 2q^ 2} h_ 1/2 + Ø(h_ 1^ 2 + h_ 2^ 2) .In the very last expression all quantities are evaluated in the same node _ 0, j_ 2, j_ 2 = 1, 2, …, N_2-1. Moreover, taking into account the boundary condition (<ref>) and assuming that Laplace equation (<ref>) (see also Remark <ref>) is fulfilled up to the boundary γ_ ℓ, we obtain thatψ_ 0, j_ 2 = Ø(h_ 1^ 2 + h_ 2^ 2) .Thus, the boundary condition is approximated to the second order in space. In a similar way we can construct finite difference approximations in boundary nodes γ_ b^ h{_ j_ 1, 0 ∈ γ^ h | j_ 1 = 1, 2, …, N_1-1} and γ_ r^ h{_ N_1, j_ 2 ∈ γ^ h | j_ 2 = 1, 2, …, N_2-1}. In the derivation of these boundary conditions one has to use the following impermeability conditions (compare with conditions (<ref>) and (<ref>)):ϝ^ 2 |_∈ γ_ b = 0 , ϝ^ 1 |_∈ γ_ r = 0 .All these difference equations have six point stencils to achieve the second order accuracy: three nodes lie in the interior of the domain ^ 0_ h and three on the boundary γ^ h. Finally, there are also two angular nodes _ 0, 0 and _ N_1, 0. The stencils contain four nodes: one interior and three on the boundaries. §.§ Difference operator In order to determine the discrete values of the velocity potential {ϕ_ ȷ}, we constructed above a finite difference problem with the uniform second order accuracy. In this difference problem one has to find the values of the velocity potential ϕ_ ȷ in all nodes except those on the free surface, where the Dirichlet-type condition is imposed:ϕ_ j_ 1, N_2 = μ_ j_ 1^ s ,j_ 1 = 0, 1, …, N_1 .In this way we have to determine only (N_1 + 1)· N_2 the values {ϕ_ ȷ}_ ȷ in the nodes {_ j_ 1, j_ 2} with j_ 1 = 0, 1, …, N_1 and j_ 2 = 0, 1, …, N_2-1. In the resulting problem we have precisely (N_1 + 1)· N_2 equations.In order to study theoretically the finite difference problem, it will be convenient to make a change of variables in order to get homogeneous Dirichlet's boundary conditions in nodes _ j_ 1, N_2, j_ 1 = 0, 1, …, N_1. For this purpose we introduce a new grid function_ j_ 1, j_ 2 μ_ j_ 1^ s , j_ 1 = 0, 1, …, N_1,j_ 2 = N_2 , 0 , j_ 1 = 0, 1, …, N_1,j_ 2 < N_2 ,and a new dependent variableϕ -.Then, in the nodes _ ȷ ∈ γ^ h_ s on the free surface the function _ ȷ will take zero values. The difference problem forwill differ from the one for ϕ by inhomogeneous right hand sides in the nodes close to the free surface pre-image γ^ h_ s. For the new problem we introduce the operator notationΛ=,whereis the vector of the right hand sides. The difference operator Λ can be decomposed in sub-operators for our convenience:Λ ≡ Λ_ 1 + Λ_ 2 ,and on one more level:Λ_ 1 ≡ Λ_ 1 1 + Λ_ 1 2 , Λ_ 2 ≡ Λ_ 2 1 + Λ_ 2 2 .Operators {Λ_ α β}_ 1 ≤ α, β ≤ 2 are defined below as follows.Λ_ 1 1 _ ȷ 1/2[ (_ 1 1 _ q^ 1^ ♮)_ q^ 1^ ♭ + (_ 1 1 _ q^ 1^ ♭)_ q^ 1^ ♮ ]_ ȷ , 0 < j_ 1 < N_1 ,0 ≤ j_ 2 < N_2 , 2/h_ 1 (_ 1 1)_1/2, j_ 2 (_ q^ 1^ ♮)_ 0, j_ 2 , j_ 1 = 0 ,0 ≤ j_ 2 < N_2 , -2/h_ 1 (_ 1 1)_N_1-1/2, j_ 2 (_ q^ 1^ ♭)_ N_1, j_ 2 , j_ 1 = N_1 ,0 ≤ j_ 2 < N_2 . Λ_ 1 2 _ ȷ 1/4 [ (_ 1 2 _ q^ 2^ ♮ + _ 1 2 _ q^ 2^ ♭)_ q^ 1^ ♮ + (_ 1 2 _ q^ 2^ ♮ + _ 1 2 _ q^ 2^ ♭)_ q^ 1^ ♭ ]_ ȷ ,0 < j_ 1 < N_1 ,0 < j_ 2 < N_2 ,1/2 h_ 1 [ (_ 1 2 _ q^ 2^ ♮ + _ 1 2 _ q^ 2^ ♭)_ 1, j_ 2 + (_ 1 2 _ q^ 2^ ♮ + _ 1 2 _ q^ 2^ ♭)_ 0, j_ 2 ] ,j_ 1 = 0 ,0 < j_ 2 < N_2 ,-1/2 h_ 1 [ (_ 1 2 _ q^ 2^ ♮ + _ 1 2 _ q^ 2^ ♭)_ N_1-1, j_ 2 + (_ 1 2 _ q^ 2^ ♮ + _ 1 2 _ q^ 2^ ♭)_ N_1, j_ 2 ] ,j_ 1 = N_1 ,0 < j_ 2 < N_2 ,1/2 [ (_ 1 2 _ q^ 2^ ♮)_ q^ 1^ ♮ + (_ 1 2 _ q^ 2^ ♮)_ q^ 1^ ♭ ]_ j_ 1, 0 ,0 < j_ 1 < N_1 ,j_ 2 = 0 , 1/h_ 1 [ (_ 1 2 _ q^ 2^ ♮)_ 1, 0 + (_ 1 2 _ q^ 2^ ♮)_ 0, 0 ] ,j_ 1 = 0 ,j_ 2 = 0 , -1/h_ 1 [ (_ 1 2 _ q^ 2^ ♮)_ N_1-1, 0 + (_ 1 2 _ q^ 2^ ♮)_ N_1, 0 ] ,j_ 1 = N_1 ,j_ 2 = 0 , Λ_ 2 1 _ ȷ 1/4 [ (_ 2 1 _ q^ 1^ ♮ + _ 2 1 _ q^ 1^ ♭)_ q^ 2^ ♮ + (_ 2 1 _ q^ 1^ ♮ + _ 2 1 _ q^ 1^ ♭)_ q^ 2^ ♭ ]_ ȷ ,0 < j_ 1 < N_1 ,0 < j_ 2 < N_2 ,1/2 [ (_ 2 1 _ q^ 1^ ♮)_ q^ 2^ ♮ + (_ 2 1 _ q^ 1^ ♮)_ q^ 2^ ♭ ]_ 0, j_ 2 ,j_ 1 = 0 ,0 < j_ 2 < N_2 ,1/2 [ (_ 2 1 _ q^ 1^ ♭)_ q^ 2^ ♮ + (_ 2 1 _ q^ 1^ ♭)_ q^ 2^ ♭ ]_ N_1, j_ 2 ,j_ 1 = N_1 ,0 < j_ 2 < N_2 ,1/2 h_ 2 [ (_ 2 1 _ q^ 1^ ♮ + _ 2 1 _ q^ 1^ ♭)_ j_ 1, 0 + (_ 2 1 _ q^ 1^ ♮ + _ 2 1 _ q^ 1^ ♭)_ j_ 1, 1 ] ,0 < j_ 1 < N_1 ,j_ 2 = 0 ,1/h_ 2 [ (_ 2 1 _ q^ 1^ ♮)_ 0, 1 + (_ 2 1 _ q^ 1^ ♮)_ 0, 0 ] ,j_ 1 = 0 ,j_ 2 = 0 ,1/h_ 2 [ (_ 2 1 _ q^ 1^ ♭)_ N_1, 0 + (_ 2 1 _ q^ 1^ ♭)_ N_1, 1 ] ,j_ 1 = N_1 ,j_ 2 = 0 , Λ_ 2 2 _ ȷ 1/2 [ (_ 2 2 _ q^ 2^ ♮)_ q^ 2^ ♭ + (_ 2 2 _ q^ 2^ ♭)_ q^ 2^ ♮ ]_ ȷ , 0 ≤ j_ 1 ≤ N_1 ,0 < j_ 2 < N_2 , 2/h_ 2 (_ 2 2)_j_ 1, 1/2 (ϕ_ q^ 2^ ♮)_ j_ 1, 0 , 0 ≤ j_ 1 ≤ N_1 ,j_ 2 = 0 .Above we used the notation(_ α α)_j_ 1+1/2, j_ 2 (_ α α)_j_ 1, j_ 2 + (_ α α)_j_ 1+1, j_ 2/2 ≡ _ α α |_ E ,(_ β β)_j_ 1, j_ 2+1/2 (_ β β)_j_ 1, j_ 2 + (_ β β)_j_ 1, j_ 2+1/2 ≡ _ β β |_ N , (_ α α)_j_ 1-1/2, j_ 2 (_ α α)_j_ 1, j_ 2 + (_ α α)_j_ 1-1, j_ 2/2 ≡ _ α α |_ W , (_ β β)_j_ 1, j_ 2-1/2 (_ β β)_j_ 1, j_ 2 + (_ β β)_j_ 1, j_ 2-1/2 ≡ _ β β |_ S ,with α, β = 1, 2. Moreover, all the operators vanish in the nodes of the upper boundary γ_ s^ h,Λ_ α β _ ȷ ≡ 0 , ∀_ ȷ ∈ γ_ s^ h , α, β = 1, 2 .Below we study the properties of the operator Λ to show that the problem (<ref>) is well-posed. §.§ Mathematical study of the finite difference problem for the velocity potentialLet us define by $̋ the set of functions defined on the grid^ 0_ hand which take zero values onγ_ s^ h ,{: ^ 0_ h ↦||__ ȷ ∈ γ_ s^ h ≡ 0 } .We introduce also on space$̋ the scalar productuv∑_j_ 1 = 1^N_1 - 1 ∑_j_ 2 = 1^N_2 - 1 u_ j_ 1, j_ 2 v_ j_ 1, j_ 2 h_1 h_2 + ∑_j_ 2 = 1^N_2 - 1 u_ 0, j_ 2 v_ 0, j_ 2 h_1 h_2/2 + ∑_j_ 1 = 1^N_1 - 1 u_ j_ 1, 0 v_ j_ 1, 0 h_1 h_2/2 + ∑_j_ 2 = 1^N_2 - 1 u_ N_1, j_ 2 v_ N_1, j_ 2 h_1 h_2/2 + u_ 0, 0 v_ 0, 0 h_1 h_2/4 + u_ N_1, 0 v_ N_1, 0 h_1 h_2/4 .This scalar product induces a natural normu^ 2uu .Let us define a new operator -Λ and study its properties. The operator Λ was introduced in (<ref>) and rigorously defined earlier.For any fixed value of index j_ 2, a function u ∈ $̋ can be considered as a function being defined on a 1D grid:_ h^ 1{q_ j_ 1^ 1 ∈ [ 0, 1 ] | j_ 1 = 0, 1, …,N_1} .The class of all grid functions defined on_ h^ 1will be denoted by_̋ 1 . For two arbitrary functionsu, v ∈ $̋ we define the scalar product on the layer j_ 2 asuv∑_j_ 1 = 1^N_1 - 1 u_ j_ 1, j_ 2 v_ j_ 1, j_ 2 h_ 1 + u_ 0, j_ 2 v_ 0, j_ 2 h_ 1/2 + u_ N_ 1, j_ 2 v_ N_ 1, j_ 2 h_ 1/2 .Then, it is not difficult to see thatuv ≡ ∑_j_ 2 = 1^N_ 2 - 1uv h_ 2 + ⟨u, v ⟩_ _̋ 1, 0 h_ 2/2 . However, we can proceed in a different way by considering instead `vertical' slices. Let _̋ 2 be the set of 1D functions defined on the grid_ h^ 2{q_ j_ 2^ 2 ∈ [ 0, 1 ] | j_ 2 = 0, 1, …,N_2} .and taking zero value for j_ 2 = N_ 2,_̋ 2{: ^ 2_ h ↦||_j_ 2 = N_ 2 ≡ 0 } .Similarly, in $̋ we can introduce a scalar product on the layerj_ 1as follows:⟨u, v ⟩_ _̋ 2, j_ 1∑_j_ 2 = 1^N_ 2 - 1 u_ j_ 1, j_ 2 v_ j_ 1, j_ 2 h_ 2 + u_ j_ 1, 0 v_ j_ 1, 0 h_ 2/2 ,and the scalar product in$̋ can be similarly expressed asuv ≡ ∑_j_ 1 = 1^N_ 1 - 1⟨u, v ⟩_ _̋ 2, j_ 1 h_ 1 + ⟨u, v ⟩_ _̋ 2, 0 h_ 1/2 + ⟨u, v ⟩_ _̋ 2, N_ 1 h_ 1/2 .It is not difficult to show that formulas (<ref>) and (<ref>) provide the same value for uv and, thus, they are nothing else than two different ways to express the scalar product in . Intuitively, we can understand it in the following way. Imagine that one wants to compute the sum of all elements of a matrix. To do it, one can move along the lines and then along the columns, which corresponds to (<ref>), or vice versa (<ref>). Since the addition is commutative, the resulting sum will be the same.§.§.§ Some discrete identitiesLet us remind that for any discrete functions {_ j}_ j = 0^ N, {u_ j}_ j = 0^ N and {v_ j}_ j = 0^ N defined on a 1D grid _ h^ 1 = { q_ j ∈ [ 0, 1 ] }_ j = 0^ N, the first discrete Green identity can be shown:⟨( u_ q^ ♭)_ q^ ♮, v ⟩ ≡ - ⟨ u_ q^ ♭,v_ q^ ♭ ] + { u_ q^ ♭ v }_ N - _ 1{u_ q^ ♮ v}_ 0 ,and also⟨( u_ q^ ♮)_ q^ ♭, v ⟩ ≡ - [u_ q^ ♮,v_ q^ ♮ ⟩ + _ N-1{ u_ q^ ♭ v }_ N - { u_ q^ ♮ v}_ 0 ,where for the sake of simplicity we introduced the following notations{ }_ j_ j _ j , {}_ j_ j _ j _ j , ⟨, ⟩h ∑_j = 1^N-1_ j _ j , ⟨,] = h ∑_j = 1^N_ j _ j , [ ,⟩ = h ∑_j = 0^N-1_ j _ j .Moreover, the following summation by parts formulas hold as well⟨ u_ q^ ♮, v⟩≡ - ⟨ u, v_ q^ ♭ ] + {u v}_ N - u_ 1 v_ 0 ,⟨ u_ q^ ♭, v⟩≡ - [ u, v_ q^ ♮ ⟩ + u_ N-1 v_ N - {u v}_ 0 . All these properties mentioned above show that the discretization proposed in the present study belongs to the class of operational finite difference schemes <cit.>, which became later known as mimetic methods <cit.>. §.§ The main resultThe operator ≡ -Λ is self-adjoint in the space .For two arbitrary grid functions u, v ∈ $̋ we consider their scalar product on a layer0 ≤ j_ 2 < N_2 :⟨Λ_ 1 1u, v⟩_ _̋ 1, j_ 2 = h_ 1 ∑_j_ 1 = 1^N_1-1{v Λ_ 1 1u}_ j_ 1, j_ 2 + {v Λ_ 1 1u}_ 0, j_ 2 h_ 1/2 + {v Λ_ 1 1u}_ N_ 1, j_ 2 h_ 1/2 .Taking into account definition (<ref>) of the operatorΛ_ 1 1and Green's identity (<ref>) we obtain⟨Λ_ 1 1u, v⟩_ _̋ 1, j_ 2 = -1/2 ∑_j_ 1 = 0^N_1 - 1{_ 1 1 u_ q^ 1^ ♮ v_ q^ 1^ ♮}_ j_ 1, j_ 2 h_ 1 + 1/2 (_ 1 1)_ N_1-1, j_ 2 {u_ q^ 1^ ♭ v}_ N_1, j_ 2- 1/2 {_ 1 1 u_ q^ 1^ ♮ v}_ 0, j_ 2 - 1/2 ∑_j_ 1 = 1^N_1{_ 1 1 u_ q^ 1^ ♭ v_ q^ 1^ ♭}_ j_ 1, j_ 2 h_ 1 + 1/2 {_ 1 1 u_ q^ 1^ ♭ v}_ N_1, j_ 2- 1/2 (_ 1 1)_ 1, j_ 2 {u_ q^ 1^ ♮ v}_ 0, j_ 2 + 2/h_ 1 (_ 1 1)_ 1/2, j_ 2 {u_ q^ 1^ ♮ v}_ 0, j_ 2 h_ 1/2- 2/h_ 1 (_ 1 1)_ N_1 - 1/2, j_ 2 {u_ q^ 1^ ♭ v}_ N_1, j_ 2 h_ 1/2 ,or simply⟨Λ_ 1 1u, v⟩_ _̋ 1, j_ 2 = -1/2 ∑_j_ 1 = 0^N_1 - 1{_ 1 1 u_ q^ 1^ ♮ v_ q^ 1^ ♮}_ j_ 1, j_ 2 h_ 1 -1/2 ∑_j_ 1 = 1^N_1{_ 1 1 u_ q^ 1^ ♭ v_ q^ 1^ ♭}_ j_ 1, j_ 2 h_ 1 .By substituting the last expression into formula (<ref>), we obtain the equalityΛ_ 1 1uv = -1/2 ∑_ j_ 2 = 1^N_2 - 1{∑_j_ 1 = 0^ N_1 - 1_ 1 1 u_ q^ 1^ ♮ v_ q^ 1^ ♮ + ∑_j_ 1 = 1^ N_1_ 1 1 u_ q^ 1^ ♭ v_ q^ 1^ ♭}_ j_ 1, j_ 2 h_ 1 h_ 2- 1/2 ∑_j_ 1 = 0^N_1 - 1{_ 1 1 u_ q^ 1^ ♮ v_ q^ 1^ ♮}_ j_ 1, 0 h_ 1 h_ 2/2 -1/2 ∑_j_ 1 = 1^N_1{_ 1 1 u_ q^ 1^ ♭ v_ q^ 1^ ♭}_ j_ 1, j_ 2 h_ 1 h_ 2/2 .The last identity is absolutely symmetric with respect to the functionsuandv ,Λ_ 1 1uv ≡ uΛ_ 1 1v .Consequently, the operatorΛ_ 1 1is self-adjoint in space$̋,Λ_ 1 1^ ∗ ≡ Λ_ 1 1 . Using the summation by parts formulas (<ref>), (<ref>) along with the definition (<ref>) of the operator Λ_ 1 2 in interior and boundary nodes, we obtain the following expression for the scalar productΛ_ 1 2uv = -1/4 ∑_j_ 2 = 1^N_2 - 1[ ∑_j_ 1 = 1^N_1 _ 1 2 (u_ q^ 2^ ♮ + u_ q^ 2^ ♭) v_ q^ 1^ ♭ + ∑_j_ 1 = 0^N_1 - 1 _ 1 2 (u_ q^ 2^ ♮ + u_ q^ 2^ ♭) v_ q^ 1^ ♮ ]_ j_ 1, j_ 2 h_ 1 h_ 2 - 1/2 ∑_j_ 1 = 1^N_1{_ 1 2 u_ q^ 2^ ♮ v_ q^ 1^ ♭}_ j_ 1, 0 h_ 1 h_ 2/2 - 1/2 ∑_j_ 1 = 0^N_1 - 1{_ 1 2 u_ q^ 2^ ♮ v_ q^ 1^ ♮}_ j_ 1, 0 h_ 1 h_ 2/2 .It is obvious thatΛ_ 1 2uv ≠ uΛ_ 1 2v .Thus, the operator Λ_ 1 2 is not self-adjoint. Nevertheless, we continue the proof without any deception. Consider right now the operator Λ_ 2 1 defined in (<ref>). Using similar techniques we obtain the following expression for the scalar product of Λ_ 2 1 u with v:Λ_ 2 1uv = -1/4 ∑_j_ 1 = 1^N_1 - 1[ ∑_j_ 2 = 1^N_2 _ 2 1 (u_ q^ 1^ ♮ + u_ q^ 1^ ♭) v_ q^ 2^ ♭ + ∑_j_ 2 = 0^N_2 - 1 _ 2 1 (u_ q^ 1^ ♮ + u_ q^ 1^ ♭) v_ q^ 2^ ♮ ]_ j_ 1, j_ 2 h_ 1 h_ 2 - 1/2 ∑_j_ 2 = 1^N_2{_ 2 1 u_ q^ 1^ ♮ v_ q^ 2^ ♭}_ 0, j_ 2 h_ 1 h_ 2/2 - 1/2 ∑_j_ 2 = 0^N_2 - 1{_ 2 1 u_ q^ 1^ ♮ v_ q^ 2^ ♮}_ 0, j_ 2 h_ 1 h_ 2/2- 1/2 ∑_j_ 2 = 1^N_2{_ 2 1 u_ q^ 1^ ♭ v_ q^ 2^ ♭}_ N_1, j_ 2 h_ 1 h_ 2/2 - 1/2 ∑_j_ 2 = 0^N_2 - 1{_ 2 1 u_ q^ 1^ ♭ v_ q^ 2^ ♮}_ N_1, j_ 2 h_ 1 h_ 2/2 .From the last formula it can be clearly seen thatΛ_ 2 1uv ≠ uΛ_ 2 1v .Thus, the operator Λ_ 2 1 as well as Λ_ 1 2 is not self-adjoint. However, we can study the sum Λ_ 2 1 + Λ_ 1 2. By regrouping the terms in (<ref>) and (<ref>) we obtainΛ_ 1 2u + Λ_ 2 1uv =-h_ 1 h_ 2/4 {∑_j_ 1 = 1^N_1 - 1∑_j_ 2 = 1^N_2 - 1[ _ 1 2 (u_ q^ 1^ ♮ + u_ q^ 1^ ♭)·(v_ q^ 2^ ♮ + v_ q^ 2^ ♭) + _ 1 2 (v_ q^ 1^ ♮ + v_ q^ 1^ ♭)·(u_ q^ 2^ ♮ + u_ q^ 2^ ♭) ]_j_ 1, j_ 2+ ∑_j_ 2 = 1^N_2 - 1[ _ 1 2 u_ q^ 1^ ♮ (v_ q^ 2^ ♮ + v_ q^ 2^ ♭) + _ 1 2 v_ q^ 1^ ♮ (u_ q^ 2^ ♮ + u_ q^ 2^ ♭) ]_0, j_ 2+ ∑_j_ 2 = 1^N_2 - 1[ _ 1 2 u_ q^ 1^ ♭ (v_ q^ 2^ ♮ + v_ q^ 2^ ♭) + _ 1 2 v_ q^ 1^ ♭ (u_ q^ 2^ ♮ + u_ q^ 2^ ♭) ]_N_1, j_ 2+ ∑_j_ 1 = 1^N_1 - 1[ _ 1 2 v_ q^ 2^ ♮ (u_ q^ 1^ ♮ + u_ q^ 1^ ♭) + _ 1 2 u_ q^ 2^ ♮ (v_ q^ 1^ ♮ + v_ q^ 1^ ♭) ]_j_ 1, 0+ [ _ 1 2 u_ q^ 1^ ♮ v_ q^ 2^ ♮ + _ 1 2 u_ q^ 2^ ♮ v_ q^ 1^ ♮ ]_ 0, 0 + [ _ 1 2 u_ q^ 1^ ♭ v_ q^ 2^ ♮ + _ 1 2 u_ q^ 2^ ♮ v_ q^ 1^ ♭ ]_ N_1, 0} .Above we used also the fact that _ 1 2 ≡ _ 2 1. From the last formula (<ref>) we readily obtain thatΛ_ 1 2u + Λ_ 2 1uv ≡ uΛ_ 1 2v + Λ_ 2 1v .Thus, the operator Λ_ 2 1 + Λ_ 1 2 is self-adjoint in the space ,(Λ_ 1 2 + Λ_ 2 1)^ ∗ ≡ Λ_ 1 2 + Λ_ 2 1 ,even if constitutive operators Λ_ 1 2 and Λ_ 2 1 are not self-adjoint if taken separately.Finally, consider the operator Λ_ 2 2, which was defined in (<ref>). We employ first the discrete Green identities (<ref>), (<ref>) to evaluate scalar products on layers j_ 1 ∈ {0, 1, …, N_1}:⟨Λ_ 2 2u, v⟩_ _̋ 2, j_ 1 = -1/2 ∑_j_ 2 = 0^N_2 - 1{_ 2 2 u_ q^ 2^ ♮ v_ q^ 2^ ♮}_ j_ 1, j_ 2 h_ 2 - 1/2 ∑_j_ 2 = 1^N_2{_ 2 2 u_ q^ 2^ ♭ v_ q^ 2^ ♭}_ j_ 1, j_ 2 h_ 2 .Using this partial result we can now compute the scalar product in the $̋ space as well:Λ_ 2 2 uv = -1/2 ∑_j_ 1 = 1^N_1 - 1[ ∑_j_ 2 = 0^N_2 - 1 _ 2 2 u_ q^ 2^ ♮ v_ q^ 2^ ♮ + ∑_j_ 2 = 1^N_2 _ 2 2 u_ q^ 2^ ♭ v_ q^ 2^ ♭ ]_ j_ 1, j_ 2 h_ 1 h_ 2- 1/2 ∑_j_ 2 = 0^N_2 - 1{_ 2 2 u_ q^ 2^ ♮ v_ q^ 2^ ♮}_ 0, j_ 2 h_ 1 h_ 2/2 - 1/2 ∑_j_ 2 = 1^N_2{_ 2 2 u_ q^ 2^ ♭ v_ q^ 2^ ♭}_ 0, j_ 2 h_ 1 h_ 2/2- 1/2 ∑_j_ 2 = 0^N_2 - 1{_ 2 2 u_ q^ 2^ ♮ v_ q^ 2^ ♮}_ N_1, j_ 2 h_ 1 h_ 2/2 - 1/2 ∑_j_ 2 = 1^N_2{_ 2 2 u_ q^ 2^ ♭ v_ q^ 2^ ♭}_ N_1, j_ 2 h_ 1 h_ 2/2 .By changing symmetrically grid functionsuandv , we obtainΛ_ 2 2uv ≡ uΛ_ 2 2v .Consequently, the operatorΛ_ 2 2is self-adjoint in the space$̋,Λ_ 2 2^ ∗ ≡ Λ_ 2 2 .Since Λ ≡ Λ_ 1 1 + (Λ_ 1 2 + Λ_ 2 1) + Λ_ 2 2 and each term is an self-adjoint operator, the operator Λ is self-adjoint as well. Hence, the same holds for ≡ -Λ. Below we are going to show also that the operatoris positive definite. The proof will be based on the uniform ellipticity property of equation (<ref>), which was established in Lemma <ref>. Another key ingredient consists in the properties of the finite difference approximation to the following mixed Boundary Value Problem (BVP) for the Laplace equation posed on _ 0:Δ ϕ = 0 ,= (q^ 1, q^ 2) ∈ _ 0 ,with boundary conditionsϕq^ 1|_q^ 1 = 1= 0 , ϕq^ 1|_q^ 1 = 0 = μ_ ℓ (q^ 2) ,ϕ|_q^ 2 = 1= μ̃ (q^ 1) , ϕq^ 2|_q^ 2 = 0 = 0 .This problem is a particular case of the BVP considered earlier for elliptic equation (<ref>) with the following coefficients:_ 1 1 = _ 2 2 ≡ 1 , _ 1 2 = _ 2 1 ≡ 0 .In this case the finite difference operator Λ admits a much simpler form since finite difference operators corresponding to mixed derivatives vanish,Λ_ 1 2 = Λ_ 2 1 ≡ 0 .The finite difference scheme for the BVP (<ref>) – (<ref>) can be written asΔ_ h ϕ =, Δ_ hΔ_ h^ (1) + Δ_ h^ (2) ,with operators Δ_ h^ (1, 2) defined asΔ_ h^ (1) ϕ_ ȷ {ϕ_ q^ 1 q^ 1^ ♭, ♮}_ j_ 1, j_ 2 , 0 < j_ 1 < N_1 - 1 ,0 ≤ j_ 2 < N_2 , 2/h_ 1 {ϕ_ q^ 1^ ♮}_ 0, j_ 2 , j_ 1 = 0 ,0 ≤ j_ 2 < N_2 , - 2/h_ 1 {ϕ_ q^ 1^ ♭}_ N_1, j_ 2 , j_ 1 = N_1 ,0 ≤ j_ 2 < N_2 , 0 , 0 ≤ j_ 1 ≤ N_1 ,j_ 2 = N_2 , Δ_ h^ (2) ϕ_ ȷ {ϕ_ q^ 2 q^ 2^ ♭, ♮}_ j_ 1, j_ 2 , 0 ≤ j_ 1 ≤ N_1 ,0 < j_ 2 < N_2 , 2/h_ 2 {ϕ_ q^ 2^ ♮}_ j_ 1, 0 , 0 ≤ j_ 1 ≤ N_1 ,j_ 2 = 0 , 0 , 0 ≤ j_ 1 ≤ N_1 ,j_ 2 = N_2 .Below we introduce also the operators _ 1-Δ_ h^ (1), _ 2-Δ_ h^ (2) and _ 1 + _ 2, - Δ_ h. By applying Theorem <ref> we conclude that the operator : ↦ $̋ is self-adjoint in the space.It is not difficult to see that the family of grid functionsψ^ (k, l)_ j_ 1, j_ 2α_ k·cos(π k q_ j_ 1^ 1)·cos(π (l + ) q_ j_ 2^ 2) ,k = 0, 1, …, N_1 ,l = 0, 1, … N_2 - 1satisfies the following identities:·ψ^ (k, l) ≡ λ^ (k, l)·ψ^ (k, l) .Henceforth, the functions{ψ^ (k, l)}_ k, lare eigenfunctions[Please, notice that each eigenfunction ψ^ (k, l) is a grid function taking in each node ȷ = (j_ 1, j_ 2) the value ψ^ (k, l)_ j_ 1, j_ 2 given in equation (<ref>).] of the operatorand numbers{λ^ (k, l)}_ k, l ∈ are its eigenvalues:λ^ (k, l)4/h_ 1^ 2 sin^ 2(π k h_ 1/2) + 4/h_ 2^ 2 sin^ 2[ (l + 1/2) π h_ 2/2 ] .If we choose the normalization factorα_ kasα_ k =√(2) , k = 0 ∧ N_1 , 2 , k = 1, 2, …, N_1 - 1 ,then the eigenfunctions{ψ^ (k, l)}_ k, lof the operatorform an orthonormal basis in,ψ^ (k, l)ψ^ (r, s) = δ^ k r·δ^ l s =1 , k = r ∧ l = s , 0 , k ≠ r ∨ l ≠ s .In other words, an arbitrary functionu ∈ $̋ can be represented as a sum of a finite Fourier series:u = ∑_k = 0^N_1 ∑_l = 0^N_2 - 1 c_ k l ψ^ (k, l) ,where the numbers {c_ k l ≡ uψ^ (k, l)}_ k, l are called the Fourier coefficients. The Parseval theorem holds in the finite dimensional setting as well:u^ 2_≡ uu = ∑_k = 0^N_1 ∑_l = 0^N_2 - 1 c_ k l^ 2 .Since all eigenvalues of the operatorare positive, the following Lemma holds:The finite difference operatoris positive definite in space $̋ and for any functionu ∈ $̋ we have the following estimationsλ_ min u^ 2_≤uu ≤ λ_ max u^ 2_,withλ_ min4/h_ 2^ 2 sin^ 2(π h_ 2/4) , λ_ max4/h_ 1^ 2 + 4/h_ 2^ 2 cos^ 2(π h_ 2/4) . Let us take an arbitrary function u ∈ $̋ and we expand it (<ref>) on the basis of orthonormal eigenfunctions of the operator . Then we haveuu = ⟨ ∑_k = 0^N_1 ∑_l = 0^N_2 - 1 c_ k l λ^ (k, l) ψ^ (k, l) , ∑_r = 0^N_1 ∑_s = 0^N_2 - 1 c_ r s ψ^ (r, s) ⟩_,where we used the fact thatψ^ (k, l)is an eigenfunction of(<ref>). Then, by taking into account the orthonormality property we can greatly simplify the scalar product:uu = ∑_k = 0^N_1 ∑_l = 0^N_2 - 1 c_ k l^ 2 λ^ (k, l) .Obviously, among a finite number of eigenvalues{λ^ (k, l)}_ k, lwe can always choose the minimal and maximal ones. Thus,∀(k, l) : λ_ min ≤ λ^ (k, l) ≤ λ_ max .Taking into account the last estimation along with the Parseval identity (<ref>), we obtain immediately the estimations (<ref>). Thanks to the property of self-adjointness and positive definiteness of the operator , we can define another scalar productuv uv , which implies the corresponding energy norm:u_ √(uu) ,generated by the operator . Below we use an equivalent representation of the energy norm, which is more suitable for our purposes:u_ ^ 2 ≡ _ 1 uu + _ 2 uu .For any function u ∈ $̋ the following identities hold:_ 1 uu = ∑_j_ 1 = 1^N_1 [ ∑_j_ 2 = 1^N_2 - 1{(u_ q^ 1^ ♭)^ 2}_ j_ 1, j_ 2 + 1/2 {(u_ q^ 1^ ♭)^ 2}_ j_ 1, 0 ] h_ 1 h_ 2 ,_ 2 uu = ∑_j_ 2 = 1^N_2 [ ∑_j_ 1 = 1^N_1 - 1{(u_ q^ 2^ ♭)^ 2}_ j_ 1, j_ 2 + 1/2 {(u_ q^ 2^ ♭)^ 2}_ 0, j_ 2 + 1/2 {(u_ q^ 2^ ♭)^ 2}_ N_1, j_ 2 ] h_ 1 h_ 2 .Taking into account equation (<ref>) and discrete identities shown in Section <ref>, we obtain the equality⟨_ 1 u, u⟩_ _̋ 1, j_ 2 = ∑_j_ 1 = 1^N_1 {(u_ q^ 1^ ♭)^ 2}_ j_ 1, j_ 2 h_ 1 .Then, from the last result and relation (<ref>) follows immediately the requested identity (<ref>). Similarly, the discrete identities along with definition (<ref>) yield⟨_ 2 u, u⟩_ _̋ 2, j_ 1 = ∑_j_ 2 = 1^N_2 {(u_ q^ 2^ ♭)^ 2}_ j_ 1, j_ 2 h_ 2 .Then, using relation (<ref>) we obtain the second requested identity (<ref>). Now we can come back to the original finite difference operatorand prove the followingThe operator ≡ - Λ is positive definite in the space , and we have the following estimations:c_ 1 λ_ min u^ 2_≤uu ≤ c_ 2 λ_ max u^ 2_,where the constants c_ 1, 2 were defined in (<ref>), (<ref>) and λ_ min, max— in (<ref>).As in the first step of the proof we take v ≡ u ∈ $̋ and substitute it into equations (<ref>). After regrouping interior and boundary nodes, we obtain:Λ_ 1 1 uu = - h_ 1 h_ 2/4 { ∑_j_ 1 = 1^N_1 - 1∑_j_ 2 = 1^N_2 - 1[ 2 _ 1 1 (u_ q^ 1^ ♮)^ 2 + 2 _ 1 1 (u_ q^ 1^ ♭)^ 2 ]_j_ 1, j_ 2 + ∑_j_ 2 = 1^N_2 - 1[ 2 _ 1 1 (u_ q^ 1^ ♮)^ 2 ]_0, j_ 2 + ∑_j_ 2 = 1^N_2 - 1[ 2 _ 1 1 (u_ q^ 1^ ♭)^ 2 ]_N_1, j_ 2 + ∑_j_ 1 = 1^N_1 - 1[ _ 1 1 (u_ q^ 1^ ♮)^ 2 + _ 1 1 (u_ q^ 1^ ♭)^ 2 ]_ j_ 1, 0 + [ _ 1 1 (u_ q^ 1^ ♮)^ 2 ]_ 0, 0 + [ _ 1 1 (u_ q^ 1^ ♭)^ 2 ]_ N_1, 0 } .We perform the same operation with equation (<ref>) as well:Λ_ 2 2 uu = - h_ 1 h_ 2/4 { ∑_j_ 1 = 1^N_1 - 1∑_j_ 2 = 1^N_2 - 1[ 2 _ 2 2 (u_ q^ 2^ ♮)^ 2 + 2 _ 2 2 (u_ q^ 2^ ♭)^ 2 ]_j_ 1, j_ 2 + ∑_j_ 2 = 1^N_2 - 1[ _ 2 2 (u_ q^ 2^ ♮)^ 2 + _ 2 2 (u_ q^ 2^ ♭)^ 2 ]_ 0, j_ 2 + ∑_j_ 2 = 1^N_2 - 1[ _ 2 2 (u_ q^ 2^ ♮)^ 2 + _ 2 2 (u_ q^ 2^ ♭)^ 2 ]_ N_1, j_ 2 + ∑_j_ 1 = 1^N_1 - 1[ 2 _ 2 2 (u_ q^ 2^ ♮)^ 2 ]_ j_ 1, 0 + [ _ 2 2 (u_ q^ 2^ ♮)^ 2 ]_ 0, 0 + [ _ 2 2 (u_ q^ 2^ ♮)^ 2 ]_ N_1, 0 + ∑_j_ 1 = 1^N_1 - 1[ 2 _ 2 2 (u_ q^ 2^ ♭)^ 2 ]_ j_ 1, N_2 + [ _ 2 2 (u_ q^ 2^ ♭)^ 2 ]_ 0, N_2 + [ _ 2 2 (u_ q^ 2^ ♭)^ 2 ]_ N_1, N_2 .Similarly, we setv ≡ uin equation (<ref>) and using the last two formulas we haveuu = h_ 1 h_ 2/4 { ∑_j_ 1 = 1^N_1 - 1∑_j_ 2 = 1^N_2 - 1[(u_ q^ 1^ ♮, u_ q^ 2^ ♮) +(u_ q^ 1^ ♮, u_ q^ 2^ ♭) +(u_ q^ 1^ ♭, u_ q^ 2^ ♮) +(u_ q^ 1^ ♭, u_ q^ 2^ ♭) ]_ j_ 1, j_ 2+ ∑_j_ 2 = 1^N_2 - 1[(u_ q^ 1^ ♮, u_ q^ 2^ ♮) +(u_ q^ 1^ ♮, u_ q^ 2^ ♭) ]_ 0, j_ 2 + ∑_j_ 2 = 1^N_2 - 1[(u_ q^ 1^ ♭, u_ q^ 2^ ♮) +(u_ q^ 1^ ♭, u_ q^ 2^ ♭) ]_ N_1, j_ 2+ ∑_j_ 1 = 1^N_1 - 1[(u_ q^ 1^ ♮, u_ q^ 2^ ♮) +(u_ q^ 1^ ♭, u_ q^ 2^ ♮) ]_ j_ 1, 0 + [(u_ q^ 1^ ♮, u_ q^ 2^ ♮) ]_ 0, 0 + [(u_ q^ 1^ ♭, u_ q^ 2^ ♮) ]_ N_1, 0+ ∑_j_ 1 = 1^N_1 - 1[ 2(0, u_ q^ 2^ ♭) ]_ j_ 1, N_2 + [(0, u_ q^ 2^ ♭) ]_ 0, N_2 + [(0, u_ q^ 2^ ♭) ]_ N_1, N_2 } .where we used the quadratic form(·, ·)defined earlier in (<ref>). By taking into account the uniform ellipticity property (<ref>), we can estimate the quantity uufrom below:uu ≥ c_ 1 h_ 1 h_ 2/4·{ ∑_j_ 1 = 1^N_1 - 1∑_j_ 2 = 1^N_2 - 1 2 [ (u_ q^ 1^ ♮)^ 2 + (u_ q^ 1^ ♭)^ 2 + (u_ q^ 2^ ♮)^ 2 + (u_ q^ 2^ ♭)^ 2 ]_ j_ 1, j_ 2 + ∑_j_ 2 = 1^N_2 - 1[ 2 (u_ q^ 1^ ♮)^ 2 + (u_ q^ 2^ ♮)^ 2 + (u_ q^ 2^ ♭)^ 2 ]_ 0, j_ 2 + ∑_j_ 2 = 1^N_2 - 1[ 2 (u_ q^ 1^ ♭)^ 2 + (u_ q^ 2^ ♮)^ 2 + (u_ q^ 2^ ♭)^ 2 ]_ N_1, j_ 2 +∑_j_ 1 = 1^N_1 - 1[ (u_ q^ 1^ ♮)^ 2 + (u_ q^ 1^ ♭)^ 2 + 2 (u_ q^ 2^ ♮)^ 2 ]_ j_ 1, 0 + [ (u_ q^ 1^ ♮)^ 2 + (u_ q^ 2^ ♮)^ 2 ]_ 0, 0 + [ (u_ q^ 1^ ♭)^ 2 + (u_ q^ 2^ ♮)^ 2 ]_ N_1, 0+ 2 ∑_j_ 1 = 1^N_1 - 1(u_ q^ 2^ ♭)^ 2_ j_ 1, N_2 + (u_ q^ 2^ ♭)^ 2_ 0, N_2 + (u_ q^ 2^ ♭)^ 2_ N_1, N_2 } _ low .Then, in the expression above for_ lowwe replace[We can do it since the right finite difference can be considered as the left one for the subsequent value of the index. Thus, this operation can be seen as a change of index by one in all summations.] all right derivativesu_ q^ α^ ♮by their left counterpartsu_ q^ α^ ♭ :_ low = c_ 1 { ∑_j_ 1 = 1^N_1 [ ∑_j_ 2 = 1^N_2 - 1 (u_ q^ 1^ ♭)^ 2_ j_ 1, j_ 2 + 1/2 (u_ q^ 1^ ♭)^ 2_ j_ 1, 0 ] + ∑_j_ 2 = 1^N_2 [ ∑_j_ 1 = 1^N_1 - 1 (u_ q^ 2^ ♭)^ 2_ j_ 1, j_ 2 + 1/2 (u_ q^ 2^ ♭)^ 2_ 0, j_ 2 + 1/2 (u_ q^ 2^ ♭)^ 2_ N_1, j_ 2 ] } h_ 1 h_ 2 .Now we can make use of Lemma <ref>, where we showed identities (<ref>) and (<ref>), which yield:_ low ≡ c_ 1 {_ 1 uu + _ 2 uu} .In other words, we just showed the following estimation from below:uu ≥ c_ 1uu .Finally, by applying Lemma <ref> we obtain thatuu ≥ λ_ min u^ 2_.The two last inequalities correspond precisely to the required lower bound in (<ref>). By departing again from equation (<ref>) and using the same techniques we can show the upper bound in (<ref>). This completes the proof of this Theorem.During the proof of Theorem <ref> we established also the following estimates:c_ 1uu ≤uu ≤ c_ 2uu , ∀ u ∈ ,or equivalentlyc_ 1≤≤ c_ 2.Such operatorsandsatisfying the inequalities above are usually referred to as energetically equivalent<cit.>. §.§.§ Final remarks From the positive definiteness property of the operatorfollows the unique solvability property of the finite difference problem (<ref>). To solve this problem numerically one can use almost any iterative technique. It is natural to use the conjugate gradient method whose convergence is guaranteed if the operator is positive definite and self-adjoint <cit.>. The speed of convergence depends essentially on the conditioning numberκof the difference operator , which can be estimated asκ ≤ c_ 2/c_ 1·λ_ max/λ_ min = c_ 2/c_ 1·4h_ 1^ 2 + 4h_ 2^ 2 cos^ 2(π h_ 24)/4h_ 2^ 2 sin^ 2(π h_ 24)_(⋆) .For larger values ofκthe convergence will be slower. As we showed earlier in Section <ref>, the first factorc_ 2c_ 1is completely determined by the properties of the geometric mapping (<ref>). This ratio becomes large for the meshes with highly elongated or highly distorted elements. The second factor(⋆)depends on the number of nodesN_1, 2chosen in the directionsO q^1, 2correspondingly and(⋆)becomes larger when we refine the grid ( whenN_1, 2increase).§ NUMERICAL ALGORITHM In this Section we describe briefly how the numerical code is organized and we provide the details on the treatment of kinematic and dynamic boundary conditions on the free surface and on the lateral fixed and moving walls.Assume that the position of the wallx = s^ non the time layert = t^ nis known along with all other fields. The gridΩ_ h^ nwith nodes_ ȷ^ n , ordered using a multi-indexȷ = (j_ 1, j_ 2)in the physical space is constructed. On this grid we know the values of grid functions{ξ_ j_ 1^ n}_ j_ 1 ,{η_ j_ 1^ n}_ j_ 1and{ϕ_ ȷ^ n}_ ȷ . The time marching requires to compute these quantities on the following time layert = t^ n+1 . It is done in several stages: *First, we compute the values of the velocity potential {ϕ_ j_ 1, N_2^ n+1}_ j_ 1 at the free surface node _ j_ 1, N_2. *These values are used as the Dirichlet data in the linear problem (<ref>) to determine the velocity potential {ϕ_ j_ 1, j_ 2^ n+1}_ j_ 1, j_ 2 in all other nodes _ ȷ with 0 ≤ j_ 1 ≤ N_1 and 0 ≤ j_ 2 < N_2. *Then, we find new positions {ξ_ j_ 1^ n+1}_ j_ 1, {η_ j_ 1^ n+1}_ j_ 1 of the free surface by using the finite difference approximation of the kinematic boundary conditions (<ref>) and (<ref>). *The new position x = s^ n+1 of the moving wall is found from the discretized version of the Equation (<ref>). *Finally, the new grid Ω̅_ h^ n+1 is constructed on the following time layer t = t^ n+1.Every step above involves grid functions on various layers and to achieve a better understanding of all the stages of our algorithm, below we provide a detailed description of every item. §.§ Approximation of the dynamic boundary condition (1) The finite difference approximation to the dynamic boundary condition (<ref>) takes the following form:ϕ_ ȷ^ n+1 - ϕ_ ȷ^ n/τ_ n - (u_ ȷ^ n· x_ t, ȷ + v_ ȷ^ n· y_ t, ȷ) +_̆ ȷ^ n^ 2 + η_ j_ 1^ n = 0 ,whereȷ = (j_ 1, j_ 2)is a multi-index withj_ 1 = 1, 2, …, N_1 - 1andj_ 2 = N_2 . In order to compute the speeds of grid nodes we employ finite differences applied to two last gridsΩ̅_ h^ nandΩ̅_ h^ n-1 :_ t, ȷ = _ ȷ^ n - _ ȷ^ n-1/τ_ n-1 ,whereτ_ n-1 ≡ t^ n - t^ n-1is the local time step,_̆ ȷ^ n^ 2 ≡ (u_ ȷ^ n)^ 2 + (v_ ȷ^ n)^ 2 . In formula (<ref>) we use only the first order approximation in time. This is done for the sake of algorithm memory efficiency. Otherwise, to achieve at least the second order accuracy inτ , we would have to keep in memory the grids for three time layers. In the current implementation we keep track of the preceding grid only.The Cartesian componentsu_ ȷ^ nandv_ ȷ^ nof the velocity vector_̆ ȷ^ nare computed after approximating formulas (<ref>) and (<ref>):u_ ȷ^ n = [ ϕ_q^ 1· y_q^ 2 - ϕ_q^ 2· y_q^ 1/ ]_ ȷ^ n ,v_ ȷ^ n = [ -ϕ_q^ 1· x_q^ 2 + ϕ_q^ 2· x_q^ 1/ ]_ ȷ^ n .Above one has to approximate also the partial derivativesϕ_ q^ 1 ,_ q^ 1over the independent variableq^ 1along the upper side of the square^ 0. These derivatives are computed with standard central finite differences up to the second order accuracy. The derivativesϕ_ q^ 2 ,_ q^ 2are computed with one-sided finite differences of the second order as well (in order to have the uniform second order accuracy in space).We use equation (<ref>) in order to compute the updated values of the velocity potential only in interior nodes of the gridγ_ s^ h . For two nodes located in the upper corners_ 0, N_2 = (0, 1)and_ N_1, N_2 = (1, 1)we use modified approximation formulas which take into account the lateral wallsγ_ ℓ, rimpermeability. In the computational domain the left wall has the fixed coordinateq^ 1 = 0 . The intersection point of the free surface with the left boundary is a triple point and it has the coordinate_ 0, N_2 . This point is permanently moving up and down the left wall.In the coordinate systemO q^ 1 q^ 2the Cartesian components= tbecome the so-called contravariant components of the velocity= twithv^ α = q^ αt = q^ αt + u q^ αx + v q^ αy , α = 1, 2 .Consequently, if a fluid particle moves such that its coordinateq^ αdoes not change, then the corresponding contravariant velocity componentv^ αhas to vanish. For instance, the fluid particles, which constitute the free surface, have the coordinateq^ 2 = 1 , consequentlyv^ 2|_ q^ 2 = 1 = 0 .Similarly, for fluid particles sliding along the left vertical wall we always haveq^ 1 = 0 , which yieldsv^ 1|_ q^ 1 = 0 = 0 .The point_ 0, N_2 ∈ γ_ s^ h ∩ γ_ ℓ^ h , thusv^ 1 (_ 0, N_2) = v^ 2 (_ 0, N_2) ≡ 0 .From formula (<ref>) and using also equations (<ref>), (<ref>) we obtain the following expressions for contravariant components of the velocity:v^ 1 = (u - x_ t)· y_ q^ 2 - (v - y_ t)· x_ q^ 2/ ,v^ 2 = -(u - x_ t)· y_ q^ 1 + (v - y_ t)· x_ q^ 1/ .From these expressions and using boundary condition[We implicitly use also the fact that the transformation (<ref>) is non-degenerate.](<ref>) for the triple point_ 0, N_2we obtain thatx_ t|_ _ 0, N_2 = u ,y_ t|_ _ 0, N_2 = v .Henceforth,^ 2 ≡ u^ 2 + v^ 2 = u x_ t + v y_ tand the free surface dynamic boundary condition (<ref>) in the corner_ 0, N_2becomes:ϕ_ t -(u· x_ t + v· y_ t) + η = 0 ,= _ 0, N_2 .The difference equation (<ref>) correspondingly takes the form:ϕ_ ȷ^ n+1 - ϕ_ ȷ^ n/τ_n -(u_ ȷ^ n· x_ t, ȷ + v_ ȷ^ n· y_ t, ȷ) + η_ j_ 1^ n = 0 , ȷ = (j_ 1, j_ 2) ≡ (0, N_2) .In a similar way one can derive the finite difference discretization of the boundary condition in the right triple point. We give only the final result. It coincides with equation (<ref>) provided that we make a substitutionj_ 1 = N_1 . §.§ Computation of the velocity potential (2) On the second stage of the numerical algorithm we solve the finite difference problem (<ref>) in order to find the values of the velocity potential{ϕ_ ȷ^ n+1}_ ȷin the nodes_ ȷ ∈ _ h^ 0 ∖ γ_ s^ h. In other words, in the nodesȷ = (j_ 1, j_ 2)with0 ≤ j_ 1 ≤ N_1and0 ≤ j_ 2 < N_2 .In order to invert the linear system (<ref>) we tested several iterative methods and no method clearly outperformed the others. In the final implementation of the code we used the Successive Over Relaxation (SOR) method <cit.> that we remind briefly here. The choice of the SOR method can be explained by two main reasons:*We would like to have a sufficiently simple to implement and sufficiently efficient method to find approximate solutions to relatively large linear systems*The method does not have to rely heavily on the matrix structure (such as Thomas's algorithm, for example). For instance, if we include obstacles in the fluid domain, its topology might change, which implies some drastic changes in the pattern of non-zero elements of the matrix.Thus, to our opinion SOR method represents the best trade-off among the efficiency, generality and ease of implementation.We use a nine-point stencil in the finite difference scheme. Thus, equationȷhas the following general form:∑_k = 0^8_ k, ȷ ϕ_ k, ȷ = _ ȷ , _ ȷ ∈ _ h^ 0 ∖ γ_ s^ h .The last condensed form is obtained from (<ref>) by regrouping all the terms in front of unknowns{ϕ_ ȷ^ n+1}_ ȷ . The corresponding coefficient is denoted by_ k, ȷ . In equation (<ref>) we used a local numeration of indices depicted in Figure <ref>(a). The right hand side is given by_ ȷ μ_ 0, j_ 2 h_ 2 ,_ ȷ ∈ γ_ ℓ^ h ,μ_ 0, 0 h_ 2 ,_ ȷ ≡ _ 0, 0 , 0 ,otherwise .Then, one step of the SOR method reads:ϕ̃_ 0, ȷ- 1/_ 0, ȷ { ∑_k ∈ {1, 2, 5, 6}_ k, ȷ ϕ_ k, ȷ^ (m+1) + ∑_k ∈ {3, 4, 7, 8}_ k, ȷ ϕ_ k, ȷ^ (m) - _ 0 } ,ϕ_ 0, ȷ^ (m+1)θ ϕ̃_ 0, ȷ + (1 - θ) ϕ_ 0, ȷ^ (m) ,where(m)is the iteration number andθis the relaxation parameter. The value of this parameter weakly influences the iterative process speed of convergence. The question of finding the optimal value of the relaxation parameterθis the key to the efficiency of the SOR method. This question can be studied theoretically for the Dirichlet problem of the Poisson equation in a square domain[ 0, ℓ ]^ 2with uniform grids. We know also that the optimal valueθ^ ⋆always belongs to the interval:θ^ ⋆ ∈ (1, 2) .For the Poisson equation on a square domain with the uniform discretizationh_ 1 = h_ 2 ≡ h , one can showθ^ ⋆ = 2/1 + sin(π hℓ) .In the last formula forℓ = 1andh = 1/30we obtain the valueθ^ ⋆ ≈ 1.81 . In our discrete problem for the velocity potentialϕ_ ȷwe cannot determine theoretically the optimal valueθ^ ⋆ . This value was determined experimentally for each problem under consideration. In numerical computations below we always tookθ^ ⋆ ∈ [ 1.85, 1.95 ] , depending on the discretization parametersh_ 1, 2as well.In boundary nodes the stencil contains six points and in corner points only four. However, it does not change anything for the SOR scheme since extra coefficients_ k, ȷcan be set to zero for our convenience. The expressions of coefficients_ k, ȷare given in Table <ref>. In this Table we use the following notation:_ 1h_ 2/h_ 1 _ 1 1 |_ W , _ 2h_ 1/h_ 2 _ 2 2 |_ S , _ 3h_ 2/h_ 1 _ 1 1 |_ E , _ 4h_ 1/h_ 2 _ 2 2 |_ N ,_ 5_ 1 2 |_ 1 + _ 1 2 |_ 2/4 , _ 6- _ 1 2 |_ 2 + _ 1 2 |_ 3/4 ,_ 7_ 1 2 |_ 3 + _ 1 2 |_ 4/4 , _ 8- _ 1 2 |_ 4 + _ 1 2 |_ 1/4 ,σ_ α_ 1 2 |_ 0 - _ 1 2 |_ α/4 , α = 1, 2 , σ_ β_ 1 2 |_ β - _ 1 2 |_ 0/4 , β = 3, 4 .In the definition of quantities_ kwe used the notation introduced in equations (<ref>)–(<ref>). Finally, the coefficient_ 0, ȷis determined as_ 0, ȷ- ∑_k = 1^4 _ k, ȷ .This concludes the description of our velocity potential solver. §.§ Free surface motion (3) The free surface position on the following time layert = t^ n+1is found by integrating in time kinematic conditions (<ref>), (<ref>). Here we consider the simplest approximation of these equations using an explicit upwind scheme:_ j_ 1^ n+1 - _ j_ 1^ n/τ_ n + v_ ȷ^ 1, n + v_ ȷ^ 1, n/2 _ q^ 1, j_ 1^ ♭, n + v_ ȷ^ 1, n - v_ ȷ^ 1, n/2 _ q^ 1, j_ 1^ ♮, n - _̆ ȷ^ n = 0 , ȷ = (j_ 1, N_2) ,where we introduced the vector(η, ξ)and indexj_ 1 = 1, 2, …, N_1 - 1 . The contravariant velocity componentv_ ȷ^ 1is approximated asv_ ȷ^ 1, n = (u_ ȷ^ n - x_ t, ȷ^ n)· y_ q^ 2, ȷ^ ♭, n - (v_ ȷ^ n - y_ t, ȷ^ n)· x_ q^ 2, ȷ^ ♭, n/_ ȷ^ n .The components of the (Cartesian) velocity vector_̆ ȷ^ nare computed using formulas (<ref>) except the fact that we use the values of the velocity potential{ϕ_ ȷ^ n+1}_ ȷon the subsequent time layert = t^ n+1 .We employ formula (<ref>) in order to compute the free surface position in `interior' nodes_ j_ 1, N_1withj_ 1 = 1, 2, …, N_1 - 1 , which lie on the upper side of the square_ h^ 0 . Thanks to impermeability conditions (<ref>), kinematic conditions (<ref>), (<ref>) take a very simple form:_ t - = 0 ,∈ {_ 0, N_2, _ N_1, N_2} .Taking into account this information, the finite difference formula (<ref>) takes a much simpler form as well:_ j_ 1^ n+1 - _ j_ 1^ n/τ_ n = _̆ j_ 1, N_2^ n ,j_ 1 ∈ {0, N_1} .The fully discrete scheme to compute the free surface transport is described. §.§ Moving wall displacement (4) The movable wall position is given by the functionx = s (t) , which is a solution to equation (<ref>). The fluid pressurepis determined from the Cauchy–Lagrange integral given in dimensionless variables in equation (<ref>). The force acting on the vertical wall is given in dimensionless variables in equation (<ref>). In curvilinear coordinates the pressure is given in equation (<ref>) and it can be used to compute the force in curvilinear coordinates as well (<ref>). The advantage of this formulation is that the wall has a fixed positionq^ 1 = 0in the transformed domain. The integral in equation (<ref>) is discretized using the trapezoidal quadrature formula:(t) ≈ ∑_j_ 2 = 0^N_ 2 - 1 p_ 0, j_ 2 + p_ 0, j_ 2 + 1/2 h_ 2 .In order to compute the time derivative of the velocity potentialϕ_ tappearing in the pressure term{p_ 0, j_ 2}_j_ 2 = 0^ N_ 2 , we use the values of the velocity potential on three time layers{ϕ_ ȷ^ n-1}_ ȷ ,{ϕ_ ȷ^ n}_ ȷand{ϕ_ ȷ^ n+1}_ ȷ . So, the time derivative of the velocity potential is approximated as:(ϕ_ t)_0, j_ 2 ≈ 3 ϕ_ 0, j_ 2^ n+1 - 4 ϕ_ 0, j_ 2^ n + ϕ_ 0, j_ 2^ n-1/2 Δ t .However, due to the change of variables in (<ref>) one has to compute also the velocities of grid nodes. This is done as specified in (<ref>). Cartesian velocities computed as specified in (<ref>) with the only difference — we use the values of velocity potential at the new time level{ϕ_ ȷ^ n+1}_ ȷ .In order to integrate numerically the nonlinear differential equation (<ref>), we rewrite this equation as a system of two first order differential equations:ṡ= υ , m υ̇ + k s= - [(t) -(0) ] ,together with initial conditions:s (0) = 0 , υ (0) = 0 .The last system of equations can be rewritten in the vectorial form for our convenience:=+(t) ,(0) =,where we introduced the following notations:(t)[ s (t); υ (t) ] , [ 0 1; - k/m 0 ] ,(t)[ 0; - 1/m [(t) -(0) ] ] .The Cauchy problem (<ref>) is solved using the so-called modified Euler method[This scheme is also called the second order explicit Runge–Kutta scheme <cit.>.]:^ ⋆ - ^ n/Δ t= ^ n + ^ n ,^ n+1 - ^ n/Δ t= 1/2 [ ^ n + ^ ⋆ ] + ^ n .If we exclude the intermediate variable^ ⋆from last equations and rewrite the obtained equations in the component-wise form, we obtain the following scheme:s^ n+1 - s^ n/Δ t= υ^ n - Δ t/2 k/m s^ n + Δ/2 m [(t) -(0) ] ,υ^ n+1 - υ^ n/Δ t= - k/m s^ n + 1/m [(t) -(0) ] - Δ t/2 k/m υ^ n .The last scheme is implemented in our numerical code. §.§ Elliptic mapping construction (5) In previous sections we assumed that mapping (<ref>) was known. In order to complete our numerical method description we have to describe how to construct this mapping in practice. Namely, we shall use the so-called equidistribution method <cit.>. A typical grid generated using this method is shown in Figure <ref>. Recently this method was successfully applied in 1D to the simulation of conservation laws <cit.>. Below we provide a detailed description of this method to two spatial dimensions.According to the equidistribution method, the coordinates of grid nodes{_ ȷ^ n+1}_ ȷ ⊆ Ω (t)in the physical space on the subsequent time layert = t^ n+1are determined by solving the following vectorial parabolic equation:t = q^ 1 [ g_ 2 2 ϖ q^ 1 ] + q^ 2 [ g_ 1 1 ϖ q^ 2 ] ,where > 0is a positive smoothing parameter whose value is chosen to minimize the oscillations in grid nodes trajectories. Aboveϖ = ϖ (, t)is the so-called monitoring function, which determines the local density of the grid nodes and{g_ α α}_ α = 1^ 2are metric tensor components defined in (<ref>).The equidistribution method based on the solution of parabolic equations for grid nodes coordinates was proposed in <cit.>. A few years later this method was applied to the simulation of 1D (compressible) gas dynamics problems in <cit.>. This method has been especially designed for non-stationary problems and its application allows to avoid (or simply reduce) abrupt changes in the nodes positions when we move from one time layer to the following. In this method one constructs the new mesh by taking into account the position of nodes at the last time layer. The new coordinates of grid nodes are determined by solving a discretized version of the parabolic problem (<ref>). This modification of the classical equidistribution method (proposed in <cit.>) of the construction of dynamically adapted moving grids has been used to solve numerous 1D problems <cit.>, this list is not being exhaustive.The system of nonlinear equations (<ref>) is solved using an iterative method of alternating directions. The monitoring functionϖwas computed at the known solution on the time layert = t^ n . As the initial guess for this iterative process we take the known mesh{_ ȷ^ n}_ ȷfrom the previous time layer as well. We mention that at the initial moment of timet = 0we employ the traditional (steady) equidistribution method as it was described in <cit.>. It is equivalent to solve the system (<ref>) with ≡ 0 .Since the position of boundary nodes varies[Indeed, we are dealing with free surface flows in the presence of a moving wall.] from one time step to another, before solving equations (<ref>) we have to determine the grid{_ ȷ^ n+1}_ ȷalong the boundary∂ _ h^ 0 . Let us assume that a portion ∈ {Γ_ ℓ, Γ_ r, Γ_ b, Γ_ s} ≡ ∂ ^ 0of the boundaryΓ^ n+1 ≡ Γ(t^ n+1)is given in the following parametric form:x = x () ,y = y () ,whereis a real parameter. Then, the coordinates of grid nodes{_ ȷ^ n+1}_ ȷ ⊆ ^ hon the boundary can be determined by formulas:x_ ȷ^ n+1 = x (_ j^ n+1) ,y_ ȷ^ n+1 = y (_ j^ n+1) .We assume that the nodes on the boundary under consideration are re-ordered with a single scalar indexjfor the sake of notation compactness. So, the problem is to determine the discrete scalar function{_ j^ n+1}_ jin the parameter space. To do it, we solve numerically the following scalar nonlinear difference equation:_ j+1^ n+1 - _ j^ n+1/τ_ n = 1/h [ {ϖ^ n ^ n+1}_ j+1/2 _ j+1^ n+1 - _ j^ n+1/h - {ϖ^ n ^ n+1}_ j-1/2 _ j^ n+1 - _ j-1^ n+1/h ] ,whereh ∈ {h_ 1, h_ 2}is the discrete step in space, which depends on the boundary under consideration. The function_ j^ n+1{√(x_ ζ^ 2 + y_ ζ^ 2)}|_ j^ n+1is the trace of the transformation Jacobian at this boundary. Please, notice that we compute the monitoring functionϖ^ non the known position of the nodes. The difference equation (<ref>) is solved on four sides (left and right walls, bottom and free surface) of the discretized rectangle_ h^ 0 . The distribution of boundary nodes obtained in this way is used as the Dirichlet-type boundary condition for the 2D grid generation, which is achieved by solving numerically equation (<ref>). By the end of this step we know the gridΩ̅_ h^ n+1nodes coordinates{_ ȷ^ n+1}_ ȷ .§.§.§ The re-computation At the next step we recompute quantities^ n+1and{ϕ^ n+1_ ȷ}_ ȷto take into account the new mesh. For this, we solve again equation (<ref>) (or an analogue of (<ref>) on the boundaries) in which the Cartesian components of the velocity (<ref>) are still computed on the previous gridΩ̅_ h^ nand using the velocity potential values from the previous time layert = t^ n . However, in formulas (<ref>) we take the new velocity of grid nodes. In other words, formula (<ref>) is replaced by_ t, ȷ = _ ȷ^ n+1 - _ ȷ^ n/τ_ n .During the second solution of system (<ref>) the right hand side member (<ref>) is modified as well to take into account the new velocity of the left wall and, obviously, the new location of mesh nodes on this boundary. Another modification concerns the velocity componentsu ,vwhen they are used in the computation of the contravariant componentv_ ȷ^ 1, nwhose discretized formula is given in (<ref>). The quantityv_ ȷ^ 1, nis needed to determine the new position of the free surface elevation^ n+1 . During the present step, the componentsu ,vare computed asu_ ȷ^ n+1 = [ ϕ_q^ 1· y_q^ 2 - ϕ_q^ 2· y_q^ 1/ ]_ ȷ^ n+1 ,v_ ȷ^ n+1 = [ -ϕ_q^ 1· x_q^ 2 + ϕ_q^ 2· x_q^ 1/ ]_ ȷ^ n+1 .The last formulas replace equations (<ref>) during this stage. §.§ Stability of the scheme In order to study the stability of the proposed scheme, we consider the linearized governing equations. Moreover, we consider only the Initial Value Problem[In fact, we have the so-called Cauchy–Poisson problem, which is of mixed type between IVP and BVP. We say that we have an IVP since there are no boundary conditions on lateral boundaries,the domain is unbounded in the horizontal directions. However, there are still boundary conditions to be satisfied on the impermeable bottom and on the (linearized) free surface.] (IVP) in order to remove the complexity of (lateral) boundary conditions treatment. The linear water wave problem is known as the Cauchy–Poisson problem <cit.> since the pioneering works of Augustin Louis Cauchy<cit.> and Siméon Denis Poisson<cit.>. Let us formulate this problem in precise mathematical terms. Consider a fluid layer of infinite horizontal extent (-∞ < x < +∞) over a solid bottom of uniform depthy = -h_ 0 =. The fluid domain is bounded from above by the free surface, whose location is assumed to be aty = 0after the linearization. The Cauchy–Poisson problem consists in finding the free surface elevationηand the velocity potentialϕby solving the Laplace equation[This equation is to be compared with equation (<ref>).] in a strip:Δ ϕ = 0 ,-∞ < x < +∞ ,- h_ 0 ≤ y ≤ 0 .The last equation has to be completed by the following free surface boundary conditions:η_ t - ϕ_ y= 0 ,y = 0 ,ϕ_ t + g η= 0 ,y = 0 ,and by one bottom impermeability condition:ϕ_ y = 0 ,y = -h_ 0 .It is not difficult to see that free surface boundary conditions given above are nothing else but linearized versions of equations (<ref>) and (<ref>). A particular solution to the Cauchy–Poisson problem can be easily obtained using some elementary Fourier analysis <cit.>:η (x, t) = η_ 0 ^ (k x - ω t) , ϕ (x, y, t) = ϕ_ 0 ^ (k x - ω t) cosh[ k (y + h_ 0) ] ,wherek2 πλis the wave number,λis the wave length,ω2 πTis the wave frequency andTis its period. Wave amplitudesη_ 0 ∈ andϕ_ 0 ∈ are some real numbers. The wave frequencyωis related to the wave numberkthrough the so-called dispersion relation of gravity waves:ω (k) = ± √(g k tanh(k h_ 0)) .For the sake of simplicity we consider only waves moving rightwards. It fixes the branch+in the relation above. We reiterate the fact that the dispersion relation (<ref>) is a necessary condition for the existence of solutions (<ref>).The numerical scheme considered above can be applied to the Cauchy–Poisson problem as well. The semi-discretization in time of free surface boundary conditions (<ref>), (<ref>) reads:η^ n+1 (x) - η^ n (x)/τ_ n - ϕ_ y^ n+1= 0 ,ϕ^ n+1 (x, 0) - ϕ^ n (x, 0)/τ_ n + g η^ n (x)= 0 .From now on, for the sake of simplicity we take the time stepτ_ n ≡ τ > 0to be constant. The elementary solution (<ref>) can be semi-discretized as well:η^ n (x) = η_ 0 ^ n ^ k x , ϕ (x, y, t) = ϕ_ 0 ^ n ^ k x cosh[ k (y + h_ 0) ] ,where we introduced the notation^-ω τ . By substituting this semi-discrete solution ansatz into relations (<ref>), (<ref>) we obtain the following linear system of equations with respect to wave amplitudesη_ 0andϕ_ 0 :g η_ 0 + ϕ_ 0 ρ - 1/τ cosh[ k h_ 0 ]= 0 ,ρ - 1/τ η_ 0 - ϕ_ 0k sinh[ k h_ 0 ]= 0 .In order to have non-trivial solutions, the determinant of this system has to vanish. It gives us the following quadratic equation with respect to the transfer coefficient :^ 2 - 2 (1 - τ^ 2 ω^ 2/2)+ 1 = 0 .To have the linear stability property of our scheme, it is necessary that both roots of this equation verify the inequalityρ^ ± ≤ 1 . To meet this requirement, it is sufficient to ask that the discriminant of the quadratic equation (<ref>) is not positive,τ^ 2 ω^ 2 (τ^ 2 ω^ 2/4 - 1) ≤ 0 ,or equivalentlyτ ≤ 2/ω ,where the wave frequency was defined in equation (<ref>) (with the sign+by our convention).LetΔ xbe the discretization step size in the horizontal direction. The minimal wave lengthλ_ inf , which can be represented on the grid with spacingΔ xisλ_ inf ≡ 2 Δ x . All other waves satisfy the inequalityλ ≥ λ_ inf . These considerations on the wave length can be translated into the language of wave numbers,k ≤ k_ supπ/Δ x .Henceforth, we can derive the following estimation for the wave frequency using the dispersion relation (<ref>):ω (k) = √(g h_ 0)·√(k/h_ 0 tanh[ k h_ 0 ]) = k √(g h_ 0)·√(tanh[ k h_ 0 ]/k h_ 0) ≤ √(g h_ 0) π/Δ x .Consequently, in order to satisfy the stability condition (<ref>), it is sufficient to impose the following restriction on the time stepτmagnitude:τ ≤ 2 Δ x/π √(g h_ 0) .An analogue of this stability condition will be used below during the simulation of fully nonlinear problems. The most important conclusion of this Section is that we have a hyperbolic-type stability Courant–Friedrichs–Lewy condition <cit.>— the time step is a linear function of the mesh spacing.§.§.§ Practical choice of the time step In the previous Section we studied the scheme stability in the linear case. However, in numerical simulations presented below, we solve the nonlinear problem. Henceforth, one may ask the question how to choose the time step in practical nonlinear simulations. Below we explain our approach to this problem.In order to cover the whole family of problems, we choose to work in scaled variables. The CFL condition (<ref>) can be rewritten in dimensionless variables asτ^ ∗ ≤ 2/π h_ 1^ ∗ ≈ 0.64 h_ 1^ ∗ ,whereh_ 1^ ∗h_ 1h_ 0 . As grid spacingh_ 1 , we take the smallest horizontal spacing along the free surface (since most important variations take place there):h_ 1 ∼ Δ x_ min^ nmin_0 ≤ j_ 1 < N_ 1x_ j_ 1 + 1, N_ 2 - x_ j_ 1, N_ 2 .Consequently, the dimensionless CFL condition can be rewritten asτ_ n^ ∗ ≤ 0.64 Δ x_ min^ n, ∗ .However, our numerical simulations show that this estimation is too pessimistic. Consequently, in all simulations presented below we took the time step according to the following less restrictive formula:τ_ n^ ∗ ≤ κ Δ x_ min^ n, ∗ ,withκ = 0.95 . Most probably, the last condition is pessimistic as well. However, it guaranteed the stability of our nonlinear computations.§ NUMERICAL RESULTS Above we described the proposed finite difference scheme and our resolution algorithm on a fixed reference domain. Below we present several validation tests and numerical experiments which show the performance and abilities of our numerical approach. §.§ Solitary wave run-up on a fixed wall In order to illustrate the applicability of our numerical algorithm, we consider the classical problem of the solitary wave/fixed wall interaction. Due to symmetry considerations, this set-up is equivalent to the head-on collision of two equal solitary waves. This problem is well-studied in the literature <cit.> and it can serve as the first validation test.Consider a solitary wave of amplitudemoving in the leftward direction. The channel has a constant depth and the wall is assumed to be fixed in order to be able to perform comparisons with previous investigations. On the right the channel is also bounded by a fixed vertical wall. The total length of the channel is equal toℓ = 20 . In this Section we provide all values in dimensionless variables as it was explained earlier in Section <ref>. Moreover, we assume in these computations that waves do not overturn. In other words, the free surface is traditionally given as the graph of a functiony = η (x, t) . The initial condition for the free surface elevationη_ 0 (x) ≡ η (x, 0)and velocity field_̆ 0 () ≡ (, 0)are given by the following approximate formulas <cit.>:η_ 0 (x) =^ 2[ κ/2 (x - x_ 0)_] , κ√(3 / + 1) ,u_ 0 (x, y) = - √(1 + ) η_ 0 (x)/1 + η_ 0 (x) + ^ 2/√(1 + ) [ 1/4 - 3/4 (y + 1/η_ 0 (x) + 1)^ 2 ]· ·[ 2 η_ 0 (x) - 1/η_ 0 (x) + 1 ^ 2+ 3 - η_ 0 (x)/η_ 0 (x) + 1 ^ 4] ,v_ 0 (x, y) = - √(3 ^ 3) (1 + y) coshsinh /( + cosh^ 2 )^ 2 ,wherex_ 0is the wave crest initial position.The interaction of a solitary wave with amplitude = 0.4with a vertical (fixed) wall is shown in Figure <ref> at different moments of time. Under a wave we show also the adapted grid. On this Figure we can see that after the reflection the solitary wave does not recover completely its initial shape. In particular, behind the main wave we observe a slight dispersive tail. The numerical simulations on refined grids show that it is not a numerical effect. This dispersive tail appears on all grids and it reflects an intrinsic property of the full Euler equations — their non-integrability <cit.>. In this example we use the following monitoring function in order to adapt the grid to the solution:ϖ (, t) = 1 +η (x, t) ,where > 0is a positive ad-hoc parameter. In computations presented in the Section we use = 10 . The main rationale behind this choice of the monitoring functionϖ (, t)is to put more nodes in areas where the waves are large.In Figure <ref> we can see that the refined area follows somehow the solitary wave crest during its motion towards and fromwards the vertical wall. From equation (<ref>) it can be seen that the monitoring function does not depend on the vertical variabley . As a result, we obtain the grids with almost vertical lines. Consequently, in accordance with stability condition (<ref>) to determine an admissible local time stepτ_ nwe can use the following formula:τ_ n =Δ x_ min^ n ,where0 <≤ 1is the security factor andΔ x_ min^ nis the minimal mesh spacing on the free surface at timet = t^ n ,Δ x_ min^ nmin_0 ≤ j_1 < N_1{x_ j_ 1+1, N_2^ n - x_ j_ 1, N_2^ n} . One of the main characteristics of the wave/wall interaction is the maximal amplitude of the wave on the wall. This quantity is called the maximal run-up and will be denoted in our study as_ max . Obviously, in our experimental conditions this quantity depends on the amplitude of the incident solitary wave,_ max = _ max () . In Figure <ref> we represent this dependence according to our numerical simulations (solid black line), experimental data (filled markers), other computations (empty markers) and the following asymptotic analytical prediction <cit.> (dashed line):_ max () = 2[ 1 + 1/4+ 3/8 ^ 2 ] + o(^ 3) ,→ 0 .In Figure <ref> we can see that there is an overall good agreement among all presented data up to the amplitude ≲ 0.4 . For higher waves some divergences start to appear. However, the agreement between our numerical model with other potential flow solvers <cit.> continues up to ≲ 0.6 . The experimental points go rather below our predictions. It can be easily explained by the neglection of viscous and friction effects in our numerical (and mathematical) model. Nevertheless, we would like to mention that our numerical results agree particularly well with experimental data reported in <cit.>. §.§ Wave generation by a numerical wave maker The study of water wave interaction with movable partially submerged bodies and objects is a very important problem of the modern computational hydrodynamics. In the previous Section <ref> the wall was considered to be fixed in order to compare our numerical predictions with available data. Starting from this Section we allow the left wall as a movable object. As the first step towards the freely moving solid boundary, we consider first the situation where left wall motion is prescribed by a given law. Physically, it corresponds to the vertical piston motion used in many experimental facilities. Traditionally, the generated waves are understood using linear or weakly nonlinear theories <cit.>. Recently this situation was modelled numerically with Boussinesq-type equations in <cit.>. The results presented below are fully nonlinear and fully dispersive.Consider a numerical wave tank with horizontal bottom and of uniform depthd = 1 𝗆(when the water is unperturbed). The left wall is initially located in the pointx = 0 . For timest > 0the wall moves according to the following law:s (t) = α (1 - ^- β t) sin(ω t) .Thus, during the initial times the wall moves to the right. In this way, the wall motion is completely determined by three parametersα > 0 ,β > 0andω . The first one (α) specifies the maximal wall oscillation amplitude (in the horizontal extent), the second parameterβcontrols the speed of the relaxation[It is easy to see that the wall oscillation amplitude tends to its maximum value α as t → +∞. This justifies the term `relaxation' associated to this parameter β. The speed of this convergence depends on the value of parameter β,bigger is faster.] towards the stationary periodic regime andωis the frequency of wall oscillations (or equivalently it controls the oscillation period).In Figure <ref> we depict a few simulations results, which were obtained by varying the parameterωfor fixed values of two other parameters:α = 0.4 𝗆 , β = 0.5 𝖧𝗓 .Figure <ref> shows clearly that the generated wave amplitude (measured at the moving wall) depends essentially on the wall oscillation period, which is determined by parameterω . This effect is illustrated in Figure <ref> for three different values of the parameterω ∈ {0.5 , 1.0 , 3.0 } . For each wall trajectory we show also the corresponding free surface excursion on the moving vertical wall. The increase inωresults in the reduction of the oscillation period. We can also witness the relaxation effect on the free surface oscillation. Indeed, as the timetgoes on, the wave amplitude increases and becomes practically stationary. The speed of relaxation depends on the parameterβ . We underline the fact that for all three (considered) values of the wall oscillation frequencyω , the amplitude of oscillationsαwas the same (α ≡ 0.4 ). Nevertheless, the wave amplitudes registered on the wall were significantly different. This is the first practical conclusion that we can draw from our numerical simulations: the piston oscillation period has a much bigger influence on the generated wave amplitude, than the amplitude of these oscillations. We can explain this observation with a simple physical argument: during `fast' piston oscillations, the energy is pumped into the fluid close to the piston faster than waves can evacuate this energy by propagating into the wave tank.The second important observation that we can make based on our simulations is that the maximal wave run-up does not take place at the moments where the wall displacement is maximal. A more detailed investigation showed that the wave amplitude on the wall is maximal when the wall acceleration is maximal. It happens when the wall passes by its initial (or mean, or unperturbed) positionx = 0 . After passing this point, the wall decelerates and the wave has time to flow away from the piston, which reduces the value of run-up.Finally, we can also observe that the wave run-up value on the wall is asymmetric with respect to the still (unperturbed) water level. This effect is much better visible for fast wall motions. Indeed, forω = 3.0 the maximal run-up is ≈ 43, while the maximal run-down is ≈ 34. §.§ Surface wave run-up on a movable wall In this Section we consider the wave/wall interaction problem, where the wall motion is not prescribed, but it is determined as a part of the problem solution. The left wall is attached to a spring system, which can deform under the wave action.§.§.§ Single pulse To begin, we consider first the case of a single localized wave impulse interacting with a moving wall. The channel depth is taken to bed = 1 and the channel length isℓ = 20. At the initial moment of timet = 0 , the velocity field is taken to be quiescent and the free surface shape is given by the following formula:η_ 0 (x) =/2 [ 1 + cos[ π/Υ (x - x_ 0) ] ] ,x - x_ 0 ≤ Υ , 0 ,x - x_ 0 > Υ .For subsequent timest > 0this initial condition is separated in two waves of the amplitude ≈ /2 , which move in opposite directions. The wave moving leftwards interacts with the moving wall. We would like to underline the fact that the initial left wall position (x = 0) does not coincide with the wall equilibrium position in the absence of water. Even if the free surface coincides with the still water level, the springs are deformed under the action of the hydrostatic pressure. Thus, the equilibrium wall positionx = x_ ewith undeformed springs is located somewhere to the right from its initial position,x_ e > 0 . We performed a series of numerical simulations by varying the parameters ,x_ 0 andΥof the initial condition (<ref>). Additionally we varied also the wall massmand the springs rigidity coefficientk .First, we conducted a series of numerical experiments for small initial wave amplitudesin view of comparisons against the numerical simulations reported in <cit.>. The Authors of the preceding study employed the Boundary Integral Equations Method (BIEM) <cit.>. Unfortunately, the Authors of <cit.> did not publish any tabulated data to perform quantitative comparisons. However, we performed qualitative comparisons of numerical results. We choose the same values as in <cit.>,x_ 0 = 0.7,Υ = 0.5 and dimensionless spring rigidity coefficientk = 10 . The wall dimensionless mass ism = 1.5 .In Figure <ref>(a) we show the free surface oscillations recorded at the initial pulse location[Here we mean the center or pulse crest.]x = x_ 0 = 0.7. We show the results for two initial wave amplitudes. The line (1) corresponds to = 1, while = 10 is depicted by line (2). Please, notice that curves presented in Figure <ref> are shown relatively to the initial wave amplitude . These curves compare quantitatively and qualitatively well to those presented in <cit.>. To the graphical resolution the oscillation decay rates and oscillation periods are the same. In Figure <ref>(b) we show moving wall trajectories in time for two initial wave amplitudes ∈ {1 , 10 } . These curves also compare well with previous computations reported in <cit.>. We can also see that for small times the solid and dashed curves shown in Figure <ref>(a,b) almost coincide. It means that for small initial wave amplitudesthe results of computations depend almost linearly on . This observation supports the applicability of the linear theory <cit.>.Let us study now the influence of the springs rigidity parameterkon the resulting wave field. We took the following parameters in this computation:x_ 0 = 2, Υ = 1,= 10,m = 1.5 .The parameterkis variable. In Figure <ref>(a) we can see that the first run-up on the wall is large. Then, the following run-ups are generally decaying in amplitude since the wave energy spreads over the whole wave tank during the dispersive (and possibly nonlinear) effects. Moreover, fork = 5(curve 2) the decay rate is the strongest since a larger part of the wave energy is converted into the wall motion. Thus, we have the wave energy transformation into the elastic energy of the spring system. When the springs rigidity is high, the wall performs small later motions, as it can be seen in Figure <ref>(b). The increase inkresults also in the increase of the wall oscillation frequency around the initial equilibrium positionx = 0 . In the same time, the wave run-up records on the wall are practically identical with the fixed wall case ( curves 3 and 4 in Figure <ref>(a)). In other words, for high rigiditiesk ≫ 1 , the `moving' wall interacts in a way very similar to the fixed wall. On the other hand, when the rigidity decreases, the wall oscillation period and amplitude both increase. Wave run-up on the wall decay rate also increases with the parameterk . However, in contrast to large values ofk , a flexible spring system can allow the maximal wave run-up value during the second (or even further subsequent) wave interactions with the wall. In Figure <ref>(a) we can see that the value of the maximal wave run-up on the wall is generally lower for weak springs comparing to quasi-fixed wall results (k ≫ 1). From these simulations we can conjecture that flexible spring systems should be used for more efficient wave run-up reduction.Let us consider now the case of the weak springs system (k = 1) and a moving wall with massm = 1 . The question we would like to investigate is how the wave run-up and wall trajectory depend on the incident wave amplitude ? In Figure <ref>(a) we show the wave run-up value on the moving wall for a moderate amplitude = 20 (curve 1) and a fairly strong amplitude = 60 (curve 2). The initial condition is given by the single pulse formula (<ref>). The initial impulse center isx_ 0 = 12 and its half-lengthΥ = 5. In Figure <ref>(b) we show the wall displacements (t) . As expected, the wall displacement becomes larger as the incident wave amplitudeincreases. It can be seen that for weak springs the wall first retracts under the incident wave action. Then, it comes back towards its initial position and slightly oscillates around this point. It is very interesting to observe the high amplitude pulse (curve 2) interaction with the moving wall depicted in Figure <ref>(a). Here the wave run-up value grows until the wall attains its left-most position. Then, on the way back the wall motion provokes the second maximal run-up value. It explains the non-monotonic behaviour of the curve 2 around its maximum. This behaviour is possible only for high amplitude waves and weak ( easily deformable) springs.We naturally come to the question of the wall massmand springs rigiditykon the maximal wave run-up value of the impulse (<ref>) on the movable wall. We use the following parameters of the incident pulse:x_ 0 = 12, Υ = 5,= 20.In Figure <ref>(a) we depict the maximal run-up value as the function of the wall massm . It can be easily seen that for a fixed rigidity parameterk , the maximal run-up is a non-monotonic function ofm . Even more, for every value ofk , one can find the valuem^ ⋆for which the maximal wave run-up will be minimal. It implies that the wave run-up can always be reduced by choosing the appropriate wall parametersk^ ⋆andm^ ⋆ . In Figure <ref>(b) we show the maximal wave run-up dependence onkfor fixed dimensionless wall massesm ∈ {1, 3, 5} . So, it can be seen that the run-up depends monotonically onk . Namely, it increases with wall rigidityk . As we increasek → +∞ , the maximal run-up tends to its limiting value, which depends on the incident wave parameters and which is independent on the dimensionless wall massm .§.§.§ Solitary wave Finally, we show some numerical results on the solitary wave interaction with a moving wall. This test case is a natural extension of the case considered earlier in Section <ref>. The initial condition is provided by the same formulas (<ref>)–(<ref>). In order to avoid any interactions of reflected waves from the right wall (during the simulation time), we increase the computational domain length toℓ = 40 (fromℓ = 20 ). The computations were performed for a solitary wave with moderate amplitude = 20. We would like to study the influence of moving wall parameters on the maximal solitary wave run-up. In Figure <ref> we show the maximal run-up as a function of the wall massm(a) and as a function of springs rigidityk(b) (the other parameter being fixed). With a dashed line we show the maximal run-up value on a fixed wall. The comparison of this result with Figure <ref> indicates that there are qualitative similarities in the behaviour of a single pulse with a solitary wave ( Figure <ref>). So, for solitary waves the run-up can be significantly reduced as well. All differences observed in Figures <ref> and <ref> can be solely explained by the incident wave amplitude and its shape.In Figure <ref>(a) we show the wave run-up record under the action of a solitary wave on the moving wall as a function of time. This computation was performed for the wall massm = 5and springs rigidity isk = 1 . In Figure <ref>(b) we report the wall displacement during the wave/wall interaction process. In Figure <ref>(a) we depict also the wave run-up record for a fixed wall as well. It can be clearly seen that the presence of a moving wall reduces significantly wave run-up height on it. We can also notice that for a movable wall the wave run-up happens twice in contrast to the fixed wall case. Indeed, the first (and usually maximal) run-up takes place when the wall is retracting under the incident wave action. Then, at some moment the springs accumulate enough elastic energy in order to push back the wall towards its equilibrium position. Right after this turning point the second run-up takes place. However, its value is usually lower than the first one.In Figure <ref> we show also the free surface evolution in space and in time. The left panel <ref>(a) shows the classical fixed wall case for the sake of comparison. In this case the wave field essentially consists of one incident and one reflected waves. Eventual inelastic collision effects are negligible for a solitary wave of amplitude = 20. A much more interesting wave field is generated by the interaction with a moving wall. It is depicted in Figure <ref>(b). The wall motion generates multiple reflected wave, which travel with lower speed (in agreement with their lower amplitude).§ DISCUSSION The present study was devoted to the simulation of free surface waves in a two-dimensional tank of variable size. Namely, a vertical wall is attached to a system of springs and it can move under the action of incident water waves. Modelling and numerical simulation of this problem was discussed in this study. Below we outline the main conclusions and perspectives of the present study. §.§ Conclusions In the present study we considered the problem of surface water wave modelling in domains with variable geometry. In general, problems on time-varying are known to be notoriously difficult <cit.> (both theoretically and numerically). Here we considered a special case of a numerical wave tank with a moving wall. This wall can move according to the prescribed law. In this case we model the wave generation process by a piston-type wave maker <cit.>. In our formulation the wall can also freely move under the wave action. It is realized by connecting the wall to a system of elastic springs. In this way, the wall massmand springs rigiditykare taken into account in our model by writing an additional nonlinear second order differential equation for the wall position. Surface waves are described mathematically using the full water wave problem,the irrotational incompressible perfect fluid flow described by Euler equations <cit.>. No approximation was made and, thus, the presented results are fully nonlinear and fully dispersive. Moreover, we proposed a numerical algorithm on adaptive moving grids. The adaptation was achieved using the so-called equidistribution principle <cit.>. The discrete problem was rigorously studied and the proposed discrete operator was shown to be self-adjoint on adaptive grids as well as the continuous problem. Thus, the numerical method preserves qualitatively some properties of the continuous operator, which is referred to as the `structure preserving' property <cit.> in modern numerical analysis.The proposed algorithm was first validated on the well-known case of a fixed rigid vertical wall. There is enough analytical, numerical <cit.> and experimental <cit.> data available to validate the solver. Then, this numerical tool was applied to generate periodic waves in silico by a moving piston-type wave maker. Finally, the wall motion and incident wave run-up on this moving wall was studied using our numerical means. Our results compared quite well with some published data <cit.>. The influence and effect of various parameters on the wave run-up was investigated. §.§ Perspectives In the present article the geometry, mathematical formulation and the numerical method have been presented for the two-dimensional configuration. In future works we plan to extend this methodology for 3D flows. A priori, this generalization is going to be tedious, but straightforward.As another possible direction for future research, we would like to underline that the moving (solid non-deformable) part in our study was a wall. This object is very important in coastal engineering. However, in future works we would like to incorporate moving objects of more general geometrical shape. As a minor point of improvement, one can notice that in all simulations presented earlier the bottom was taken to be flat. It was done on purpose to isolate the effect of the moving wall on incident waves. However, it is not a limitation of the method. The formulation presented earlier works for general bottoms and the simultaneous effect of the moving boundary and local bathymetric features on the wave run-up on this wall has to be investigated.Finally, the formulation we used was fully conservative. In other words, no dissipation mechanism was included in our study except for the motion of the wall under the action of water waves (however, the whole wave/wall system was conservative). Of course, this model is an idealization of the reality. In real flows some dissipative effects are also present due to the molecular viscosity, friction, turbulence or other mechanisms. For this purpose a suitable visco-potential formulation has been proposed <cit.>, which occupies the intermediate level between the potential flows and Navier–Stokes equations. §.§ Side effects Our work has also at least one nice side effect. Namely, if we simplify the problem considered in the present study by fixing the wall position,s(t) ≡ 0 , we obtain a robust numerical wave tank for 2D surface waves similar to <cit.>. However, the technique presented here is fully adaptive. By the way, we are not aware of any recent study using similar adaptive redistribution methods for the full Euler equations with free surface. So, this aspect might be new. §.§ Acknowledgments tocsubsectionAcknowledgmentsThis research was supported by RSCF project No 14-17-00219. D. Dutykh acknowledges the support of the CNRS under the PEPS InPhyNiTi project FARA and projectEDC26179 —“Wave interaction with an obstacle” as well as the hospitality of the Institute of Computational Technologies SB RAS during his visit in October 2015. D. Dutykh would like to acknowledge the help of his colleague Tom Hirschowitz who timely provided tea leaves in cases of emergency.§ TRANSFORMATION OF COORDINATES In this Appendix we explain some calculation details of coordinate transformations which allow to pass from the system (<ref>)–(<ref>) posed on a variable domainΩ (t)to the system (<ref>)–(<ref>) on the fixed domain^ 0 . Consider a smooth bijective map (<ref>) (see also Figure <ref> for an illustration):{[ x = x (q^ 1,q^ 2,t) ,; y = y (q^ 1,q^ 2,t) . ]. ⟺{[ q^ 1 = q^ 1 (x,y,t) ,; q^ 2 = q^ 2 (x,y,t) . ].We consider that the time variable is the same in both coordinate systems. Two maps written above are mutually inverse. So, let us write this conditionx≡ x [q^ 1 (x,y,t),q^ 2 (x,y,t),t] , y≡ y [q^ 1 (x,y,t),q^ 2 (x,y,t),t] .Then, we differentiate (<ref>) and (<ref>) with respect tox . We thus obtain two relations{[x_ q^ 1 q^ 1x + x_ q^ 2 q^ 2x= 1 ,; [0.75em] y_ q^ 1 q^ 1x + y_ q^ 2 q^ 2x= 0 . ].The last two relations can be considered as a system of two linear equations with respect toq^ 1xandq^ 2xas unknowns. By trivially applying the Cramer rule we obtainq^ 1x = y_q^ 2/ , q^ 2x = -y_q^ 1/ ,wherex_q^ 1 y_q^ 2 - x_q^ 2 y_q^ 1is the Jacobian defined in (<ref>). The Jacobian can be also seen as the determinant of a matrix:= [ 1 0 0;x_ t x_ q^ 1 x_ q^ 2;y_ t y_ q^ 1 y_ q^ 2 ] ≡ [ x_ q^ 1 x_ q^ 2; y_ q^ 1 y_ q^ 2 ] ≡ x_q^ 1 y_q^ 2 - x_q^ 2 y_q^ 1 . Similarly, by differentiating (<ref>) and (<ref>) with respect toyand using Cramer's rule, one can show thatq^ 1y = -x_q^ 2/ , q^ 2y = x_q^ 1/ .Now, by differentiating (<ref>) and (<ref>) with respect totwe obtain the following two equations{[ x_ q^ 1 q^ 1t + x_ q^ 2 q^ 2t = -x_ t ,; [0.75em] y_ q^ 1 q^ 1t + y_q^ 2 q^ 2t= - y_ t . ].Solving the last linear system with respect toq^ 1tandq^ 2tyieldsq^ 1t = y_ t· x_ q^ 2 - x_ t· y_ q^ 2/ , q^ 2t = x_ t· y_ q^ 1 - y_ t· x_ q^ 1/ .This concludes the computation of partial derivatives of the inverse mapping =(, t) .§ TRANSFORMATION OF THE LAPLACE OPERATOR Now we can rewrite the Laplace operator (<ref>) in curvilinear coordinates(, t) . The idea consists in viewing the velocity potential asϕ (, t) ≡ ϕ ( (, t),t) .The first derivatives can be easily computed:ϕx = ϕq^ 1·q^ 1x + ϕq^ 2·q^ 2x ,and using the just derived relation (<ref>) we can express everything in terms of derivatives solely with respect toq_1,q_2:ϕx (, t) = 1/ [ y_ q^ 2·ϕq^ 1 - y_ q^ 1·ϕq^ 2 ] .Similarly, we can compute the other partial derivative of the velocity potentialϕwith respect toyin new coordinates:ϕy (, t) = 1/ [ - x_ q^ 2·ϕq_1 + x_ q^ 1·ϕq^ 2 ] .Now we can apply recursively the just obtained results forϕx (, t)andϕy (, t)to compute the second derivatives as well:^ 2 ϕx^ 2 = x(ϕx) = q^ 1 (ϕx)·q^ 1x + q^ 2 (ϕx)·q^ 2x = q^ 1 (ϕq^ 1·q^ 1x + ϕq^ 2·q^ 2x)·q^ 1x + q^ 2 (ϕq^ 1·q^ 1x + ϕq^ 2·q^ 2x)·q^ 2x .By taking into account the relations derived earlier, we obtain that^ 2 ϕx^ 2 (, t) = 1/ { q^ 1 [ ϕ_ q^ 1 y_ q^ 2^ 2 - ϕ_ q^ 2 y_ q^ 1 y_ q^ 2/ ] + q^ 2 [ - ϕ_ q^ 1 y_ q^ 1 y_ q^ 2 + ϕ_ q^ 2 y_ q^ 1^ 2/ ] } .Using similar methods we can obtain the following expression for^ 2 ϕy^ 2 (, t) :^ 2 ϕy^ 2 (, t) = 1/ { q^ 1 [ ϕ_ q^ 1 x_ q^ 2^ 2 - ϕ_ q^ 2 x_ q^ 1 x_ q^ 2/ ] + q^ 2 [ - ϕ_ q^ 1 x_ q^ 1 x_ q^ 2 + ϕ_ q^ 2 x_ q^ 1^ 2/ ] } .The Laplace operator can be particularly compactly written in new variables if we introduce the metric tensor components:g_ 1 1x_ q^ 1^ 2 + y_ q^ 1^ 2 ,g_ 1 2 ≡ g_ 2 1x_ q^ 1· x_ q^ 2 + y_ q^ 1· y_ q^ 2 ,g_ 2 2x_ q^ 2^ 2 + y_ q^ 2^ 2 ,along with the implied anisotropic diffusion coefficients:_ 1 1g_ 2 2/ , _ 1 2 ≡ _ 2 1 - g_ 1 2/ , _ 2 2g_ 1 1/ .Using this notation we can finally write the Laplace operator in transformed coordinates:^ 2 ϕx^ 2 + ^ 2 ϕy^ 2 ≡ 1/ [ q^ 1 (_ 1 1 ϕ_ q^ 1 + _ 1 2 ϕ_ q^ 2) + q^ 2 (_ 2 1 ϕ_ q^ 1 + _ 2 2 ϕ_ q^ 2) ] .As a result, the Laplace equation reads:q^ α [ _ α β ϕq^ β ] = 0 , α, β = 1, 2 ,where we employed an implicit summation over repeated indices[In the literature this rule is referred to as the Einstein summation convention.]. This result coincides exactly with equation (<ref>) presented earlier in the manuscript.§ BOUNDARY CONDITIONS TRANSFORMATION Finally, let us derive some relations which turn out to be useful in the transformation of boundary conditions. In particular, in this Appendix we focus on the kinematic condition (<ref>). First of all, during the derivation of boundary conditions one needs time derivatives of the inverse mapping in (<ref>):q^ 1t = 1/ [ x_ q^ 2 yt - y_ q^ 2 xt ] , q^ 2t = 1/ [ y_ q^ 1 xt - x_ q^ 1 yt ] .These formulas were given also earlier in (<ref>) with more details on their derivation. Let us compute Cartesian components of the velocity vector:u (, t)= ϕx (, t) = ϕ_ q^ 1 q^ 1x + ϕ_ q^ 2 q^ 2x (<ref>)= ϕ_ q^ 1 y_ q^ 2 - ϕ_ q^ 2 y_ q^ 1/ , v (, t)= ϕy (, t) = ϕ_ q^ 1 q^ 1y + ϕ_ q^ 2 q^ 2y (<ref>)= - ϕ_ q^ 1 x_ q^ 2 + ϕ_ q^ 2 x_ q^ 1/ .The free surface elevationηon the transformed domain becomes a function of the variableq^ 1and timet :η = η̃ (q^ 1, t) ≡ η (x (q^ 1, q^ 2 ≡ 1, t), t) .Now, we can compute partial derivatives of the functionη , which appear in the boundary conditions:η_ x= η̃_ q^ 1 q^ 1x + η_ q^ 2_≡ 0 q^ 2x ,η_ t= η̃_ t + η̃_ q^ 1 q^ 1t .By substituting all these elements into equation (<ref>), we obtain the following condition:η̃_ t + v^ 1 η̃_ q^ 1 - v = 0 ,wherev^ 1is a contravariant component of the velocity, which can be computed according to this formula:v^ 1 = q^ 1t + u q^ 1x + v q^ 1y .The last equation (<ref>) completes the derivation of the free surface kinematic boundary conditions. Other boundary conditions can be derived in a similar manner. Above, in the main text of our manuscript we do not use the tilde notation η̃ in order to avoid overloading of the text with multiple notations. It is assumed that the reader can make the difference between η (x, t) and η̃ (q^ 1, t) depending on the context. § PISTON MOTION MODELLING Let us introduce a Cartesian coordinate system with the horizontal axisO χlooking rightwards such that the left moving wall is located at the pointχ = 0in the absence of any kind of loading (only the hydrostatic atmospheric pressure that we neglect here). In this case the wave tank is dry. Now we fill it with still water of the uniform depthh_ 0 > 0 . The hydrostatic force (0) > 0(defined in equations (<ref>)) appears on the left wall and the springs accumulate some elastic deformation. It results in the wall displacement of magnitude (0) > 0in the negative direction of the axisO χ . The displacement (0)can be found by applying the Hook law <cit.>:k(0) =(0) ,wherekis the springs rigidity. Fort > 0we have another force (t)acting on the left wall due to the wave motion. This force acts against the wall. Thus, in the second law of Newton we take it with the opposite sign <cit.>:m χ̈ + k χ = -(t) ,wheremis the wall mass. The initial conditions for this ODE are:χ (0) = -(0) , χ̇ (0) = 0 .Now, we make a change of coordinates0 χ ⇝ O xso that the wall is located initially at the pointx = 0 ,x = χ +(0) .Then, the initial conditions become by construction:x (0) = 0 , ẋ (0) = 0 .Equation of motion (<ref>) transforms to:m ẍ + k x - k(0)_≡(0) = -(t) .The last equation can be rewritten asm ẍ + k x = - [(t) -(0) ] .Finally, by denoting the wall displacement bys (t)x (t)we recover equation (<ref>). tocsectionReferencesabbrv10Bell1983 J. B. Bell and G. R. Shubin. An adaptive grid finite difference method for conservation laws.J. Comp. Phys., 52(3):569–591, dec 1983.Brebbia1980 C. A. Brebbia. 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Samarskii. The Theory of Difference Schemes.CRC Press, New York, 2001.Samarskii1981 A. A. Samarskii, V. F. Tishkin, A. P. Favorskii, and M. Y. Shashkov. Operational finite-difference schemes.Differential Equations, 17(7):854–862, 1981.Shokin1982 Y. I. Shokin and A. I. Urusov. On the construction of adaptive algorithms for unsteady problems of gas dynamics in arbitrary coordinate systems.InEighth International Conference on Numerical Methods in Fluid Dynamics, pages 481–486. Springer, Berlin, Heidelberg, 1982.Shokin1985 Y. I. Shokin and N. N. Yanenko. Method of Differential Approximation. Application to Gas Dynamics.Nauka, Novosibirsk, 1985. (in Russian)Stoker1957 J. J. Stoker. Water Waves: The mathematical theory with applications.Interscience, New York, 1957.Su1980 C. H. Su and R. M. Mirie. On head-on collisions between two solitary waves.J. Fluid Mech., 98:509–525, 1980.Tikhonov1963 A. N. Tikhonov and A. A. Samarskii. Equations of Mathematical Physics.Dover Publications, Inc., New York, 1963.Wang2011 K.-H. Wang, Z. Dai, and H. S. Lee. Modeling Wave Run-up along a Sloping or a Moving Wall Boundary.J. Coast. Res., 27(6):1159–1169, nov 2011.Yang1992 S. A. Yang and A. T. Chwang. An experimental study of nonlinear waves produced by an accelerating plate.Phys. Fluids A, 4(11):2456, 1992.Young1950 D. M. Young. Iterative Methods for Solving Partial Difference Equations of Elliptic Type.PhD, Harvard University, 1950.Zagryadskaya1980 N. N. Zagryadskaya, S. V. Ivanova, L. S. Nudner, and A. I. Shoshin. Action of long waves on a vertical obstacle.Bulletin of VNIIG, 138:94–101, 1980. (in Russian)Zegeling1992 P. A. Zegeling and J. G. Blom. An evaluation of the gradient-weighted moving-finite-element method in one space dimension.J. Comp. Phys., 103(2):422–441, dec 1992. | http://arxiv.org/abs/1706.08790v2 | {
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S.X. Yi GW-Cherenkov radiation from photons Department of Physics, The University of Hong Kong, Pokfulam Road, Hong [email protected] On gravitational wave-Cherenkov radiation from photons when passing through diffused dark matters Shu-Xu Yi December 30, 2023 ================================================================================================= Analogy to Cherenkov radiation, when a particle moves faster than the propagation velocity of gravitational wave in matter (v>c_g), we expect gravitational wave-Cherenkov radiation (GWCR). In the situation that a photon travels across diffuse dark matters, the GWCR condition is always satisfied, photon will thence loss its energy all the path. This effect is long been ignored in the practice of astrophysics and cosmology, without justification with serious calculation. We study this effect for the first time, and shows that this energy loss time of the photon is far longer than the Hubble time, therefore justify the practice of ignoring this effect in astrophysics context.Keywords: Gravitational waves, Cherenkov radiation, diffused dark matters PACS Nos.: 04.30.Db § INTRODUCTION TO GRAVITATIONAL WAVE-CHERENKOV RADIATIONGenerally speaking, when a source of perturbation travels faster than the speed of propagation of that perturbation in medium, the front surfaces of influence from the source at different instances will coherently add up to form a shock wave. In the case that a stream of particles rams into another bulk of gas, if the velocity of the stream is larger than the sound speed in the gas, shock wave will occur; In the case of an electromagnetic field source, e.g. a charged particle, if the velocity of the particle is faster than the speed of light in the medium, an electromagnetic shock wave, i.e., Cherenkov radiation will arise.When specific to gravitational perturbation, the source is a package of energy-momentum tensor and the propagation speed is the speed of gravitational wave (GW) c_g. If some how the speed of the source can surpass c_g, we would also expect the GW-shock wave, also known as the GW Cherenkov radiation (GWCR). However since the speed of GW is generally believed to be the speed of light in vacuum (c_g=c), GWCR was thought impossible to occur in our ordinary physical world.Some researchers <cit.> considered GWCR emitted by particles faster than c (Tachyons). Since the existence of Tachyons is not wildly believed, those work receive limited attention. Another way to realize GWCR is to consider slower c_g. Pioneered by ref. 1980AnPhy.125...35C, and recently ref. 2001JHEP...09..023M reconsidered the case of possible GWCR from ultra-energetic cosmic rays when c_g<c due to 4-D Lorentz violation (Rosen's bimetric theory of gravity in ref. 1980AnPhy.125...35C). They argued that in this case, GWCR sets a maximum travel time of a particle with given momentum. With the observation of energetic protons from >10 kpc, they set the upper bound of the difference between c_g and c.In fact, even in ordinary framework, c_g is less than c when passing through matters due to dispersion. It is interesting to study the GWCR in diffused dark matters, which dominates mass of the universe in large scale. For c_g in dark matters, we use the formula for dust <cit.>:c^2/c^2_g=1+4π Gρ/3ω_g^2, (ω_g≫√(4π Gρ/3)).where ω_g is the circular frequency of the GW. For a particle with mass m, Lorentz factor Γ and energy E, the range of the GWCR spectrum is limited by two conditions: * the particle's velocity is larger than the phase velocity of GW at ω_g, which gives:ω_g<√(Γ4π Gρ/3)and* the energy of each graviton ħω_g<E, where ħ is the Plank constant divided by 2π.Since Γ=E/mc^2, for a given E, m→0 gives Γ→∞. In this case, condition 1 is always satisfied and therefore the GWCR power is max.Therefore, GWCR from photons traveling through diffused dark matters is the most important scenario to study this effect. This effect is completely ignored in the practice of astrophysics and cosmology, but without serious discussion.§ PHOTONS AS SOURCES OF GWCRThe energy spectrum of the Cherenkov Gravitational Wave Radiation is <cit.>:P(ω_g)=Gω_gE^2_γ/n^2c^5(n^2-1)^2.where n≡ c/c_g is the refractive index of the GW. Although this formula is derived with non-zero mass particles, we assume it also applies to photons <cit.>. The energy losing rate of the photon is:dE/dt=-KE_γ^2,whereK ≡ G/c^5∫_ω_g,-^ω_g,+ω_g/n^2(n^2-1)^2dω_g= G/2c^5∫_ω_g,-^ω_g,+(n^2-1)^2/n^2dω_g^2.From equation (<ref>) we know that:ω^2_g=2π Gρ/3(n-1).take above formula into the integration in equation (<ref>):K = G^2ρπ/3c^5∫^n_+_n_-(n^2-1)^2/n^2(n-1)^2dn≈ 2G^2ρπ/3c^5(n_+-n_-)< 2G^2ρπ/3c^5.In the universe, galactic clusters are places with most ambient dark matters, where the average density of dark matters can be up to ρ=10^12 M_⊙/Mpc^3, where M_⊙ is the mass of the sun. Thus the upper limit of K is:K<4×10^-107 eV^-1s^-1.The time life of the photon is:τ ≡ E/Ė> 10^106(eV/E) s.From the limit of photon-photon scattering process with the cosmology microwave back ground (CMB) and Extragalactic background light (EBL), the energy of photons cannot access ∼10^14eV. Therefore the life time of the photon under the GWCR is much longer than Hubble time, i.e., the age of our universe. As a conclusion, a lthough GWCR from photons when passing through diffuse dark matters is nonzero, it hardly leaves clues for its existence.§ ACKNOWLEDGEMENTThe author appreciates the support from the department of physics, university of Hong Kong. The author thanks helpful discussion with Prof. Wu Kinwah and comments from anonymous reviewer. 0 1980AnPhy.125...35C Caves, C. M. 1980, Annals of Physics, 125, 351972NPhS..235....6L Lapedes, A. S., & Jacobs, K. C. 1972, Nature Physical Science, 235, 6 1972NPhS..236...79W Wimmel, H. K. 1972, Nature Physical Science, 236, 79 Schwartz2011 Schwartz, C.2011, Modern Physics Letters A, 26, 2223 2001JHEP...09..023M Moore, G. D., & Nelson, A. E. 2001, Journal of High Energy Physics, 9, 023 2012Optik.123..814G Grado-Caffaro, M. A., & Grado-Caffaro, M. 2012, Optik, 123, 814 1980grg2.conf..393G Grishchuk, L. P., & Polnarev, A. G. 1980, General Relativity and Gravitation II, 2, 393 1994PhLB..336..362P Pardy, M. 1994, Physics Letters B, 336, 362 | http://arxiv.org/abs/1706.08722v2 | {
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^1Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley CA 94720, USA ^2Center for Quantum Coherent Science, Department of Physics, University of California, Berkeley CA 94720, USA ^3Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA ^4Schmid College of Science and Technology, Chapman University, Orange, CA 92866, USAThe quantum Zeno effect is the suppression of Hamiltonian evolution by repeated observation, resulting in the pinning of the state to an eigenstate of the measurement observable. Using measurement only, control of the state can be achieved if the observable is slowly varied such that the state tracks the now time-dependent eigenstate. We demonstrate this using a circuit-QED readout technique that couples to a dynamically controllable observable of a qubit. Continuous monitoring of the measurement record allows us to detect an escape from the eigenstate, thus serving as a built-in form of error detection. We show this by post-selecting on realizations with arbitrarily high fidelity with respect to the target state. Our dynamical measurement operator technique offers a new tool for numerous forms of quantum feedback protocols, including adaptive measurements and rapid state purification. Incoherent qubit control using the quantum Zeno effect I. Siddiqi^1,2 December 30, 2023 ======================================================In the field of quantum control, two essentially distinct resources are available for state manipulation. Application of a time-dependent Hamiltonian via external driving enables state preparation given a known initial state. In contrast, measurement and dissipation provide a uniquely quantum resource, owing to the stochastic back-action that necessarily accompanies acquisition of information. In addition, measurement-based, or incoherent control <cit.> also extracts entropy from a system, this information can be used to detect and correct for errors and imperfections. While incoherent and Hamiltonian control are often used in conjunction <cit.>, full control is also possible using measurement alone <cit.>. Measurement-only manipulation has been demonstrated using a fixed measurement basis <cit.>, but unlike Hamiltonian-based methods, implementation of a time-dependent measurement basis is lacking. Such a capability is a versatile additional degree of freedom for measurement based protocols, such as rapid state purification <cit.> and state manipulation <cit.>, for control by projection into a subspace referred to as quantum Zeno dynamics <cit.>, and for measurement-based quantum computation <cit.>.In this Letter, we present a method to dynamically tune the measurement operator in a circuit-QED system, and use this capability to deterministically and incoherently manipulate the state of an effective qubit. Our method relies on the suppression of coherent evolution via strong measurement, known as the quantum Zeno effect (QZE), which has been observed in many systems <cit.>.Essentially it pins a quantum state to an eigenstate of the measurement operator.Changing the operator at a rate slow compared to the rate of measurement-induced dephasing Γ_D, we effectively `drag' the state using measurement alone <cit.>. This method does not require the measurement record or feedback to achieve control. However by monitoring the record with a quantum-limited Josephson parametric amplifier (JPA), we characterize the dynamics and verify good agreement with theory. In the fast-driving limit, where the Zeno effect breaks down, we observe a characteristic arcing effect in which the state maintains relatively high purity even as is transitions to the unwanted measurement eigenstate. Using the measurement record to post-select, we show that we can achieve arbitrarily high fidelity with respect to the target state. Thus measurement serves a dual role, both controlling the state and providing real-time information on its performance. Our system setup is similar to the one used in Ref. <cit.>. It consists of a transmon <cit.> qubit dispersively coupled to the modes of a 3D superconducting cavity. We apply a tone resonant with the qubit frequency that drives Rabi oscillations on the qubit at a frequency of Ω_R, so that its Hamiltonian becomes that of an effective qubit with energy splitting determined by Ω_R/2 π=40 MHz.The new energy eigenstates in this dressed basis are |±⟩=(|g⟩±|e⟩)/√(2), where |g⟩ and |e⟩ are the ground and excited states of the bare qubit respectively. It is within the frame of this effective qubit that we demonstrate the ability to drag the state. We then apply a pair of sideband tones detuned above and below the cavity frequency by Ω_R, as illustrated in Fig. <ref>, which gives us the following Hamiltonian for our effective low frequency qubit <cit.>,H=χa̅_0/2 (a+a^†) σ_δ(t),where a̅_0 is the amplitude of sideband tones, a, a^† are the cavity ladder operators, and χ is the qubit dispersive frequency shift. The measurement operator σ_δ(t)≡σ_x cosδ(t) +σ_y sinδ(t) is set by the relative sideband phase δ(t). This Hamiltonian is a resonant cavity drive, the sign of which depends on the qubit state along the σ_δ axis. Detecting the cavity output field yields a measurement of the qubit at a rate Γ_M=Γ_D η = 2 χ^2 a̅_0^2 η/κ in the σ_δ basis <cit.>, where κ is the cavity mode decay rate and η=0.49 is the detection quantum efficiency. We detect the cavity displacement using a JPA operated in phase-sensitive mode, choosing the amplified axis to align with the displaced quadrature. The full system calibration procedure can be found in Ref. <cit.>.We start by initializing the effective qubit in the |y=+1⟩ state, whichcorrespondsto the ground state of the non-driven transmon qubit. We then continuously measure the effective qubit while changing the measurement axis. This is followed by one of seven pulses {I, x_π/2, -x_π/2, y_π/2, -y_π/2, x_π, -x_π}, and a strong projective measurement for tomography. The dephasing rate during the continuous measurement is fixed, and set to Γ_D/2 π = 0.13 MHz. We repeat the runs for measurement rotation speeds relative to the effective qubit spanning from v=0.01 MHz to v=0.18 MHz, and perform tomography at intervals from 1μs to 5μs for each rotation speed.The thermal population of the transmon qubit was about 15%, so before each measurement we perform a 1μs projective measurement heraldingthe preparation state.We also use the projective readout at the end to ensure that the transmon qubit is still within the two-level subspace after the run. The tomography for the ensemble average behavior is shown in Fig. <ref>a. The colored dots show the tomography from ∼20000 traces per dot, and the lines are theory for the following parameters: initial state with ⟨ y ⟩=0.94, ⟨ x ⟩ = ⟨ z ⟩=0, Γ_D/2 π = 0.13 MHz, and an additional pure dephasing, which we attribute mainly to instabilities in the Rabi drive, at a rate Γ_ϕ/2 π=0.005 MHz (corresponding to the decay time of the Rabi oscillations of the bare qubit). The statistical errors are negligible and the small discrepancy of the tomography data with theory is most likely due to systematic drifts of the measurement rate (amplitude of the side band tones) and leakage tone at the cavity mode frequency (LO leakage - see methods in Ref <cit.>).We now focus on the conditional dynamics of the state as it is being dragged. For this, we reconstruct the quantum trajectories <cit.> from the continuous traces (see also the supplemental material).Our system is in a regime where κ≫Γ_D, in which we can infer the diffusive nature of the quantum jumps. Because we operate the JPA such that it amplifies the optimal (informational) quadrature, the qubit evolution due to the measurement is not affected by phase back-action <cit.>. Then the dynamics of the system can be described by the following master equation, in Itô form <cit.>:dρ = Γ_D/2ℒ[σ_δ(t)] ρ dt + √(Γ_D/2η) ℋ[σ_δ(t)] ρ dW,where ℒ[X]ρ = Xρ X^† - (X^† Xρ + ρ X^† X)/2 is the Lindblad dissipation superoperator, ℋ[X]ρ = Xρ + ρ X^†-⟨ Xρ + ρ X^†⟩ρ, and dW is a Gaussian distributed variable with a variance dt <cit.>, which is itself extracted from the measurement record.We use the POVM that generates this equation with additional corrections to account for extra dephasing on the effective qubit (at a rate Γ_ϕ) to reconstruct the trajectories as function of time from the continuous traces (see supplemental material). Fig. <ref>b shows two example trajectories for a dragging velocity of v=50KHz, with one trajectory showing a state that was successfully dragged, while the other illustrates a `quantum jump'.Note that after the jump the measurement process continues to drag the state on the opposite side of the Bloch sphere.The dynamics of the whole ensemble can be visualized by plotting the distribution of the state of the qubit in the Bloch sphere as function of time, as shown in Fig. <ref>.There are several prominent qualitative features in these plots.As expected, the rate at which the qubit jumps is larger for faster dragging velocities.Strikingly, these quantum jumps always diffuse in an arc that extendsopposite to the direction of rotation. This can be understood from the form of the back-action,which is zero at the poles of the measurement axis, and maximal in-between. Hence, when the state gets `pushed forward' (that is, in the direction of the rotation) by the back-action, it is pushed towards a region of lower back-action. At the same time, it cannot go past the measurement axis because the back-action goes to zero at the pole. On the other hand, if the state gets `pulled back' by the back-action, it is towards a region of higher back-action, thus having an increased probability of `escaping' and undergoing a transitionto the other side of the Bloch sphere, i.e. a quantum jump.Due to the relatively high quantum efficiency of our system, the state remains close to the surface of the Bloch sphere, and trajectories that jump arc out before arriving at the other side.A consequence of the arcing feature in the dynamics is the lagging of the average of the state behind the measurement axis. For our specific experiment the ensemble averaged dynamics can be solved analytically by going into a frame rotating at the dragging velocity v, where the measurement axis is fixed and the qubit is driven by the Hamiltonian H = (Ω / 2) σ_z,with Ω = 2 π v.In this measurement-axis frame the average qubit state evolvesaccording to dρ = -i Ω/2[σ_z,ρ] dt + Γ_D/2ℒ[σ_y] ρ dt,where the measurement axis is now fixed along the y direction, and for simplicity we drop the negligible purely dephasing term Γ_ϕ.The solutions display two characteristically different regimes: (i) Γ_D < 2 |Ω| –oscillatory with λ_± complex, and(ii) Γ_D ≥ 2 |Ω| – overdamped with λ_± real. λ_± = (-Γ_D±√(Γ_D^2 - 4 Ω^2))/2 and V⃗_± = ( 1 , (λ_± + Γ_D)/Ω) are the eigenvalues and eigenvectors respectively. In the oscillatory regime the state of the qubit oscillates with respect to the measurement axis, and thus is not dragged by the measurement. In the overdamped regime, or Zeno regime, the oscillatory behaviour vanishes and is replaced by exponential decay along the axes defined by the eigenvectors V_±.As Γ_D →∞, the eigenvalue λ_+ goes to zero, which means that if the qubit starts near a pole of V⃗_+ it will remain pinned to it for an arbitrarily long time.The slow decay for Γ_D < ∞ can be attributed to quantum jumps between the poles of V⃗_+. In each realization of the experiment these jumps can be observed. The jump axis is identified with the direction in which the damping rate is smallest, since the fast damping in the orthogonal direction aligns the poles of the jump with the slow axis. Since λ_+ ≥λ_-, the jump axis is the eigenvector V⃗_+, with a characteristic angle relative to the measurement axisθ =arctan( 2 Ω/Γ_D+√(Γ_D^2 - 4 Ω^2)).This angle characterizes the direction along which the population of the qubit concentrates, and is only defined within the Zeno regime, where dragging occurs.In such regime a qubit state close to a pole of the jump axis eventually jumps to the other pole at a rate γ_J =|λ_+|/2.Note that for slow dragging velocities, in the limit Γ_D ≫ 2|Ω|, the jump axis aligns with the measurement axis, and the jump rate converges to the familiar form Ω^2/(2Γ_D) <cit.>. Fig. <ref>a illustrates the jump axis, indicated by a red line, laggingbehind the measurement axis at an angle θ. Moreover, Fig. <ref>b and Fig. <ref>c show good agreement between theoretical and experimental ensemble dynamics in the frame of the jump axis for dragging velocities in the Zeno regime.We can see the exponentially decaying behaviour in the direction perpendicular to the jump axis, indicating that the population is aligning with it.The lagging angle between the average state and the measurement axis can be understood to arise from competition between the stochastic back-action and rotation. Without observing the measurement outcome there is an optimal initial measurement axis and rotation velocity that maximizes the fidelity with respect to a target state <cit.>. However, as the magnitude of the back-action depends on the measurement outcome, its relative size can be inferred from the measurement record. As a larger positive measurement outcome induces a larger change toward the measurement axis, one can use this effect to post-select on trajectories in which the state was pulled closer to the measurement axis. In Fig. <ref>, we show fidelity with respect to the target measurement eigenstate for various post-selection criteria. One can see that the more aggressively one post-selects on the integrated voltage, the higher the resulting fidelity. The apparent degradation in fidelity for the most negative postselections is due to insufficient statistics. Thus, measurement allows us not only to drag the state, but also to monitor its dynamics and herald arbitrarily high fidelity. The above dynamics suggest that given a `runaway' state, or an `error', the measurement axis could rotate and drive it back via a feedback protocol, achieving improved control. A feedback protocol achieving such a result has been shown <cit.>. The idea is to feedback on the measurement axis such that it is always half way between the current state and the target state.This dynamical control of the measurement operator enables novel capabilities for qubit control, such as the incoherent control demonstrated here, improved incoherent control with feedback <cit.>, rapid state purification <cit.>, and adaptive measurements <cit.>. This measurement scheme also generalizes to multi-level systems.In such multi-level settings, fast measurement rates of certain operators restrict the system to evolve within a particular subspace of the total Hilbert space, which is known as Quantum Zeno Dynamics <cit.>. Such restriction has been recently shown to enable universal quantum computation within that subspace <cit.>. Changing these subspaces dynamically through the evolution of the monitored operators is an avenue that has yet to be explored.Acknowledgments We thank Alexander Korotkov, Andrew Jordan, Juan Atalaya, Emanuel Flurin, Machiel Blok and Howard Wiseman for discussions. L.S.M. acknowledges support from the National Science Foundation (graduate fellowship grant 1106400). and from a Berkeley Fellowship for Graduate Study. This work was supported by the Air Force Office of Scientific Research (grant FA9550-12-1-0378) and the Army Research Office (grant W911NF-15-1-0496). § SUPPLEMENTAL MATERIAL§.§ Device parametersThe qubit has a transition frequency of ω_q/2π = 4.262 GHz, an energy-decay timescale of T_1 = 60 μs, and a dephasing (Ramsey decay) timescale of T_2^* = 30 μs. For this experiment we use the second lowest cavity mode, with a frequency of ω/2 π = 7.391 GHz, and a linewidth of κ/2 π = 4.3 MHz. The qubit dispersive frequency shift is χ/2 π = -0.23 MHz. §.§ Trajectory reconstructionThe stochastic master equation given in the main text is generated by the following measurement operatorΩ(V)= exp[-Γ_Dη/2(V(t)-σ_δ(t))^2 dt ]ρ(t+dt)= ℰ_1-ηΩρ(t) Ω^†/Tr[Ωρ(t) Ω^†],where ℰ_1-η_i is a superoperator which models dephasing due to finite quantum efficiency and small additional dephasing taken from the finite measured Rabi time. To ensure positivity of the state when dt is taken to be finite, we use Eq. (<ref>) to numerically propagate the quantum trajectories. The parameters Γ_D and η are calibrated independently. The former we measure by preparing |+⟩ and then performing a Ramsey measurement. We measure the quantum efficiency by preparing states |y=± 1⟩. Histograms of the integrated measurement records yield a pair of Gaussians which separate as a function of time. The quantum efficiency is given byη = (μ_y=+1 - μ_y=-1)^2/8 τσ^2 Γ_D,where μ_y=± 1 is the mean of the Gaussian for the |y=± 1⟩ state preparation, σ is the average standard deviation of the Gaussians and τ is the measurement duration <cit.>.When reconstructing the quantum trajectories and comparing the average of the trajectories to the solution for the master equation for the the average state we found a disagreement between theory and experiment. This seemed to be a systematic discrepancy due to a small offset in the detector voltage. We corrected this using an informed `guess' offset, which was calibrated from a different experiment <cit.> performed using the same setup. The offset value used is 0.17V where the separation of the mean of the Gaussians for this detector was 1.74V. In Fig. <ref> we show the comparison of the average of the trajectories with theory for 3 dragging velocities, for processing with and without the correction. In the main text we use the corrected data. | http://arxiv.org/abs/1706.08577v1 | {
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"published": "20170626200035",
"title": "Incoherent qubit control using the quantum Zeno effect"
} |
Department of Materials Science and Engineering,Kyoto University, Sakyo, Kyoto 606-8501, JapanDepartment of Materials Science and Engineering,Kyoto University, Sakyo, Kyoto 606-8501, JapanDepartment of Materials Science and Engineering,Kyoto University, Sakyo, Kyoto 606-8501, Japan For classical many-body systems, our recent study reveals thatexpectation value of internal energy, structure, and free energy can be well characterized by a single specially-selected microscopic structure. This finding relies on the fact that configurational density of states (CDOS) for typical classical system before applying interatomic interaction can be well characterized by multidimensional gaussian distribution. Although gaussian distribution is an well-known and widely-used function in diverse fields, it is quantitatively unclear why the CDOS takes gaussian when system size gets large, even for projected CDOS onto a single chosen coordination. Here we demonstrate that for equiatomic binary system, one-dimensional CDOS along coordination of pair correlation can be reasonably described by gaussian distribution under an appropriate condition, whose deviation from real CDOS mainly reflects the existence of triplet closed link consisting of the pair figure considered. The present result thus significantly makes advance in analytic determination of the special microscopic states to characterized macroscopic physical property in equilibrium state.Landscape of Configurational Density of States for Discrete Large Systems Kazuhito Takeuchi today ============================================================================§ INTRODUCTIONFor classical many-body system where internal energy is the sum of kinetic and potential energy, physical quantity (especially, dynamical variables) in equilibrium state can be obtained through thermodynamicaverage (the so-called canonical average), which includes summation taken over all microscopic states at provided composition on phase space.Since number of possible microscopic states astronominally increases with increase of system size, direct evaluation macroscopic physical property from the thermodynamic average is far from practical. To avoid such problem, several theoretical approaches have been amply developed to effectively sample important microscopic states to estimate dynamical variables, including Metropolis algorism, entropic sampling and Wang-Landau sampling.<cit.>Whereas physical quantities in equilibrium state can be reasonablly predicted by the existing theoretical approaches, a set of microscopic state to dominantly characterize equilibrium properties is generally unknown a priori, without providing information about many-body interactions or temperature. This is a natural outcome because probability to find a chosen microscopic state i is proportional to Boltzmann factor, exp(-β E_i). Despite these facts, we recently find a special set of microscopic structure (called "projection state: PS") constructed independently of information about many-body interaction and temperature, where their physical quantity can well characterize equilibrium properties including internal energy and macroscopic structure.<cit.> This finding relies on the fact that configurational density of states (CDOS) before applying many-body interaction to the system is well-characterized by multidimensional gaussian when system size gets large.<cit.> In our previous study, although condition of structure for PS is explicitly provided, analytical expression of the structure of PS on practical lattice remains unclear: The PS has been constructed based on the numerical simulation with special random sampling on configuration space. Therefore, in order to overcome such problem, we should clarify the quantitative landscape of CDOS for large systems. So far, it is not quantitatively clear why even one-dimensional CDOS along chosen coordination of pair correlation can be characterized by a gaussian-like distribution for typical periodic lattice. In the present study, we provide analytical expression of one-dimensional CDOS along selected pair figure on equiatomic binary system, by clarifying system-size dependence of any given moment of the CDOS. We demonstrate the validity of the derived expression by comparing the moment of CDOS obtained by numerical simulation for representative periodic lattices. § DERIVATION AND APPLICATIONSRecently, we reveal analytical expression for composition dependence of 2nd order moment of CDOS for pair correlation on provided lattice. In that study, we employ generalized Ising model<cit.> (GIM) to quantitatively describe microscopic structure (i.e., atomic arrangement) on lattice, whose occupation at lattice site i by A (B) element is given by the so-called spin variable, σ_i = +1 (-1). Briefly, correlation for microscopic structure σ⃗ along chosen pair m in GIM is given by ξ_m( σ⃗) = σ_i σ_k_m,lattice,where ·_m,lattice denotes taking linear average over all symmetry-equivalent pair to m on given lattice. With these preparations, we here extend our previous derivation for 2nd order moment of CDOS to any higher-order moment. In the same way to our previous approach, we start from rewriting pair correlation of Eq. (<ref>) asξ_m( σ⃗) = ( 2D_mN )^-1∑_i,k g_m( i,k ) σ_i( σ⃗) σ_k( σ⃗),where D_m and N denotes number of pair m per site and number of lattice points in the system, respectively, and summation is taken over all lattice points. g_m( i,k ) takes 1 (0) if site i and k forms pair m (for otherwise). Based on Eq. (<ref>), r-th order moment of the CDOS along pair m can be given by μ_r^(m) = ξ_m^r_σ⃗ = 1/( 2D_mN )^r∑_p_1,p_2∑_p_3,p_4⋯∑_p_2r-1,p_2r g_m( p_1,p_2 ) ⋯ g_m( p_2r-1, p_2r) σ_p_1σ_p_2⋯σ_p_2r_σ⃗, where ·_σ⃗ represents taking linear average over possible microscopic structure on given lattice. We have shown that at equiatomic composition, linear average of four spin product σ_p_1σ_p_2σ_p_3σ_p_4_σ⃗ can be treated by taking product over independently occupied spin variables depending only on composition, i.e., x=0.5: Based on the idea, we have successfully provide composition dependence of2nd order moment of CDOS for given pair. Not only in terms of considering the landscape of CDOS, but also of statistical independence of CDOS, treating multisite spin product as product of independent-occupation: Our previous study showed that density of eigenvalues for covariance matrix of practical CDOS at equiatomic composition become numerically identical to that obtained from random matrix with gaussian orthogonal ensemble, where each element of the matrix independently takes normal random numbers, which directly means that for large systems, global landscape of CDOS become close to the DOS with independently-occupied random states.When we extend the idea to higher-order moment, non-zero contribution to μ_r from 2r-spin product, σ_p_1σ_p_2⋯σ_p_2r_σ⃗, should always consists of even-times spin product for all constituent lattice points included in a set of { p_1,p_2,⋯,p_2r}, which naturally comes from symmetric definition of spin variable σ=± 1. For instance, when we estimate third order moment, six-spin product should be considered. Figure <ref> shows the possible combination of six-spin products σ_i σ_k σ_p σ_q σ_t σ_u, where lattice points connected with vertical broken line denotes that they corresponds to the same lattice point: e.g., in Fig. <ref> (a), i=t, k=p and q=u, and in (b), i=p=t and k=q=u. Among the six combinations, only (a) has non-zero contribution to μ_3, since other five contains at least one odd-times spin product in the same lattice point. For instance, (c) has one even-time product at p and t, while it has two odd-time products at i, q u and k.With this approach, we can significantly decrease the number of terms considered in Eq. (<ref>).However, when order of moment r goes infinity, the number of terms with non-zero contribution to μ_r diverges. Therefore, additional strategy should be required to quantitatively determine μ_r for any given order. When we consider thermodynamic limit of N→∞, terms containing maximum power of N in the summation ∑_p_1,p_2∑_p_3,p_4⋯∑_p_2r-1,p_2r only contributes to μ_r, since maximum power of N in the summation for r-order moment is always less than r. Based on the idea in Fig. <ref>, such maximum contributions can be straightforwardly illustrated, for even- and odd-order moment as shown in Fig. <ref>. As shown, maximum contributions to each moment should be composed of (n-1) two-pairs whose individual lattice point is the same (connected by dashed lines), and a single three-pairs (for odd moment) or a single two-pairs (for even moment). It is obvious that any other combinations of pairs always results in contribution of lower power of N to the moment. Note that open circles that do not connected by dashed lines within the figure always correspond to different lattice point in the system.Therefore, when the system size N gets large, moments for CDOS can be explicitly given byμ_2α = μ_2^α·(2α -1)!!μ_2α + 1 = μ_3·μ_2^(α -1)·_2α+1C_3 ·(2α -3)!!,where μ_2^α and μ_3·μ_2^(α -1) for even- and odd-order moment in the equation respectively comes from the maximum contribution of Fig. <ref>, and the rest terms of (2α -1)!! and _2α+1C_3 ·(2α -3)!! corresponds to the number of possible permutations to assign constituent pairs to the figure for maximum contribution of Fig. <ref>. It has been shown<cit.> that μ_2 = (D_mN)^-1 at equiatomic composition. Thus, we here should determine the rest unknown term, μ_3, as a function of D_m and N. μ_3 for m-th pair can be expressed asμ_3^(m) = 1/( 2D_mN )^3∑_i,k∑_p,q∑_t,u g_m( i,k )g_m( p,q ) g_m( t,u )σ_iσ_kσ_pσ_qσ_tσ_q_σ⃗. From Figs. <ref> and <ref>, non-zero, maximum contribution to μ_3 corresponds to the Fig. <ref> (a), therebyμ_3^(m) =2D_mN· 4 · 2 · M_m /( 2D_mN )^3 = 2 M_m/( D_mN )^2,where in the numerator of the first equation, 2D_mN corresponds to the number of ways to choose the first pair among three (e.g., i-k pair), 4· 2 to the possible permutation of lattice points for the rest two pairs in the considered figure (i.e., Fig. <ref> (a)), and M_m denotes the number of triples consisting of three m-th pairs where one of the three pair is kept fixed. Using Eqs. (<ref>) and (<ref>), we can get final expression for the moments of CDOS for m-th pair, namelyμ_2α^(m) = ( 2α-1)!!/(D_mN)^α μ_2α + 1^(m) = 2M_m·_2α+1C_3·(2α-3)!!/(D_mN)^α+1. From Eq. (<ref>), we can provide relationships between different order of moments by the following reccurence formula:μ_2α^(m) = μ_2α-2^(m)·(2α-1)·μ_2^(m) μ_2α+1^(m) = μ_2α-1^(m)·α(2α+1)/(α-1)·μ_2^(m). It is now clear that the first equation of Eq. (<ref>) is identical to the relationship of even-order moment for single-variate gaussian. Therefore, when all the odd-order moments are zero for chosen pair m, it is reasonable that the corresponding CDOS for large systems can be characterized by the gaussian. Such condition is satisfied when M_m takes zero, i.e., there is no triplet consisting of three symmetry-equivalent m-th pairs on given lattice. For instance, 1NN pair on fcc lattice have M_m=4, while 2NN have M_m=0.Finally, in order to demonstrate the validity of the derived analytical expression of the moments in Eq. (<ref>), we perform Monte Carlo (MC) simulation for equiatomic binary system along 1NN on fcc and 1NN and 2NN pair on bcc lattices, where possible microscopic structures are uniformely sampled without any statistical weight.Based on the simulation, we estimate from third- to eight-order moment as a function of number of lattice points in the system, N. The results are compared with the derived expression of Eq. (<ref>), as shown in Fig. <ref>. Note that pairs that have M_m=0 are excluded in Fig. <ref>, since theorical prediction always show N-independence, while numerical simulation should always exhibit N dependence. We can clearly see the excellent agreement of N-dependence of all the moments shown, indicating that the present theoretical approach can reasonably capture the landscape of CDOS in terms of simple geometric information of coordination number, number of specially-selected triplets and of the system size. § CONCLUSIONSFor classical discrete systems at equiatomic composition, we propose analytical expression of any-order moments of configurational density of states (CDOS) for non-interacting system.Validity of the derived expression is demonstrated by comparing the moments obtained by numerical simulation as a function of the system size.We confirm that for large systems, landscape of CDOS can be reasonablly characterized by simple geometric information such as coordination number, number of specially-selected triplets and of the system size.§ ACKNOWLEDGEMENTThis work was supported by a Grant-in-Aid for Scientific Research (16K06704) from the MEXT of Japan, Research Grant from Hitachi Metals·Materials Science Foundation, and Advanced Low Carbon Technology Research and Development Program of the Japan Science and Technology Agency (JST).9 mc1 N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Tellerand, and E. Teller, J. Chem. Phys. 21, 1087 (1953). mc2 A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989).mc3 J. Lee, Phys. Rev. Lett. 71, 211 (1993). wl F. Wang and D.P. Landau, Phys. Rev. Lett. 86, 2050 (2001). em1 K. Yuge, J. Phys. Soc. Jpn.84, 084801 (2015). em2 K. Yuge, J. Phys. Soc. Jpn. 85, 024802(2016). em3 K. Yuge, T. Kishimoto and K. Takeuchi, Trans. Mat. Res. Soc. Jpn. 41, 213 (2016). em4 T. Taikei, T. Kishimoto, K. Takeuchi and K. Yuge, J. Phys. Soc. Jpn. (submitted). ce J.M. Sanchez, F. Ducastelle, and D. Gratias, Physica A 128, 334 (1984). sqs S.-H. Wei, L. G. Ferreira, J. E. Bernard, and A. Zunger, Phys. Rev. B 42, 9622 (1990). cdos K. Yuge, T. Taikei and K. Takeuchi, Trans. Mat. Res. Soc. Jpn. (in press). | http://arxiv.org/abs/1706.08796v1 | {
"authors": [
"Koretaka Yuge",
"Tetsuya Taikei",
"Kazuhito Takeuchi"
],
"categories": [
"cond-mat.dis-nn"
],
"primary_category": "cond-mat.dis-nn",
"published": "20170627115219",
"title": "Landscape of Configurational Density of States for Discrete Large Systems"
} |
An optimization technique on pseudorandom generators based on chaotic iterations Jacques M. Bahi, Xiaole Fang, and Christophe Guyeux* FEMTO-ST Institute, UMR 6174 CNRSUniversity of Franche-Comté, Besançon, FranceEmail:{jacques.bahi, xiaole.fang, christophe.guyeux}@univ-fcomte.frDecember 30, 2023 =================================================================================================================================================================================================================Internet communication systems involving cryptography and data hiding often require billions of random numbers. In addition to the speed of the algorithm, the quality of the pseudo-random number generator and the ease of its implementation are common practical aspects. In this work we will discuss how to improve the quality of random numbers independently from their generation algorithm. We propose an additional implementation technique in order to take advantage of some chaotic properties. The statistical quality of our solution stems from some well-defined discrete chaotic iterations that satisfy the reputed Devaney's definition of chaos, namely the chaotic iterations technique. Pursuing recent researches published in the previous International Conference on Evolving Internet (Internet 09, 10, and 11), three methods to build pseudorandom generators by using chaotic iterations are recalled. Using standard criteria named NIST and DieHARD (some famous batteries of tests), we will show that the proposed technique can improve the statistical properties of a large variety of defective pseudorandom generators, and that the issues raised by statistical tests decrease when the power of chaotic iterations increase. Internet security; Pseudorandom Sequences; Statistical Tests; Discrete Chaotic Iterations; Topological Chaos. § INTRODUCTION Chaos has recently attracted more and more interests from researchers in the fields of mathematics, physics, and computer engineering, among other things due to its connection with randomness and complexity <cit.>. In particular, various research works have recently regarded the possibility to use chaos in random number generation for Internet security. Indeed, the security of data exchanged through the Internet is highly dependent from the quality of the pseudorandom number generators (PRNGs) used into its protocols. These PRNGs are everywhere in any secure Internet communication: in the keys generation of any asymmetric cryptosystem, in the production of any keystream (symmetric cryptosystem), the generation of nonce, in the keys for keyed hash functions, and so on.Numerous pseudorandom number generators already exist, but they are either secure but slow, or fast but insecure. This is why the idea to mix secure and fast PRNGs, to take benefits from their respective qualities, has emerged these last years <cit.>. Chaotic dynamical systems appear as good candidates to achieve this mixture for optimization. Indeed, chaotic systems have many advantages as unpredictability or disorder-like, which are required in building complex sequences <cit.>. This is why chaos has been applied to secure optical communications <cit.>. But chaotic systems of real-number or infinite bit representation realized in finite computing precision lead to short cycle length, non-ideal distribution, and other deflation of this kind. This is the reason of that chaotic systems on an infinite space of integers have been looked for these last years, leading to the proposition to use chaotic iterations (CIs) techniques to reach the desired goals. More precisely, we have proposed in INTERNET 2009 <cit.> to mix two given PRNGs by using chaotic iterations, being some particular kind of discrete iterations of a vectorial Boolean function. This first proposal has been improved in INTERNET 2010 <cit.> and INTERNET 2011 <cit.>, to obtain a new family of statistically perfect and fast PRNGs. A short overview of these previous researches is given thereafter.In <cit.>, CIs have been proven to be a suitable tool for fast computing iterative algorithms on integers satisfying the topological chaotic property, as it has been defined by Devaney <cit.>. A first way to mix two given generators by using these chaotic iterations, called Old CIPRNGs, has been proposed in Internet 09 <cit.> and further investigated in <cit.>. It was chaotic and able to pass the most stringent batteries of tests, even if the inputted generators were defective.This claim has been verified experimentally, by evaluating the scores of the logistic map, XORshift, and ISAAC generators through these batteries, when considering them alone or after chaotic iterations. Then, in <cit.>, a new version of this family has been proposed. This “New CIPRNG” family uses a decimation of strategies leading to the improvement of both speed and statistical qualities. Finally, efficient implementations onGPU using a last family called Xor CIPRNG have been designed in <cit.>, showing that a very large quantity of pseudorandom numbers can be generated per second (about 20 Gsamples/s). In this paper, the statistical analysis of the three methods mentioned above are carried out systematically, and the results are discussed. Indeed PRNGs are often based on modular arithmetic, logical operations like bitwise exclusive or (XOR), and on circular shifts of bit vectors. However the security level of some PRNGs of this kind has been revealed inadequate by today's standards. Since different biased generators can possibly have their own side effects when inputted into our mixed generators, it is normal to enlarge the set of tested inputted PRNGs, to determine if the observed improvement still remains. We will thus show in this research work that the intended statistical improvement is really effective for all of these most famous generators.The remainder of this paper is organized in the following way. In Section <ref>, some basic definitions concerning chaotic iterations are recalled. Then, four major classes of general PRNGs are presented in Section <ref>. Section <ref> is devoted to two famous statistical tests suites. In Section <ref>, various tests are passed with a goal to achieve a statistical comparison between our CIPRNGs and other existing generators. The paper ends with a conclusion and intended future work. § CHAOTIC ITERATIONS APPLIED TO PRNGSIn this section, we describe the CIPRNG implementation techniques that can improve the statistical properties of any generator. They all are based on CIs, which are defined below. §.§ NotationsS^n → the n^th term of a sequence S=(S^1,S^2,) v_i → the i^th component of a vector v=(v_1,, v_n)f^k → k^th composition of a function fstrategy → a sequence which elements belong in 1;𝖭 𝕊 → the set of all strategies 𝐂_n^k → the binomial coefficient nk = n!/k!(n-k)!⊕ → bitwise exclusive or ≪and≫ → the usual shift operators (𝒳, d) → a metric spaceLCM(a, b) → the least common multiple of a and b§.§ Chaotic iterationsThe set B denoting {0,1}, let f:B^𝖭 ⟶B^𝖭 be an “iteration” function and S∈𝕊 be a chaotic strategy. Then, the so-called chaotic iterations are defined by x^0∈B^𝖭, and[ ∀ n∈N^∗,∀ i∈1;𝖭 ,x_i^n={[x_i^n-1 if S^n≠ i; f(x^n-1)_S^n if S^n=i. ]. ] In other words, at the n^th iteration, only the S^n-th cell is “iterated”. §.§ The CIPRNG family §.§.§ Old CIPRNG Let 𝖭 = 4. Some chaotic iterations are fulfilled to generate a sequence (x^n)_n∈N∈(B^4)^N of Boolean vectors: the successive states of the iterated system. Some of these vectors are randomly extracted and their components constitute our pseudorandom bit flow <cit.>. Chaotic iterations are realized as follows. Initial state x^0 ∈B^4 is a Boolean vector taken as a seed and chaotic strategy (S^n)_n∈N∈ 1, 4 ^N is constructed with PRNG_2. Lastly, iterate function f is the vectorial Boolean negation. At each iteration, only the S^n-th component of state x^n is updated. Finally, some x^n are selected by a sequence m^n, provided by a second generator PRNG_1, as the pseudorandom bit sequence of our generator.The basic design procedure of the Old CI generator is summed up in Algorithm <ref>. The internal state is x, the output array is r. a and b are those computed by PRNG_1 and PRNG_2. §.§.§ New CIPRNG The New CI generator is designed by the following process <cit.>. First of all, some chaotic iterations have to be done to generate a sequence (x^n)_n∈N∈(B^32)^N of Boolean vectors, which are the successive states of the iterated system. Some of these vectors will be randomly extracted and our pseudo-random bit flow will be constituted by their components. Such chaotic iterations are realized as follows. Initial state x^0 ∈B^32 is a Boolean vector taken as a seed and chaotic strategy (S^n)_n∈N∈ 1, 32 ^N is an irregular decimation of PRNG_2 sequence, as described in Algorithm <ref>.Another time, at each iteration, only the S^n-th component of state x^n is updated, as follows: x_i^n = x_i^n-1 if i ≠ S^n, else x_i^n = x_i^n-1. Finally, some x^n are selected by a sequence m^n as the pseudo-random bit sequence of our generator. (m^n)_n ∈N∈ℳ^N is computed from PRNG_1, where ℳ⊂N^* is a finite nonempty set of integers.The basic design procedure of the New CI generator is summarized in Algorithm <ref>. The internal state is x, the output state is r. a and b are those computed by the two input PRNGs. Lastly, the value g_1(a) is an integer defined as in Eq. <ref>. m^n = g_1(y^n)= {[ 0if 0 ⩽y^n<C^0_32,;1if C^0_32⩽y^n<∑_i=0^1C^i_32,; 2if ∑_i=0^1C^i_32⩽y^n<∑_i=0^2C^i_32,;⋮ ⋮; Nif ∑_i=0^N-1C^i_32⩽y^n<1.;].§.§.§ Xor CIPRNG Instead of updating only one cell at each iteration as Old CI and New CI, we can try to choose a subset of components and to update them together. Such an attempt leads to a kind of merger of the two random sequences. When the updating function is the vectorial negation, this algorithm can be rewritten as follows <cit.>: {[ x^0 ∈ 0, 2^𝖭-1 , S ∈ 0, 2^𝖭-1 ^N;∀ n ∈N^*, x^n = x^n-1⊕ S^n, ]. The single basic component presented in Eq. <ref> is of ordinary use as a good elementary brick in various PRNGs. It corresponds to the discrete dynamical system in chaotic iterations.§ ABOUT SOME WELL-KNOWN PRNGS §.§ Introduction Knowing that there is no universal generator, it is strongly recommended to test a stochastic application with a large set of different PRNGs <cit.>. They can be classified in four major classes: linear generators, lagged generators, inversive generators, and mix generators:* Linear generators, defined by a linear recurrence, are the most commonly analyzed and utilized generators. The main linear generators are LCGs and MLCG. * Lagged generators have a general recursive formula that use various previously computed terms in the determination of the new sequence value. * Inversive congruential generators form a recent class of generators that are based on the principle of congruential inversion. * Mixed generators result from the need for sequences of better and better quality, or at least longer periods. This has led to mix different types of PRNGs, as follows: x^i=y^i⊕ z^iFor instance, inversive generators are very interesting for verifying simulation results obtained with a linear congruential generator (LCG), because their internal structure and correlation behavior strongly differs from what LCGs produce. Since these generators have revealed several issues, some scientists refrain from using them. In what follows, chaotic properties will be added to these PRNGs, leading to noticeable improvements observed by statistical test.Let us firstly explain with more details the generators studied in this research work (for a synthetic view, see Fig. <ref>). §.§ Details of some Existing Generators Here are the modules of PRNGs we have chosen to experiment.§.§.§ LCGThis PRNG implements either the simple or the combined linear congruency generator (LCGs). The simple LCG is defined by the recurrence:x^n = (ax^n-1 + c) mod mwhere a, c, and x^0 must be, among other things, non-negative and less than m <cit.>. In what follows, 2LCGs and 3LCGs refer as two (resp. three) combinations of such LCGs. For further details, see <cit.>.§.§.§ MRGThis module implements multiple recursive generators (MRGs), based on a linear recurrence of order k, modulo m <cit.>:x^n = (a^1x^n-1+ ... +a^kx^n-k) mod mCombination of two MRGs (referred as 2MRGs) is also be used in this paper.§.§.§ UCARRYGenerators based on linear recurrences with carry are implemented in this module. This includes the add-with-carry (AWC) generator, based on the recurrence:[ x^n = (x^n-r + x^n-s + c^n-1) mod m,;c^n= (x^n-r + x^n-s + c^n-1) / m, ]the SWB generator, having the recurrence:[ x^n = (x^n-r - x^n-s - c^n-1) mod m,; c^n={[ 1 if (x^i-r - x^i-s - c^i-1)<0; 0 else, ]. ]and the SWC generator designed by R. Couture, which is based on the following recurrence:[ x^n = (a^1x^n-1⊕ ... ⊕ a^rx^n-r⊕ c^n-1) mod 2^w,; c^n = (a^1x^n-1⊕ ... ⊕ a^rx^n-r⊕ c^n-1) / 2^w. ] §.§.§ GFSRThis module implements the generalized feedback shift register (GFSR) generator, that is:x^n = x^n-r⊕ x^n-k §.§.§ INVFinally, this module implements the nonlinear inversive generator, as defined in <cit.>, which is: [ x^n={[ (a^1 + a^2 / z^n-1) mod mif z^n-1≠ 0; a^1if z^n-1 = 0 . ]. ] § STATISTICAL TESTS Considering the properties of binary random sequences, various statistical tests can be designed to evaluate the assertion that the sequence is generated by a perfectly random source. We have performed some statistical tests for the CIPRNGs proposed here. These tests include NIST suite <cit.> and DieHARD battery of tests <cit.>. For completeness and for reference, we give in the following subsection a brief description of each of the aforementioned tests. §.§ NIST statistical tests suite Among the numerous standard tests for pseudo-randomness, a convincing way to show the randomness of the produced sequences is to confront them to the NIST (National Institute ofStandards and Technology) statistical tests, being an up-to-date tests suite proposed by the Information Technology Laboratory (ITL). A new version of the Statistical tests suite has been released in August 11, 2010.The NIST tests suite SP 800-22 is a statistical package consisting of 15 tests. They were developed to test the randomness of binary sequences produced by hardware or software based cryptographic pseudorandom number generators. These tests focus on a variety of different types of non-randomness that could exist in a sequence.For each statistical test, a set of P-values (corresponding to the set of sequences) is produced. The interpretation of empirical results can be conducted in various ways. In this paper, the examination of the distribution of P-values to check for uniformity (P-value_T) is used. The distribution of P-values is examined to ensure uniformity. If P-value_T⩾ 0.0001, then the sequences can be considered to be uniformly distributed.In our experiments, 100 sequences (s = 100), each with 1,000,000-bit long, are generated and tested. If the P-value_T of any test is smaller than 0.0001, the sequences are considered to be not good enough and the generating algorithm is not suitable for usage. §.§ DieHARD battery of testsThe DieHARD battery of tests has been the most sophisticated standard for over a decade. Because of the stringent requirements in the DieHARD tests suite, a generator passing this battery of tests can be considered good as a rule of thumb.The DieHARD battery of tests consists of 18 different independent statistical tests. This collectionof tests is based on assessing the randomness of bits comprising 32-bit integers obtained from a random number generator. Each test requires 2^23 32-bit integers in order to run the full set of tests. Most of the tests in DieHARD return a P-value, which should be uniform on [0,1) if the input file contains truly independent random bits.These P-values are obtained by P=F(X), where F is the assumed distribution of the sample random variable X (often normal). But that assumed F is just an asymptotic approximation, for which the fit will be worst in the tails. Thus occasional P-values near 0 or 1, such as 0.0012 or 0.9983, can occur. An individual test is considered to be failed if the P-value approaches 1 closely, for example P>0.9999. § RESULTS AND DISCUSSION Table <ref> shows the results on the batteries recalled above, indicating that almost all the PRNGs cannot pass all their tests. In other words, the statistical quality of these PRNGs cannot fulfill the up-to-date standards presented previously. We will show that the CIPRNG can solve this issue.To illustrate the effects of this CIPRNG in detail, experiments will be divided in three parts: * Single CIPRNG: The PRNGs involved in CI computing are of the same category.* Mixed CIPRNG: Two different types of PRNGs are mixed during the chaotic iterations process.* Multiple CIPRNG: The generator is obtained by repeating the composition of the iteration function as follows: x^0∈B^𝖭, and ∀ n∈N^∗,∀ i∈1;𝖭,[ x_i^n={[x_i^n-1 if S^n≠ i; ∀ j∈1;𝗆,f^m(x^n-1)_S^nm+j if S^nm+j=i. ]. ]m is called the functional power.We have performed statistical analysis of each of the aforementioned CIPRNGs. The results are reproduced in Tables <ref> and <ref>. The scores written in boldface indicate that all the tests have been passed successfully, whereas an asterisk “*” means that the considered passing rate has been improved. §.§ Tests based on the Single CIPRNG The statistical tests results of the PRNGs using the single CIPRNG method are given in Table <ref>. We can observe that, except for the Xor CIPRNG, all of the CIPRNGs have passed the 15 tests of the NIST battery and the 18 tests of the DieHARD one. Moreover, considering these scores, we can deduce that both the single Old CIPRNG and the single New CIPRNG are relatively steadier than the single Xor CIPRNG approach, when applying them to different PRNGs. However, the Xor CIPRNG is obviously the fastest approach to generate a CI random sequence, and it still improves the statistical properties relative to each generator taken alone, although the test values are not as good as desired.Therefore, all of these three ways are interesting, for different reasons, in the production of pseudorandom numbers and, on the whole, the single CIPRNG method can be considered to adapt to or improve all kinds of PRNGs.To have a realization of the Xor CIPRNG that can pass all the tests embedded into the NIST battery, the Xor CIPRNG with multiple functional powers are investigated in Section <ref>. §.§ Tests based on the Mixed CIPRNG To compare the previous approach with the CIPRNG design that uses a Mixed CIPRNG, we have taken into account the same inputted generators than in the previous section. These inputted couples (PRNG_1,PRNG_2) of PRNGs are used in the Mixed approach as follows:{[ x^0 ∈ 0, 2^𝖭-1 , S ∈ 0, 2^𝖭-1 ^N; ∀ n ∈N^*, x^n = x^n-1⊕ PRNG_1⊕ PRNG_2, ]. With this Mixed CIPRNG approach, both the Old CIPRNG and New CIPRNG continue to pass all the NIST and DieHARD suites. In addition, we can see that the PRNGs using a Xor CIPRNG approach can pass more tests than previously. The main reason of this success is that the Mixed Xor CIPRNG has a longer period. Indeed, let n_P be the period of a PRNG P, then the period deduced from the single Xor CIPRNG approach is obviously equal to:n_SXORCI= {[n_P if x^0=x^n_P; 2n_P if x^0≠ x^n_P.;]. Let us now denote by n_P1 and n_P2 the periods of respectively the PRNG_1 and PRNG_2 generators, then the period of the Mixed Xor CIPRNG will be:n_XXORCI= {[LCM(n_P1,n_P2) if x^0=x^LCM(n_P1,n_P2); 2LCM(n_P1,n_P2) if x^0≠ x^LCM(n_P1,n_P2).; ]. In Table <ref>, we only show the results for the Mixed CIPRNGs that cannot pass all DieHARD suites (the NIST tests are all passed). It demonstrates that Mixed Xor CIPRNG involving LCG, MRG, LCG2, LCG3, MRG2, or INV cannot pass the two following tests, namely the “Matrix Rank 32x32” and the “COUNT-THE-1's” tests contained into the DieHARD battery. Let us recall their definitions: * Matrix Rank 32x32. A random 32x32 binary matrix is formed, each row having a 32-bit random vector. Its rank is an integer that ranges from 0 to 32. Ranks less than 29 must be rare, and their occurences must be pooled with those of rank 29. To achieve the test, ranks of 40,000 such random matrices are obtained, and a chisquare test is performed on counts for ranks 32,31,30 and for ranks ≤29.* COUNT-THE-1's TEST Consider the file under test as a stream of bytes (four per2 bit integer).Each byte can contain from 0 to 8 1's, with probabilities 1,8,28,56,70,56,28,8,1 over 256.Now let the stream of bytes provide a string of overlapping5-letter words, each “letter” taking values A,B,C,D,E. The letters are determined by the number of 1's in a byte: 0,1, or 2 yield A, 3 yields B, 4 yields C, 5 yields D and 6,7, or 8 yield E. Thus we have a monkey at a typewriter hitting five keys with various probabilities (37,56,70,56,37 over 256).There are 5^5 possible 5-letter words, and from a string of 256,000 (over-lapping) 5-letter words, counts are made on the frequencies for each word. The quadratic form in the weak inverse of the covariance matrix of the cell counts provides a chisquare test: Q5-Q4, the difference of the naive Pearson sums of (OBS-EXP)^2/EXP on counts for 5- and 4-letter cell counts. The reason of these fails is that the output of LCG, LCG2, LCG3, MRG, and MRG2 under the experiments are in 31-bit. Compare with the Single CIPRNG, using different PRNGs to build CIPRNG seems more efficient in improving random number quality (mixed Xor CI can 100% pass NIST, but single cannot). §.§ Tests based on the Multiple CIPRNGUntil now, the combination of at most two input PRNGs has been investigated. We now regard the possibility to use a larger number of generators to improve the statistics of the generated pseudorandom numbers, leading to the multiple functional power approach. For the CIPRNGs which have already pass both the NIST and DieHARD suites with 2 inputted PRNGs (all the Old and New CIPRNGs, and some of the Xor CIPRNGs), it is not meaningful to consider their adaption of this multiple CIPRNG method, hence only the Multiple Xor CIPRNGs, having the following form, will be investigated.{[ x^0 ∈ 0, 2^𝖭-1 , S ∈ 0, 2^𝖭-1 ^N; ∀ n ∈N^*, x^n = x^n-1⊕ S^nm⊕ S^nm+1…⊕ S^nm+m-1 , ]. The question is now to determine the value of the threshold m (the functional power) making the multiple CIPRNG being able to pass the whole NIST battery. Such a question is answered in Table <ref>.§.§ Results Summary We can summarize the obtained results as follows. * The CIPRNG method is able to improve the statistical properties of a large variety of PRNGs.* Using different PRNGs in the CIPRNG approach is better than considering several instances of one unique PRNG.* The statistical quality of the outputs increases with the functional power m. § CONCLUSION AND FUTURE WORK In this paper, we first have formalized the CI methods that has been already presented inprevious Internet conferences. These CI methods are based on iterations that have been topologically proven as chaotic. Then 10 usual PRNGs covering all kinds of generators have been applied, and the NIST and DieHARD batteries have been tested. Analyses show that PRNGs using the CIPRNG methods do not only inherit the chaotic properties of the CI iterations, they also have improvements of their statistics. This is why CIPRNG techniques should be considered as post-treatments on pseudorandom number generators to improve both their randomness and security.In future work, we will try to enlarge this study, by considering a larger variety of tests. The CIPRNG's chaotic behavior will be deepened by using some specific tools provided by the mathematical theory of chaos. Finally, a large variety of Internet usages, as cryptography and data hiding, will be considered for applications.plain | http://arxiv.org/abs/1706.08773v1 | {
"authors": [
"Jacques M. Bahi",
"Xiaole Fang",
"Christophe Guyeux"
],
"categories": [
"cs.CR",
"nlin.CD"
],
"primary_category": "cs.CR",
"published": "20170627110244",
"title": "An optimization technique on pseudorandom generators based on chaotic iterations"
} |
UWThPh-2017-13 MIT-CTP 4915a,b]André H. Hoang a]Christopher Lepenik a,c]Moritz Preisser[a]University of Vienna, Faculty of Physics Boltzmanngasse 5, A-1090 Wien, Austria [b]Erwin Schrödinger International Institute for Mathematical Physics,University of Vienna, Boltzmanngasse 9, A-1090 Wien, Austria [c]Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, [email protected] [email protected] [email protected] We provide a systematic renormalization group formalism for the mass effects in the relation of the pole mass m_Q^ pole and short-distance masses such as themass _Q of a heavy quark Q, coming from virtual loop insertions of massive quarks lighter than Q.The formalism reflects the constraints from heavy quark symmetry and entails a combined matching and evolution procedure that allows to disentangle and successively integrate out the corrections coming from the lighter massive quarks and the momentum regions between them and to precisely control the large order asymptotic behavior. With the formalism we systematically sum logarithms of ratios of the lighter quark masses and m_Q, relate the QCD corrections for different external heavy quarks to each other, predict the O(α_s^4) virtual quark mass corrections in the pole- mass relation, calculate the pole mass differences for the top, bottom and charm quarks with a precision of around 20 MeV and analyze the decoupling of the lighter massive quark flavors at large orders. The summation of logarithms is most relevant for the top quark pole mass m_t^ pole, where the hierarchy to the bottom and charm quarks is large. We determine the ambiguity of the pole mass for top, bottom and charm quarks in different scenarios with massive or masslessbottom and charm quarks in a way consistent with heavy quark symmetry, and we find that it is 250 MeV. The ambiguity is larger than current projections for the precision of top quark mass measurements in the high-luminosity phase of the LHC.On the Light Massive Flavor Dependence of the Large Order Asymptotic Behavior and the Ambiguityof the Pole Mass [ December 30, 2023 =================================================================================================================§ INTRODUCTIONThe masses of the heavy charm, bottom and top quarks belong to the most important input parameters in precise theoretical predictions of the Standard Model and models of new physics. Due to the effects of quantum chromodynamics (QCD) and because quarks are states with color charge, however, the mass of a heavy quark Q is not a physical observable and should, in general, be better thought of as a renormalized and scheme-dependent parameter of the theory. This concept is incorporated most cleanly in the so-calledmass _Q(μ), which is defined through the same renormalization prescription as theQCD coupling α_s(μ). It can be measured from experimental data very precisely, but does not have any kinematic meaning, and it can be thought of incorporating short-distance information on the mass from scales larger than μ. On the other hand, the so-called pole mass m_Q^ pole is defined as the single particle pole in correlation functions involving the massive quark Q as an external on-shell particle, and it determines the kinematic mass of the quark Q in the context of perturbation theory. It is therefore unavoidable that the pole mass scheme appears in one way or another in higher order QCD calculations involving external massive quarks. For perturbative predictions involving the production of top quarks at hadron colliders, the pole mass scheme is therefore the main top quark mass scheme used in the literature, and switching scheme is cumbersome since these computations are predominantly numerical where the pole scheme provides the most efficient approach for the computations. In Refs. <cit.> the relation between theand the pole mass has been computed up to O(α_s^4) in the approximation that all quarks lighter than Q are massless. Assuming the values _t≡_t(_t)=163 GeV, _b≡_b(_b)=4.2 GeV and _c≡_c(_c)=1.3 GeV we obtain[ We assume α^(5)(M_Z)=0.1180 for M_Z=91.187 GeV for theQCD coupling and account for 5-loop evolution <cit.> and flavor matching at the scales _c,b,t <cit.>, which gives α_s^(6)(_t)=0.10847, α_s^(5)(_b)=0.22430, α_s^(4)(_c)=0.38208.]m_t^ pole = 163+7.5040 + 1.6005+ 0.4941+(0.1944± 0.0004),m_b^ pole = 4.2+0.3998 + 0.1986+ 0.1443+(0.1349± 0.0002),m_c^ pole = 1.3+0.2108 + 0.1984+ 0.2725+(0.4843± 0.0005),where the terms show the series in powers of the strong coupling α_s(_Q) in thescheme that includes Q as a dynamical flavor. The fourth order coefficient displays the numerical uncertainties from <cit.>, which are, however, much smaller than other types of uncertainties considered in this paper. The pole mass renormalization scheme is infrared-safe and gauge-invariant <cit.>, but suffers from large corrections in the QCD perturbation series. This is because the pole mass scheme involves subtractions of on-shell quark self energy corrections containing virtual gluon and massless quark fluctuations which are linearly sensitive to small momenta. The on-shell approximation of the self energy diagrams entails that this sensitivity increases strongly with the order. The effect this has for the form of the corrections can be seen in Eqs. (<ref>)–(<ref>), which in the asymptotic large order limit have the formm_Q^ pole-_Q(_Q)∼μ ∑_n=0^∞16/3(2β_0^())^n n! (α_s^()(μ)/4π)^n+1 ,in the β_0/LL approximation, which means that the terms in the QCD β-function,α_s^()(μ)/logμ=β^()(α_s(μ)) = - 2 α_s^()(μ)∑_n=0^∞β_n^()(α_s^()(μ)/4π)^n+1 ,beyond the leading logarithmic level (i.e. β_n>0) are neglected. Hereis the number of massless quark flavors. The factorially diverging pattern of the perturbation series and the linear dependence on the renormalization scale μ of the strong coupling displayed in Eq. (<ref>) are called the O(Λ_ QCD) renormalon of the pole mass <cit.>. The form of the series on the RHS of Eq. (<ref>) implies that at asymptotic large orders, and up to terms suppressed by inverse powers of n, the series becomes independent of its intrinsic physical scale m_Q. This and the n-factorial growth is an artifact of the pole mass scheme itself and not related to any physical effect. Technically this issue entails that for computing differences of series containing O(Λ_ QCD) renormalon ambiguities using fixed-order perturbation theory one must consistently expand in powers of the strong coupling at the same renormalization scale such that the renormalon can properly cancel. The O(Λ_ QCD) renormalon problem of the pole mass has received substantial attention in the literature as it turned out to benot just an issue of pedagogical interest, but one that is relevant phenomenologically <cit.>. This is because for μ=_Q the known coefficients of the series in Eqs. (<ref>)–(<ref>) agree remarkably well with the corresponding large order asymptotic behavior already beyond the terms of O(α_s) (so that the terms of the series are known quite precisely to all orders) and because even for orders where the QCD corrections still decrease with order they can be very large numerically and make phenomenological applications difficult. The pole mass scheme has therefore been abandoned in high precision top, bottom and charm quark mass analyses in favor of quark mass schemes such asor low-scale short distance masses such as the kinetic mass <cit.>, the potential-subtracted (PS) mass <cit.>, the 1S mass <cit.>, the renormalon-subtracted (RS) mass <cit.>, the jet mass <cit.> or the MSR mass <cit.>. These mass schemes do not have an O() renormalon and are called short-distance masses.It is commonly agreed from many studies that it is possible to determine short-distance masses with theoretical uncertainties of a few 10 MeV <cit.>, and we therefore neglect any principle ambiguity in their values in this paper.Using the theory of asymptotic series one can show that the best possible approximation to the LHS of Eq. (<ref>) is to truncate the series on the RHS at the minimal term at order n_ min which is approximately n_ min≈ 2π/(β_0^()α_s^()(μ)). The size of the correction of the minimal term is approximately Δ(n_ min)≈(4πα_s^()(μ)/β_0^())^1/2Λ_ QCD^(), and there is a region in the orders n around n_ min of width Δ n≈(2π^2/(β_0^()α_s^()(μ)))^1/2 in which all series terms have a size close to the minimal term. At orders above n_ min+Δ n/2 the series diverges quickly and the series terms from these orders are useless even if they are known through an elaborate loop calculation. The uncertainty with which the pole mass can be determined in principle given the full information about the perturbative series is called the pole mass ambiguity. It is universal, independent on the choice of the renormalization scale μ and exists in equivalent size in any context without the possibility to be circumvented. However, the μ-dependence of n_ min, Δ(n_ min) and Δ n indicates that the way how the renormalon problem appears in practical applications based on perturbative QCD can differ substantially depending on the physical scale of the quantity under consideration and the corresponding choice of the renormalization scale μ. Using the method of Borel resummation the pole mass ambiguity can be estimated to be of order Λ_ QCD^(), where the superscript () stands for the dependence of the hadronization scale on the number of massless quark flavors. The norm of the ambiguity,which we call N_1/2^() in this paper, and the resulting pattern of the large order asymptotic behavior of the series can be determined very precisely and have been studied in many analyses (see e.g. the recent work of Refs. <cit.>). However, when quoting a concrete numerical size of the ambiguity, criteria common for converging series cannot be applied, and it is instrumental to consider more global aspects of the series and the quantity it describes. An essential aspect of the low-energy quantum corrections in heavy quark masses is heavy quark symmetry (HQS) <cit.> on which we put particular focus in this work.An issue that has received less attention in the literature so far is how the masses of the lighter massive quarks affect the large order asymptotic behavior of the pole- mass relation, where we refer to the effects of quarks with masses that are larger than Λ_ QCD.These corrections come from insertions of virtual quark loops and are known up to O(α_s^3) <cit.> from explicit loop calculations. It is known that the masses of lighter massive quarks provide an infrared cutoff and effectively reduce the number n_ℓ of massless flavors governing the large order asymptotic behavior <cit.>. Due to the -dependence of the QCD β-function the finite bottom and charm quark masses lead to an increased infrared sensitivity of the top quark pole mass and a stronger divergence pattern of the series, as can be seen from Eq. (<ref>). The ambiguity therefore inflates following the -dependent increase of Λ_ QCD. In Refs. <cit.> it was pointed out that the O(α_s^2) and O(α_s^3) virtual quark mass corrections are already dominated by the infrared behavior related to the O(Λ_ QCD) renormalon. In Ref. <cit.> it was further observedthat the O(α_s^3) charm mass corrections in the bottom pole- mass relation can be rendered small when the series is expressed in terms of α_s^(=3) rather than α_s^(=4), i.e. the charm quark effectively decouples. A systematic and precise understanding of the intrinsic structure of the lighter massive quark effects from the point of view of disentangling the different momentum modes and their interplay has, however, not been provided so far in the literature. The task is complicated since apart from being a problem in connection with the behavior of perturbation theory at large orders, it also represents a multi-scale problem with scales given by the quark masses as well as Λ_ QCD and where, for the top quark, logarithms of mass ratios can be large. It is the main purpose of this paper to present a formalism that can do exactly that. It is based on the concept of the renormalization group (RG) and allows to successively integrate out momentum modes from the pole- mass relation of a heavy quark Q in order to disentangle the contributions coming from the lighter massive quarks and to systematically sum logarithms of the mass ratios. The approach allows to quantify and formulate precisely the effects the masses of the lighter massive quarks have on the pole- mass relation and therefore on the pole mass itself and may find interesting applications in other contexts. As the essential new feature the RG formalism entails linear scaling with the renormalization scale. The common logarithmic scaling, as known for the strong coupling, cannot capture the linear momentum dependence of QCD corrections to the heavy quark mass for scales below m_Q.The formalism is in particular useful since it fully accounts for all aspects of HQS. It can be used to concretely formulate and study in a transparent way two important properties of the heavy quark pole masses following from HQS:(1) The pole mass ambiguity is independent of the mass of the heavy quark and (2) the ambiguities of all heavy quarks are equal up to power corrections of order Λ_ QCD^2/m_Q. The essential technical tool to set up the formalism is the MSR mass m_Q^ MSR(R) <cit.>. Like the perturbative series for the pole- mass relation, the pole-MSR mass relation is calculated from on-shell heavy quark self energy diagrams, but has also linear dependence on R. It is the basis of the RG formalism we propose, allows to precisely capture the QCD corrections from the different quark mass scales and, in particular, to encode and study issue (1) coming from HQS. The renormalization group evolution in the scale R is described by R-evolution <cit.>, which is free of the O(Λ_ QCD) renormalon, and allows to sum large logarithms of ratios of the quark masses in the evolution between the quark mass scales. Using the concepts of the MSR mass and the R-evolution it is then possible to relate the pole- masses of the top, bottom and charm quarks to each other. This allows to systematically encode and study issue (2) coming from HQS, and to interpret the small effects of HQS breaking as matching corrections in a renormalization group flow that connects the QCD correction of the top, bottom and charm quarks.The resulting formula can be used to specify the heavy quark pole mass ambiguity in the context of lighter massive quarks and to derive a generalized expression for the large order asymptotic behavior accounting accurately for the light massive flavor dependence. Concerning the accuracy of our description of the virtual quark loop mass effects in the large order asymptotic behavior we reach a precision of a few MeV, which applies equally for top, bottom and charm quarks.The second main purpose of this paper is to use the RG formalism to specify concretely the ambiguity of the top quark pole mass and also the pole mass of the bottom and charm quarks assuming thattheirmasses are given. We in particular address the question how the outcome depends on different scenarios for treating the bottom and charm quarks as massive or massless, and we explicitly take into account the consistency requirements of HQS. The aim is to provide a concrete numerical specification of the ambiguity of the top quark pole mass beyond the qualitative statement that the ambiguity is “of order Λ_ QCD^()” and to make a concrete statement up to which principle precision the top quark pole mass may still be used as a meaningful phenomenological parameter. We stress that in this context we adopt the view that the pole masses have well-defined and unique meaning, so that the pole mass ambiguity acquires the meaning of an intrinsic numerical uncertainty. This differs from the view sometimes used in high-precision analyses, where the pole mass is employed as an intrinsic order-dependent parameter to effectively parameterize the use of a short-distance mass scheme. Apart from specifying the ambiguity of the pole masses we are also interested in studying the dependence of their value on the different scenarios for treating the bottom and charm quarks as massive or massless.The issue is of particular interest for the top quark pole mass which is still widely used for theoretical predictions and phenomenological studies in top quark physics. The top quark pole mass is, due to its linear sensitivity to small momenta, also linearly sensitive to the masses of the lighter massive quarks. Since many short-distance observables used for top quark pole mass determinations are at most quadratically sensitive to small momenta, the dominant effects of the bottom and charm masses may well come from the top quark pole definition itself. A large dependence of the top quark pole mass value on whether the bottom and charm quarks are treated as massive or massless would therefore affect the ambiguity estimate if one considers the top quark pole mass as a globally defined mass scheme (valid for any scenario for the bottom and charm quark masses).We can address this question precisely because the RG-formalism we use allows for very accurate numerical calculations of the lighter quark mass effects. Within the size of the ambiguity, we do not find any such dependence. The outcome of our analysis is that the top quark pole mass ambiguity, and the ambiguity of the bottom and charm quark pole masses, is around 250 MeV. Prior to this work the best estimate and the ambiguity of the top quark pole mass were studied in Ref. <cit.>. They analyzed the top quark pole- mass series of Eq. (<ref>) for μ=_t and massless bottom and charm quarks and in an extended analysis also for massive bottom and charm quarks.They argued that the ambiguity of the top, bottom and charm quark pole masses amounts to 110 MeV. We believe that their ambiguity estimate of 110 MeV is too optimistic, and we explain this in detail from the requirements of HQS. They also quantified the bottom and charm mass effects coming from beyond the known corrections atO(α_s^2) and O(α_s^3) by using a heuristic prescription based on an order-dependent reduction of the flavor number. This does not represent a systematic calculation, but we find it to be an adequate approximation for the task of estimating the top quark pole mass renormalon ambiguity.The paper is organized as follows: In Sec. <ref> we review the explicitly calculated corrections up to O(α_s^4) for the pole- and the pole-MSR mass relations for the case that all quarks lighter than quark Q are massless and we explain our notation for parameterizing the virtual quark mass corrections due to the light massive quarks. This notation is essential for our setup of the flavor number dependent RG evolution of the MSR mass, which we also review to the extend needed for our studies in the subsequent sections. We also review known basic issues about the large order asymptotic behavior and the renormalon ambiguity of the pole- and the pole-MSR mass relations, including their dependence on the number of massless quarks. In Sec. <ref> we explain details about the matching procedures that allow to integrate out the virtual corrections coming from the heavy quark Q and the lighter massive quarks, and to relate the pole-MSR mass relation of quark Q to the pole- mass relation of the next lighter massive quark, which is based on heavy quark symmetry. These considerations and the numerical analysis of the latter matching corrections allow us to derive a prediction for the yet uncalculated O(α_s^4) virtual quark mass corrections and to discuss the large order asymptotic form of the virtual quark mass corrections. As an application of the RG formalism devised in our work we compute the difference of the pole masses of the top, bottom and charm quarks. Since their differences are short-distance quantities we can compute them with a precision of around 20 MeV. We also analyze the validity of the effective flavor decoupling at large orders in the context of the top quark pole mass. In Sec. <ref> we finally discuss in detail the best possible estimate of the top quark pole mass and in particular its ambiguity in the context of three different scenarios for the bottom and charm quark masses. We discuss these three scenarios separately because the pole mass concept, strictly speaking, depends on the setup for the lighter quark masses, and we also discuss our results in the context of adopting the view that the top quark pole mass is a general concept.Finally, in Sec. <ref> we conclude. InApp. <ref> we provide explicit results for the virtual quark mass corrections at O(α_s^3) in our notation, using the results from Ref. <cit.>, and we complete them concerning the corrections coming from the insertion of two quark loops involving quarks with two arbitrary masses.§ PRELIMINARIES AND NOTATION§.§ MS-bar MassThe perturbative series of the difference between themass _Q(μ) at the scale μ=_Q(_Q) and the pole mass _Q of a heavy quark Q is the basic relation from which we start our analysis of the renormalon ambiguity of the pole mass. To be more specific we consider _Q ≡_Q^(+1)(_Q^(+1)) ,which is themass defined for (+1) active dynamical flavors, where ≡.In this work we use these two definitions for all massive quarks, and depending on the context we also use the lower case letter q for massive quarks. We also define≡,which we strictly treat in the massless approximation. Assuming that q_1,…,q_n are the massive quarks lighter than Qin the order of decreasing mass (i.e. m_Q > m_q_1 > … > m_q_n> with n < n_Q and =-n), the pole- mass relation for the heavy quark Q can be written in the form_Q= _Q + _Q ∑_n=1^∞ a_n(+1,0) (α_s^(+1)(_Q)/4π)^n +_Q[_Q^(Q,q_1,…,q_n)(1,r_q_1Q,…,r_q_nQ) + _Q^(q_1,…,q_n)(r_q_1Q,…,r_q_nQ) + … + _Q^(q_n)(r_q_nQ)] ,witha_1(,n_h)= 16/3 , a_2(,n_h)= 213.437 + 1.65707n_h - 16.6619, a_3(,n_h)= 12075. + 118.986n_h + 4.10115n_h^2 - 1707.35+ 1.42358n_h+ 41.7722 ^2 , a_4(,n_h)= (911588.± 417.) + (1781.61± 30.72) n_h - (60.1637± 0.6912) n_h^2 - (231.201± 0.102) n_h- (190683.±10.)+ 9.25995 n_h^2+ 6.35819 n_h^3 + 4.40363 n_h ^2 + 11105. ^2 - 173.604 ^3 ,where α_s^(+1) is the strong coupling that evolves with (+1) active dynamical flavors, see Eq. (<ref>).The coefficients a_n(,n_h) encode the QCD corrections to _Q - _Q for the case that thequarks lighter than Q are assumed to be massless, and n_h=1 is just an identifier for the corrections coming from virtual loops of the quark Q. The coefficients a_1,2,3 are known analytically from Refs. <cit.>, and a_4 was determined numerically in Refs. <cit.>, where the quoted numerical uncertainties have been taken from Ref. <cit.>. In Ref. <cit.> an approach was suggested to further reduce the uncertainties of the -dependent terms. The numerical uncertainties of the coefficient a_4 are, however, tiny and irrelevant for the analysis carried out in this work. We quote them just for completeness throughout this work.The terms _Q^(q,q^',…)(r_qQ,r_q^' Q,…) contain the mass corrections coming from the quark Q on-shell self-energy Feynman diagrams with insertions of virtual massive quark loops. We remind the reader that the quarks with mass below the hadronization scale are taken as massless and do not contribute. The superscript (q,q^',…) indicates that each diagram contains at least one insertion of the massive quark q and in addition all possible insertions of the (lighter) massive quarks q^',… as well as of massless quark and gluonic loops. From each diagram the corresponding diagram with all the quark loops in the massless limit is subtracted in the scheme compatible with the flavor number scheme for the strong coupling α_s. The fractionr_qq^' ≡ _q/_q^' ,stands for the ratio ofmasses for massive quarks q and q^' as defined in Eq. (<ref>). In the pole- mass relation for the heavy quark Q only mass ratios with respect to the heavy quark mass _Q arise.By construction, the sum of all virtual quark mass corrections contained in the functions _Q^(q,q^',…)(r_qQ,r_q^' Q,…) are RG-invariant and do not contain effects from quarks heavier than the external quark Q. The effects on the mass of the quark Q related to quarks heavier than Q are accounted for in the renormalization group evolution of themass _Q(μ) for scales μ>m_Q and are not considered here. The virtual quark mass corrections satisfy the following two relations to all orders of perturbation theory_Q^(q_1,q_2,…,q_n)(0,0,…,0)= 0 , _Q^(Q,q_1,…,q_n)(1,0,…,0)= ∑_n=2^∞ [ a_n(,1) - a_n(+1,0) ](α_s^(+1)(_Q)/4π)^n.Due to Eq. (<ref>) the pole- mass relation of Eq. (<ref>) can be rewritten in the alternative form_Q= _Q + _Q ∑_n=1^∞ a_n(,1) (α_s^(+1)(_Q)/4π)^n +_Q[_Q^(Q,q_1,…,q_n)(1,r_q_1Q,…,r_q_nQ) - _Q^(Q,q_1,…,q_n)(1,0,…,0) ..+ _Q^(q_1,…,q_n)(r_q_1Q,…,r_q_nQ) + … + _Q^(q_n)(r_q_nQ)] .In the limit that all quarks lighter than Q are massless, allterms cancel or vanish in Eq. (<ref>), and only the first line involving the a_n coefficients remains. The perturbative expansion of the virtual quark mass corrections in the pole- mass relation of Eq. (<ref>) and (<ref>) can be written in the form_Q^(q,q^',…)(r_qQ,r_q^' Q,…)=δ_2(r_qQ)(α_s^(+1)(_Q)/4π)^2 + ∑_n=3^∞ δ_Q,n^(q,q^',…)(r_qQ,r_q^' Q,…)(α_s^(+1)(_Q)/4π)^n ,which together with Eq. (<ref>) implies thatδ_2(1)= a_2(,1) - a_2(+1,0)= 18.3189, δ_Q,n^(Q,q,q^',…)(1,0,0,…)= a_n(,1) - a_n(+1,0) .The O(α_s^2) correction comes from the on-shell self energy diagram of quark Q with the insertion of a loop of the massive quark q. The result was determined analytically in Ref. <cit.>. At O(α_s^3), in Ref. <cit.>, the virtual quark mass corrections were determined in a semi-analytic form for arbitrary quark masses for insertions of loops of the quark Q and one other massive quark q. The expressions for these virtual quark mass corrections are for convenience collected in App. <ref> after adapting the results of Ref. <cit.> to our notation. We also provide the O(α_s^3) result for insertions of loops with two arbitrary massive quarks, which were not given in Ref. <cit.>. The O(α_s^4) virtual quark mass corrections have not been determined through an explicit loop calculation.One can interpret themass _Q= _Q^(+1)(_Q^(+1)) as the pole mass minus all self-energy corrections coming from scales at and below_Q. So _Q only contains mass contributions from momentum fluctuations from above _Q, which illustrates that it is a short-distance mass that is strictly insensitive to issues related to low momentum fluctuations at the hadronization scale Λ_ QCD. See Fig. <ref> for illustration.§.§ MSR Mass and R-EvolutionIn order to integrate out high momentum contributions and formulate the renormalization group flow of momentum contributions in the heavy quark masses we use the MSR mass _Q(R) introduced in Ref. <cit.>[In Ref. <cit.> the natural and the practical MSR masses were introduced. In this paper we employ the natural MSR mass and call it just the MSR mass for convenience.], extending its definition to account for the mass effects of the lighter massive quarks.The MSR mass for the heavy quark Q is derived from on-shell self-energy diagrams just like the pole- mass relation of Eq. (<ref>), but it does not include any diagrams involving virtual loops of the heavy quark Q, i.e. the contributions from heavy quark Q virtual loops are integrated out. Like themass, the MSR mass is a short-distance mass, and since the corrections from the heavy quark Q are short-distance effects, its relation to the pole mass fully contains the pole mass O(Λ_ QCD) renormalon (just as the pole- mass relation of Eqs. (<ref>) and (<ref>)). Furthermore the MSR mass depends on the arbitrary scale R≲ m_Q to describe contributions in the mass from the momenta below the scale m_Q, and therefore represents the natural extension of the concept of themass for scales below m_Q.Assuming that q_1,…,q_n are the massive quarks lighter than Q in the order of decreasing mass (i.e. m_Q > m_q_1 > … > m_q_n> with n < n_Q and =-n), the MSR mass _Q(R) is defined by the relation_Q= _Q(R) + R ∑_n=1^∞ a_n(,0)(α^()_s(R)/4π)^n + _Q [δ_Q^(q_1,…,q_n)(r_q_1Q,…,r_q_nQ) + … + δ_Q^(q_n)(r_q_nQ)] ,where the coefficients a_n are given in Eqs. (<ref>) and the perturbative expansion is in powers of the strong coupling in the n_Q-flavor scheme since the quark Q is integrated out. The R-dependence of the strong coupling entails that the scale R has to be chosen sufficiently larger than Λ_ QCD to stay away from the Landau pole. The definition generalizes the one already provided in Ref. <cit.>, which only consideredmassless quarks.The notation used for the virtual quark mass corrections involving the functions δ_Q^(q,q^',…)(r_qQ,r_q^' Q,…) is the same as the one for themass described above, and their sum is by construction RG-invariant. Their perturbative expansion has theformδ_Q^(q,q^',…)(r_qQ,r_q^' Q,…)=δ_2(r_qQ)(α_s^()(_Q)/4π)^2 + ∑_n=3^∞ δ_Q,n^(q,q^',…)(r_qQ,r_q^' Q,…)(α_s^()(_Q)/4π)^n ,where the coefficient functions δ_Q,n^(q,q^',…)(r_qQ,r_q^' Q,…) are identical to the ones appearing in Eq. (<ref>). In our definition of the MSR mass, the virtual quark mass corrections are independent of R. This entails that the renormalization group evolution of the MSR mass in R does not depend on the masses of the n_Q lighter quarks. So_Q(R) is defined in close analogy to the μ-dependentstrong coupling and themasses, whose renormalization group evolution only depends on the number of active dynamical quarks (which is typically the number of quarks lighter than μ) and where mass effects are implemented by threshold corrections when μ crosses a flavor threshold. Moreover, because the O() renormalon ambiguity of the series proportional to R is independent of R and because the corrections from the virtual loops of the heavy quark Q are short-distance effects, the series of the pole-MSR mass relation in Eq. (<ref>) suffers from the same O() renormalon ambiguity as the pole- mass relation of Eqs. (<ref>) and (<ref>). It can therefore also be used to study and quantify the O() renormalon of the pole mass m_Q^ pole.As explained below Eq. (<ref>), in order to expand the difference of MSR masses at two scales R and R^' in the fixed-order expansion in powers of α_s^() it is necessary to do that at a common renormalization scale μ so that the renormalon in the R-dependent corrections of Eq. (<ref>) cancels order by order. This unavoidably leads to large logarithms if the scale separation is large, similarly to when considering the fixed-order expansion of the difference of the strong coupling at widely separated scales. To sum the logarithms in the difference of MSR masses we use its RG-evolution equation in R, which readsR/ Rm_Q^(R)=- R γ^R,(n_Q)(α^()_s(R))=- R∑_n=0^∞γ_n^R,(n_Q)(α^()_s(R)/4π)^n+1 ,where the coefficients are known up to four loops and given by <cit.>γ_0^R,(n_Q)= 16/3 , γ_1^R,(n_Q)= 96.1039 - 9.55076, γ_2^R,(n_Q)= 1595.75 - 269.953- 2.65945 ^2 , γ_3^R,(n_Q)= (12319.±417.) - (9103.±10.)+ 610.264 ^2 - 6.515 ^3 . The difference of MSR masses at two scales R^' and R can then be computed from solving the evolution equationΔ m^(n_Q)(R,R^') = _Q(R^') - _Q(R) = ∑_n=0^∞ γ_n^R,(n_Q)∫_R^'^RR (α^()_s(R)/4π)^n+1 ,which accounts for the RG-evolution in the presence ofactive dynamical quark flavors.The RG-equation of the MSR mass has a linear as well as logarithmic dependence on R and thus differs from the usual logarithmic RG-equations for α_s and themass. Since its linear dependence on R allows to systematically probe linear sensitivity to small momenta it can be used to systematically study the O() renormalon behavior of perturbative series <cit.>. Since this is impossible for usual logarithmic RG-evolution equations, Eq. (<ref>) was called the R-evolution equation in Refs. <cit.>. Continuing on the thoughts made at the end of Sec. <ref> we note that one can interpret the MSR mass _Q(R) as the pole mass minus all self-energy contributions coming from scales below R and all virtual quark mass corrections from quarks lighter than Q, see Fig. <ref>. This also illustrates that the MSR mass _Q(R) is a short-distance mass. The negative overall sign on the RHS of Eq. (<ref>) expresses that self-energy contributions are added to the MSR mass when R is evolved to smaller scales, and that Δ m^(n_Q)(R,R^') for R>R^' is positive and represents the self-energy contributions to the mass in the presence of n_Q active dynamical flavors coming from the scales between R^' and R. This is illustrated in Fig. <ref>. In the context of the analyses in this work the essential property is that the O() renormalon ambiguity in the series on the RHS of Eq. (<ref>) is R-independent. This entails that the R-evolution equation is free of the O() renormalon, and solving the R-evolution equation in Eq. (<ref>) allows to relate MSR masses at different scales in a way that is renormalon free and, in addition, systematically sums logarithms ln(R/R^') to all orders in a way free of the O() renormalon. So the R-evolution equation resolves the problem of the large logarithms that arise when computing MSR mass differences in the fixed-order expansion. The integral of Eq. (<ref>) can be readily computed numerically, and an analytic solution has been discussed in detail in <cit.>. The analytic solution also allows to derive the large-order asymptotic form of the perturbative coefficients a_n. To implement renormalization scale variation in Eq. (<ref>) one expands α_s^()(R) as a series in α_s^()(λ R), and by varying λ in some interval around unity. We note that in our analysis we consider the top, bottom and charm mass scales, and using the R-evolution equation is instrumental for our discussion of the top quark pole mass.In Tab. <ref> we show numerical results for various MSR mass differences Δ m^(n_Q) relevant in our examinations below for =3,4,5. We display the results obtained from using the R-evolution equation at O(α_s^n) for n=1,2,3,4. The uncertainties are from λ variations in the interval [0.5,2] for the cases where scales above the charm mass scale 1.3 GeV are considered, and in the interval [0.6,2.5] for cases which involve the charm mass scale. We see an excellent convergence and stability of the results and a significant reduction of scale variation with the order, illustrating that the mass differences Δ m^(n_Q)(R,R^') are free of an O(Λ_ QCD) renormalon ambiguity. For our analyses below we use the most precise O(α_s^4) results shown in the respective lowest lines.§.§ Asymptotic High Order Behavior and BorelTransform for Massless Lighter QuarksIn this section we review a number of known results relevant for the analyses in the subsequent parts of the paper. The results are already known since Refs. <cit.>. We adapt them according to our notation and present updated numerical results accounting for the recent perturbative calculations of the pole- mass relation and the QCD β-function.The Borel transform of an α_s power seriesf(α_s(R)) =R ∑_n=0^∞ a_n+1(α_s(R)/4π)^n+1 ,is defined asB[f](u) = R ∑_n=0^∞ a_n+1 u^n/n! β_0^n+1 ,where β_0 is the one-loop β-function coefficient in the flavor number scheme of α_s. For the approximation that all quarks lighter than the heavy quark Q are massless (i.e. =) the Borel transform of the series for the pole-MSR mass readsB [_Q-_Q(R)](u)=N_1/2^() R 4π/ ∑_k=0^∞ g_k^()Γ(1+b̂_1^()-k)/Γ(1+b̂_1^()) (1-2 u)^-1-b̂_1^()+k + … ,where the non-analytic (and singular) terms multiplied by the normalization factor N_1/2^() single out the O() renormalon behavior of the pole-MSR mass series and the ellipses stand for contributions not affected by an O() renormalon. Their form is unambiguously determined by the coefficients β_n^() of the QCD β-function in Eq. (<ref>), and the sum over k parametrizes the subleading effects due to the higher order coefficients of the QCD β-function.The coefficients g^()_k can be determined from the recursion formulae <cit.> b̂_n+1 = 2∑_i = 0^n b̂_n-i β_i+1/(-2β_0)^i+2 , g_n+1 = 1/1+n∑_i=0^n (-1)^i b̂_i+2 g_n-iwith b̂_0=g_0=1, where we dropped the superscript () for simplicity. Currently, coefficients g^()_k are known up to k=3. The factor N_1/2^() precisely quantifies the overall normalization of the O() renormalon behavior and can be determined quite precisely from the coefficients a_n(,0) known from explicit computations. Accounting for the coefficients up to O(α_s^4) the normalization was determined with very small errors for the relevant flavor numbers = 3,4,5 in Refs. <cit.>, all of which are in agreement. We use the results from Ref. <cit.>:N_1/2^(=3) = 0.526±0.012, N_1/2^(=4) = 0.492±0.016, N_1/2^(=5) = 0.446±0.024.The uncertainties are not essential for the outcome of our analysis and quoted for completeness. Their small size reflects that the large-order asymptotic behavior of the series is known very precisely. The inverse Borel transform∫_0^∞duB[f](u) e^-4π u/β_0α_s(R),has the same α_s power series as the original series f(α_s(R)) and provides the exact result if it can be calculated unambiguously from the Borel transform B[f](u). However, for the case of Eq. (<ref>), due to the singularity at u=1/2 and the cut along the positive real axis for u>1/2, the integral cannot be computed without further prescription and an ambiguity remains. Using an iϵ prescription (1-2u)^α→(1-2u-iϵ)^α to shift the cut to the lower complex half plane, the resulting imaginary part of the integral isΔ m_Borel^() ≡|Im∫_0^∞du exp(-4π u/β_0^()α_s^()(R)).. ×[N_1/2^()R 4π/β_0^()∑_k=0^∞ g_k^()Γ(1+b̂_1^()-k)/Γ(1+b̂_1^()) (1-2u)^-1-b̂_1^()+k] |=N_1/2^() 2π^2/β_0^()Γ(1+b̂_1^())Λ_QCD^() , and represents a quantification of the ambiguity of the pole mass, where Λ_QCD^() isgiven by the expression (t_R=-2π/β_0^()α_s^()(R))Λ_QCD^()=R exp(t_R+b̂_1^()log(-t_R)-∑_k=2^∞b̂_k^()/(k-1)t_R^k-1).In this work we use this expression as the definition of Λ_QCD formassless flavors. The RHS is R-independent, and truncating at k=4 provides the resultsΛ_ QCD^(=3) =253 MeV , Λ_ QCD^(=4) =225 MeV , Λ_ QCD^(=5) =166 MeV ,with uncertainties below 0.5 MeV. Λ_QCD^() increases for smaller flavor numberssince the scale-dependence of α_s, and thus also the infrared sensitivity of QCD quantities, increases with . The expressions forΔ m_Borel^() for the size of the imaginary part of the inverse Borel transform in Eq. (<ref>) provide a parametric estimate for the ambiguity of the pole mass. Using Eqs. (<ref>) and (<ref>) they give Δ m_Borel^(3,4,5)=(329± 8,295± 10,213± 11) MeV which are around a factor 1.3 larger than the corresponding values for Λ_ QCD^(). From the expression for the Borel transform given in Eq. (<ref>) one can derive the large order asymptotic form of the perturbative coefficients a_n of the pole-MSR mass series (which describe the case that all quarks lighter than Q are massless, i.e. =):a_n^asy(,n_h)= a_n^asy(,0)=4π N_1/2^()(2β_0^())^n-1∑_k=0^∞ g_k^()Γ(n+b̂_1^()-k)/Γ(1+b̂_1^()) ,where the value of n_h is insignificant because the virtual effects of quark Q do not affect the large order asymptotic behavior. The sum in k is convergent, and truncating at k=3 one can use the results for n>4 as an approximation for the yet uncalculated series coefficients. The results up to n=12 for =3, 4, 5 using the values for the N_1/2^() from Eq. (<ref>) are displayed in Tab. <ref>.With the normalization factors N_1/2^(), which are known to a precision of a few percent and which also entails the same precision for Δ m_Borel^() and the asymptotic coefficients a_n^ asy, the series for the pole-MSR and also for the pole- mass relation are essentially known to all orders for the case of =. The task to determine the ambiguity of the pole mass involves to specify how this precisely known pattern limits the principle capability to determine the pole mass numerically, see the discussion in Sec. <ref>. In other words, the ambiguity of the pole mass is known to be proportional to Δ m_Borel^() or Λ_ QCD^(), but the factor of proportionality has to be determined from an additional dedicated analysis.§ INTEGRATING OUT HARD MODES FROM THE HEAVY QUARK POLE MASS §.§ MSR-MS-bar Mass MatchingUsing the MSR mass we can successively separate off, i.e. integrate out, hard momentum contributions from the pole- mass difference, _Q - _Q. We start with the matching relation between the MSR and themasses at the common scale μ=R=_Q, which can be obtained by eliminating the pole mass from Eqs. (<ref>) and (<ref>). The matching relation accounts for the virtual top quark loop contributions and can be written in the form_Q(_Q) - _Q = Δ m_Q^(+1→)(_Q) + δ m_Q,q_1,…,q_n^(+1→)(_Q).The term Δ m_Q^(+1→)(_Q) contains the virtual top quark loop contributions in the approximation that all n_Q quarks lighter than quark Q are massless and has the form <cit.>Δm_Q^(+1→)(_Q) = _Q{1.65707 (α_s^(+1)(_Q)/4π)^2 + [110.05 + 1.424 ] (α_s^(+1)(_Q)/4π)^3 . .+ [352.±31. - (111.59±0.10)+ 4.40 ^2](α_s^(+1)(_Q)/4π)^4 + …} ,where we expressed the series in powers of the strong coupling in the (+1) flavor scheme. The series only contains the hard corrections coming from the virtual heavy quark Q and therefore does not have any O(Λ_ QCD) ambiguity, see Fig. <ref> for illustration.In Tab. <ref> the numerical values for Δ m_Q^(+1→)(_Q) are shown at O(α_s^2,3,4) for the top, bottom, and charm quarks for (_t,_b,_c) = (163,4.2,1.3) GeV. Also shown is the variation due to changes in the renormalization scale in the range 0.5 _Q≤μ≤2 _Q, for the top and bottom quark and 0.65 _c≤μ≤2.5 _c for the charm quark. The O(α_s^3) corrections are quite sizable compared to the O(α_s^2) contributions, but the O(α_s^4) corrections are small indicating that the O(α_s^4) result and the uncertainty estimate based on the scale variations can be considered reliable. Overall, the matching corrections amount to 32,4 and 5 MeV for the top, bottom and charm quarks, respectively with an uncertainty at the level of 1 to 2 MeV. The numerical uncertainties of the O(α_s^4) coefficients displayed in Eq. (<ref>) are smaller than 0.1 MeV for all cases and therefore irrelevant for practical purposes.The term δ m_Q,q_1,…,q_n^(+1→)(_Q) represents the virtual top quark loop contributions arising from the finite masses of the lighter massive quarks q_1,…,q_n. Since at O(α_s^2) only the loop of quark Q can be inserted, the series for δ m_Q,q_1,…,q_n^(+1→)(_Q) starts at O(α_s^3), where only self energy diagrams with one insertion of a loop of quark Q and one insertion of a loop of one of the lighter massive quarks q_1,…,q_n can contribute. At O(α_s^3) δ m_Q,q_1,…,q_n^(+1→)(_Q) has the formδ m_Q,q_1,…,q_n^(+1→)(_Q) = _Q {[δ_Q,3^(Q,q_1,…,q_n)(1,r_q_1 Q,…,r_q_n Q)- δ_Q,3^(Q,q_1,…,q_n)(1,0,…,0)](α_s^(+1)(_Q)/4π)^3 + …}= _Q {∑_i=1^n[14.2222r_q_i Q^2 - 18.7157r_q_i Q^3 +(7.3689-11.1477 ln(r_q_i Q))r_q_i Q^4. +. …](α_s^(+1)(_Q)/4π)^3 + …} ,where r_q Q = _q/_Q, and for simplicity we suppress the masses of the quarks q_1,…,q_n in the argument of δ m_Q,q_1,…,q_n^(+1→). Starting at O(α_s^4) the finite quark mass corrections in δ m_Q,q_1,…,q_n^(+1→)(_Q) become also dependent on the flavor threshold corrections relating α_s^()(_Q) and α_s^(+1)(_Q). In Eq. (<ref>) we have also displayed the first terms of the expansions in the mass ratios r_q_i Q. They start quadratically in the r_q_i Q indicating that the corrections are governed by the scale _Q just like the matching term Δ m_Q^(+1→)(_Q) and do not have any linear sensitivity to small momenta and the lighter quark masses, in particular. This feature is realized at any order of perturbation theory.Because the finite mass corrections δ m_Q,q_1,…,q_n^+1→(_Q) start at O(α_s^3) and are quadratic in the mass ratios r_q_i Q they are extremely small and never exceed 0.01 MeV for the top quark (due to the finite bottom or charm masses) and the bottom quark (due to the finite charm mass). We can expect that this is also exhibited at higher orders, so that δ m_Q,q_1,…,q_n^(+1→)(_Q) can be neglected for all practical purposes and will not be considered and discussed any further in this work. §.§ Top-Bottom and Bottom-Charm Mass MatchingComparing the pole- mass relation (<ref>) for the heavy quark Q to the pole- mass relation (<ref>) for the next lighter massive quark q, one immediately notices that for R=_q the correctionsare identical in the approximation that in the virtual quark loops alllighter quarks (i.e. including the quark q) are treated as massless. This identity is a consequence of heavy quark symmetry which states that the low-energy QCD corrections to the heavy quark masses coming from massless partons are flavor-independent.For the top MSR and the bottommasses (i.e. for Q=t and q=b) the resulting matching relation reads[_t-_t(_b)] - [_b-_b] = δ m_b,c^(t→ b)(_b,_c),where δ m_b,c^(t→ b)(_b,_c) encodes the heavy quark symmetry breaking corrections coming from the finite virtual charm and bottom quark masses. Their form can be extracted directly from Eqs. (<ref>) and (<ref>) and written in the form (r_q q^'=_q/_q^')δ m_b,c^(t→ b)(_b,_c) = _t [ δ_t^(b,c)(r_bt,r_ct) + δ_t^(c)(r_ct)] - _b [ δ_b^(b,c)(1,r_cb) + δ̅_b^(c)(r_cb)],where the first term on the RHS (multiplied by _t) represents the virtual bottom and charm mass effects from the top quark self energy and the second term (multiplied by _b) represents the virtual bottom and charm mass effects from the bottom quark self energy. Their explicit form up to O(α_s^3) reads_t[δ_t^(b,c)(r_bt, r_ct) +δ_t^(c)(r_ct)] = _t [ δ_2(r_bt) + δ_2(r_ct)] (α_s^(5)(μ)/4π)^2 + _t [ δ_t,3^(b,c)(r_bt,r_ct) +δ_t,3^(c)(r_ct) + 4 β_0^(5)ln(μ/_t)[ δ_2(r_bt) + δ_2(r_ct)] ] (α_s^(5)(μ)/4π)^3 + … ,and_b[δ_b^(b,c)(1, r_cb) +δ_b^(c)(r_cb)] = _b [ δ_2(1) + δ_2(r_cb)] (α_s^(5)(μ)/4π)^2 + _b [ δ_b,3^(b,c)(1,r_cb) +δ_b,3^(c)(r_cb) + 4 β_0^(5)ln(μ/_b)[ δ_2(1) + δ_2(r_cb)] ] (α_s^(5)(μ)/4π)^3 + … .It is important that the quark mass corrections in (<ref>) are expressed coherently in powers of α_s at the common scale μ because the individual δ_n terms carry contributions that modify the infrared sensitivity and therefore each contain O() renormalon ambiguities. In Eq. (<ref>) these renormalon ambiguities mutually cancel. We also note that δ m_b,c^(t→ b)(_b,_c) also depends on the top quark mass _t. We have suppressed _t in the argument since δ m_b,c^(t→ b)(_b,_c) encodes symmetry breaking corrections due to the finite bottom and charm quark masses.For the bottom MSR and the charmmasses the corresponding matching relation reads[_b - _b(_c)] - [_c - _c] = δ m_c^(b→ c)(_c),withδ m_c^(b→ c) (_c) = _b δ_b^(c)(r_cb) - _c δ_c^(c)(1),where the first term on the RHS (multiplied by _b) represents the virtual charm mass effects from the bottom quark self energy and the second term (multiplied by _c) represent the virtual charm mass effects from the charm quark self energy. Their explicit form up to O(α_s^3) reads_b δ_b^(c)(r_cb)= _bδ_2(r_cb) ( α_s^(4)(μ)/4π)^2+ _b [ δ_b,3^(c)(r_cb) + 4β_0^(4)δ_2(r_cb)ln(μ/_b)]( α_s^(4)(μ)/4π)^3 + … ,and_c δ_c^(c)(1)= _cδ_2(1) ( α_s^(4)(μ)/4π)^2 + _c [ δ_c,3^(c)(1) + 4β_0^(4)δ_2(1)ln(μ/_c)]( α_s^(4)(μ)/4π)^3 + … ,where again we expanded both terms consistently for a common renormalization scale μ in the strong coupling.In Fig. <ref> the top-MSR bottom- mass matching correction δ m_b,c^(t→ b)(_b,_c) of Eq. (<ref>) is displayed as a function of the renormalization scale μ at O(α_s^2) (red dashed line) and O(α_s^3) (red solid line) for (_t,_b,_c) = (163,4.2,1.3) GeV. The matching correction at O(α_s^3) amounts to 6 MeV and has a scale variation of only 1 MeV for _b≤μ≤_t. Compared to the O(α_s^2) result we see a strong reduction of the scale-dependence at O(α_s^3). The final numerical results at O(α_s^2) and O(α_s^3) are shown in the second column of Tab. <ref> where the uncertainties are obtained from variations of the renormalization scale in the range _b≤μ≤_t and the central values are the respective mean of the largest and smallest values obtained in the scale variation. The corresponding results for a vanishing charm quark mass are shown in Fig. <ref> and the third column of Tab. <ref>. We see that the charm mass effects in the top-MSR bottom- mass matching correction δ m_b,c^(t→ b) (_b,_c) are only around 1 MeV, and the stability for _c→ 0 shows that the matching correction is governed by scales of order _b and higher, which reconfirms the range _b≤μ≤_t for the variation of the renormalization scale.In Fig. <ref> the bottom-MSR charm- mass matching correction δ m_c^(b→ c)(_c) of Eq. (<ref>) is displayed as a function of the renormalization scale μ for _b=4.2 GeV and _c=1.3 GeV at O(α_s^2) and O(α_s^3) using the same color coding and curve styles as for Figs. <ref> and <ref>. In the fourth column of Tab. <ref> the final numerical results at O(α_s^2) and O(α_s^3) are shown using _c≤μ≤_b for the renormalization scale variation. The stability and convergence is again excellent, and at O(α_s^3) the matching correction amounts to 4 MeV with an uncertainty of 1 MeV.Given that the heavy quark symmetry breaking matching corrections δ m_b,c^(t→ b)(_b,_c) and δ m_c^(b→ c)(_c) amount to only 4 to 6 MeV, we note that they may be simply neglected in practical applications where they yield contributions that are much smaller than other sources of uncertainties. In fact, this also applies to our subsequent studies of the top, bottom and charm quark pole masses. However, we include them here for completeness. Due to their small size, we have not explicitly included the heavy quark symmetry breaking matching corrections in the graphical illustration of Fig. <ref>. §.§ Light Virtual Quark Mass Corrections at 4-Loop Order and BeyondThe excellent perturbative convergence of the top-MSR bottom- mass matching correction δ m_b,c^(t→ b) (_b,_c) and of the bottom-MSR charm- mass matching correction δ m_c^(b→ c) (_c) discussed in the previous section illustrates that they both are short-distance quantities and free of an O() renormalon ambiguity. This is also expected theoretically due to heavy quark symmetry. However, the facts that the overall size of the matching corrections only amounts to a few MeV, and that the O(α_s^3) corrections are only around 1 MeV allows us to draw interesting conceptual implications for the large order asymptotic behavior of the virtual quark mass corrections in the mass relations of Eqs. (<ref>), (<ref>) and (<ref>). We discuss these implications in the following. As a consequence we can predict the yet uncalculated virtual quark mass corrections at O(α_s^4) to within a few percent without an additional loop calculation and draw important conclusions on their properties for the orders beyond.To be concrete, we consider the matching correction δ m_q^(Q→ q) (_q) between the MSR mass of heavy quark Q and themass of the next lighter massive quark q assuming the massless approximation for all quarks lighter than quark q i.e. = n_q+1 = +1 and n_ℓ=n_q being the number of massless quarks. This situation applies to the matching relation for the top-MSR and the bottommasses for a massless charm quark or to the matching relation between the bottom-MSR and the charm- masses.In Fig. <ref> we have displayed separately the virtual bottom and charm mass effects to the top quark self energy of Eq. (<ref>) (green curves) and the virtual bottom and charm mass effects to the bottom quark self energy of Eq. (<ref>) (blue lines) at O(α_s^2) (dashed) and O(α_s^3) (solid). In Fig. <ref> the charm quark is treated as massless in the same quantities. In Fig. <ref> the virtual charm mass effects to the bottom quark self energy of Eq. (<ref>) and the virtual charm mass effects to the charm quark self energy of Eq. (<ref>) are shown at O(α_s^2) and O(α_s^3) with the analogous line styles and colors. We see that both types of contributions each are quite large and furthermore do not at all converge. The O(α_s^3) corrections are even bigger than the O(α_s^2) corrections, which indicates that the corresponding asymptotic large order behavior already dominates the O(α_s^2) and O(α_s^3) corrections.The origin of this behavior has been already mentioned and is understood: The mass of the virtual quark q acts as an infrared cutoff and therefore modifies the infrared sensitivity of the self energy diagrams (of quark Q and of quark q) with respect to the case where the virtual loops of quark q are evaluated in the massless approximation. As a consequence these corrections individually carry an O() renormalon ambiguity. Moreover, at large orders in perturbation theory the sensitivity of the self energy diagrams to infrared momenta increases due to high powers of logarithms from gluonic and massless quark loops. As a consequence, at large orders, the finite mass effects of the virtual loops of quark q in the self energy diagrams of quark Q and the self energy diagrams of quark q become equivalent due to heavy quark symmetry. The strong cancellation in the sum of both types of corrections in δ m_q^(Q→ q)(_q) (∼75% at O(α_s^2) and ≳90% at O(α_s^3) for the cases displayed in Fig. <ref>) thus confirms that the known O(α_s^2) and O(α_s^3) self energy corrections coming from virtual quark masses are already dominated by their large order asymptotic behavior.From the observations that the series for δ m_b,c^(t→ b)(_b,_c) and δ m_c^(b→ c)(_c) converge very well and that their O(α_s^3) corrections amount to only about 1 MeV, we can therefore expect that the two types of corrections that enter δ m_b,c^(t→ b)(_b,_c) as well as δ m_c^(b→ c)(_c) agree to even better than 1 MeVat O(α_s^4) and beyond. This allows us to make an approximate prediction for the yet uncalculated O(α_s^4) finite mass corrections from virtual loops of quark q in the pole- mass relations of quark Q of Eqs. (<ref>) and (<ref>) by setting the O(α_s^4) correction in δ m_q^(Q→ q)(_q) to zero:δ_Q,4^(q)(r_qQ) ≈ r_qQ[ δ_q,4^(q)(1) + (6 δ_q,3^(q)(1) + 4 δ_2(1)) ln(μ/_q) + 12 δ_2(1)(ln(μ/_q))^2] - ( 6 δ_Q,3^(q)(r_qQ) + 4 δ_2(r_qQ))ln(μ/_Q) - 12 δ_2(r_qQ)(ln(μ/_Q))^2 .The prediction has a residual μ-dependence, which would vanish in the formal limit that the virtual quark q mass corrections are entirely dominated by their large order asymptotic behavior. Therefore the dependence on the scale μ can be used as an uncertainty estimate of our approximation.In Fig. <ref> we show the prediction for δ_Q,4^(q)(r_qQ) for _q≤μ≤_Q (green bands) for =n_q+1=+1=5 (lower band) and =n_q+1=+1=4 (upper band). The prediction satisfies exactly the required boundary condition δ_Q,4^(q)(0) = 0 and Eq. (<ref>) for r_qQ=1 and provides an interpolation for 0<r_qQ<1 with an uncertainty of ±3% (for r_qQ≲0.1) or smaller (for r_qQ>0.1). To judge the quality of the prediction we apply the same method at O(α_s^3)to “predict” δ_Q,3^(q)(r_qQ) which givesδ_Q,3^(q)(r_qQ) ≈ r_qQ[ δ_q,3^(q)(1) + 4 δ_2(1)ln(μ/_q)] - 4δ_2 (r_qQ)ln(μ/_Q).The result for the prediction of δ_Q,3^(q)(r_qQ) is shown in Fig. <ref> for =n_q+1=+1=5. The green band illustrates again the range of predictions for μ-variations_q≤μ≤_Q, and represents an uncertainty of ±10% (for r_qQ≲0.1) or smaller (for r_qQ>0.1). Compared to the O(α_s^4) result, the larger μ variation we observe at O(α_s^3) is expected because the infrared sensitivity is weaker and the large order asymptotic behavior is less dominating at the lower order. The red curve is the exact result for δ_Q,3^(q)(r_qQ) obtained from the results in Ref. <cit.>, see also Eq. (<ref>). We see that the prediction is fully compatible with the exact result and that the uncertainty estimate based on the μ-variation is reliable. The prediction for δ_Q,3^(q)(r_qQ) for =n_q+1=+1=4 has the same good properties but is not displayed since it is numerically very close to the prediction for =n_q+1=+1=5.Overall, the examination shows that the prediction and the uncertainty estimate for δ_Q,4^(q)(r_qQ) can be considered reliable. We can also provide a very simple closed analytic expression by evaluating Eq. (<ref>) for μ=_Q, which gives δ_Q,4^(q)(r_qQ) ≈ r_qQ[ δ_q,4^(q)(1) - (6 δ_q,3^(q)(1) + 4 δ_2(1) )ln(r_qQ) + 12 δ_2(1)(ln(r_qQ))^2] = r_qQ[ (203915.±32.) - 22962.+ 525.2 ^2 + (-130946. +13831.-328.5 ^2) ln(r_qQ)+ (26599.1 - 3224.1+ 97.70 ^2 ) ln(r_qQ)^2 ].The expression depends via the boundary condition of Eq. (<ref>) entirely on the coefficients a_n(n_q,n_h) of Eq. (<ref>), which for this case describe the corrections to the heavy quark q self energy for the case that all lighter quarks are massless, and the coefficients of the β-function. The expression is shown as the black dashed lines in Fig. <ref> for =+1=5 (lower line) and =+1=4 (upper line). This approximation for δ_Q,4^(q)(r_qQ)has a simple overall linear behavior on the mass ratio r_qQ=_q/_Q. The behavior is just a manifestation of δ_Q,4^(q)(r_qQ) being dominated by the large order asymptotic behavior due to its O(Λ_ QCD) renormalon ambiguity which is related to linear sensitivity to small scales. The overall linear dependence of δ_Q,4^(q)(r_qQ) on _q arises since the mass of quark q represents an infrared cut and thus represents the characteristic physical scale that governs δ_Q,4^(q)(r_qQ).This also explains the origin of the logarithms shown in Eq. (<ref>): They arise because all virtual quark mass corrections inEqs. (<ref>), (<ref>) and (<ref>) are defined in an expansion in α_s(_Q). We note that for the O(α_s^3) virtual massive quark correction δ_Q,3^(q)(r_qQ) these aspects were already discussed in Ref. <cit.> and later in Ref. <cit.>, where a direct comparison to the explicit calculations from Ref. <cit.> could be carried out. These analyses were, however, using generic considerations and were not carried out within a systematic RG framework.The expression of Eq. (<ref>) is a special case of the general statement that the asymptotic large order behavior of the coefficients δ_Q,n^(q)(r_qQ) can be obtained from the relationδ_2(r_qQ) (α_s^()(_Q)/4π)^2 + ∑_n=3^∞ δ_Q,n^(q)(r_qQ)(α_s^()(_Q)/4π)^n ≈r_qQ δ_q^(q)(1) =r_qQ [ δ_2(1) ( α_s^()(_q)/4π)^2 + ∑_n=3^∞δ_q,n^(q)(1)(α_s^()(_q)/4π)^n ] ,where on the RHS of the approximate equality α_s^()(_q) has to be expanded in powers of α_s^()(_Q), and we have δ_2(1)=18.3189, δ_q,3^(q)(1)=1870.79 - 82.1208 and. The terms δ_q,n^(q)(1) for n>4 can be obtained fromusing Eqs. (<ref>) and (<ref>) together with the large order asymptotic form of the coefficients a_n shown in Eq. (<ref>), givingδ_q,n>4^(q)(1)≈a_n^asy(n_q) - a_n^asy(n_q+1) =a_n^asy(n_Q-1) - a_n^asy(n_Q) ,where we would like to remind the reader that for the case we consider here we have n_Q = n_q+1 = n_ℓ+1. Our examination at O(α_s^3) and O(α_s^4) above showed that this relation provides an approximation for δ_Q,4^(q) within a few percent. For the higher-order terms δ_Q,n^(q) with n>4 it should be even more precise, and we therefore believe that it should be sufficient for essentially all future applications in the context of studies of the pole mass scheme. To conclude we note that it is straightforward to extend Eq. (<ref>) from the case of having only one massive quark q being lighter than heavy quark Q, i.e. =n_q+1=+1, to the case of having a larger number of lighter massive quarks. For example for the case that there are two massive quarks lighter than quark Q (let's say q and q^', in order of decreasing mass) with =n_q+1=n_q^'+2=+2, the generalization of the approximation formula (<ref>) readsδ_Q,4^(q,q^')(r_qQ,r_q^' Q) + δ_Q,4^(q^')(r_q^' Q) ≈ r_qQ { δ_q,4^(q,q^')(1,r_q^' q) + δ_q,4^(q^')(r_q^' q) + [6 (δ_q,3^(q,q^')(1,r_q^' q) + δ_q,3^(q^')(r_q^' q)) + 4 ( δ_2(1) + δ_2(r_q^' q) )] ln(μ/_q)+ 12 (δ_2(1) + δ_2(r_q^' q)) (ln(μ/_q))^2}- [6 (δ_Q,3^(q,q^')(r_qQ,r_q^' Q) + δ_Q,3^(q^')(r_q^' Q)) + 4 ( δ_2(r_qQ) + δ_2(r_q^' Q))] ln(μ/_Q)- 12 (δ_2(r_qQ) + δ_2(r_q^' Q)) (ln(μ/_Q))^2 . §.§ Pole Mass DifferencesUsing the MSR mass we have set up a conceptual framework to systematically quantify the contributions to the pole mass of a heavy quark coming from the different momentum regions contained in the on-shell self energy diagrams. The pole mass of a heavy quark Q contains the contributions from all momenta, while themass _Q(μ) and the MSR mass _Q(R) contain the contributions from above the scales μ and R, respectively (see Fig. <ref>). The MSR mass is the natural extension of themass, which is applied for scales μ>m_Q, to scales R<m_Q, and obeys a RG-evolution equation that is linear in R, called R-evolution <cit.>. The R-evolution equation quantifies in a way free of the O(Λ_ QCD) renormalon the change in the MSR mass when contributions from lower momenta are included into the mass when R is decreased, as long as R>.In Sec. <ref> we discussed the matching corrections Δ m_Q^(+1→)(_Q) that arise when the virtual loop contributions of quark Q are integrated out by switching from _Q to _Q(_Q). In Sec. <ref> we discussed the MSR mass difference Δ m^(n_Q)(R,R^') = _Q(R^') - _Q(R), which is determined from solving the R-evolution equation of the MSR mass and which systematically sums logarithms of R/R^'. In Sec. <ref> we examined the matching between the QCD corrections to the MSR mass of the heavy quark Q and themass of the next lighter massive quark q, δ m_q,q^',…^(Q→ q)(_q,_q^',…) accounting for the mass effects of the quarks q,q^',…. This matching is based on heavy quark symmetry and the small numerical size of δ m_q,q^',…^(Q→ q)(_q,_q^',…) reflects that the symmetry breaking effects due to the finite quark masses are quite small. These two types of matching corrections and the R-evolution of the MSR mass each are free of O() renormalon ambiguities and show excellent convergence properties in QCD perturbation theory.An interesting application is the determination of the difference of the pole masses of two massive quarks. Due to heavy quark symmetry, the differences of two heavy quark pole masses are also free of O() renormalon ambiguities and can therefore be determined to high precision. The matching corrections discussed above and the R-evolution of the MSR mass allow us to systematically sum logarithms of the mass ratios that would remain unsummed in a fixed-order calculation, and to achieve more precise perturbative predictions <cit.>. Taking the example of the top and bottom mass one can then write the difference of the top quark pole- mass relation and the bottom quark pole- mass relation in the form[ _t - _t ] - [ _b - _b ] = Δ m_t^(6→ 5)(_t) + Δ m^(5)(_t,_b) + δ m_b,c^(t→ b)(_b,_c).The analogous relation for the bottom and charm quarks reads[ _b - _b ] - [ _c - _c ] = Δ m_b^(5→ 4)(_b) + Δ m^(4)(_b,_c) + δ m_c^(b→ c)(_c).Each of the mass differences is the sum of universal matching and evolution building blocks which each can be computed to high precision, as shown in Tabs. <ref>, <ref>, <ref>.The resulting relations between the top, bottom and charm quark pole masses read_t - _b= [ _t - _b ] + Δ m_t^(6→ 5)(_t) + Δ m^(5)(_t,_b) + δ m_b,c^(t→ b)(_b,_c), _b - _c= [ _b - _c ] + Δ m_b^(5→ 4)(_b) + Δ m^(4)(_b,_c) + δ m_c^(b→ c)(_c), _t - _c= [ _t - _c ] + Δ m_t^(6→ 5)(_t) + Δ m^(5)(_t,_b) + δ m_b,c^(t→ b)(_b,_c) + Δ m_b^(5→ 4)(_b) + Δ m^(4)(_b,_c) + δ m_c^(b→ c)(_c),and can be readily evaluated from the highest order results given in Tabs. <ref>, <ref>, <ref> for the case (_t,_b,_c) = (163,4.2,1.3) GeV:_t - _b= 158.800 + (0.032±0.001) + (9.331±0.016) + (0.006±0.001) GeV= 168.169±0.016 GeV ,_b - _c= 2.9 + (0.004±0.001) + (0.423±0.017) + (0.004±0.001) GeV= 3.331±0.017 GeV , _t - _c= 171.500 ± 0.024 GeV ,where we have added all uncertainties quadratically.We can compare our results for the bottom-charm pole mass difference _b - _c to the result obtained in Ref. <cit.> using a fixed-order expansion at O(α_s^3) for the mass difference. Their result was based on a linear approximation for the virtual charm quark mass effects derived in Ref. <cit.> which is similar to Eq. (<ref>), but used a numerical calculation of the coefficient linear in r_qQ from Ref. <cit.>. In this analysis the pole mass difference was used to eliminate the charm quark mass as a primary parameter in the predictions. They determined_b - _c=3.401± 0.013 GeV and obtained _c=1.22± 0.06 GeV from the fits using _b=4.16± 0.05 GeV as input. Their result for _b - _c is consistent with ours, but one should keep in mind thatlogarithms of _c/_b were not systematically summed and thattheir result also included nontrivial QCD corrections to semileptonic B-meson decay spectra for B→ X_cℓν and B→ X_sγ which were only known to O(α_s^2). The mutual agreement is reassuring (also for the theoretical approximations made in the context of the B meson analyses) and in particular shows that the summation of logarithms of _c/_b is not essential for bottom and charm masses, which is expected, and that the O(α_s^4) corrections are tiny, which can also be seen explicitly in our results. The larger error we obtain in our computation of _b - _c arises from the renormalization scale scale variation in Δ m^(4)(_b,_c) which includes scales as low as 0.6_c while in their analysis the lowest renormalization scale was _c.Similar determinations of bottom and charm quark masses from B-meson decay spectra were carried out in Ref. <cit.>, and they are also consistent with our result for _b - _c. For the case (_t,_b,_c) = (163,4.2,0) GeV, the difference between the top and bottom pole masses reads_t - _b= 158.800 + (0.032±0.001) + (9.331±0.016) + (0.005±0.001) GeV= 168.168±0.016 GeV .This result differs from Eq. (<ref>) by only 1 MeV showing that the effects of the finite charm quark mass are tiny in the difference of the top and bottom pole masses.The uncertainties in the pole mass differences are between 16 and 24 MeV and should be considered as conservative estimates of the theoretical uncertainties due to missing higher order corrections. §.§ Lighter Massive Flavor DecouplingAnother very instructive application of the RG framework to quantify and separate the contributions to the pole mass of a heavy quark coming from the different physical momentum regions is to examine the effective massive flavor decoupling at large orders. It was observed in Ref. <cit.> that the sum of the known O(α_s^2) and O(α_s^3) charm quark mass effects in the bottom quark pole- mass series expressed in four flavor coupling α_s^(4)(_b) (where they amount to about 35 MeV) are essentially fully captured simply by expressing the series in the three flavor coupling α_s^(3)(_b) (where they amount to only -2 MeV). This observation entails that one can simply neglect the charm quark mass corrections by computing the bottom quark pole- mass relation right from start in the three flavor theory without any charm quark (which corresponds to an infinitely heavy charm quark). This effective decoupling of lighter massive quarks is obvious and truly happening at asymptotic large orders. The importance of the observation made in Ref. <cit.> was that the finite charm quark mass corrections in the decoupled calculation at O(α_s^2) and O(α_s^3) were so tiny that there was no need to compute them explicitly in the first place. If this decoupling property would be true in general (i.e. the remaining light quark mass correction become negligible) it would represent a great simplification because it may make an explicit calculation of the lighter massive quark corrections and also the summation of the associated logarithms irrelevant. Using the RG framework for the lighter massive flavor dependence of the pole mass we can examine systematically in which way this effective lighter massive quark decoupling property is realized. In the following we analyze this issue for (_t,_b,_c)=(163,4.2,1.3) GeV. We start with the effects of the charm quark mass in the bottom pole- mass relation examined in Ref. <cit.>. Applying the same considerations as for the pole mass differences in Sec. <ref> for this case we can write down the relation_b-[ _b+_b∑_n=1^∞ a_n(=3,0)(α_s^(3)(_b)/4π)^n ] =Δ m_b^(5→ 4)(_b)+Δ m^(4)(_b,_c)+δ m_c^(b→ c)(_c)+Δ m_c^(4→ 3)(_c) -Δ m^(3)(_b,_c)=(0.004±0.001) + (0.423±0.017) + (0.004±0.001) + (0.005±0.002) - (0.434±0.020)GeV= 0.002±0.026 GeV .The RHS represents a computation of the charm quark mass corrections that remain within a calculation where the charm mass effects are approximated by making the charm infinitely heavy (i.e. =3). The individual numerical results have been taken from the highest order results in Tabs. <ref>, <ref> and <ref>, and for the final numerical result we have conservatively added all uncertainties quadratically. We see that these remaining corrections are essentially zero, fully confirming the observation of Ref. <cit.>. This is not surprising since the bottom and charm quark masses are similar in size and the ratio _c/_b does not lead to large logarithms. So the summation of these logarithms which is contained in our computation does not make an improvement, and the agreement with Ref. <cit.> simply represents a computational cross check of both calculations. The scale uncertainty is larger than the one shown in Ref. <cit.> because we considered variations of the renormalization scale down to μ=0.6 _c, which were not considered by them, and because we do not attempt to eliminate the strong correlation in scale-dependence between Δ m^(4)(_b,_c) and Δ m^(3)(_b,_c) from these low scales here. Let us now investigate the case of the bottom quark mass corrections in the top quark pole- mass relation assuming a massless charm quark. We can simply adapt Eq. (<ref>) through trivial modifications and obtain the relation_t- [ _t+_t∑_n=1^∞ a_n(=4,0)(α_s^(4)(_t)/4π)^n ] =Δ m_t^(6→ 5)(_t)+Δ m^(5)(_t,_b)+δ m_b,c^(t→ b)(_b,0)+Δ m_b^(5→ 4)(_b)-Δ m^(4)(_t,_b)= (0.032±0.001) + (9.331±0.016) + (0.005±0.001)+ (0.004±0.001)- (9.114±0.014)GeV= 0.258±0.021 GeV .We see that using the approximation of an infinitely heavy bottom quark for a calculation of the bottom mass effects in the top quark pole- mass relation gives a result that is about 260 MeV too small.We can now go one step further and also consider the case where the masses of both the bottom and charm quark are accounted for. Generalizing the previous two calculations to this case is straightforward and we obtain_t- [ _t+_t∑_n=1^∞ a_n(=3,0)(α_s^(3)(_t)/4π)^n ] =Δ m_t^(6→ 5)(_t)+Δ m^(5)(_t,_b)+δ m_b,c^(t→ b)(_b,_c)+Δ m_b^(5→ 4)(_b)+Δ m^(4)(_b,_c)+δ m_c^(b→ c)(_c)+Δ m_c^(4→ 3)(_c) -Δ m^(3)(_t,_c).= (0.032±0.001) + (9.331±0.016) + (0.006±0.001)+ (0.004±0.001) + (0.423±0.017) + (0.004±0.001) + (0.005±0.002) - (9.111±0.032)GeV= 0.694±0.040 GeV .In this case using the approximation of infinitely heavy bottom and charm quarks for a calculation of the bottom and charm mass effects in the top quark pole- mass relation gives a result that is almost 700 MeV too small.Our results show that the approximation of computing thelighter heavy flavor mass corrections in a theory where these heavy flavors are decoupled is an excellent approximation for the charm mass corrections in the bottom quark pole mass, but it is considerably worse for the top quark, where the discrepancy even reaches the 1 GeV level. The reason is that the decoupling limit can in general not capture the true size of the lighter quark mass effects if the hierarchy of scales is large. One should therefore not use this approximation to determine bottom or charm quark mass effects for the top quark. § THE TOP QUARK POLE MASS AMBIGUITY §.§ General Comments and Estimation MethodIn this section we address the question of the best possible approximation and the ambiguity of the top quark pole mass m_t^ pole using the RG formalism for the top mass described in the earlier sections. As a reminder and for illustration we show in Fig. <ref> _t as a function of the order obtained from the series for _t - _t(_t) in powers of α_s^(5) given in Eq. (<ref>) for massless bottom and charm quarks, where the central dots are obtained for the default choice of renormalization scale μ=_t in the strong coupling and the error bars represent the scale variation _t/2≤μ≤2 _t. The corresponding results from the series for _t-_t given in Eq. (<ref>) in powers of α_s^(6), also for massless bottom and charm quarks, are shown in gray. We have used the asymptotic form of the perturbative coefficients shown in Tab. <ref> for the series coefficients beyond 𝒪(α_s^4)[The uncertainties of the normalization factors N_1/2^() are about an order of magnitude smaller than the renormalization scale variation of the series beyond O(α_s^4) and therefore not significant for our analysis.].We note that focusing on the approximation of massless bottom and charm quarks by itself is phenomenologically valuable because it is employed for most current predictions in the context of top quark physics, and since the analytic expressions are most transparent for this case.The graphics illustrates visually the problematic features associated to the top quark O() pole mass renormalon, and in particular the specific properties of the series for μ∼ m_t already mentioned in Sec. <ref>: The minimal term of the series is obtained at order n_ min = 8, which according to the theory of asymptotic series is the order that provides the best possible approximation for the top quark pole mass. Furthermore, the corrections are numerically close to the eighth order correction for the orders in the range 6 to 10, i.e. Δ n ≈ 5, for which the partially summed series increases linearly with the order. According to the theory of asymptotic series it is this region of orders that is relevant for the size of the principle uncertainty of this best approximation.We also see two very important practical issues appearing already at lower orders which can make dealing with the pole mass in mass determinations difficult: First, the higher order corrections are much larger than indicated by usual renormalization scale variations of the lower order prediction and, second, the common renormalization scale variation at any given truncation order is not an appropriate tool to estimate the perturbative uncertainty. In this context it is easy to understand that specifying a concrete numerical value for the principle uncertainty of the top quark pole mass is non-trivial even if the series is known precisely to all orders. So to obtain a top quark pole mass determination with uncertainties close to the principle uncertainty within a phenomenological analysis based on a usual truncated finite order calculation may be quite difficult. As a comparison let us recall the much better perturbative behavior of a series that is free of an O() renormalon ambiguity such as the MSR mass differences Δ m^(n_Q)(R,R^') of Eq. (<ref>) with numerical evaluations given in Tab. <ref>.Prior to this work the issue of the best possible estimate and the ambiguity of the top quark pole mass were already studied in Ref. <cit.>. They examined the pole- mass relation of Eq. (<ref>) for massless bottom and charm quarks (i.e. =n_t==5) and their analysis addressed the numerical uncertainty of the top quark pole mass accounting for all series terms displayed in Fig. <ref> for μ=_t. They adopted a prescription given in Ref. <cit.>, which defined the top quark pole mass uncertainty as the imaginary part of the inverse Borel integral of Eq. (<ref>), Δ m_ Borel^(=5), divided by π, which gives about 65 MeV. Since this agrees in size with the minimal series term[ In Ref. <cit.> the order of the minimal series term n_ min and the size of the minimal term Δ(n_ min) were not chosen from the set of the actual series terms but computed from the minimum of a quadratic fit to the series terms in the vicinity of the minimum, so that their n_ min was a non-integer value and theirΔ(n_ min) value is slightly smaller than the minimal term in the series.There are neither practical nor conceptual advantages of this procedure, and the numerical results are unchanged within their errors if Δ(n_ min) is taken as the minimal terms in the series.],which arises at order α_s^8,they argued that Δ m_ Borel^(=5)/π (or the size of the minimal term) is a reliable quantification of the top quark pole mass ambiguity, which they finally specified as 70 MeV. Interpreting the specification like a numerical uncertainty, this gives m_t^ pole=173.10± 0.07, which is shown in Fig. <ref> as the thin gray horizontal band. The uncertainty band is about the same size as the renormalization scale variation of the series truncated at the eighth order.We believe that quoting 70 MeV for the top quark pole mass ambiguity for massless bottom and charm quarks is too optimistic.Given (i) the overall bad behavior of the series,(ii) that there is a sizable range of orders where the corrections have very similar size and (iii) that the partially summed series increases linearly with the order in the range 6 to 10 (Δ n ≈ 5), we see no compelling reason to truncate precisely at the order n_ min=8 and to quote a number at the level of the scale variation of the truncated series or the size of the correction at this order as the principle uncertainty. Our view is also supported by heavy quark symmetry (HQS) <cit.> which states that the pole mass ambiguity is independent of the mass of the heavy quark up to power corrections of O(Λ_ QCD^2/m_Q). This is the first aspect following from HQS we discussed in Sec. <ref>. HQS requires that the criteria and the outcome of the method used to determine the top quark pole mass ambiguity are independent of the top mass value (as long as it is sufficiently bigger than Λ_ QCD). So it is straightforward to carry out a test concerning HQS by changing the value of _t while keeping μ/_t=1 and checking whether the approach to estimate the ambiguity provides stable results.Concerning Ref. <cit.> this check is best carried out in the five-flavor scheme for the strong coupling, and we therefore evaluate the size of the minimal term in the series for m_t^ pole-m_t^ MSR(_t). Adopting the values 163, 20, 4.2, 2 and 1.3 GeV for _t we obtain 62, 75, 91, 113 and 131 MeV for the minimal term Δ(n_ min). This behavior is roughly described by the approximate formula Δ(n_ min)≈(4πα_s^(=5)(μ)/β_0^(=5))^1/2Λ_ QCD^(=5), already mentioned in Sec. <ref> and shows that the basic dependence on μ is logarithmic. We can even render the minimal term arbitrarily small if we adopt for _t values much larger than 163 GeV.We see thatΔ m_ Borel^(=5)/π, which is independent of the top mass value and therefore proportional to the ambiguity, agrees with the size of the minimal term only for μ∼ 163 GeV, but disagrees for other choices. So theline of reasoning used for the analysis of the top quark pole mass ambiguity inRef. <cit.> is not independent of the top quark mass value, and one has to conclude that the ambiguity must be larger than Δ m_ Borel^(=5)/π and certainly larger than 130 MeV, which is the size of the minimal term for a very small value of _t. Concerning the quoted numbers, we emphasize that we still discuss the case of massless bottom and charm quarks. From the relationΔ n×Δ(n_ min)∝π^2 Λ_ QCD^()/β_0∝Δ m_ Borel we see in particular that a reliable method consistent with HQS has to explicitly account for the range n_ min±Δ n/2 in ordersfor which the terms in the series have values close to Δ(n_ min). We stress that the latter issue is not at all new and has been known since the work of Refs. <cit.>. It was also argued in <cit.> that their approach to estimate the size of the top quark pole mass ambiguity is consistent concerning that issue. However, their approach did not account for the actual size of Δ n, which is about 5 for the case discussed in <cit.> and also shown in Fig. <ref>.In the following subsections we apply a method to determine the best possible estimate and the ambiguity of the top quark pole mass which explicitly accounts for the range n_ min±Δ n/2 in orders where the Δ(n) are very close to Δ(n_ min). It also accounts for the practical problems in an order-by-order determination of the pole mass from a series containing the O(Λ_ QCD) renormalon which we discussed above in the context of Fig. <ref>. To describe the method we define, for a given series to calculate the top quark pole mass,Δ(n) ≡_t(n) - _t(n-1),where _t(n) is the partial sumat O(α_s^n) of the series for the top quark pole mass that contains the O() pole mass renormalon, and thus Δ(n) is the n-th order correction. The method we use is as follows:* We determine the minimal term Δ(n_ min) and the set of orders {n}_f≡{n: Δ(n)≤ f Δ(n_ min)} in the series for a default renormalization scale, where f is a number larger but close to unity. * We use half of the range of values covered by _t(n) with n∈{n}_f evaluated for this setup and include renormalization scale variation in a given range as an estimate for the ambiguity of the top quark mass. We use the midpoint of the covered range as the central value.While n_ min, Δ(n_ min) and Δ n each can vary substantially depending on which setup one uses to determine m_t^ pole, the method provides results that are setup-independent and is therefore consistent with HQS. Through the RG formalism we developed in the previous sections we can explicitly implement the other important requirement of HQS, namely that the ambiguities of the pole masses of all heavy quarks agree. To do this we apply our method for three different scenarios which differ on whether the bottom and charm quarks are treated as massive or massless and we furthermore study the pole-MSR mass difference for different values of R.§.§ Massless Bottom and Charm QuarksFor the case that the bottom and charm quarks are treated as massless we can calculate the top quark pole mass from the top MSR mass _t(R) at different scales R≤_t. Using the -MSR mass matching contribution Δ m_t^(6→ 5)(_t) of Eq. (<ref>) and R-evolution from the scale _t to R of Eq. (<ref>) with n_t=5 active dynamical flavors one can write the top quark pole mass as_t = _t + Δ m_t^(6→ 5)(_t) + Δ m^(5)(_t,R) + R ∑_n=1^∞ a_n(n_ℓ=5,0)(α_s^(5)(R)/4π)^n,where the sum of the second and third term on the RHS is just _t(R)-_t. The terms Δ m_t^(6→ 5)(_t) and Δ m^(5)(_t,R) are free of an O() renormalon ambiguity and can be evaluated to the highest order given in Tabs. <ref> and <ref>. We can then determine the best estimate of the top quark pole mass and its O() renormalon ambiguity from the R-dependent series which is just equal to _t-_t(R). The outcome of the analysis using the method described in Sec. <ref> for _t=163 GeV and R=163,20,4.2 and 1.3 GeV and f=5/4 is shown in the upper section of Tab. <ref>.The entries are as follows: The second column shows _t(R)-_t = Δ m_t^(6→ 5)(_t) + Δ m^(5)(_t,R) at the highest order. The third and fourth column show the order n_ min and Δ(n_ min) for the default renormalization scale μ=R for the cases R=163,20 and 4.2 GeV and μ=2_c for R=1.3 GeV. The values for Δ(n_ min) for R=163 and 20 GeV have an uncertainty because for these cases n_ min>4 and the values for Δ(n>4) are determined from the asymptotic large order values given in Tab. <ref> which have a numerical uncertainty from the normalization factor N_1/2^(5) in Eqs. (<ref>). The fifth column shows the sum of the perturbative corrections beyond the explicitly calculated O(α_s^4) terms up to order n_ min showing the amount of extrapolation needed to obtain the best possible top quark mass based on the asymptotic approximation. The sixth column shows the set of orders {n}_f=5/4 for which Δ(n)≤ f Δ(n_ min) and which are used for determining the best estimate and the uncertainty of the top quark pole mass. The seventh column then contains the best estimate and the ambiguity of the series for _t - _t(R) using the method from Sec. <ref>. To obtain the uncertainties we used renormalization scale variation for α_s^(5)(μ) in the range R/2≤μ≤ 2 R for the cases R=163,20,4.2 GeV and in the range 1.5GeV≤μ≤ 5 GeV for R = 1.3 GeV.For R=1.3 GeV we always use renormalization scales μ of the strong coupling that are larger than 1.5 GeV because the dependence on the renormalization scale grows rapidly for smaller scales. The last column contains the final result for _t combining the results for _t(R) - _t and _t - _t(R) where the uncertainties of both are added quadratically to give the final number for the ambiguity of m_t^ pole. These results are also displayed graphically in Figs. <ref>-<ref> as the gray hatched horizontal bands.In Figs. <ref> we have also shown in black the results for _t(n) over the order n for the different setups where the dots are the results for the default renormalization scales that are used to determine n_ min, Δ(n_ min) and {n}_f. The error bars represent the range of values at each order of the truncated series coming from the variations of the renormalization scale of the strong coupling. The black dot at n=0 visible in Figs. <ref>, <ref> shows the highest order result for _t(R). We see that the results for the top quark pole mass m_t^ pole for the different R values are fully compatible to each other. In particular, the ambiguity estimates based on our method agree within ± 15% and average to 182 MeV. Furthermore, the central values for the best estimates vary by at most 110 MeV and average to 173.150 GeV. It is reassuring that the spread of the central values is smaller than the size of the ambiguity. We emphasize that the consistency of our results for the different R values to each other cannot be interpreted in any way statistically since the analyses for different R values are not theoretically independent. The agreement just shows that our method is consistent since the best estimate (and also the ambiguity) of the top quark pole mass is independent of R.Interestingly our estimate for the ambiguity of the top quark pole mass agrees quite well with ^(=5) = 166 MeV given in Eq. (<ref>).As already pointed out in Sec. <ref>, the minimal correction Δ(n_ min) increases from around 60 MeV for R=163 GeV to about100 MeV[ This number is obtained for the default renormalization scale μ=2 _c=2.6 GeV. In the short analysis of Sec. <ref> we quoted 131 MeV for the size of the minimal term for R=1.3 GeV, which was obtained for μ=1.3 GeV.]for R=1.3 GeV. At the same time, the order n_ min where the minimal correction Δ(n_ min) arises decreases from n_ min=8 at R=_t down to n_ min=4 and 3 for R=4.2 and 1.3 GeV. Moreover, the contribution in the best estimate for m_t^ pole from orders beyond n=4 until order n_ min decreases from about 310 MeV at R=_t to about 150 MeV at R=20 GeV. For R scales around the bottom quark mass and below, where n_ min≤ 4, there is no need any more to extrapolate beyond the explicitly calculated four orders to get the best value for m_t^ pole. This information is not just of academic importance but it is also relevant for phenomenology: The MSR mass m_t^ MSR(R) for some low scale R can serve as a low-scale short-distance mass for a physical application where the characteristic physical scale is R. Typical examples include the top pair inclusive cross section at the production threshold where R ∼ m_tα_s ∼ 25 GeV <cit.>, or the reconstructed invariant top quark mass distribution where R is in the range of 5 to 10 GeV <cit.>. The behavior of the series for m_t^ pole-m_t^ MSR(R) thus reflects the typical behavior of the QCD corrections to the mass for the respective physical applications.The observations we make for the R-dependence of the behavior of the series show that the best possible determination of the top quark mass from an observable characterized by a low characteristic physical scale can in general be achieved at a lower order and also involves smaller perturbative corrections compared to an observable characterized by high characteristic physical scales (such as inclusive top pair cross sections at high energies or virtual top quark effects). This general property is also reflected visually in the graphical illustrations shown in Fig. <ref>.We note that our numerical analysis has a rather weak overall dependence on the choice of f and that the results change by construction in a non-continuous way. Using f=4/3 only the outcome for R=20 GeV is modified to _t-_t(R) = 2.100±0.206. Using f=6/5 only the outcome for R=163 GeV is modified to _t-_t(R) = 10.088±0.123. This leaves the overall conclusion about the ambiguity of the top quark pole mass unchanged and we therefore consider f=5/4 as a reasonable default choice.Comparing our results to those of Ref. <cit.>, we find that our estimate of the top quark pole mass ambiguity of 180 MeV exceeds theirs of 70 MeV by a factor of 2.5. The discrepancy arises since their result was only related to the size of the minimal term Δ(n_ min) for an R value close to 163 GeV and did not account for the number of orders Δ n for which the Δ(n) are close to the minimal term Δ(n_ min). For R = 163 GeV we have Δ n = 4 for f=5/4 and we see the discrepancy is roughly compatible with Δ n/2.Since for other choices of R the values of Δ(n_ min) and Δ n vary individually substantially (while their product is stable) we believe that a specification of the top quark pole mass ambiguity of 70 MeV is not consistent with heavy quark symmetry. §.§ Massless Charm QuarkFor the case of a massive bottom quark and treating the charm quark as massless we can calculate the top quark pole mass from the bottom MSR mass m_b^(R≤_b) using the top-bottom mass matching contribution δ m_b,c^(t→ b)(_b,0) of Eq. (<ref>) for _c=0 in combination with the top and bottom -MSR mass matching contributions, Δ m_t^(6→ 5)(_t) and Δ m_b^(5→ 4)(_b) of Eq. (<ref>) and R-evolution, see Eq. (<ref>), with n_t=5 active dynamicalflavors from _t to _b and with n_b=4 active dynamical flavors from _b to R. The resulting expression for the top quark pole mass systematically sums all logarithms log(_b/_t) and uses that the bottom quark pole-MSR mass relation, which specifies the bottom quark pole mass ambiguity, fully encodes the top quark pole mass ambiguity due to heavy quark symmetry. The expression for the top quark pole mass we use readsm_t^=_t+Δ m_t^(6→ 5)(_t)+Δ m^(5)(_t,_b)+δ m_b,c^(t→ b)(_b,0) +Δ m_b^(5→ 4)(_b) +Δ m^(4)(_b,R) + R∑_n=1^∞a_n(=4,0)(α_s^(4)(R)/4π)^n ,where the sum of the first four terms on the RHS is just m_t^-m_b^+_b, using Eq. (<ref>), and the sum of the fifth and sixth term is the difference of the bottom MSR andmasses m_b^(R)-_b. Both quantities are free of an () renormalon ambiguity and can be evaluated to the highest order given in Tabs. <ref>, <ref> and <ref>. We can then study the uncertainty of the top quark pole mass and its () renormalon ambiguity from the R-dependent series which is just equal to m_b^-m_b^(R). The outcome of the analysis using the method described in Sec. <ref> for (_t,_b)=(163,4.2) GeV as well as R=163, 20, 4.2, 1.3 GeV and f=5/4 is shown in the middle section of Tab. <ref>. Except for the second and seventh column the entries are analogous to the analysis for _b=_c=0 in Sec. <ref>. Here, the second column shows m_t^-m_b^+m_b^(R)-_t and the seventh shows m_b^-m_b^(R), which contains the () renormalon ambiguity. The default choices and the ranges of variation for the renormalization scale in the strong coupling in the series for m_b^-m_b^(R) are the same as for our analysis for _b=_c=0 in Sec. <ref> for the corresponding R values. The last column contains again the final result for m_t^ combining the results for m_t^-m_b^+m_b^(R)-_t and m_b^-m_b^(R) where the uncertainties of both are added quadratically. The results are also displayed graphically in Figs. <ref>- <ref> as the light red hatched horizontal bands.In the upper section of Tab. <ref> we also show the best estimate for the bottom quark pole mass m_b^ pole obtained for the respective R values, which can be obtained using Eq. (<ref>) and the result for the top-bottom pole mass difference of Eq. (<ref>). In Figs. <ref> we have shown in red the results for m_t^(n) over the order n for the different setups where the dots are again the results for the default renormalization scales that are used to determine n_min, Δ(n_min) and {n}_f. The error bars are the range of values coming from the variations of the renormalization scale of the strong coupling. The red dots at n=0 visible in Figs. <ref> and <ref> show the highest order results for m_t^-m_b^+m_b^(R).We again see that the results for the top quark pole mass for the different R values are compatible each other. The ambiguity estimates average to 217 MeV. Interestingly this estimate for the ambiguity of the top quark pole mass roughly agrees with ^(=4)=225 MeV given in Eq. (<ref>). This is larger than ^(5)=166 MeV since the infrared sensitivity of the top quark pole mass increases when the number of massless quarks is decreased (i.e. β_0^(4)>β_0^(5)). Furthermore, we observe that the central values for the top quark pole mass cover a range that is compatible with case of a massless bottom quark. The central values average to 173.168 GeV which is about 20 MeV larger than for a massless bottom quark, which is, however, insignificant given the range of values covered by the central values or even the size of the ambiguity. So the bottom quark mass does essentially not affect the overall value of the top quark pole mass. We also note that the minimal corrections Δ(n_min) are all larger than the corresponding terms for the case of massless bottom and charm quarks. For R=4.2 and 1.3 GeV they amount to about 130 MeV. §.§ Massive Bottom and Charm QuarksWe now, finally, consider the case that both the bottom and the charm quark masses are accounted for. Since this situation involves three scales, it is the most complicated concerning matching and evolution that systematically sums logarithms log(_t/_b) and log(_b/_c). However, the case can be treated in a straightforward way by iterating the top-bottom mass matching procedure of the previous section one more time concerning the bottom-charm mass matching. The resulting formula for the top quark pole mass reads_t=_t +Δ m_t^(6→ 5)(_t)+Δ m^(5)(_t,_b)+δ m_b,c^(t→ b)(_b,_c)+Δ m_b^(5→ 4)(_b)+Δ m^(4)(_b,_c)+δ m_c^(b→ c)(_c)+Δ m_c^(4→ 3)(_c)+Δ m^(3)(_c,R)+ R∑_n=1^∞ a_n(=3,0)(α_s^(3)(R)/4π)^n .The expression combines the top-bottom and bottom-charm mass matching contributions δ m_b,c^(t→ b)(_b,_c) and δ m_c^(b→ c)(_c) from Eqs. (<ref>) and (<ref>), respectively, and the top, bottom and charm -MSR mass matching contributions Δ m_t^(6→ 5)(_t), Δ m_b^(5→ 4)(_b) and Δ m_c^(4→ 3)(_c) of Eq. (<ref>). Furthermore it contains contributions from R-evolution with n_t=5 active dynamical flavors from _t to _b, with n_b=4 active dynamical flavors from _b to _c and with n_b=3 active dynamical flavors from _c to R. We do not employ any evolution to scales below _c due to instabilities of perturbation theory for the charm pole-MSR mass relation at such low scales but we can explore scales above _c using the R-evolution.On the RHS of Eq. (<ref>) the sum of the first seven terms is just m_t^-m_c^+_c, using Eq. (<ref>), and the eighth term is the charm -MSR matching contribution. Both quantities are free from an () renormalon ambiguity and can be evaluated to the highest order given in Tabs. <ref>, <ref> and <ref>. We can then study the ambiguity of the top quark pole mass due to the () renormalon from the R-dependent series which is just equal to m^_c-_c(R).This relation specifies the charm quark pole mass ambiguity, and it fully encodes the top and bottom quark pole mass ambiguities due to heavy quark symmetry. We note that among all the terms shown in Eq. (<ref>) the contributions from the MSR mass differences Δ m^(5)(_t,_b), Δ m^(4)(_b,_c) and Δ m^(3)(_c,R), determined with R-evolution, and the series proportional to R, which contains the () renormalon, constitute the numerically most important terms. They exceed by far the contributions from the matching corrections, which amount to only 50 MeV and, therefore, fully encode the large order asymptotic behavior of the top quark pole- mass series m_t^ pole-_t as defined in Eq. (<ref>) in the presence of finite bottom and charm quark masses. The large order asymptotic form of the coefficients in the expansion in powers of α_s^(6)(_t) may then be determined directly from these terms for R=_c using the analytic solution for the MSR mass differences provided in Eq. (4.2) of Ref. <cit.> and expanding inα_s^(6)(_t). However, the resulting series suffers from the large logarithms involving the ratios of the top, bottom and charm quark masses, and is therefore less reliable for applications than the result shown in Eq. (<ref>). The outcome of the analysis using the method described in Sec. <ref> for (_t,_b,_c)=(163,4.2,1.3) GeV, as well as R=163, 20, 4.2, 1.3 GeV and f=5/4 is shown in the lower section of Tab. <ref>. Except for the second and seventh column the entries are analogous to the previous two analyses in Secs. <ref> and <ref>. Here, the second column shows m_t^-m_c^+_c(R)-_t and the seventh shows m_c^-_c(R), which contains the () renormalon ambiguity of the top quark pole mass. The default choices and the ranges of variation for the renormalization scale in the strong coupling in the series for m_c^-m_c^(R) are the same as for the two previous analyses in Secs. <ref> and <ref> for the corresponding R values. The last column contains again the final result for m_t^ combining the results for m_t^-m_c^+_c(R)-_t and m_c^-_c(R) where all uncertainties are added quadratically. These results are also displayed graphically in Fig. <ref>-<ref> as the light blue hatched horizontal bands. In the lower section of Tab. <ref> we also show the best estimate for the charm and bottom quark pole masses m_c^ pole and m_b^ pole, respectively, for the different R values, which can be obtained using Eq. (<ref>) and the result for the top-bottom and top-charm pole mass difference of Eqs. (<ref>) and (<ref>).In Fig. <ref> we have also shown in blue the results for m_t^(n) over the order n for the different setups where the dots are again the results for the default renormalization scales that are used to determine n_min, Δ(n_min) and {n}_f. The error bars are the range of values coming from the variations of the renormalization scale of the strong coupling. The blue dots visible in Figs. <ref> and <ref> at n=0 shows the highest order result for m_t^-m_c^+_c(R).We see that the results for the top quark pole mass for the different R values are again fully consistent to each other. The ambiguity estimates average to 253 MeV, which is more than twice the 110 MeV ambiguity obtained in Ref. <cit.>. The reason for the discrepancy is the same as for the analysis for massless bottom and charm quarks already explained in Secs. <ref> and <ref>, and we therefore do not discuss it here further.Concerning the size of the minimal corrections Δ(n_min), we find that they reach116, 154 and 128 MeV for R=20, 4.2 and 1.3 GeV, respectively, each of which is larger than 110 MeV. As in the two previous analyses our result for the ambiguity agrees very well with the corresponding value of , given in Eq. (<ref>), which in this case is also ^(=3)=253 MeV. This is larger than the uncertainties we obtained for the cases discussed in the two previous analyses, where either the bottom and charm quarks were massless or just the charm quark, and thus again follows the pattern that the infrared sensitivity of the top quark pole mass increases when the number of massless quarks decreases (i.e. β_0^(3)>β_0^(4)>β_0^(5)).Furthermore, we find that the central values for the top quark pole mass cover a range that is within errors in agreement with the two previous analyses. The range is, however, shifted slightly upwards by about 70 MeV with respect to the case of massless bottom and charm quarks. For the value of the average we have 173.186 GeV which is about 40 MeV higher than the average 173.150 GeV we obtained for massless bottom and charm quarks. This shift may represent a slight trend, but it is overall insignificant compared to the range of values covered by the central values or the size of the ambiguity.This shows that the charm quark mass, like the bottom quark mass, does not affect the value of the top quark pole mass. We can compare to the result of Ref. <cit.>, where they found that the finite bottom and charm quark masses increase the top quark pole mass by 80± 30 MeV, where the 30 is their estimate for the uncertainty in their computation of the bottom and charm mass effects. This is consistent with the dependence on the bottom and charm masses we find in our analysis. Their prescription was based on a successive order-dependent reduction of the effective flavor number in the series motivated by the decoupling property observed in Ref. <cit.>. It incorporated some basic features of the bottom and charm mass corrections beyond the third order but is otherwise heuristic and does not systematically sum logarithms of _b/_t and _c/_t. The consistency shows that concerning the estimate of the top quark pole mass ambiguity and within errors their prescription provides an adequate approximation.§.§ Overall Assessment for the Pole Mass Ambiguity The overall outcome of the analyses above concerning the best possible estimates (and the ambiguities) of the top quark pole mass and the pole masses of the bottom and charm quarks is summarized as follows: * Heavy quark symmetry states that the ambiguity of a heavy quark pole mass is independent of the mass of the heavy quark and that the ambiguities of the pole masses of all heavy quarks are equivalent.Our method for estimating the ambiguity is insensible to the masses of the heavy quarks and, within any given setup for the heavy quark mass spectrum, obtains the same ambiguities for all heavy quark pole masses. It is therefore fully consistent with heavy quark symmetry. * Our examinations for different setups for the spectrum of the masses of the bottom and charm quarks show that the top quark pole mass ambiguity increases when the numberof massless quarks is decreased (which arises when the number of lighter massive quarks is increased). The numerical size we find agrees very well with ^() defined in Eqs. (<ref>). So our studies show that the well-accepted statement that “heavy quark pole masses have an ambiguity of order ” can be specified to the more precise statement that “the ambiguity of the heavy quark pole masses is ^(), whereis the number of massless quarks”.* Considering the value of the top quark pole mass (and not its ambiguity) we find essentially no dependence on whether the bottom and charm quarks are treated massive or massless. This also implies that there is no dependence on actual values of the bottom and charm quark masses (which are know to a precision of a few 10 MeV in thescheme). Likewise we also find that the value of the bottom quark pole mass has no dependence on whether the charm quark is treated massive or massless.These observations are important because, although the pole mass concept depends, due to the linear sensitivity to small momenta, intrinsically on the spectrum of the lighter massive quarks, they imply that one can give the top and the bottom quark pole masses a unique global meaning irrespective which approximation is used for the bottom and charm masses. In such a global context, however, one has to assign the largest value foras the ambiguity of the pole mass. This value is obtained for finite bottom and charm quark masses and amounts to 250 MeV which we adopt as our final specification of the top quark pole mass ambiguity.§ CONCLUSIONSIn this work we have provided a systematic study of the mass effects of virtual massive quark loops in the relation between the pole mass _Q and short-distance masses such as themass _Q(μ) and the MSR mass _Q(R) <cit.> of a heavy quark Q, where we mean virtual loop insertions of quarks q with < m_q < m_Q. In this context it is well-known that the virtual loops of a massive quark act as an infrared cut-off on the virtuality of the gluon exchange that eliminates the effects of that quark from the large order asymptotic behavior of the series. This effect arises from the O(Λ_ QCD) renormalon contained in the pole mass which means that the QCD corrections have a linear sensitivity of small momenta that increases with the order in the perturbative expansion. The primary aim of this work was to study this effect in detail at the qualitative and quantitative level. We established a renormalization group formalism that allows to discuss the mass effects coming from virtual quark loops in the on-shell self energy diagrams of heavy quarks in a coherent and systematic fashion. We in particular examined (i) how the logarithms of mass ratios that arise in this multi-scale problem can be systematically summed to all orders, (ii) the large order asymptotic behavior and structure of the mass corrections themselves and (iii) the consequences of heavy quark symmetry (HQS). The basis of our formalism is that the difference of the pole mass and a short-distance mass contains the QCD corrections from all momentum scales between zero and the scale at which the short-distance mass is defined, which is μ for themass _Q(μ) or R for the MSR mass m_Q^ MSR(R). The MSR mass _Q(R), which is derived from self energy diagrams like themass, is particularly suited to describe the scale-dependence for momentum scales R<m_Q since its renormalization group (RG) evolution is linear in R, called R-evolution <cit.>. When the finite masses of lighter heavy quarks are accounted for, the MSR mass concept allows to establish a RG evolution and matching procedure where the number of active dynamical flavors governing the evolution changes when the evolution crosses a mass threshold and where threshold corrections arise when a massive flavor is integrated out. This follows entirely the common approach of logarithmic RG equations as known from the n_f flavor dependent μ-evolution of the strong coupling α_s^(n_f)(μ) and reflects the properties of HQS.Due to heavy quark symmetry, the procedure allows for example to relate the QCD corrections in the top quark pole- mass difference _t-_t(_t) that are coming from scales smaller than the bottom mass, to the bottom quark pole- mass difference _b-_b(_b). This relation can be used to generically study and determine the large order asymptotic behavior and the structure of the lighter virtual quark mass corrections in the pole- mass difference of a heavy quark Q. Within the RG framework we have proposed, we find that the bulk of the lighter virtual quark mass corrections is determined by their large order asymptotic behavior already at 𝒪(α_s^3) (very much like the QCD corrections for massless virtual quarks), which confirms earlier observations made in Refs. <cit.> and <cit.>. Using our RG framework and heavy quark symmetry we used this property to predict the previously unknown 𝒪(α_s^4) lighter virtual quark mass corrections to within a few percent from the available information on the 𝒪(α_s^4) corrections for massless lighter quarks without an additional loop computation, see Eq. (<ref>). Furthermore we calculated the differences of the top, bottom and charm quark pole masses with a precision of around 20 MeV, and we analyzed in detail the quality of the coupling approximation of Ref. <cit.>, which works in an excellent way for the charm mass effects in the bottom quark pole mass, where in the context of the top quark, it fails.The second aim of the paper was to use the formalism to determine a concrete numerical specification of the ambiguities of the heavy quark pole masses and in particular of the top quark pole mass. This is of interest because the top quark pole mass is still the most frequently used mass scheme in higher order theoretical predictions for the LHC top physics analyses. The ambiguity of the pole mass is the precision with which the pole mass can be determined in principle given that the complete series is known. This ambiguity is universal (i.e. it exists in equivalent size in any context and cannot be circumvented) and its size can therefore be quantified from the relation of the pole mass and any short-distance mass alone for which all terms in the series can be determined to high precision. With the renormalization group formalism we have proposed we carried out an analysis accounting explicitly for the constraints coming from HQS. HQS states (i) that the ambiguity of a heavy quark is independent of its mass, and (ii) that the QCD effects in the heavy quark masses coming from momenta below the lightest massive quark are all equivalent, which implies that the ambiguities of all heavy quarks are equal.With our formalism both aspects were incorporated and validated in detail at the qualitative and quantitative level. We considered different scenarios for the treatment of the bottom and charm quark masses and employed a method to estimate the ambiguity that does not depend on the mass of the heavy quark in a way that is consistent with heavy quark symmetry. For the caseof massless bottom and charm quarks we found that the ambiguity of the top quark pole mass is 180 MeV,when the charm quark ismassless we found 215 MeV and when the finite masses of both the bottom and charm quarks are accounted for we obtained 250 MeV. Numerically, the ambiguity turns out be essentially equal to the hadronization scale Λ_ QCD^(n_ℓ), defined in Eq. (<ref>), whereis the number of massless quarks.Thus, our analysis allows to specify the well-known qualitative statement “the heavy quark pole masses have an ambiguity of order ” to the more specific statement “the ambiguity of heavy quark pole masses is ^(), whereis the number of massless quarks”. This dependence of the top quark pole mass ambiguity on the number of massless flavors is fully consistent with the behavior expected from the pole mass renormalon. Furthermore, we have found that there is no significant dependence of the central value of the top quark pole mass on whether the bottom and charm quarks are treated as massive or massless. Our results for the ambiguities differ considerably from those of Ref. <cit.>. They estimated the top quark pole mass ambiguity as 70 MeV for the case that bottom and charm masses are neglected and as 110 MeV when the bottom and charm masses are accounted for. We have shown in detail in which ways these values are incompatible with heavy quark symmetry and why our ambiguity estimates should be considered more reliable.If one considers the top quark pole mass as a globally defined mass scheme valid for all choices of approximations for the bottom and charm quark masses, one should assign it an intrinsic principle ambiguity due to the O() renormalon of 250 MeV.We stress, that this intrinsic uncertainty refers to the best possible precision with which one can in principle theoretically determine the top quark pole mass, and does not account in any way for issues unrelated to the pole mass renormalon in applications for actual phenomenological quantities, which typically involve NLO, NNLO or even NNNLO corrections from perturbative QCD. Furthermore, in order to achieve this theoretical precision it is required to have access to orders where the corrections (in the relation involving the pole mass) become minimal. The order where this happens in an actual phenomenological analysis also depends on the typical physical scale (i.e. the value of R) governing the examined quantity. If the top quark mass is determined from a quantity which has a low characteristic physical scale (e.g. top pair production close to threshold, kinematic endpoints, reconstructed top invariant mass distributions) then the minimal term is reached at very low orders, which may well be within the orders that can be calculated explicitly. If the top quark mass is determined from a quantity which has a high characteristic scale of the order or the top quark mass (e.g. total inclusive cross sections at high energies, virtual top quark effects) then the minimal term is reached only at high orders, which are not accessible to full perturbative computations. This also explains why top mass sensitive observables involving low characteristic physical scales are more sensitive for top quark mass determinations than observables involving high characteristic physical scales. So reaching the uncertainties in top quark pole mass determinations that come close to the ambiguity limit is in general much harder for observables governed by high physical scales.Currently, the most precise measurements of the top quark mass from the D0 and CDF experiments at the Tevatron <cit.> and the ATLAS and CMS collaborations at the LHC <cit.> use the top reconstruction method and already reach the level of 500 to 700 MeV. Projections for LHC Run-2 further indicate that this uncertainty can be reduced significantly in the future and may reach the level of 200 MeV for the high-luminosity LHC run <cit.>.The outcome of our analysis disfavors the top quark pole mass as a practically adequate mass parameter in the theoretical interpretation of these measurements.As a final comment we would like to remind the reader that all tricky issues concerning the convergence of the perturbative series andthe way how to properly estimate the ambiguity of top quark pole mass become irrelevant if one employs an adequate short-distance mass definition. This may of course not mean in general that switching to a short-distance mass scheme will automatically lead to smaller uncertainties simply because other unresolved issues may then dominate. The outcome of our analysis, however, implies that even reaching a 250 MeV uncertainty for the top quark pole mass in a reliable way within a practical application is difficult. This is because the O(Λ_ QCD) renormalon prevents using common ways such as scale variation for the truncated series to estimate theoretical uncertainties, and can affect the behavior of the series already at low orders where the corrections still decrease.It is therefore advantageous to abandon the pole mass scheme in favor of an adequately chosen short-distance mass at latest when the available QCD corrections for a mass sensitive quantity yield perturbative uncertainties in the pole mass that become of the order of its ambiguity, which we believe is when they approach 0.5 GeV. § ACKNOWLEDGMENTSWeacknowledge partial support by the FWF Austrian Science Fund under the Doctoral Program No. W1252-N27 and the Project No. P28535-N27 and the U.S. Department of Energy under the Grant No. DE-SC0011090. We also thank the Erwin-Schrödinger International Institute for Mathematics and Physics forpartial support. § VIRTUAL QUARK MASS CORRECTIONS UP TO 3-LOOP ORDER The virtual quark mass corrections of O(α_s^2) were determined in Ref. <cit.> and readδ_2(1) = 8(π^2/3-1)= 18.3189δ_2(r)= 8/9π^2 +16/3ln^2 r - 16/3 r^2(3/2 + ln r) + 16/3(1+r)(1+r^3)(π^2/6 - 1/2ln^2 r + ln rln(1+r) +Li_2(-r))+ 16/3(1-r)(1-r^3)(-π^2/3 - 1/2ln^2 r + ln rln(1-r) +Li_2(r)).The expansion of δ_2 for small r has the form δ_2(r)=(8π^2/3) r -16 r^2+(8π^2/3) r^3+…. At O(α_s^3) the virtual quark mass corrections were determined semi-analytically in Ref. <cit.> for the case of one more massive quark q in the heavy quark Q self-energy. The corrections from the insertions of virtual loops of two different massive quarks q and q^' were not provided and are given in Eq. (<ref>). In the following we provide the results for the full set ofO(α_s^3) virtual quark mass corrections using the results from Ref. <cit.> in the expansion for _q/_Q≪1 adapted to our notation. The expressions for general _q/_Q, which are extensive, can be downloaded at <https://backend.univie.ac.at/fileadmin/user_upload/i_particle_physics/publications/hpw.m>. We consider the O(α_s^3) virtual quark mass corrections to the pole- mass relation of the heavy quark Q coming from n lighter massive quarks q_1, q_2, …q_n in the order of decreasing mass and n_ℓ additional quarks lighter than Λ_ QCD, which we treat as massless. So, the number n_Q of quark flavors lighter than quark Q is n_Q=n+. The expressions for the functions δ_Q,3 defined in Eqs. (<ref>) and (<ref>) can be written in the formδ_Q,3^(Q,q_1,q_2,…,q_n)(1,r_q_1 Q,…,r_q_n Q)= h(1) + (+1) p(1) + ∑_i=1^n w(1,r_q_i Q), δ_Q,3^(q_1,q_2,…,q_n)(r_q_1 Q,r_q_2 Q,…,r_q_n Q)= h(r_q_1 Q) +p(r_q_1 Q) + ∑_i=2^n w(r_q_1 Q,r_q_i Q), δ_Q,3^(q_m,q_m+1,…,q_n)(r_q_m Q,r_q_m+1 Q,…,r_q_n Q)= h(r_q_m Q) + (-m+1) p(r_q_m Q) + ∑_i=m+1^n w(r_q_m Q,r_q_i Q).All three formulae follow the same general scheme, where the number multiplying the function p(r) is just the number of massive quarks in the superscript plus the number of massless quarks, . We have displayed them nevertheless for clarity. The explicit form of the functions h, p and w ish(1)= 1870.7877,h(r)= r (1486.55 - 1158.03 ln r)+ r^2 (-884.044-683.967ln r ) + r^3 (906.021 - 1126.84 ln r) + r^4 (225.158 + 11.4991ln r - 80.3086 ln^2 r + 21.3333 ln^3 r)+ r^5 (126.996 -182.478ln r) + r^6 (-22.8899 + 38.3536ln r - 54.5284 ln^2 r)+ r^7 (15.3830 - 34.8914ln r) + r^8 (2.52528 - 3.82270ln r - 20.4593 ln^2 r) +O(r^9),andp(1)= -82.1208,p(r)= 32/27∫_0^∞ z[z/2 + (1-z/2)√(1+4/z) ]P(r^2/z)(ln z-5/3)= r (-66.4668 + 70.1839 ln r) + r^214.2222 + r^3 (15.4143 + 70.1839 ln r)+ r^4 (-23.1242 + 18.0613ln r + 15.4074ln^2 r - 4.74074ln^3 r) - 31.5827 r^5+ r^6 (11.9886 - 1.70667ln r) - 4.17761 r^7 + r^8 (2.40987 - 0.161088ln r)+O(r^9) ,as well as w(1,1)= 6.77871,w(1,r)= r^2 14.2222- 18.7157 r^3 + r^4 (7.36885 - 11.1477ln r)+ r^6 (3.92059 - 3.60296ln r + 1.89630ln^2 r)+ r^8 (0.0837382 - 0.0772789 ln r + 0.457144 ln^2 r) +O(r^9) , w(r_1,r_2)= p(r_2) + 32/27∫_0^∞ z[z/2 + (1-z/2)√(1+4/z) ]P(r_1^2/z) P(r_2^2/z) ,whereΠ(x)= 1/3 - (1-2 x)[2-√(1+4 x)ln(√(1+4 x)+1/√(1+4 x)-1)], P(x)= Π(x) + ln x + 5/3 .JHEP | http://arxiv.org/abs/1706.08526v2 | {
"authors": [
"Andre H. Hoang",
"Christopher Lepenik",
"Moritz Preisser"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170626180001",
"title": "On the Light Massive Flavor Dependence of the Large Order Asymptotic Behavior and the Ambiguity of the Pole Mass"
} |
[email protected] Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio),RJ, 22453-900, [email protected] Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio),RJ, 22453-900, BrazilInstituto de Física, Universidade Federal Fluminense, 24210-346 Niterói, RJ, BrazilDepartamento de Física, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio),RJ, 22453-900, Brazil The SU(4)-SU(2) crossover, driven by an external magnetic field h, is analyzed in a capacitively-coupled double-quantum-dot device connected to independent leads. As one continuously charges the dots from empty to quarter-filled, by varying the gate potential V_g, the crossover starts when the magnitude of the spin polarization of the double quantum dot, as measured by ⟨ n_↑⟩ -⟨ n_↓⟩, becomes finite.Although the external magnetic field breaks the SU(4) symmetry of the Hamiltonian, the ground state preserves it in a region of V_g, where ⟨ n_↑⟩ -⟨ n_↓⟩ =0.Once the spin polarization becomes finite, it initially increases slowly until a sudden change occurs, in which ⟨ n_↓⟩ (polarization direction opposite to the magnetic field) reaches a maximum and then decreases to negligible values abruptly, at which point an orbital SU(2) ground state is fully established.This crossover from one Kondo state, with emergent SU(4) symmetry, where spin and orbital degrees of freedom all play a role, to another, with SU(2) symmetry, where only orbital degrees of freedom participate, is triggered by a competition between gμ_Bh, the energy gain by the Zeeman-split polarized state and the Kondo temperature T_K^SU(4), the gain provided by the SU(4) unpolarized Kondo-singlet state.At fixed magnetic field, the knob that controls the crossover is the gate potential, which changes the quantum dots occupancies.If one characterizes the occurrence of the crossover by V_g^max, the value of V_g where ⟨ n_↓⟩ reaches a maximum, one finds that the function f relating the Zeeman splitting, B_max, that corresponds to V_g^max, i.e., B_max=f(V_g^max), has a similar universal behavior to that of the function relating the Kondo temperature to V_g. In addition, our numerical results show that near the SU(4) Kondo temperature and for relatively small magnetic fields the device has a ground state that restricts the electronic population at the dots to be spin polarized along the magnetic field.These two facts introduce very efficient spin-filter properties to the device, also discussed in detail in the paper.This phenomenology is studied adopting two different formalisms: the Mean Field Slave Bosons Approximation, which allows an approximate analysis of the dynamical properties of the system, and a Projection Operator Approach, which has been shown to describe very accurately the physics associated to the ground state of Kondo systems.The SU(4)-SU(2) crossover and spin filter properties of a double quantum dot nanosystem E. V. Anda December 30, 2023 =======================================================================================§ INTRODUCTION The discovery in 1998 of the Kondo effect in artificial atoms <cit.>, so-called quantum dots (QDs), has greatly motivated the study of this phenomenon in nanostructures in the last two decades. Since the pioneering works where QDs were shown to possess all the properties of real atoms <cit.>, very many investigations were done to determine the behavior of different structures of QDs associated to the Kondo effect <cit.>.It has been shown that nano-systems with QDs are powerful tools to experimentally investigate a variety of properties of highly correlated electrons <cit.>. QDs have proven as well to have very interesting applications as quantum gates <cit.>, spin filters <cit.>, and thermal conductors <cit.>.Transport properties as a function of temperature, magnetic field, and gate potential, have been analyzed in systems with lateral QDs <cit.>, carbon nanotubes <cit.>, molecular transistors <cit.>, etc. The major reason the interest in these studies has increased is due to advances in experimental techniques and in the fabrication of nano-devices, which have raised the prospect of many applications in areas like nano-electronics <cit.>, spintronics <cit.>, and quantum computation <cit.>.All this experimental activity has greatly promoted the development of new theoretical studies and formalisms to analyze this phenomenology. A large amount of theoretical predictions related to electronic transport have been obtained using numerical methods. Among the most largely utilized we can mention the Numerical Renormalization Group (NRG)<cit.>, the Density Matrix Renormalization Group (DMRG)<cit.>, and the Logarithmic Discretized Embedded Cluster Approximation (LDECA)<cit.>. Other algebraic approaches have as well been used as the various Slave Boson Approximations<cit.> and Projection Operator Approach (POA)<cit.> and others based on the Green's function formalism as the Non-Crossing and One-Crossing Approximation (NCA, OCA)<cit.>, and the equation of motion method<cit.>.In addition, it should be mentioned the use of the Perturbative Renormalized Group Approach <cit.> aswell as extensions of Noziere's Fermi liquid-like theories <cit.>. In the last years, several studies have appeared in the literature related to the Kondo effect in which, in addition to the spin degree of freedom, the nanostructure presents degenerate orbital degrees of freedom, such that the complete symmetry of the system corresponds to the SU(N) Lie group, for N>2. This was the case, for N=4, of a single atom transistor <cit.>, in carbon nanotubes <cit.>, and in capacitively-coupled double QDs <cit.>. Several theoretical interpretations have been proposed <cit.> and, in particular, more closely related to our work, it was theoretically shown that there is an SU(4)-SU(2) crossover when the SU(4) symmetry is broken by either introducing a different gate voltage V_g in each dot or by connecting them to the leads by different hopping matrix elements V. In addition, it was shown that by manipulating the parameters of the system, without explicitly restoring the broken symmetry, the ground state mightdisplay, as an `emergent' property, the SU(4) symmetry <cit.> (see also Ref. Keller2014). Here, it should be noticed that Ref. Nishikawa2016 has shown that the conclusionsof Tosi et al. <cit.> regarding the emergent SU(4) symmetry areasymptotically achieved if the intradot and interdot Coulomb repulsionsare larger than the half-bandwidth (see also Ref. Nishikawa2013). Finally, recent theoretical studies of a double quantum dot (DQD) device, connected to twoindependent channels, under the effect of a magnetic field, was shown to exhibitan exotic SU(2) Kondo state with the property of having spin polarized currents(of opposite polarization) through each QD <cit.>. In this work, we will concentrate on two main subjects: (i) the Kondo SU(4)-SU(2) crossover, driven byan external magnetic field h, occurring in a capacitively-coupled DQD device and (ii) the associated spin-filterproperties of this capacitively-coupled DQD device that emerge in the SU(2) side of the crossover. Althoughsome aspects of related problems have already been studied (see Ref. Busser2014 for (i)and Refs. Busser2012,Vernek2014 for (ii), and references therein), there are very importantproperties of this crossover that were not analyzed yet and will be discussed here. The main ideas behind the SU(4)-SU(2) crossover can be summarized as follows.The crossover is driven by the magnetic field h (causing a Zeeman splitting B) that decreases thesymmetry of the Hamiltonian from SU(4) to SU(2).Despite the presence of a finite magnetic field, our results show thatthe symmetry of the ground state changes from SU(4) to SU(2) when the gate potentialapplied to the DQD is reduced. That the ground state of the DQD may have a higher symmetry, SU(4), than itsSU(2)-symmetric Hamiltonians —a manifestation of an effect dubbed an `emergent'SU(4) Kondo ground state <cit.> — is by itself an interesting result. Indeed, we show in Sec. <ref>, that the SU(4)-SU(2) crossover can be studied by taking the value of thespin polarization, i.e., the difference ⟨ n_↑⟩ -⟨ n_↓⟩,evaluated in the ground state of the DQD system, as playing a similar role to an order parameter thatdefines the transition between two phases, although in this case we are dealing with a crossover process.At a particular value of the external field h, which produces a Zeeman splitting B=gμ_Bh,the crossover is characterized as occurring at the gate potential valueV_g^max where the electronic spin-down occupation, ⟨ n_↓⟩,has a very well-defined maximum, denoted ⟨ n_↓⟩^max (see Fig. <ref>).We name the Zeeman splitting corresponding to this maximum as B_max.If we then analyze the functional relation between B_max and V_g^max, i.e.,B_max=f(V_g^max), our results show that, within the Kondo regime, f hasa similar universal behavior to that the Kondo temperature has as a function of the gate potential.It should be noted that the crossover, as defined here, occurs even when thesystem is deep inside the charge fluctuation regime, in which case it cannot properly besaid that the system has a Kondo ground state.The existence of this clear maximum, irrespective of the regime the system is in,allows B_max to be characterized as the energy scale controlling the crossover. Regarding subject (ii) mentioned above, i.e., the spin-filter properties of the DQD system studied here,our results show that, in the SU(2) side of the crossover theelectronic population at the QDs is already clearly polarized along the magnetic field. As to the important question, regarding what is the minimum temperature and minimum magnetic field neededfor the DQD to operate as a spin-filter device, our results show that, as B_max is much smaller than theKondo SU(4) temperature T_K^SU(4), it could operate at temperatures around 10 K, with a fieldh ≈ 0.1 Tesla.These two facts introduce very efficient spin-filter properties to the device, also discussed in detail in the paper. This phenomenology is studied adopting two different formalisms: (i) the Mean Field Slave Bosons Approximation (MFSBA) <cit.>, which allows an approximate analysis of the dynamical properties of the system, and (ii) the POA, which has been shown to describe, almost exactly, the static properties associated to the ground state of the Anderson Impurity Hamiltonian <cit.>. Note that we have extended the POA, originally derived to study single-impurity Kondo problems, to the analysis of two capacitively-coupled local levels. As it was the case for single-impurity problems, this extension can be considered to provide almost exact results, as far as the static zero-temperature properties are concerned. In Ref. POA_1 the POA results for various Kondo static properties agree quite well with the Bethe Anzats <cit.> exact results. It is important to mention that both approaches used to study the system, the MFSBA and the POA, provide the same qualitative and semi-quantitative physical description.The rest of the paper is organized as follows: In section <ref>, we provide a description of the capacitively-coupled DQD system; in section <ref> we present the MFSBA and the POA used to study the properties of the system; section <ref> is dedicated to the analysis of the SU(4)-SU(2) crossover; section <ref> describes the spin filter characteristics of the DQD device. We end the paper in section <ref> with the conclusions. The theoretical methods used are discussed in detail in appendixes A and B.§ DESCRIPTION OF THE SYSTEM The system is composed by two parallel QDs, each one connected to two independent contacts (see Fig. <ref>).These QDs, besides an intra-QD Coulomb interaction U, are also capacitively coupled by an inter-QD Coulomb interaction U^'. In addition, they are under the influence of an external magnetic field h, as shown in Fig <ref>.On one hand, the two configurations shown in panels (a) and (b) in Fig. <ref> give identical results from the point of view of the SU(4)-SU(2) crossover and related physics. On the other hand, whether the QDs are embedded [Fig. <ref>(a)] or side-coupled [Fig. <ref>(b)] to the contacts plays a fundamental role in the transport properties of the system, and the differencein these properties will be explicitly analyzed below when we study the conductance.The general discussion regarding the SU(4)-SU(2) crossover is presented for the side-coupled QDs geometry <cit.>.Note that similar physics can be obtained using instead a single carbon nanotube QD, where the extra degree of freedom, besides spin, is provided by the valley quantum number present in the graphene honeycomb lattice <cit.>. The system will be described by an extension of the Anderson Impurity Model (AIM) Hamiltonian <cit.>, appropriate for two impurities, plus the Zeeman term, given by H_tot= H_band + H_DQD + H_hyb + H_Zeeman,where H_band =∑_j,k_j,σϵ_k_jc_k_jσ^†c_k_jσ, H_DQD =∑_j,σ( V_gn_jσ+U/2n_jσn_jσ̅ )+ U^'∑_σ,σ^'n_1σn_2σ^', H_hyb =∑_j,k_j,σ V_k_j(c_djσ c_k_jσ^†+c_djσ^†c_k_jσ), H_Zeeman = ∑_j gμ_BS_j^zh,where j=1,2 labels the QDs and the corresponding attached contacts (after a symmetric/antisymmetrictransformation between left and right contacts), which are modeled in eq. (<ref>) as non-interacting Fermi seas with dispersion ϵ_k_j, where c_k_jσ^† ( c_k_jσ) creates (annihilates) an electron with spin σ in contact j. Equation (<ref>) models the QDs, introducing a Coulomb repulsion U between electrons in the same QD, as well as an inter-QD repulsion U^', where c_djσ^† ( c_djσ) creates (annihilates) an electron with spin σ in QD j,n_jσ=c_djσ^† c_djσ is the number operator in QD j, and we assume that the same gate potential V_g is applied to each QD. In eq. (<ref>), V_k_j couples each QD to the corresponding lead (see Fig. <ref>).As usual, we take the matrix element V_k_j=V to be independent of momentum k_j.Note that, unless stated otherwise, for the sake of brevity, as n_1σ=n_2σ, we will from now on drop the j sub-index when referring to the spin occupation number of the QDs.Finally, eq. (<ref>) describes the effect of an applied magnetic field h acting on spins with magnetic moment gμ_B in both QDs (where μ_B is the Bohr magneton and g is the gyromagnetic factor of the electrons in the QD).Rigorously speaking, at h=0, the system only has SU(4) symmetry when both the gate potential V_g and the hybridization matrix element V are independent of j, and, in addition, U^'=U. In particular, we assume U and U^' to be infinite, which restricts the QDs occupations to be either zero or one, a condition that simplifies significantly the numerical calculations. However, within the context of the MFSBA, we consider a case where U^' is finite in order to show that in the appropriate region of the parameter space the physical properties of the system do not depend upon the particular value of the inter-QD Coulomb repulsion. § THE MEAN FIELD SLAVE BOSONS APPROXIMATION AND THE PROJECTION OPERATOR APPROACH In this section, we will briefly discuss the two formalisms used to study the properties of the DQD system. A more detailed presentation of these two treatments is given in appendixes A and B. Although most of the discussion is restricted to the case where the inter-QD repulsion is equal to the intra-QD one, i.e., U=U^'→∞, the case of finite U^' is explicitly treated in the MFSBA calculations. Although, as mentioned above, this could be a more realistic situation, we will see that, in the region of parameter space where |V_g|< U^', the results do not qualitatively depend upon the particular value of U^'/U. §.§ Mean Field Slave Bosons ApproximationAs already mentioned we assume that U→∞, which simplifies the treatment, as it eliminates double occupied intra-QD states from the Hilbert space. However, as just mentioned above, we will also present results for double inter-QD occupation, taking a finite value for U^'. Following the MFSBA formalism <cit.>, it is necessary to introduce new bosonic operators. As discussed in detail in appendix A, seven auxiliary operators are introduced, each one associated to a different eigenstate of the isolated DQD system, as shown in Table <ref>. A new Hamiltonian can be written with the help of these operators. Restrictions on the Hilbert space are necessary in order to remove additional non-physical states, which is accomplished by imposing relationships among these operators [eqs. (A1) and (A2)]. The boson operators, within the mean-field approximation <cit.>, are replaced by their respective expectation values: e→⟨ e ⟩, p_σ →⟨ p_j^σ⟩, d_12 →⟨ d^σσ̅_12⟩, d_1σ →⟨ d^1_σ⟩. The restrictions on the mean values of the bosonic operators are incorporated through the Lagrange multipliers λ and λ_jσ.Following this procedure and in order to simplify the notation,we assume that the bosons operators denote their mean values. In this case, the effective Hamiltonian can be written:H_eff =∑_j,k_j,σϵ_k_jn_k_j,σ +∑_j,σ( V_g - σ B) c^†_dj,σc_dj,σ + U^'∑_σd^σσ̅†_12d^σσ̅_12 + ∑_σ(U^'± 2B)d^1†_σd^1_σ + ∑_j,σV_j(c^†_k_j,σc_dj,σ + h.c)Z_jσ + λ(I-1)+ ∑_j,σλ_j,σ(c^†_dj,σc_dj,σ - Q_j,σ). The effective Hamiltonian corresponds to a one-body quasi-fermionic system in which the local energylevels in each QD are renormalized by its respective spin dependent Lagrangemultiplier: ϵ_σ =V_g -σ B + λ _σ. As discussed in appendix A, the bosonic operator expectation values and the Lagrange multipliers (λ_j,σ→λ_σ), necessary to impose the charge conservation conditions, are determined by minimizing the total energy and the free energy of the system. This requires the self-consistent solution of a system of nine equations, thus obtaining the parameters that define the effective one-body Hamiltonian, eq. (<ref>), which can then be solved by applying a standard Green's function method. §.§ Projection Operator Approach The ground state energy, E, of our N-particle system satisfies the eigenvalue Schrödinger equationH|Ψ⟩=E|Ψ⟩ ,where |Ψ⟩ represents the ground state eigenvector of the model Hamiltonian, eq. (<ref>). We proceed by projecting its Hilbert space into two subspaces, S_1 and S_2, and constructing a renormalized Hamiltonian H_ ren that operates in just one of them <cit.>. For the case of subspace S_1, H_ ren can be written as <cit.>, H_ ren=H_11+H_12(E-H_22) ^-1H_21,where,H_ ij=| i⟩⟨ i| H| j⟩⟨ j|,and state | i ⟩ belongs to subspace S_i. In our case, subspace S_1 contains only state | 1 ⟩, consisting of the tensor product of the ground state of the two Fermi seas with the uncharged DQD. All the other states are contained in subspace S_2, which can be accessed from subspace S_1 through successive applications of the H_21 operator.It is convenient to define Δ E, as the difference between the ground state energy E and 2ϵ_T, the sum of the energies of the two uncoupled contact Fermi seas,Δ E=E-2ϵ_T,where ϵ_T is given by ϵ_T =2∫ _-2t^0 ωρ(ω) dω,and ρ(ω) is the density of states of the Fermi sea. As shown in appendix B, Δ E can be found by solving Δ E=f_1(Δ E),where f_1(ξ) and f_0(ξ), given by f_1(ξ) =∑_σ∫_-2t^0{ρ( ω) ×2V^2ξ +ω -V_g+σ B-f_0(ξ+ω) }dω,and f_0(ξ)=∫_0^2t{ρ(ω)V^2ξ-ω-f_1(ξ-ω)} dω,are obtained self-consistently.As briefly described above, the POA results depend on the choice of a convenient S_1 subspace, where the model Hamiltonian will be projected, resulting in an effective Hamiltonian. In our case, consisting of two identical QDs with infinite intra-QD Coulomb repulsion, two auxiliary functions have to be self-consistently obtained. Although this requires only a moderate numerical effort, it becomes more involved, and therefore computationally more expensive, in a more general situation of two different QDs and finite intra-QD Coulomb repulsion, as the number of functions to be self-consistently determined increases accordingly.§ THE SU(4)-SU(2) CROSSOVER In this section, we study the SU(4)-SU(2) crossover driven by an external magnetic field applied to theDQD system. The QD occupation numbers are used to characterize the crossover. With this objective,⟨ n_↑⟩ and ⟨ n_↓⟩ at each QD is calculated as afunction of the gate potential using both methods, the MFSBA and the POA. Unless stated otherwise, the parameters taken toperform the calculations (in units of Δ, see below) are as follows:the coupling between each QD and the corresponding contact isV=8.0, the half-bandwidth of the contacts is D=64.0, and the Zeeman splitting is given by B= 3.2 × 10^-3. Taking typical values for GaAs,for instance, this corresponds to a magnetic field h ≲ 0.1 Tesla. Our unit of energy, Δ, isthe broadening of the localized QD levels, i.e., Δ = π V^2 ρ( ϵ_F), whereρ( ϵ_F) is the density of states at the Fermi energy. We discuss first the results obtained using the MFSBA. The renormalized spin dependent QD local energy ϵ̃_σ, shown in Fig. <ref>(a) as a function of the gate potential, is the same for both QDs, but is nevertheless spin dependent due to the applied external magnetic field.Results for σ=↑ and σ=↓ are given by the solid (red) and the dashed (blue) curves, respectively.As V_g decreases, starting around the Fermi energy ϵ_F=0, the renormalized energies (for different spin projections) are undistinguishable down to V_g ≈ -8, where they split (ϵ̃_↓ > ϵ̃_↑). This indicates that a change in the ground state occurs for V_g ≲ -8, region in the parameter space where the ground state SU(4) symmetry is lost.In particular, continuously reducing V_g, the renormalized energy ϵ̃_↑ displays a typical Kondo behavior, within the MFSBA approach, being almost independent of the gate potential and taking a value in the immediate vicinity of the Fermi energy, representing the SU(2) Kondo peak, while ϵ̃_↓ maintains its value above the Fermi energy. This spin-dependent splitting also occurs for the parameter Z^2_σ, which renormalizes the matrix elements that connect the QDs to the electron reservoirs Ṽ_σ=VZ_σ, as shown in Fig. <ref>(b), where Z^2_σ decreases with the gate potential, and takes different values for different spin orientations for V_g < -8.0, in agreement with Fig. <ref>(a).As Ṽ_σ controls the width of the peak associated to ϵ̃_σ, one expects that the peak for σ=↑, which reaches the Fermi level (ϵ_F=0) as V_g decreases [solid (red) curve in Fig. <ref>(b)], and therefore determines the properties of the Kondo ground state, such as the Kondo temperature, will get narrower as the (SU(4)-SU(2) transition occurs, implying that T_K^SU(4)>>T_K^SU(2).This will be shown to be indeed the case by an explicit calculation of the width of the QD levels, as shown next, in Fig. <ref>. The results shown in Figs. <ref>(a) and (b) can be better understood by comparing the QD's local density of states (LDOS), for each spin projection, for gate potential values above and below V_g = -8, where the ϵ̃_σ splitting occurs. The LDOS results for the two identicalQDs are shown in Fig. <ref>, for V_g=-6.4 [panel (a)], -8.0 [(b)], -9.6 [(c)], -11.84 [(d)], and -13.76 [(e)], for σ=↑ [solid (red) curves] and σ=↓ [dashed (blue) curves]. Figure <ref>(a) illustrates the situation for gate potential values above the splitting, where the LDOS peaks for both spin projections are essentially superposed, showing that although the magnetic field has broken the SU(4) symmetry, the ground state preserves it, as this better minimizes its energy.Although not explicitly shown, this situationprevails in the interval -8.0 < V_g < 0.As V_g keeps decreasing, the LDOS peak narrows and splits up, both of the resulting peaks still located above the Fermi energy, as shown in panel (b) of Fig. <ref>.Therefore, below V_g = -8, the ground state responds to the Zeeman splitting, caused by the magnetic field, by explicitly taking the Hamiltonian's SU(2) symmetry, as now this better minimizes its energy.This SU(2)-Kondo is an orbital-Kondo state, its degenerate DQD states being (using notation from Table <ref>) |0;↑⟩ and |↑;0⟩. Further decreasing V_g leads to further narrowing of both peaks, accompanied by a larger splitting between them, which is achieved by the σ=↑ peak accelerating its shift towards the Fermi energy, while the σ=↓ peak moves slightly up in energy.The narrowing of the peaks, as first discussed in relation to the variation of Z^2_σ with V_g [see Fig. <ref>(b)], is compatible with the fact that the Kondo temperatures of the SU(2) and SU(4) Kondo ground states satisfy T_K^SU(4)>>T_K^SU(2) (see Ref. Lim2006).This is clearly illustrated by the sizable narrowing of the solid (red) peak from panel (a) to panel (e) in Fig. <ref>.As will be discussed below in detail, the spin-dependent renormalization reflects the high spin filter efficiency of the device and it is also critical to understand, within the MFSBA, the abrupt changes in the QD's occupation as a function of the gate potential.Taking the same parameters as in Figs. <ref> and <ref>, the spin dependent electronoccupation in each QD ⟨ n_σ⟩, as a function of V_g, is calculatedusing POA and MFSBA, as shown in Figs. <ref>(a) and (b). In the case of MFSBA, theoccupation numbers are calculated by integrating the density of states at the QDs obtained fromthe corresponding Green's function. To calculate the same quantities in the POA formalism, wetake the derivative of the ground state energy with respect to the gate potential V_g.The B=3.2 × 10^-3 results in Fig. <ref>(a) show a semi-quantitative agreement betweenPOA (symbols) and MFSBA (solid lines). Inspecting the ⟨ n_σ⟩ POA results in Fig. <ref>(b), for four different Zeeman splitting values,B=3.2 × 10^-4, 3.2 × 10^-3, 3.2 × 10^-2, and 3.2 × 10^-1,shows that while the QD level V_g is near the Fermi level and, as a consequence, the QDs arestill in the charge fluctuating regime, the two smaller Zeeman splittings(B=3.2 × 10^-4 and 3.2 × 10^-3) are notable to minimize the energy of the system (thus polarizing the QDs) when compared to the gain inenergy brought by the SU(4)-Kondo-singlet ground state.Therefore, in this regime, the magnetic field is not a relevant quantity, as the ground state doesnot reflect the broken SU(2) symmetry introduced by the field, as already discussed above(see also Ref. Keller2014).However, as the gate potential is further reduced, and the Kondo temperature T_K^SU(4) exponentially decreases, eventually becoming smaller than B, asudden change in the behavior of the occupation numbers ⟨ n_σ⟩occurs: ⟨ n_↓⟩ reaches a maximum and undergoes a sharp drop, tending tozero as V_g is further reduced, while ⟨ n_↑⟩ keeps increasing,eventually saturating at ⟨ n_↑⟩ =1. Obviously, this occurs because theZeeman splitting B has overtaken T_K^SU(4).On the other hand, for the larger Zeeman splittings (B=3.2 × 10^-2 and 3.2 × 10^-1),the polarization starts to occur for considerably larger values of V_g, asa small decrease in V_g will be enough to make T_K^SU(4)≲ B.It should be clear, however, that the discussion above does not imply thatB should be compared to the zero-field T_K^SU(4), as a finite magneticfield does suppress the Kondo temperature, as shown in Fig. <ref>(c) <cit.>. The inflexion point in the function ⟨ n_↓⟩(V_g) (where d ⟨ n_↓⟩/dV_g=0) will be used to characterize the SU(4)-SU(2) crossover.The results in Fig. <ref>(b) indicate that V_g^max, the value where the maximum for ⟨ n_↓⟩ occurs, as expected, strongly depends upon the magnetic field: for larger B values, the split between ⟨ n_↑⟩ and ⟨ n_↓⟩ occurs for values of V_g^max nearer to the Fermi energy. On one hand, this reflects the fact that, as the field increases for a fixed value of gate potential, a Zeeman-split ground state will eventually have a lower energy than an SU(4)-Kondo-singlet ground state. On the other hand, the lower is B, more charging of the QDs will be required to achieve a splitting, thus resulting in a lower value of V_g^max. At this point, it is interesting to mention that the qualitative results for the occupation numbers do not depend upon taking U^'→∞. As the MFSBA calculations are not restricted to the condition U=U^', we show in Fig. <ref> the variation of ⟨ n_↑⟩ and ⟨ n_↓⟩ with V_g for U^'=64.0, keeping U→∞.The results obtained qualitatively agree with results for U=U^'→∞. As mentioned above, although in this case the Hamiltonian does not have an explicit SU(4) symmetry (not only because of the presence of a finite magnetic field, but also because U^'≠ U), the ground state of the DQD system still preserves this symmetry (up to V_g ≈ -7.0), as an emergent property <cit.>, and an SU(4) - SU(2) crossover still occurs (compare with Fig. <ref>). It is believed that, in the presence of a magnetic field, a broken SU(4) symmetry will beclearly observable only when B ≈ T_K^SU(4)<cit.>.In order to clarify this point, inFig. <ref> we present a semi-log plot with POA results for the Zeeman splitting B_maxin the left axis (in logarithmic scale), and the corresponding values ofV_g^max, at which the maximum in ⟨ n_↓⟩ occurs, in the horizontal axis.The variation in B_max spans more than four orders of magnitude. The main panel results are forD=64.0 [(red) circles], with two extra sets of results plotted in the inset, for D=44.4 [(green) squares]and 16.0 [(blue) triangles]. The results in the main panel and in the inset clearly show an exponential dependence ofB_max on V_g^max, therefore a least squares fitting was done, using the expression B_max=Dexp(aV_g^max),and the results of these fittings were plotted as solid lines.The value of the Zeeman splitting, B_max, is the relevant energy scale that controls the SU(4)-SU(2) crossover,which, according to our definition, occurs when V_g=V_g^max.This energy scale has a universal behavior in the Kondo regime, as described by eq. <ref>,extending into the charge fluctuating regime as well, although it loosesits universal character in the neighborhood of the Fermi energy. This is illustrated in Fig. <ref> bythe fact that the two nearest points to the Fermi energy no longer coincide with the straight line givenby eq. <ref>. The loss of universality is an expected result, clearly showing that the universalbehavior is restricted to the Kondo regime, as it is the case for the Kondo temperature. Anyhow, itis important to emphasize that, for larger values of B, as illustrated for B=3.2 × 10^-1 in Fig. <ref>(b),⟨ n_↓⟩ reaches a maximum along the entire chargefluctuation region, therefore defining the energy scale B_max as controlling the SU(4)-SU(2) crossover also in this regime.In addition, the results in the inset for three different values of D (keeping Δ, our unit of energy, constant)clearly show that the parameter a ∼ 1.23, from eq. <ref>, is independent of D, llustrating the universalityof the Zeeman splitting scale of energy that characterizes the SU(4)-SU(2) crossover. Finally, the least squares fittingof the POA results (points) using eq. <ref> also shows that the choice of the band half-width D as prefactoris correct, as the fitting recovers, with good numerical accuracy, the values of D used forthe POA calculations <cit.>.Also shown in the same plot (right axis, in logarithmic scale too) are the Kondo temperatures T_K^SU(2) (dashed line)and T_K^SU(4) (dotted line) obtained through the expression <cit.> T_K^SU(N)=Dexp(π V_g/N), which was obtainedthrough a U →∞ variational wave function for the ground state of the system <cit.>,which coincides as well with the mean-field solution of a slave boson formalism (also in the same limit) <cit.>.These curves are shown in order to facilitate the comparison of their exponential dependence on V_g,as shown in eq. <ref>, with the exponential dependence of the Zeeman splitting B_max(V_g^max),as described in eq. <ref>. These two Kondo temperatures aredisplayed just for values of V_g^max<-7, which roughly corresponds to the Kondo regime,to emphasize that the expression above is not valid in the charge fluctuation regime. Surprisingly enough, the Zeeman splitting exponent factor a ∼ 1.23 ineq. <ref> has an intermediate value between thoseof the T_K^SU(4) and T_K^SU(2) Kondo states (see eq. <ref>):π/4< a < π/2. Moreover,a simple inspection of Fig. <ref> shows that the value of B_max is between oneto two orders of magnitude less than T_K^SU(4) and equally greater than T_K^SU(2), the largerdifference occurring for larger values, in magnitude, of V_g, deep into the Kondo regime.As the value of the exponent factor controlling the Zeeman splitting is between those correspondingto T_K^SU(4) and T_K^SU(2), it is possible, under the effect of small magnetic fields,the operation of the DQD system in a regime of high spin polarization(⟨ n_↑⟩≫ ⟨ n_↓⟩), with important consequencesfor its spin filter performance, as discussed in the next section. To properly characterize the SU(4)-SU(2) crossover, it is interestingto do the opposite of what was done up to now, i.e., instead of fixing the external fieldand analyzing how ⟨ n_σ⟩ depends upon V_g, westudy the variation of ⟨ n_σ⟩, at fixed V_g, as a function of magnetic field.This analysis is done using POA.The main idea is to use V_g to place the system, at zero field, either well inside the SU(4) Kondo regimeor closer to the charge fluctuation region, and then analyze how does theapplication of a magnetic field change the system's properties.We study the spin occupation numbers ⟨ n_↑⟩ and ⟨ n_↓⟩, which are shown in Fig. <ref>(a) (where solid lines indicate ⟨ n_↑⟩ anddashed ones ⟨ n_↓⟩) for four different values of gate potential:V_g=-19.2 [(blue) up triangles] places it well inside the SU(4) Kondo regime, V_g=-9.6 [(red) circles] places the system nearer to the charge fluctuation regime, while V_g=-14.4 [(green) squares] places it halfway between these two.These three data sets were obtained for D=64.0 and we add a fourth one[(magenta) down triangles] at V_g=-11.1, with a smaller D=44.4,to analyse the effect of a different half-bandwidth D on the results obtained, as discussed below. The results in Fig. <ref>(a) indicate that, closer to the charge fluctuation regime frontier, (V_g=-9.6 and -11.1), and even well inside the Kondo SU(4) regime (V_g=-14.4), the spin polarization, as measuredby ⟨ n_↑⟩ -⟨ n_↓⟩, is gradually raised in response toan increasing (from zero) magnetic field (see the circles, down triangles, and squares curves).This behavior can be explained by the larger values of T_K^SU(4) for V_g values closer to the Fermi energy(see dotted curve in Fig. <ref>) as it will take a larger value of field to force the systemto transition from the SU(4) to the SU(2) regime. This is specially evident for the V_g=-9.6results [(red) circles, with the highest T_K^SU(4)], where a larger field is needed togenerate a sizable spin polarization. One would expect then that the system will require justa very small magnetic field to transition from the SU(4) Kondo regime to theorbital SU(2) Kondo regime once T_K^SU(4) decreases substantially. This is exactly what is observed forV_g=-19.2 [(blue) up triangles], where T_K^SU(4) is much smaller(see Fig. <ref>) and the system responds much more abruptly to the magnetic field.In reality, even results for V_g=-14.4 [(green) squares], where T_K^SU(4) is not so low, show that a small externalmagnetic field h ≈ 0.1 Tesla (corresponding to B ≈ 0.0022, if one takes, for instance, thegyromagnetic factor for GaAs), is enough to obtain a sizable spin polarization, as shown in Fig. <ref>(a). The results in Fig. <ref>(a), despite being interesting, were somewhat expected.What makes them more relevant are the results presented in panel (b), where it is shown that if the⟨ n_↑⟩ and ⟨ n_↓⟩ data in panel (a) are plottedagainst B/T_K^SU(4) (with T_K^SU(4) as obtained from eq. <ref>), instead of against just B,all the curves for different parameters collapse into each other. This is true even for the V_g=-11.1 data[(magenta) down triangles], which has a different value of D in relation to the other data sets.This universality result shows that there is a deep connection between the spin polarizationand the B/T_K^SU(4) ratio when an external magnetic field is applied. It is important toemphasize that this universality is obtained when adoptingeq. <ref> to calculate T_K^SU(4), which gives additional support to the useof eq. <ref> to describe the SU(4) Kondo state in the U →∞ limit. In Fig. <ref>(c) we reproduce (left axis) the ⟨ n_σ⟩results for V_g=-11.1 and D=44.4, as a function of Zeeman splitting B [(magenta) down triangles],together with (right axis, in log scale) the Kondo temperatures T_K^SU(4) (dotted line)and T_K^SU(2) (dashed line) at zero magnetic field (thus, shown as horizontal lines), obtainedfrom eq. <ref>.As previously discussed, in the crossover region the system is in a Kondo ground state that is goingthrough a transformation from SU(4) to SU(2) symmetry.An estimation of the Kondo temperature of this `crossover state',and its dependence on the magnetic field, can be obtained from a variational calculation that interpolates,as a function of the magnetic field, betweenT_K^SU(4) at B=0 and T_K^SU(2) obtained for B →∞ <cit.>.This interpolated Kondo temperature, denoted as T_K^Var, is shown in Fig. <ref>(c) as a black solid curve.Obviously, it starts at T_K^SU(4), decreases with B, and, for the small intervalof field variation in the figure, it stays at least three orders of magnitude above T_K^SU(2).In addition, for B ≈ 0.0022 (which corresponds to h ≈ 0.1 Tesla, as mentioned above), for example,T_K^Var is almost equal to T_K^SU(4), which, for the parameter values taken,results to be of the order of 10K. These values of field and temperature are perfectly accessibleexperimental conditions for operation of the DQD as a spin-filter, as described in the next section. § SPIN FILTER Besides the natural intrinsic interest in systems whose properties depend on spin orientation, they are also important because, under adequate control, they can have very significant applications. The spin filter properties of a QD, or structures of QDs, is one of these very interesting aspects that have been studied in the last years<cit.>.The proposal of producing polarized lead currents as they go through a QD is based on the idea that the Zeeman splitting can be made much stronger in the QD than in the leads, thus creating a spin filter.Spin filter phenomena are obtained when the QD spin-up sublevel is located in the transport window, while the spin-down one can be manipulated to be just outside of it. This requires high magnetic fields (even considering renormalized g factors for the QD) and weak coupling of the QD to the leads, therefore resulting in very sharp localized states, thus properly separating in energy the spin-up from the spin-down level.The first restriction introduces experimental limitations to the applicability of the device, while the last condition reduces significantly the intensity of the current circulating through it. Neither of these difficulties are present in our case because our DQD system, being in the Kondo regime, has a very sharp Kondo spin-polarized level, tuned to be at the vicinity of the Fermi energy, well separated from the other spin polarization [see, for example, Fig. <ref>(a)].As the device is required to be in the Kondo regime, the temperature should be below the Kondo temperature, which is a limitation.Fortunately, however, the Zeeman splitting required to separate ⟨ n_↑⟩ from ⟨ n_↓⟩, as already discussed, although below T_K^SU(4), can be taken to be very near it, much larger than T_K^SU(2).In order to clarify these points and to show the spin-filter potentialities of our DQD system, we calculate the current as a function of the relevant parameters. The quantum conductance, a dynamical property, can be obtained, within the context of the MFSBA, using the Keldysh formalism <cit.>. The currentthrough one of the QDs is given by <cit.>,J_c=2eh∫_-∞^∞T(ϵ)[f(ϵ-ϵ_L )-f(ϵ-ϵ_R)] dϵwhere T(ϵ) is the transmission, f(ϵ) the Fermi-Dirac distribution and ϵ_L,R are the Fermi energies of the left and right reservoirs, respectively. For an infinitesimal bias potential (thusin the linear regime, where inelastic processes can be neglected <cit.>), from eq. (<ref>) one obtains thefamiliar expression for the conductance G=2e^2hT(ϵ_F), where the transmission, at the Fermi energy, is given by <cit.>,T(ϵ_F )=4π^2V_e^4ρ_1(ϵ_F) ρ_1̅(ϵ_F) | G_00^σ(ϵ_F)| ^2,where ρ_1(ϵ_F) = ρ_1̅(ϵ_F) is the LDOS at the first site of the leads, (see labeling in Fig. <ref>).For an embedded QD configuration [see Fig. <ref>(a)], the Green's function G_00^σ(ϵ _F) is given by G_dd^σ(ϵ _F), which is the dressed Green's function at the QD, and V_e=V.In the case of side-coupled QDs [Fig. <ref>(b)], V_e is the nearest-neighbor hopping matrix element in the tight-binding representation of the leads, i.e., V_e = t, and G_00^σ(ϵ _F) is given byG_00^σ(ϵ _F)=g_0 + g^2_0V^2G_dd^σ(ϵ _F), where g_0 = -i/√(4t^2 - w^2) corresponds to the Green's function at the first site of a semi-infinite tight-binding chain. This calculation is straightforward for the MFSBA, as the Green's functions can be obtained directly.From the perspective of POA, their values at the Fermi energy have to be calculated from the previously obtained electronic occupations at the QDs, using the Friedel sum rule <cit.>.In the next few paragraphs we briefly describe how to do that.The Green's function for a QD connected to an electron reservoir can be written as G_dd^σ(ω) =1/ω -V_g-Σ _1B(ω) -Σ _MB(ω) +iη,where Σ _1B(ω) and Σ _MB(ω) are the one- and many-body self-energies, respectively; and η is a small displacement in the imaginary plane to regularize the Green's function for values of ω outside the band defined by the Fermi sea.For simplicity, we assume a flat band to describe the leads density of states. Using the identity,∂/∂ωln[ G_dd^σ(ω) ] ^-1=G_dd^σ(ω)(1-∂∂ωΣ _1B(ω) -∂∂ωΣ _MB(ω)),then integrating both sides, using that ⟨ n_σ⟩ =-2π∫ _-∞^ϵ _F{ G_dd^σ(ω)} dω(where {⋯} means taking the imaginary part) and imposingthe Fermi liquid conditions<cit.>, we obtain that {-1πln[(G_dd^σ(ω)) ^-1]}_-∞^ϵ _F =⟨ n_σ⟩/2. Now, we explicitly introduce the phase of the Green's function, G_dd^σ(ω) =| G_dd^σ(ω)| e^iϕ(ω).The asymptotic behavior of the one-body propagator,G_dd^σ(ω→∞)=1/(ω +iη), and somealgebra, allows us to write thatϕ(-∞) =π,andϕ(ϵ _F)=π(1-⟨ n_σ⟩/2) . Then, from the definition of ϕ and eq. (<ref>), it is possible to obtain| G_dd^σ(ϵ _F) | ^2=sin ^2[π/2⟨ n_σ⟩]Δ ^2.From eqs. (<ref>), (<ref>), and (<ref>) the conductance can be written in terms of the occupations numbers ⟨ n_σ⟩, for the case of the embedded QDs, resulting inG_σ( e^2/h) =sin ^2[π/2⟨ n_σ⟩]. For side-coupled QDs it is possible to relate | G_00^σ(ϵ _F) | ^2 with the electronic ocuppations at the QDs ⟨ n_σ⟩ through eq. (<ref>). Reasoning in an analogous way as just done above, the conductance results to be G_σ( e^2/h) =1-sin ^2[π/2⟨ n_σ⟩]. Using the equations just obtained, we show in Fig. <ref>(a) MFSBA (lines) and POA (symbols) conductance results obtained for the case of embedded QDs, under the effect of an external magnetic field, as a function of V_g.An inspection of the figure allows us to conclude that both approaches provide qualitatively equivalent results for the transport properties. In the region V_g < -12.0 (for both panels) the spin-up conductance is almost 2e^2/h, while it is close to zero for spin-down.This is an interesting result, showing that even for relatively low magnetic fields B=3.2 × 10^-3 (h<0.1 Tesla, for the case of GaAs), in the appropriate region of the parameter space, the DQD device operates as a very effective spin filter. It is interesting to notice that, in the case of side-coupled QDs, Fig. <ref>(b), the role of the electron spin is interchanged, i.e., the transmitted electrons are down-spins (opposing the field direction), while for embedded QDs the transmitted electrons are up-spins (along the field direction).For the side-coupled QD configuration [Fig. <ref>(a)], when the system is in a Kondo regime, an up-spin electron circulating through the system has two channels to go through, one connecting the leads directly, and another channel that visits the side-coupled QD.As they have opposite phases, the destructive interference between them gives rise to a typical Fano anti-resonance.This destructive interference, regarding spin polarization, results in the opposite effect (polarization opposite to field direction) in comparison to embedded QDs. In this case, the spin down electron is the one that is transmitted, while the spin-up conductance rapidly vanishes for decreasing V_g, asshown in Fig. <ref>(b). § CONCLUSIONS We studied the SU(4)-SU(2) crossover driven by an external magnetic field fortwo capacitively coupled QDs connected to metallic leads.The crossover is characterized by the Zeeman splitting B_max at which the ⟨ n_↓⟩has a well-defined maximum as a function of the gate potential for a value denoted as V_g^max.The functional dependence of B_max = f(V_g^max), turns out to have a universal character,B_max = D exp( a V_g^max),in the Kondo regime, as discussed in detail in Fig <ref>.This universality is lost as one enters into the charge fluctuating regime,the same way as it happens to the Kondo temperature.However, it is important to emphasize that the occurrence of the maximum extends into the valence fluctuating regime, what permits to define the energy scale B_max as the magnitude thatcontrols the SU(4)-SU(2) crossover independently of the system regime. We were able to show that already in the crossover region, in an SU(2) ground state, foran effective Kondo temperature near the SU(4) one, the electronic populations at the QDsare significantly spin polarized along the magnetic field. Moreover, depending upon the parametersof the system, this can be obtained even for small magnetic fields (h ≲ 0.1 Teslafor the case of GaAs and a Kondo temperature that could be of the order of several degrees Kelvin).In that respect, we should mention that, in comparison to a similar device proposedin Ref. <cit.>, our device can operate at considerably lower field.In addition, this DQD structure was studied adopting the MFSBA and a POA formalisms, which were able todescribe the mentioned properties, giving qualitatively equivalent results. With this purpose,it was necessary to extend the POA, originally derived to study one Kondo impurity, to theanalysis of two capacitively coupled local levels.This extension provides almost exact results, as far as the staticzero-temperature properties are concerned. We conclude that this DQD system, under the influence of a magnetic field, has very interestingcross-over properties and, studying its conductance, that it could also operate as an effectivespin-filter, with potential applications in spintronics. § ACKNOWLEDGMENT V.L. and R.A.P. acknowledge a PhD studentship from the brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and E.V.A. acknowledges the financial support from (CNPq) and the brazilian agency Fundação de Amparo a Pesquisa e Desenvolvimento do Estado do Rio de Janeiro (FAPERJ).§ THE MEAN FIELD SLAVE BOSONS APPROXIMATION In the slave bosons approximation, extra bosonic operators are introduced to represent all the possible states of charge occupation of our DQD system.In our case these operators are defined in Table <ref> in the main text.The charge conservation condition for each QD and the completeness condition impose relations that the boson operators should fulfill, given by Q_jσ =p^†_jσp_jσ + d^σσ̅†_12d^σσ̅_12δ_1j + d^σ̅σ†_12d^σ̅σ_12δ_2j + d^1†_σd^1_σ = c^†_dj,σc_dj,σ,andI=e^†e + ∑_j,σp^†_jσp_jσ+∑_σd^σσ̅†_12 d^σσ̅_12+∑_σd^1†_σd^1_σ = 1,where Q_jσ is the charge per spin in QD j = 1,2, for σ = ↑/↓, I=1 defines the completeness condition, and δ_ij is the Kronecker delta.The fermionic operators of the impurity, in the context of the slave bosons formalism, transform as follows: c^†_dj,σ→Z^†_j,σc^†_dj,σ, where the Z_jσ operator, consisting of all bosonic operators associated with processes in which an electron with spin σ is annihilated, is defined as Z_jσ=Q^-1/2_jσ(e^†p_jσ + p^†_j̅σ̅(d^σσ̅_12δ_1j + d^σ̅σ_12δ_2j) + p^†_j̅σd^1_σ)(1 -Q^-1/2_jσ). The mean field approximation of this formalism, the so-called MFSBA, consists in replacing the bosonic operators by their mean values.For the sake of simplicity, they are named by the same letter as the operators themselves.These mean values and the Lagrange multipliers λ and λ_σ, incorporated to satisfy the slave boson conditions, are determined by minimizing the free energy of the system.These conditions create a set of nine non-linear equations (one for each of the six bosonic operators and three Lagrange multipliers), which should be self-consistently solved to obtain the parameters of the effective one-body Hamiltonian:∂⟨H_eff⟩/∂e= 2∑_σV∂Z_σ/∂e(⟨c^†_k,σc_d,σ⟩+ h.c) + 2λe = 0,∂⟨H_eff⟩/∂p_↑= 2∑_σV∂Z_σ/∂p_↑(⟨c^†_k,σc_d,σ⟩+ h.c) + 4(λ- λ_↑) p_↑= 0, ∂⟨H_eff⟩/∂p_↓= 2∑_σV∂Z_σ/∂p_↓(⟨c^†_k,σc_d,σ⟩+ h.c) + 4(λ- λ_↓) p_↓= 0, ∂⟨H_eff⟩/∂d_12= 2∑_σV∂Z_σ/∂d_12(⟨c^†_k,σc_d,σ⟩+ h.c) +4(λ-λ_↑- λ_↓+ U^') d_12 = 0, ∂⟨H_eff⟩/∂d_1↑= 2∑_σV∂Z_σ/∂d_1↑(⟨c^†_k,σc_d,σ⟩+ h.c) + 2(λ-2λ_↑+ U^'- 2μ_BB) d_1↑ = 0, ∂⟨H_eff⟩/∂d_1↓= 2∑_σV∂Z_σ/∂d_1↓(⟨c^†_k,σc_d,σ⟩+ h.c) + 2(λ-2λ_↓+ U^'+ 2μ_BB) d_1↓ = 0, ∂⟨H_eff⟩/∂λ= e^2 + 2p^2_↑+ 2p^2_↓+ 2d^2_12 + d^2_1↑ + d^2_1↓ -1 = 0,∂⟨ H_eff⟩/∂λ_↑ = ⟨ c^†_d↑ c_d↑⟩ - p^2_↑ - d^2_12 - d^2_1↑ = 0,∂⟨ H_eff⟩/∂λ_↓ = ⟨ c^†_d↓ c_d↓⟩ - p^2_↓ - d^2_12 - d^2_1↓ = 0,where H_eff is given by eq. (<ref>), and e^2, p^2_σ, d^2_12, d^2_1σ, as previously mentioned, are taken to be the mean values of the corresponding bosonic operators. Fig. <ref> shows results of all these mean values, as functions of V_g, for U = U^'→∞ and B = 10^-4.For positive values of V_g, the empty QD state, represented by the meanvalue e^2, is dominant, but rapidly decreases as V_g approaches the Fermi level.We can observe the splitting of the spin dependent occupancy p_σ^2, for V_g ≈ -4.0, indicating the SU(4)-SU(2) crossover.The double occupancy state |↑,↑⟩ has probability d^2_1↑=0, as it costs an infinite energy to simultaneously populate the QDs with two electrons due to the infinite U^' inter-dot Coulomb repulsion.For a finite value of U^', the occupation numbers (not shown), in the parameter region V_g > -U^', are almost identical to those for U^'=U→∞.This indicates that in this region of parameter space the value of U^' does not change the results qualitatively.§ THE PROJECTION OPERATOR APPROACH As discussed in the main text, the central idea of the POA is to separate the Hilbert space of the system of interest, which ground state |Ψ⟩ obeys,H|Ψ⟩=E|Ψ⟩ ,into two different subspaces: (i) the subspace S_1, containing a single state, denoted | 1⟩ and (ii) subspace S_2, containg the rest of the states in the Hilbert space, which are generically denoted as | 2⟩.The idea is to choose | 1⟩ so that, by operating in S_1 with a renormalized Hamiltonian, one can obtain not only the ground state energy E, but also some of its static properties <cit.>.The renormalized Hamiltonian that operates in the S_1 subspace can be written as,H_ ren = H_11+H_12( E-H_22)^-1H_21 ,where, H_ij=| i⟩⟨ i| H| j⟩⟨ j|,such that the renormalized Hamiltonian satisfies,H_ ren| 1⟩ = E | 1⟩that permits trivially to obtain, ⟨ 1| H_ ren| 1⟩ = E. The self-consistent solution of this last equation, -the renormalized Hamiltonian depends explicitly upon the energy E-, permits to find the ground state energy E of the system. It is important to adequately choose the state | 1⟩. We take it as given by the ground state of the two Fermi seas and the two uncharged QDs.All other states that belong to subspace S_2 can be obtained by successive applications of the Hamiltonian H_21 on state | 1⟩.To obtain the ground state energy it is necessary to calculate ⟨ 1| H_ ren| 1⟩. The first term is the expected value of H_11, given by,ϵ_T=⟨ 1| H_11| 1⟩ =2∑_ϵ_k<ϵ_Fϵ_k The contribution to the energy of subspace S_2 is calculated assuming the QDs to be connected to identical leads through matrix elements V_k_j=V that are taken to be independent of the momentum k_j.The energy can be written as<cit.>, E=Δ E+2ϵ_TΔ E=f_1(Δ E)f_0(ξ)=∑_ϵ_K>ϵ_FV^2ξ-ϵ_K-f_1(ξ-ϵ_K)f_1(ξ)=∑_σ,ϵ_k<ϵ_F2V^2ξ +ϵ_k-V_g+σ B-f_0( ξ +ϵ_k) In the thermodynamic limit these equation can be written as, f_0(ξ)=∫_0^2t{ρ(ω) V^2ξ-ω-f_1(ξ-ω)} dωf_1(ξ) =∑_σ∫_-2t^0{ρ(ω)×2V^2ξ+ω-V_g+σ B-f_0(ξ+ω)}dωwhere ρ(ω) is the density of states of the leads. It can be written as,ρ(ω)=ρ_LC(ω)=1π√(4t^2-ω^2)orρ(ω)=ρ_SC(ω)=√(4t^2-ω^2)2π t^2that corresponds to a one dimensional linear chain, equation (<ref>), or to two linear semi-chains, equation (<ref>), depending on the geometry of the system.The behavior of the function f_1(ξ) is represented on Fig. <ref> for three values of V_g. The ground state solution corresponds to the lesser value of the intersection between the straight line and the f_1(ξ) curves, that occurs on ξ=Δ E.It can be shown that the derivative of the function f_1(ξ) is singular at the point, Δ E=f_1(Δ E), from which the energy is determined <cit.>. As we decrease V_g, the peak with a minimum value becomes sharper and other solutions with greater energy are possible.However we are interest only in the ground state energy of the system. | http://arxiv.org/abs/1706.08618v1 | {
"authors": [
"V. Lopes",
"R. A. Padilla",
"G. B. Martins",
"E. V. Anda"
],
"categories": [
"cond-mat.str-el",
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.str-el",
"published": "20170626224021",
"title": "The $SU(4)-SU(2)$ crossover and spin filter properties of a double quantum dot nanosystem"
} |
naist]Masaki Oguracor [email protected] [naist]Graduate School of Information Science, Nara Institute of Science and Technology, Ikoma, Nara 630-0192, Japan [cor]Corresponding authorpenn]Victor M. Preciado [email protected] [penn]Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104-6314, USAIn this paper, we study the dynamics of contagious spreading processes taking place in complex contact networks. We specifically present a lower-bound on the decay rate of the number of nodes infected by a susceptible-infected-susceptible (SIS) stochastic spreading process. A precise quantification of this decay rate is crucial for designing efficient strategies to contain epidemic outbreaks. However, existing lower-bounds on the decay rate based on first-order mean-field approximations are often accompanied by a large error resulting in inefficient containment strategies. To overcome this deficiency, we derive a lower-bound based on a second-order moment-closure of the stochastic SIS processes. The proposed second-order bound is theoretically guaranteed to be tighter than existing first-order bounds. We also present various numerical simulations to illustrate how our lower-bound drastically improves the performance of existing first-order lower-bounds in practical scenarios, resulting in more efficient strategies for epidemic containment. Complex networks, spreading processes, stochastic processes, stability.§ INTRODUCTION Understanding the dynamics of spreading processes taking place in complex networks is one of the central questions in the field of network science, with applications in information propagation in social networks <cit.>, epidemiology <cit.>, and cyber-security <cit.>. Among various quantities characterizing the asymptotic behaviors of spreading processes, the decay rate (see, e.g., <cit.>) of the spreading size (i.e., the number of nodes affected by the spread) is of fundamental importance. Besides quantifying the impact of contagious spreading processes over networks <cit.>, the decay rate has been used to measure the performance of containment strategies to control epidemic outbreaks <cit.>. In this direction, the authors in <cit.> presented an optimization-based approach for distributing a limited amount of resources to efficiently contain spreading processes by maximizing their decay rate towards the disease-free equilibrium. This framework was later extended to the cases where the underlying network in which the spreading process is taking place is uncertain <cit.>, temporal <cit.>, and adaptively changing <cit.>. Recently, the authors in <cit.> presented an approach for achieving an optimal resource allocation in order to maximize the decay rate under sparsity constraints.However, finding the decay rate of a spreading process is, in general, a computationally hard problem. Even for the case of the susceptible-infected-susceptible (SIS) model <cit.>, which is one of the simplest models of spread, the exact decay rate is given in terms of the eigenvalues of a matrix whose size grows exponentially fast with respect to the number of nodes in the networks <cit.>. In order to avoid this computational difficulty, it is common in the literature <cit.> to use a lower-bound on the decay rate based on first-order mean-field approximations of the spreading processes. However, this first-order approximation is not necessarily accurate; in other words, its approximation error can be significantly large for several important social and biological networks, as we will demonstrate later in this paper. Therefore, the design of strategies for epidemic containment based on mean-field approximations can result in inefficient control policies.The aim of this paper is to present a tighter lower-bound on the decay rate of the stochastic SIS process based on a second-order moment closure. Specifically, we show that the decay rate is bounded from below by the maximum real eigenvalue of a Metzler matrix whose size grows quadratically with respect to the number of nodes in the network. In order to derive our lower-bound, we describe the stochastic dynamics of the SIS process using a system of stochastic differential equations with Poisson jumps. This approach allows us to conveniently evaluate the dynamics of the first and the second-order moments of random variables relevant for the spreading processes. Furthermore, we prove theoretically and illustrate numerically that our lower-bound strictly improves the one based on first-order approximations.Weremark that, although improved decay rates for the discrete-time SIS model were presented using second-order analysis in <cit.>, their bounds are applicable only to the special case where the transmission and recovery rates of nodes are homogeneous and, furthermore, satisfy restrictive algebraic conditions in terms of nonnegativity of infinitely many matrices. Likewise, the second-order analysis of the continuous-time SIS model by the authors in <cit.> uses mean-field approximations and, hence, it is not clear how the analysis relates to the dynamics of the original stochastic SIS process. Moreover, their analysis is valid only when a dominant eigenvalue of a certain matrix (i.e., an eigenvalue having the maximum real part) is real. In contrast with these limitations of the results in the literature, our framework applies to the heterogeneous SIS model without any restrictions, and is supported by rigorous proofs instead of approximations.This paper is organized as follows. In Section <ref>, we state the problem studied in this paper. In Section <ref>, we present our lower-bound on the decay rate, and show that this bound strictly improves the one based on first-order approximations. The effectiveness of our lower-bound is numerically illustrated in Section <ref>. §.§ Mathematical preliminaries We denote the identity and the zero matrices by I and O, respectively. For a vector u, we denote by u_\{i} the vector that is obtained after removing the ith element from u. Likewise, for a matrix A, we let A_i, \{j} denote the row vector that is obtained after removing the jth element from the ith row of A. We say that a square matrix A is irreducible if no similarity transformation by a permutation matrix transforms A into a block upper-triangular matrix. The block-diagonal matrix containing matrices A_1, …, A_n as its diagonal blocks is denoted by ⊕_i=1^n A_i. If the matrices A_1, …, A_n have the same number of columns, then the matrix obtained by stacking A_1, …, A_n in vertical is denoted by _1≤ i≤ n A_i.A directed graph is defined as the pair 𝒢 = (𝒱, ℰ), where 𝒱 is a finite ordered set of nodes and ℰ⊂𝒱×𝒱 is a set of directed edges. By convention, if (v, v') ∈ℰ, we understand that there is an edge from v pointing towards v', in which case v is said to be an in-neighbor of v'. A directed path from v to v' in 𝒢 is an ordered set of nodes (v_0, …, v_ℓ) such that v_0 = v, v_ℓ = v', and (v_k, v_k+1) ∈ℰ for k = 0, …, ℓ-1. We say that 𝒢 is strongly connected if there exists a directed path from v to v' for all v, v'∈𝒱. The adjacency matrix of 𝒢 is defined as the square matrix, having the same dimension as the number of the nodes, such that its (i, j)th entry equals 1 if the jth node is an in-neighbor of the ith node, and equals 0 otherwise. It is well known that a directed graph is strongly connected if and only if its adjacency matrix is irreducible.A real matrix A (or a vector as its special case) is said to be nonnegative, denoted by A≥ 0, if all the entries of A are nonnegative. Likewise, if all the entries of A are positive, then A is said to be positive. For another matrix B having the same dimensions as A, the notation A≤ B implies B-A≥ 0. If A≤ B and A≠ B, we write A⪇ B. For a square matrix A, we say that A is Metzler <cit.> if the off-diagonal entries of A are nonnegative. It is easy to see that e^At≥ 0 if A is Metzler and t≥ 0 (see, e.g., <cit.>). For a Metzler matrix A, the maximum real part of the eigenvalues of A is denoted by λ_max(A). In this paper, we use the following basic properties of Metzler matrices: The following statements hold for a Metzler matrix A: *λ_max(A) is an eigenvalue of A. Moreover, if A is irreducible, then there exists a positive eigenvector corresponding to the eigenvalue λ_max(A). *If A ≤ B, then λ_max(A) ≤λ_max(B). Furthermore, if A is irreducible and A≠ B, then λ_max(A) < λ_max(B). *Assume that A is irreducible. If there exist a positive vector u and a positive constant ρ such that Au ⪇ρ u, then λ_max(A) < ρ. The first claim is part of the Perron-Frobenius theorem for Metzler matrices (see, e.g., <cit.>). The second claim follows from the Perron-Frobenius theory and the monotonicity of the maximum real eigenvalue of nonnegative matrices <cit.>. To prove the last statement, let ϵ = ρ u - Au and define A' = A + ⊕(ϵ_1/u_1, …, ϵ_n/u_n), where n is the length of the vector u. Since A'u = A u + ϵ = ρ u, A' is irreducible, and v is positive, it follows that λ_max(A') = ρ from the Perron-Frobenius theorem for irreducible Metzler matrices <cit.>. Since A is irreducible and A⪇ A', the second statement of the lemma shows that λ_max(A) < λ_max(A')= ρ.§ PROBLEM STATEMENT We start by giving a brief overview of the SIS model <cit.>. Let 𝒢 = (𝒱, ℰ) be a strongly connected directed graph with nodes v_1, …, v_n. In the SIS model, at a given (continuous) time t ≥ 0, each node can be in one of two possible states: susceptible or infected. When a node v_i is infected, it can randomly transition to the susceptible state with an instantaneous rate δ_i > 0, called the recovery rate of node v_i. On the other hand, if an in-neighbor of node v_i is in the infected state, then the in-neighbor can infect node v_i with an instantaneous rate β_i, where β_i > 0 is called the infection rate of node v_i. It is easy to see that the SIS model is a continuous-time Markov process and has a unique absorbing state at which all the nodes are susceptible. Since this absorbing state is reachable from any other state, the SIS model reaches this infection-free absorbing state in a finite time with probability one. The aim of this paper is to study the stability of this infection-free absorbing state, defined as follows: Let ϵ>0 and define the probability p_i(t) = (v_i is infected at time t).We say that the SIS model is ϵ-exponentially mean stable if there exists a constant C>0 such that, for all nodes v_i and t≥ 0, we have p_i(t) ≤ C e^-ϵ t for any set of initially infected nodes at time t=0. Then, we define the decay rate of the SIS model as ρ = sup{ϵSIS model is ϵ-exponentially stable}. The notion of the decay rate was studied in, e.g., <cit.> and <cit.> for the cases of continuous- and discrete-time problem settings, respectively, and is closely related to other important quantities on spreading processes such as epidemic thresholds <cit.> and mean-time-to-absorption <cit.>. Specifically, a basic argument from the theory of Markov processes shows that the SIS model is ϵ-exponentially mean stable for a sufficiently small ϵ>0 (with a possibly large C) and, therefore, it always has a positive decay rate. However, exact computation of the decay rate is hard in practice. Even in the homogeneous case, where all nodes share the same infection and recovery rates, the decay rate equals the modulus of the largest real-part of the non-zero eigenvalues of a 2^n× 2^n matrix representing the infinitesimal generator of the SIS model <cit.>. An alternative approach for analyzing the decay rate is via upper bounds on the dynamics of the SIS model based on first-order mean-field approximations. An example of such a first-order upper bound is described below. Let us define the vector p(t) = _1≤ i≤ np_i(t) containing the infection probabilities of the nodes. Also, let A be the adjacency matrix of 𝒢 and define the diagonal matrices B = ⊕(β_1, …, β_n) and D = ⊕(δ_1, …, δ_n). Then, we can show <cit.> the inequality p(t) ≤ e^(BA-D)tp(0), which gives the following lower-bound on the decay rate:ρ≥ρ_1 = -λ_max(BA-D).Although this lower-bound is computationally efficient to find, there are several cases in which we can observe a large gap between this bound and the exact decay rate, as illustrated in the following example:Let us consider the SIS model over a romantic and sexual network in a high school (Jefferson, n=288) taken from <cit.>. For simplicity, we assume a homogeneous transmission rate β_i = 0.9/λ_max(A) and a (normalized) recovery rate δ_i = 1 for all nodes. In order to approximately compute the true decay rate ρ, we generate [group-separator=,]10000 sample paths of the SIS model over the time interval [0, 100] starting from the initial state at which all nodes are infected. From this computation, we obtain a decay rate of ρ≈ 0.454. On the other hand, the first-order approximation in (<ref>) equals ρ_1 = 0.1, whose relative error from ρ is around 78%.We can in fact show that the strict inequality ρ > ρ_1 holds in (<ref>). Let v_i be a node having the minimum recovery rate δ_min = min_1≤ i≤ nδ_i > 0 among all nodes, and consider the situation where only the node v_i is initially infected. Since p_i(t)≥ e^-δ_i t for every t≥ 0, we haveρ≥δ_i = δ_min.Then, let us take an arbitrary positive constant b < min_1≤ i≤ nβ_i. Since δ_min = -λ_max(-D) and -D ≤ b A -D, Lemma <ref>.<ref> shows-δ_min≤λ_max(b A - D).On the other hand, Since b A - D is irreducible (by the strong connectivity of 𝒢) and bA-D ⪇ BA-D, we also haveλ_max(bA - D) < λ_max(BA-D) = -ρ_1by Lemma <ref>.<ref>. Inequalities (<ref>)–(<ref>) prove the strict inequality ρ > ρ_1, as desired.§ MAIN RESULTAs we have demonstrated in Example <ref>, there may be a large gap between the true decay rate ρ and its first-order approximation ρ_1 for the SIS model. The aim of this paper is to fill in this gap by providing a better lower-bound on the decay rate. Specifically, the following theorem presents an improved lower-bound on the decay rate and is the main result of this paper:Define the n^2× n^2 Metzler matrix 𝒜=[ -D⊕_i=1^n (β_i A_i, \{i});_1≤ i≤ n(δ_i V_i) ⊕_i=1^n( - Γ_i + _j≠ iβ_j A_j, \{i}) ],where V_i∈ℝ^(n-1)× n is the matrix satisfying V_i x = x_\{i} for all x∈ℝ^n, and Γ_i= ⊕_j≠ iγ_ij for γ_ij = δ_i + δ_j + a_ijβ_i. Define ρ_2 = -λ_max(𝒜). Then, we haveρ≥ρ_2 > ρ_1.In order to prove this theorem, we use a representation of the SIS model as a system of stochastic differential equations with Poisson jumps (see, e.g., <cit.>). For this purpose, we define the variable x_i(t) as x_i(t)=1 if node v_i is infected at time t, and x_i(t)=0 otherwise. Then, we can see that these variables obey the following stochastic differential equations with Poisson jumps:dx_i = -x_i dN_δ_i + ∑_k=1^n a_ik (1-x_i)x_k dN_β_i,where a_ik is the (i, k)th entry of the adjacency matrix and N_δ_i and N_β_i denote stochastically independent Poisson counters <cit.> with rates δ_i and β_i, respectively. The rest of this section is devoted to the proof of the theorem. We divide the proof into two parts, namely, the proof of ρ≥ρ_2 (Subsection <ref>) and ρ_2>ρ_1 (Subsection <ref>). §.§ Proof of ρ≥ρ_2 From the system (<ref>) of stochastic differential equations, we can easily show <cit.> that the expectation p_i = E[x_i] obeys the differential equationdp_i/dt = -δ_i E[x_i] + ∑_k=1^n a_ik E[(1-x_i)x_k]β_i= -δ_i p_i + β_i∑_k≠ i a_ik q_ik,where q_ij(t) = E[(1-x_i(t))x_j(t)] is a second-order variable representing the probability that v_i is susceptible and v_j is infected at time t. Since q_ii≡ 0 by definition, in the sequel we do not consider the variable of the form q_ii. We next derive differential equations to characterize the second-order variables q_ij. Applying Ito's formula (see, e.g., <cit.>) for stochastic differential equations with Poisson jumps to the variable (1-x_i)x_j, we can showd((1-x_i)x_j) = x_ix_j dN_δ_j - (1-x_i)x_j ∑_k=1^n a_ikx_k dN_β_i - (1-x_i)x_j dN_δ_j + (1-x_i)(1-x_j)∑_k=1^n a_jkx_k dN_β_jfor any distinct pair (v_i, v_j) of nodes. To proceed, we define the probabilities p_ij(t) = E[x_i(t) x_j(t)] and p_ijk(t) = E[x_i(t) x_j(t) x_k(t)] for nodes v_i, v_j, and v_k. Then, from (<ref>), we can compute the derivative of q_ij asdq_ij/dt =δ_j E[x_ix_j] - β_i ∑_k=1^n a_ikE[(1-x_i)x_j x_k]- δ_j E[(1-x_i)x_j] + β_j ∑_k=1^n a_jk E[(1-x_i)(1-x_j)x_k]= δ_i p_ij - β_i a_ij(p_j-p_ij)- δ_j q_ij + β_j ∑_k=1^n a_jk (p_k - p_ik) - ϵ_ij,where the function ϵ_ij = β_i ∑_k≠ j a_ik(p_jk - p_ijk) + β_j ∑_k=1^n a_jk (p_jk-p_ijk) is nonnegative because p_jk≥ p_ijk for all nodes v_i, v_j, and v_k. Using the identity p_j - p_ij = q_ij and defining the variable γ_ij = δ_i + δ_j + a_ijβ_i, we obtaindq_ij/dt= -γ_ij q_ij + δ_i p_j + β_j ∑_k≠ i a_jkq_ik - ϵ_ij. In order to upper-bound the infection probabilities of the nodes, we define the vector variables q_i = _j≠ i q_ij and q = _1≤ i≤ nq_i having dimensions n-1 and n(n-1), respectively. Then, we can rewrite the differential equation (<ref>) as dp_i/dt = -δ_i p_i + β_i A_i, \{i} q_i. Stacking this differential equation in vertical with respect to i, we obtaindp/dt=-D p + (⊕_1=1^nβ_iA_i, \{i})q,where D = ⊕(δ_1, …, δ_n). Also, from (<ref>), it follows that dq_ij/dt = -γ_ij q_ij + δ_i p_j + β_j A_j, \{i}q_i - ϵ_ij. Stacking this differential equation with respect to j ∈{1, …, n}\{i}, we see that dq_i/dt = - Γ_i q_i+ δ_i V_i p + ( _j≠ i (β_j A_j, \{i}))q_i - _j≠ iϵ_ij. By stacking this differential equation with respect to i = 1, …, n, we obtain the following differential equationdq/dt = (_1≤ i≤ nδ_i V_i )p + ⊕_i=1^n (- Γ_i +_j≠ iβ_j A_j, \{i}) q - ϵ,where ϵ= _1≤ i≤ n_j≠ iϵ_ij is a vector-valued nonnegative function. This differential equation and (<ref>) show that the variable r = (p, q) satisfies dr/dt = 𝒜 r-(0, ϵ).We are now at the position to prove the inequality ρ≥ρ_2. Since 𝒜 is Metzler and ϵ(t) is entry-wise nonnegative for every t≥ 0, we can obtain the upper bound r(t) = e^𝒜 t r(0) - ∫_0^t e^𝒜(t-τ)(0, ϵ(τ)) dτ≤ e^𝒜 tr(0). This inequality clearly implies that the SIS model is ρ_2-exponentially mean stable. This completes the proof of the inequality. §.§ Proof of ρ_2>ρ_1 If ρ_2 ≥δ_min, then we trivially have ρ_2>ρ_1 because we know δ_min >ρ_1 from (<ref>) and (<ref>). Let us consider the case of ρ_2 < δ_min. Let u be a non-zero vector of 𝒜 corresponding to the eigenvalue -ρ_2 = λ_max(𝒜), i.e., assume that 𝒜 u = -ρ_2 u. We split the matrix 𝒜 as 𝒜 = ℳ - 𝒩 using the matricesℳ = [O ℳ_12;O ℳ_22 ], 𝒩 =[DO; 𝒩_21 𝒩_22 ],where ℳ_12=⊕_i=1^n β_i A_i, \{i},ℳ_22=⊕_i=1^n ( _j≠ i (β_j A_j, \{i}) - ⊕_j≠ i a_ijβ_i ) ,𝒩_21 = -_1≤ i≤ nδ_i V_i,𝒩_22 = ⊕_i=1^n ⊕_j≠ i(δ_i + δ_j).Then, we have ℳ u = (𝒩 -ρ_2 I)u and, hence, (𝒩 -ρ_2 I)^-1ℳ u = u, where the inversion of 𝒩 -ρ_2 I is allowed by our assumption ρ_2 < δ_min. Therefore, the matrix (𝒩 -ρ_2 I)^-1ℳ has an eigenvalue equal to 1. Since this matrix has the following upper-triangular structure(𝒩 -ρ_2 I)^-1ℳ =[ O *; O ℒ ]for the Metzler matrix ℒ defined by ℒ= (𝒩_22 -ρ_2 I)^-1 ( -𝒩_21(D -ρ_2 I)^-1ℳ_12 + ℳ_22 ),it follows that ℒ has an eigenvalue equal to 1. This specifically implies that λ_max(ℒ) ≥ 1. On the other hand, we can obtain an upper bound on λ_max(ℒ) as follows. The irreducible matrix BA-D has a positive eigenvector v corresponding to the eigenvalue -ρ_1 by Lemma <ref>.<ref>. Define the positive vector w = _1≤ i≤ nv_\{i}. Then, it is easy to see that ℳ_12 w = BAv and ℳ_22w = _1≤ i≤ n(BAv)_\{i} - _1≤ i≤ n_j≠ iβ_j a_ijv_i. Using these equalities and the eigenvalue equation (BA-D)v = -ρ_1 v, we can show that ℒ w= (𝒩_22 -ρ_2 I)^-1_i( δ_i V_i (D -ρ_2 I)^-1(D-ρ_1 I) v )+(𝒩_22 -ρ_2 I)^-1_i ((D-ρ_1 I) v)_\{i} - ϕ= _i_j≠ i(δ_j -ρ_1/δ_j -ρ_2v_j) - ϕ≤(max_1≤ i≤ nδ_i-ρ_1/δ_i-ρ_2)w -ϕfor the nonzero and nonnegative vector ϕ = _1≤ i≤ n_j≠ ia_ij(β_i v_j +β_j v_i)/δ_i+δ_j-ρ_2.Since ℒ is irreducible (for the proof, see Appendix A), the inequality (<ref>) and Lemma <ref>.<ref> show λ_max(ℒ) < max_1≤ i≤ n (δ_i-ρ_1)/(δ_i-ρ_2). Since we have already shown (<ref>), there must exist an i such that 1< (δ_i-ρ_1)/(δ_i-ρ_2). This inequality is equivalent to ρ_2>ρ_1 because both of δ_i-ρ_1 and δ_i-ρ_2 are positive. This completes the proof.§ NUMERICAL SIMULATIONSIn this section, we illustrate the effectiveness of our results with numerical simulations. The simulations are performed using Python 3.6 on a 4.2 GHz Intel Core i7 processor. In our simulations, we consider the SIS model over several complex networks with a homogeneous transmission rate β_i = β and a recovery rate δ_i = δ for all nodes. We normalize δ = 1 without loss of generality. We first consider the following three random graphs: 1) Erdös-Rényi (ER), 2) Barabási-Albert (BA), and 3) Newman-Watts-Strogatz (NWS) graphs. For each of the networks and various network sizes, we compute the first-order bound ρ_1, our second-order bound ρ_2, and an approximation of the true decay rate ρ (by the same procedure used in Example <ref>). We present the sample averages of the relative errors e_1 =(ρ-ρ_1)/ρ and e_2 = (ρ-ρ_2)/ρ in Fig. <ref> (20 realizations of random graphs for each data point) for β = 0.9 /λ_max(A), 0.7/ λ_max(A), and 0.5/ λ_max(A). We can observe that our second-order bound remarkably improves the first-order bound, specifically for the cases of BA and NWS networks.We then consider the SIS model over several real-world networks <cit.>. Specifically, we compute lower-bounds on the decay rates for 1) a bipartite network from participation of women in social events (Davis, n=32), 2) a social network of the Zachary's Karate club (Karate, n=34), 3) the connectivity network of states in the USA (USA, n = 49), 4) a network of bottlenose dolphins (Dolphin, n=62), 5) a network of protein-protein interactions (PDZBase, n=212), 6) the high-school network described in Example <ref> (Jefferson, n=288), 7) an email communication network at the University Rovira i Virgili (University, n = 1133), 8) a friendship network on hamsterster.com (Hamsterster, n = 1858), and 9) a friendship network in Facebook (Facebook, n = 2888). We consider the homogeneous case as in the above simulations for random graphs, and use the transmission rate β = 0.9/λ_max(A) and the recovery rate δ=1. We show the relative errors e_1 and e_2 in Fig. <ref>. These simulations confirm that our second-order lower-bound can remarkably improve the first-order bound.§ CONCLUSION In this paper, we have presented an improved lower-bound on the decay rate of the SIS model over complex networks. We have specifically derived a lower-bound on the decay rate in terms of the maximum real eigenvalue of an n^2× n^2 Metzler matrix, and have shown that our lower-bound improves existing lower-bound based on mean-field approximations of the SIS model. For deriving our lower-bound, we have used a linear upper-bounding model for the first and second-order moments on the SIS model. In our simulations, we have shown that our lower-bound significantly improves on the first order lower-bound, in the cases of both random and realistic networks. This improvement suggests that incorporating second-order moments could allow us to drastically improve the performance of existing strategies for spreading control <cit.>. § ACKNOWLEDGMENTS This work was supported in part by the NSF under Grants CAREER-ECCS-1651433 and IIS-1447470.§ IRREDUCIBILITY OF ℒ Since 𝒩_22-ρ_2I is a diagonal matrix, it is sufficient to show the irreducibility of ℒ' = (𝒩_22-ρ_2I)ℒ. We can show that the matrix ℒ' can be represented as the block matrix [ℒ'_ij]_i, j having the block elementsℒ'_ij = _j≠ iβ_j A_j, \{i} - ⊕_j≠ i a_ijβ_i, if i=j,δ_i β_jδ_j - ρ_2 V_j(e_j⊗ A_j, \{j}),otherwise.Therefore, the irreducibility of ℒ' is equivalent to that of the block matrix ℒ” = [ℒ”_ij]_i, j having the block elementsℒ”_ij = _j≠ i A_j, \{i}, if i=j, V_j(e_j⊗ A_j, \{j}), otherwise.To prove the irreducibility of ℒ”, we notice that ℒ” equals the adjacency matrix of the directed graph 𝒢” = (𝒱”, ℰ”) having the n(n-1) nodes {v_1,2, …, v_1,n,v_2,1, v_2,3, …, v_2,n,…, v_n, 1, …, v_n,n-1}and edges ℰ” = ℰ”_1 ∪ℰ”_2, where ℰ”_1 = { (v_i, j, v_j, k) (j, k) ∈ℰ} and ℰ”_2 = { (v_i, j, v_i, k) (j, k) ∈ℰ}. Let us show that 𝒢” is strongly connected. Take two arbitrary nodes v_i_0, j_0 and v_i_1, j_1. Since 𝒢 is strongly connected, we can find a directed-path of the form (v_i_0, v_k_1, …, v_k_ℓ_1-1, v_i_1, v_k_ℓ_1) in 𝒢. Then, from the definition of ℰ”_1, we see that the ordered set( v_i_0, j_0, v_j_0, k_1, v_k_1, k_2, …, v_k_ℓ-2, k_ℓ-1, v_k_ℓ-1, i_1, v_i_1, k_ℓ ) is a directed path in (𝒱”, ℰ”_1). In order to continue this directed path to v_i_1, j_1, we take another directed path (v_k_ℓ, v_k_ℓ+1, …, v_k_ℓ', v_j_1) in 𝒢. Then, from the definition of ℰ”_2, we can see that the ordered set ( v_i_1, k_ℓ, …, v_i_1, k_ℓ', v_i_1, j_1 ) is a directed path in (𝒱”, ℰ”_2). We have thus shown the existence of a directed path from v_i_0, j_0 to v_i_1, j_1 in 𝒢”. This shows the strong connectivity of 𝒢” because v_i_0, j_0 and v_i_1, j_1 were taken arbitrarily. This proves the irreducibility of ℒ” and, therefore, the irreducibility of ℒ, as desired. 10 url<#>1urlprefixURL href#1#2#2 #1#1Lerman2010 K. Lerman, R. 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Kunegis, KONECT – the Koblenz network collection, in: Proceedings of the 22nd International Conference on World Wide Web, ACM Press, New York, New York, USA, 2013, pp. 1343–1350. | http://arxiv.org/abs/1706.08602v2 | {
"authors": [
"Masaki Ogura",
"Victor M. Preciado"
],
"categories": [
"cs.SI",
"q-bio.PE"
],
"primary_category": "cs.SI",
"published": "20170626212610",
"title": "Second-Order Moment-Closure for Tighter Epidemic Thresholds"
} |
[email protected] Department of Earth, Planetary, and Space Sciences, UCLA, Los Angeles, CA 90095, USA Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand Department of Earth, Planetary, and Space Sciences, UCLA, Los Angeles, CA 90095, USA Dipartimento di Fisica, Università della Calabria, Cosenza I-87036, Italy The magnetic topology and field line random walk properties of a nanoflare-heated and magnetically confined corona are investigated in the reduced magnetohydrodynamic regime. Field lines originating from current sheets form coherent structures, called Current Sheet Connected (CSC) regions, extended around them. CSC field line random walk is strongly anisotropic, with preferential diffusion along the current sheets' in-plane length. CSC field line random walk properties remain similar to those of the entire ensemble but exhibit enhanced mean square displacements and separations due to the stronger magnetic field intensities in CSC regions. The implications for particle acceleration and heat transport in the solar corona and wind, and for solar moss formation are discussed.§ INTRODUCTIONThe stochastic properties of magnetic fluctuations in turbulent plasmas are reflected in the stochastic character of magnetic field lines, giving rise to field line random walk <cit.> that strongly affects the propagation and cross-field transport of energetic particles. Additionally the intense electric fields associated to turbulent coherent structures, such as current sheets (and the related in- and out-flows), strongly contribute to particle acceleration <cit.>.Because in current sheets particles are energized and plasma heated, the topology of field lines that originate from them determines how these accelerated particles and heat are transported. Furthermore in strong magnetic fields, where particle diffusion perpendicular to field lines is small and thermal conduction highly anisotropic (essentially parallel), heat and particles are in first approximation transported along field lines.It is therefore key to understand the interplay between current sheets and magnetic field lines.It has become increasingly clear that the effective heating and particles acceleration occur at scales of the order of the ion (proton) inertial length d_i <cit.>, that in the solar corona, for an ion density n_i ∼ 10^8 cm^-3, is d_i = c/ω_pi∼ 23 m (ω_pi = √(4π n_i e^2/m_i) is the proton plasma frequency, c the speed of light, e the electron charge, and m_i the proton mass). For typical hot coronal loops with temperatures T ∼ 10^6 K and magnetic field intensities B ∼ 50 G the ion gyroradius is much smaller than d_i (reaching d_i only in the higher β regions typical of the solar wind).In situ measurements in Earth's magnetotail <cit.> and magnetosheath <cit.>, and laboratory experiments <cit.> show that current sheet thickness is generally somewhat larger than the ion inertial length, with activity increasing for thinner current sheets as their width approaches d_i.Additionally PIC simulations of fully developed turbulence have shown that clustering ofcurrent sheet thickness occurs at scales of ∼ d_i, with substructures down to the electron inertial length d_e <cit.>. These are the natural scales at which kinetic effects will convert the energy coming from large scales into the different species thermal and non-thermal energies.Note that these findings are also consistent with the recent understanding that thin current sheets are strongly unstable under the plasmoid instability <cit.>, with growth rates reaching fast “ideal” Alfvén values (γτ_A ∼ 1) for sufficiently small thicknesses <cit.>. Including the Hall effect the instability becomes explosive as the current sheet thickness approaches d_i <cit.>.Although the aforementioned studies include at most a weak guide magnetic field, the formation of current sheets with the exponentially thinning widths have been observed in fully nonlinear 2D and 3D MHD simulations <cit.>, and line-tied simulations with a strong guide field and vanishing initial velocity <cit.>. Although kinetic simulations with a strong guide field are still computationally challenging, we expect that the overall phenomenology and current sheet structure is not substantially modified in the strong guide field case of interest to the solar corona and inner heliosphere.We then consider the gyroradii of bulk ions and electrons to be generally smaller than the current sheet thickness, and the initial stage of their acceleration is thus strongly affected by the field line topology.FLRW in turbulent fields is a topic of intense research <cit.>, but little attention has been dedicated to the effects of spatial intermittency and coherent structures <cit.>. However, the plasma thermodynamical properties are strongly affected by the topology of field lines originating in current sheets, both in the corona and solar wind. For instance the thermodynamics and high-energy radiative emission of coronal loops are determined by the temporal and spatial properties of energy dissipation along the field lines <cit.>. Additionally, energetic particles and heat transporttoward the transition region at coronal loops footpoints give rise to a reticulated spongy pattern in X-rays and EUV, so-called moss, that could be explained by the complex trajectories of energetic particles in a stochastic magnetic field <cit.>.Here we investigate the magnetic field lines random walk in a nanoflare-heated and magnetically confined corona to advance our understanding of transport of heat and energetic particles in stochastic magnetic fields, its relationship with coherent structures, and discuss its impact on coronal and solar wind dynamics.§ MODEL Our model coronal loops are “straightened-out” in a Cartesian elongated box with axial length L (along the z-direction) and orthogonal square cross section of size ℓ (x-y planes), with aspect ratio L/ℓ=10. The system, with uniform density ρ_0,is threaded by a strong axial magnetic field 𝐁_0 = B_0 ê_z, and its dynamics are well described by the reduced magnetohydrodynamic (RMHD) equations <cit.>, valid in the limit of a large loop aspect ratio (ϵ = ℓ /L ≪ 1) and of a small ratio of orthogonal to axial magnetic field (b/B_0 ≤ϵ). The velocity (𝐮) and fluctuating magnetic field (𝐛) have only components perpendicular to the axial direction z, and indicating their potentials with φ and ψ, they can be written as 𝐮 = ∇φ×ê_z and 𝐛 = ∇ψ×ê_z, with the current density j = -∇^2 ψ, and vorticity ω = -∇^2 φ. In non-dimensional form they are given by: ∂_t ψ = [ φ, ψ] +B_0 ∂_z φ +η∇^2 ψ,∂_t ω = [ j, ψ] - [ ω, φ] +B_0 ∂_z j+ν∇^2 ω, where the magnetic field has been expressed as an Alfvén velocity (i.e., b → b/√(4πρ_0)), and then all velocities normalized to u^∗ = 1 km s^-1 (the photospheric granulation velocity rms). The Poisson bracket is defined as, e.g.,[φ,ψ] = ∂_x φ∂_y ψ - ∂_y φ∂_x ψ = - 𝐮·∇ψ, and the Laplacian operator has only orthogonal components. Lengths are normalized to the orthogonal box length, thus ℓ=1 and L=10. Normalized resistivity and viscosity coefficients are set equal with η=ν=1/R, where the Reynolds numberR=800, numerical resolutions is 2048^2×512, and the guide field intensity B_0=10^3.As in previous simulations field lines are line-tied to the top and bottom plates z=0,10 where they are shuffled by a prescribedphotospheric granulation-mimicking velocity constant in time with length-scale ∼1/4 <cit.>, while in x-y planes periodic boundary conditions are implemented. In the x-y planes a pseudo-spectral scheme with periodic boundary conditions is implemented, time is advanced with a third-order Runge-Kutta and an adaptive time-step. More details on the model and numerical code can be found in <cit.>.The numerical integration of Eqs. (<ref>)-(<ref>) cannot implement enough grid points to attain a realistic description of the internal structure of current sheets (that would additionally require the inclusion of the Hall term, or the integration of a kinetic model, that in turn could not describe properly the large scale dynamics). Nevertheless, while a good representation of the small-scale structure is very important for the acceleration of particles <cit.>, since FLRW is affected mostly by the large scale components of the magnetic field, its properties should not depend critically on the small-scale structure of the current sheets, and an MHD model represents a good starting point. For these reasons we adopt an empirical approach, selecting the value of resistivity for essentially numerical reasons, but subsequently associating the resulting dissipative scale with the ion inertial length. § RESULTS Our simulations start with the guide field B_0 (directed along z) and no magnetic or velocity fluctuations in the computational box. The imposed large-scale velocity at the boundaries z=0, and L twists the field lines and, once the twist exceeds a small critical threshold, the orthogonal magnetic field line tension is no longer balanced.Thus, as proposed by <cit.>, the magnetic field 𝐛 transitions to non-equilibrium <cit.>, bringing about turbulent dynamics that transfers energy towards the small scales where it is dissipated in nanoflares <cit.>. Line-tying keeps the velocity field in the computational box smaller than the magnetic field (far from equipartition). Nevertheless a cascade with preferential energy transfer in the x-y planes orthogonal to B_0 and a broad-band power-law magnetic energy spectrum develop, but the enhanced field lines stiffness introduced by line-tying gives rise to steeper magnetic energy spectra E_M(k_⊥) ∝ k_⊥^-αwith α∈ [5/3, 3], with the steepest spectra corresponding to stronger guide fields B_0 <cit.>. In the simulations considered here α∼ 3.The magnetic field structure is characterized by approximately field-aligned current sheets. Although the overall physical conditions are markedly different between the line-tied and fully periodic reduced MHD, we hypothesize that the FLRW properties of field lines traced from current sheets are qualitatively the same, since in both cases current sheets are field aligned <cit.> and their axial extension must be linked to the parallel correlation length. Clearly the current sheet extension in the axial direction can change depending on the particular type of forcing, but our conclusions on field line random walk and diffusion properties may be tentatively extended to the fully periodic case, as discussed inour Conclusions, section <ref>.§.§ Field line diffusion from a single pointSince in reduced MHD the z-component (B_0) of the magnetic field is constant, the magnetic field line equation can bewritten as d𝐱/d z( z ) =1/B_0𝐛( 𝐱(z), z ) →d/d z𝐱^2 =2/B_0 𝐱·𝐛,where 𝐱 = (x, y) indicates the orthogonal coordinates. The mean square displacement⟨𝐗^2(z) ⟩ =⟨ [𝐱(z) - 𝐱_0]^2 ⟩ =⟨𝐱^2(z) ⟩ - 𝐱_0^2, where ⟨…⟩ indicates ensemble average,is shown in Fig. <ref> (top panel). After an initial ballistic stage with ⟨𝐗^2 ⟩∝ z^2, it subsequently exhibits diffusion with ⟨𝐗^2 ⟩ = 4D z.To understand this behavior, from Eq. (<ref>) we can write d/d z⟨𝐱^2 (z) ⟩ = 2/B_0^2∫_0^z d z' ⟨𝐛( 𝐱(z'), z' )·𝐛( 𝐱(z),z ) ⟩. Although the position vectors 𝐱 (z) are random functions determined by the trajectory, the integrand in Eq. (<ref>) is linked tothe magnetic field two-point correlation function C(𝐱_1, z_1, 𝐱_2, z_2 ) = ⟨𝐛(𝐱_1, z_1) ·𝐛(𝐱_2, z_2) ⟩. For homogeneity and isotropy this depends only on the relative parallel and orthogonal distances of the two points, i.e., indicating withℓ_⊥ = 𝐱_2 - 𝐱_1, and Δ z = z_2 - z_1: C(ℓ_⊥, |Δ z| ) = ⟨𝐛(0, 0) ·𝐛( ℓ_⊥, Δ z)⟩, independent of the origination point (as long as both points are within the z-span). As shown in Fig. <ref> (center), the correlation decreases at larger ℓ_⊥ and z. But while it vanishes in the perpendicular direction at the correlation length λ_⊥∼ 0.11, it does not vanish in the parallel direction (for ℓ_⊥=0). Namely the parallel correlation length λ_∥ is larger than the box size L, i.e., the turbulent field has a strong 2D component. Clearly this is due to the low frequency of photospheric motions. Indeed, for typical hot loops, the field line footpoints are shuffled slowlycompared to the fast Alfvén crossing timescale at which the induced magneticfield twist propagates along the loop axis. The correlation in Eq. (<ref>) is Lagrangian, i.e., it is computed along the field lines: C( z, Δ z ) = ⟨𝐛( 𝐱(z-Δ z), z-Δ z ) ·𝐛( 𝐱(z),z ) ⟩,with 0 ≤Δ z ≤ z, and it is shown in Fig. <ref> (bottom panel).Introducing the change of variableΔ z = z - z' we can then write d/d z⟨𝐱^2 (z) ⟩ = 2/B_0^2∫_0^z dΔ z C( z, Δ z ). Since the mean square displacement between two points along a field line at a parallel distance Δ z is to a good approximation given by⟨𝐗^2(Δ z) ⟩, the two correlations are then approximately linked by C( Δ z ) ∼ C( ⟨𝐗^2(Δ z) ⟩^1/2, Δ z ). Additionally the correlation function is connected to the second-order structure function as C(0,0)-C(ℓ_⊥,0)= ⟨δ b_ℓ_⊥^2⟩/2 <cit.>, that in turn, for values of ℓ_⊥ in the inertial rangeis linked to the magnetic energy spectrum by E_ℓ_⊥∝ℓ_⊥δ b_ℓ_⊥^2, consequently E_ℓ_⊥∝ℓ_⊥^α⟶δ b_ℓ_⊥^2 ∝ℓ_⊥^α -1, with α∈ [5/3, 3] for our boundary forced coronal loop model <cit.>. Therefore extending the power-law behavior beyond the inertial range for all ℓ_⊥≤λ_⊥, and taking into account that the correlation vanishes at λ_⊥, we can approximate the magnetic correlation function with C(ℓ_⊥,0)/⟨ b^2 ⟩∼ 1 - (ℓ_⊥/λ_⊥)^α-1 for ℓ_⊥≤λ_⊥, 0for ℓ_⊥≥λ_⊥, with the exponent ranging from 2/3 for α =5/3 up to 2 for α = 3. This function is plotted in Fig. <ref> (middle) for α = 3. Here we neglect the parallel variation of C when used in Eq. (<ref>) because ⟨𝐗^2(Δ z) ⟩ increases monotonically with Δ z, and as shown in Fig. <ref> (middle) the curves then tend to overlap quickly becoming approximately independent of Δ z.The behavior of the mean square displacement ⟨𝐗^2(z) ⟩ can then be readily understood from the correlation function C(ℓ_⊥,Δ z) (Fig. <ref>). For small values of z the integral in Eq. (<ref>) can be Taylor-expanded, and sinceC(0,0) = ⟨ b^2 ⟩, to the first order we obtain ⟨𝐗^2(z) ⟩≈z^2 ⟨ b^2 ⟩/B_0^2. On the other hand as soon as⟨𝐗^2(z) ⟩^1/2≳λ_⊥exceeds the orthogonal correlation lengthλ_⊥∼ 0.11, the integral in Eq. (<ref>) remains approximately constant, because the largest contribution comes fromℓ_⊥ < λ_⊥, hence⟨𝐗^2 (z) ⟩∼ 4Dz diffuses linearly. The transition from the ballistic to the diffusive stage occurs for ⟨𝐗^2(z) ⟩∼ z^2 ⟨ b^2 ⟩ / B_0^2 ∼λ_⊥^2, i.e., for z_D ∼λ_⊥ B_0/⟨ b^2 ⟩^1/2. In our case, since λ_⊥∼ 0.11, B_0=10^3, and ⟨ b^2 ⟩^1/2∼ 20, the transition occurs at z_D ≈ 5.5, as confirmed in Fig. <ref> (top).We can estimate the diffusion coefficient D by using in Eq. (<ref>) the approximation for the mean square displacement outlined in the previous paragraph (i.e., ⟨𝐗^2(Δ z) ⟩≈Δ z^2 ⟨ b^2 ⟩ /B_0^2 for z ≤ z_D). The Lagrangian correlation along the field lines can then be approximated from Eqs. (<ref>) and (<ref>) with C(Δ z)/⟨ b^2 ⟩ = 1- ( Δ z/z_D)^α -1 forΔ z ≤ z_D, 0forΔ z ≥ z_D.Substituting in Eq. (<ref>), and integrating it toobtain ⟨𝐗^2(z) ⟩, the diffusioncoefficient D is then given by D ∼α -1/2α λ_⊥⟨ b^2⟩^1/2/B_0, a functional form characteristic of Bohm diffusion <cit.>, with the coefficient (α-1)/2α ranging in the narrow interval [1/5, 1/3] as α∈ [5/3, 3]. Since ⟨ b^2 ⟩^1/2∼ 20, B_0=10^3, λ_⊥∼ 0.11 and α∼ 3, we obtain D ∼ 7.3× 10^-4, corresponding toD ∼ 2.9 km in conventional units, compatible with D ∼ 3.1 km computed from our simulation (Fig. <ref>, top panel). The approximated Lagrangian correlation function C (Δ z) (Eq. (<ref>)) with α=3 and z_D ∼ 5.5 is shown in Fig. <ref> (bottom panel). §.§ Current sheet connected regionsThe current density j is intermittently distributed in space, as typical of turbulent systems. Its Probability density function (pdf), shown in Fig. <ref>, is not gaussian and exhibits typical large tails where the current is strong, corresponding to current sheets in physical space. The noticeable skewness in Fig. <ref> results from the use of a single snapshot, and it is a fluctuation that vanishes when averaging over several snapshots, i.e., the time-averaged distribution is symmetric. We define as current sheets all those spatial regions where current is larger than two standard deviations |j| ≥ 2σ, with σ = ⟨ j^2 ⟩^1/2, shown in yellow for the representative planes z=0, L/2 and L in Fig. <ref>.The relationship between magnetic field topology and current sheets is then investigated by tracing field lines originating in current sheets. Specifically field lines are traced from all the grid points where |j| ≥ 2σ in 9 equispaced x-y planes (from z=0 up to z=L, separated by L/8). For grid points that are not at the boundaries z=0 or z=L the respective field lines are traced both forward and backward with respect to the z-direction. We trace a total of 1,107,242 field lines, extending from the bottom to the top plate, and their intersection with the selected plane is shown as a black dot in Fig. <ref>. Clearly field lines are present also in the yellow regions, both those traced from there plus others originating from current sheets in different planes.Current sheets are elongated in the guide field direction z, and the field lines traced from them form similarly shaped coherent structures, that we indicate as Current Sheet Connected (CSC) regions. Although current sheets in reduced MHD have a complex structure with a cross-shear magnetic field component and mostly external X-points <cit.>, noticeably the presence of a strong magnetic shear in correspondence of current sheets makes the field line random walk strongly anisotropic, with field lines diffusing preferably along the in-plane sheet length and very little across it (Fig. <ref>). Since diffusion increases with distance its effects are most apparent in planes z=0 and L.The coherence and strong anisotropy of the CSC regions are in stark contrast with the homogeneity of the stochastic properties typically associated with FLRW, and their well-know tendency to fragment flux tubes <cit.>.Nevertheless the mean square displacement ⟨𝐗^2(z) ⟩ of the CSC field lines has properties similar to those of the entire ensemble, as shown in Fig. <ref> (top). The higher value of the diffusion coefficientD_ CSC/D_ All∼ 1.2 is due to the higher magnetic field intensity in the CSC region as ⟨ b^2 ⟩_CSC^1/2 / ⟨ b^2 ⟩_All^1/2∼ 1.2 in agreement with Eq. (<ref>). §.§ Pair separation To further understand the magnetic topology we considerthe separation of field line pairs. From Eq. (<ref>) their orthogonal separation in the x–y planeξ (z) = 𝐱_2(z) - 𝐱_1(z), given the initial separation ξ(0) = ξ_0, is determined as a function of z by dξ/d z(z) =1/B_0[ 𝐛( 𝐱_2(z), z ) -𝐛( 𝐱_1(z), z ) ].Similarly to the single field line case we obtain d⟨ξ^2(z) ⟩/d z = 2/B_0^2∫_0^z d z' ⟨[ 𝐛( 𝐱_2(z'), z') -𝐛( 𝐱_1(z'), z' ) ] · [ 𝐛( 𝐱_2(z), z ) -𝐛( 𝐱_1(z), z ) ]⟩, that following <cit.> can be written as d/d z⟨ξ^2(z) ⟩ = 4/B_0^2( I_11 - I_12), where: I_11 = ∫_0^z d z' ⟨𝐛( 𝐱_1(z'), z' ) ·𝐛(𝐱_1(z),z)⟩, I_12 = ∫_0^z d z' ⟨𝐛( 𝐱_1(z'), z' ) ·𝐛( 𝐱_2(z), z ) ⟩. I_11 is same as the integral in Eq. (<ref>) because it refers to single field lines, therefore we already understand its behavior. But I_12 differs as it considers a pair.As indicated by <cit.> for the hydrodynamic case, for z sufficiently small the mean-squareseparation will grow quadratically with z(see inset in Fig. <ref>, left panel) and it will beproportional to the second-order structure function. Indeed from Eq. (<ref>), Taylor-expanding Eqs. (<ref>)-(<ref>) in z, and sinceC(0,0)-C(ξ_0,0)= ⟨δ b_ξ_0^2⟩/2 we obtaind/d z⟨ξ^2 (z) ⟩ = 2z ⟨δ b_ξ_0^2 ⟩/B_0^2, hence ⟨ξ^2 (z) ⟩ - ξ^2_0 = z^2 ⟨δ b_ξ_0^2 ⟩/B_0^2, where as usual ξ_0 = |ξ_0|.Additionally we can now approximate the second order structure function as in Eq. (<ref>) with⟨δ b^2_ξ_0⟩∼ 2 ⟨ b^2 ⟩ (ξ_0/λ_⊥)^α-1 for ξ_0 ≤λ_⊥ and⟨δ b^2_ξ_0⟩∼ 2 ⟨ b^2 ⟩ for ξ_0 ≥λ_⊥, with the scaling relation more accurate for values of ξ_0 in the inertial range. We can then write ⟨ξ^2 (z) ⟩ - ξ^2_0 ≈2/λ_⊥^α-1 ⟨ b^2 ⟩/B_0^2 z^2 ξ_0^α-1,for ξ_0 ≤λ_⊥ 2 ⟨ b^2 ⟩/B_0^2 z^2,for ξ_0 ≥λ_⊥ with α∈ [5/3,3].We plot (⟨ξ^2(z) ⟩ - ξ^2_0)/z^2in Fig. <ref> (right panel) as a function of ξ_0 for 11 values of z ∈ [0,2] separated by 0.2, almost perfectly overlapping and showing that in the inertial range (ξ_0 ≲ 0.1) it scales approximately as ξ_0^2, compatible with α∼ 3 in our simulations, and saturates correctly to ∼ 2⟨ b^2 ⟩/B_0^2 ∼ 8 × 10^-4 with our parameters (⟨ b^2 ⟩∼ 400, B_0=10^3, λ_⊥∼ 0.11).The small departure from ξ_0^2 in the inertial range shown in Fig <ref> occurs because the second order structure function⟨δ b_ξ_0^2/2 ⟩ = C(0,0)-C(ξ_0,0) considers zero separation in z and is calculated in the bottom boundary plane z=0 (since the coefficients in the Taylor expansion in z of Eqs. (<ref>)-(<ref>) are calculated for z=0). Indeed the 2D magnetic energy spectrum averaged over the whole box E_ℓ_⊥∝ℓ_⊥δ b_ℓ_⊥∼ℓ_⊥^3, corresponding to δ b_ℓ_⊥∼ℓ_⊥^2, and the same behavior is observed for the spectra in all z-planes, except those in proximity of the boundaries z=0 and L where line-tying boundary conditions are applied. At these boundaries the velocity field is prescribed and therefore the dynamics does not follow the same equations as in the interior, with the effect of slightly modifying the magnetic energy spectrum for the planes in their close proximity. Nevertheless the scaling for the separation remains very close to a ∝ξ_0^2 scaling, departing strongly in the inertial range from ξ_0^2/3 expected for a standard Kolmogorov spectrum with α = 5/3 <cit.>. Therefore the ratio of mean square separations for field lines with relative larger initial separations ξ'_0 > ξ_0 are increasingly bigger forsteeper spectral indices, indeed from Eq. (<ref>) the ratio of their separations in the ballistic range (∝ z^2) grows like (ξ'_0/ξ_0)^α-1. In general, as ξ_0 → 0 the two field linestend to the same field line, i.e.,𝐱_2(z) →𝐱_1(z), consequently I_12→ I_11, and the mean square separation vanishes in this limit.On the other hand the mean square separation in I_12, i.e.ξ_12(z',z)=⟨ [𝐱_2(z') - 𝐱_1(z)]^2 ⟩^1/2 is always larger than ξ_0. For initial separations larger than λ_⊥ the correlation is small ∀ z ∈ [0,L]so that I_12≈ 0. In this case the diffusion coefficient forpair separation is double that of single field line diffusion, i.e.,⟨ξ^2 (z) ⟩ - ξ^2_0 = 4D_pz with D_p = 2D, as shown in Fig. <ref> (left panel) for 0.1 < ξ_0 < 1 by the continuous black line (we average these curves since they overlap).For any ξ_0 < λ_⊥ there is always a critical height z_D(ξ_0) above which the mean separation between the two fieldlines in I_12 is larger than λ_⊥. Hence the increasingly larger negative contribution of the I_12 termto mean square pair separation will display diffusion at progressively lower heights, with a smaller total diffusion coefficients D ≤ D_p ≤ 2D,as shown in Fig. <ref> by the red lines, that consider 5 bins with ξ_0 ∈ [0.015,0.095] with Δξ_0 =0.02. Clearly, for sufficiently small initial separations, field lines will not be able to display diffusion because our system is bounded in the axial direction z to maximum length L and separation cannot grow up to the perpendicular correlation length, as shown well by the blue continuous lines in Fig. <ref>. A ballistic stage ∝ z^2 is always present initially for z ≲ 3, as shown in Fig. <ref> (insets), but field lines with initial separation within the dissipative range (ξ_0 ≲ 10 dx, where dx=1/nx=1/2048 is the numerical grid step size) exhibit subsequently a Richardson-like superdiffusive stage <cit.> with mean square separation up to ∝ z^4 (blue continuous lines in Fig. <ref>, with respectively ξ_0=2.5, 6.5, and 10.5 dx), while for larger separations they transition to the diffusive regime.Even though FLRW in CSC regions is strongly anisotropic, CSC field line pair separation (shown in Fig. <ref> with dashed lines) exhibit similar properties to the whole ensemble (shown with continuous lines, color code is the same for both line types). Their statistics are degraded for larger ξ_0 because in CSC regions the number of field line couples diminishes at larger separations so that the averages do not saturate yet to their ensemble value. This can be seen in the left inset, where the red dashed curves become negative, because the relation⟨ξ^2(z) ⟩ - ξ_0^2 = ⟨ (ξ(z)-ξ_0)^2 ⟩≥ 0 is valid only for a sufficiently high number of field lines, when ⟨ξ(z) ⟩ = ξ_0.The main difference between CSC pair separations and the whole ensemble is that for same initial separation ξ_0 the CSC field lines exhibit higher separations. Similarly to mean square displacement (Fig. <ref>, top panel) the higher values are due to the greater magnetic field intensity in CSC regions(⟨ b^2 ⟩_CSC/⟨ b^2 ⟩∼ 1.5), and indeed the separation in the ballistic range is proportional to ⟨ b^2 ⟩ (Eq. (<ref>)). Notice that the coefficient in Eq. (<ref>) includes also the perpendicular correlation length λ_⊥ that is not readily computable in the non-Cartesian CSC region, but from the data we can estimate that the separation for field lines with same initial separation is about 10 times larger in CSC regions than for the whole ensemble.As mentioned previously the statistics are degraded for CSC field lines with larger initial separations. Therefore while the larger separations for CSC field lines is very well demonstrated for smaller initial separations (blue dashed lines, and first red dashed line), we cannot yet fully verify this conclusion for larger initial separations, as the averages have not yet saturated to their ensemble values (this point will be further investigated in upcoming work). § CONCLUSIONS AND DISCUSSIONTo gain insight into particle acceleration and heat transport in coronal loops, solar wind, and more in general for plasmas in the reduced MHD regime, we have investigated the magnetic topology of field lines originating from current sheets. We have found that they form coherent structures, dubbed Current Sheet Connected (CSC) regions, similarly to the current sheets they originate from. Field lines in these regions perform highly anisotropic FLRW, with diffusion occurring preferentially along the current sheet in-plane length. Nevertheless FLRW and diffusion coefficients have similar properties for CSC field lines and the whole ensemble, with larger displacement and separations occurring in CSC regions where the magnetic field intensity is higher.This emerging picture has strong implications for particle acceleration and heat transport, particularly in the low corona, where all protons and electrons with temperatures below 10^6 K have gyroradii smaller than the current sheet thickness. It indeed implies that in coronal loops particle and heat are transported almost exclusively within the CSC region, a small volume of plasma around current sheets with a small filling factor, while most of the volume is topologically disconnected from current sheets and the associated flow of particles and heat.This picture is fully consistent with observations <cit.> and recent thermodynamical 3D simulations <cit.> strongly suggesting that coronal loops cannot be modeled with single isothermal flux tubes, as their radiative properties can only be explained by the presence of both hot and cold plasmas at observational sub-resolution scales (multi-temperature loops). Also, the complex topology in CSC regions, with enhanced magnetic field line displacements and separations, points to a complex stochastic nature for the heating function along the field lines.Additionally the structure of the CSC regions in the top and bottom plates z=0 and L is consistent with that of so-called moss, the spongy reticulated pattern in X-rays and EUV formed at the coronal base of hot loops, confirming that the FLRW can play a strong role in the formation of these structures as recently proposed <cit.>.The reduced MHD FLRW properties and topology strongly support the results of recent test-particle simulations <cit.> with initial gyroradii smaller than current sheet widthspropagated in similar magnetic fields to those discussed here. Particles are at first strongly accelerated along the z-direction in current sheets by the strong electric field associated with the current, until they pitch-angle scatter thus increasing their gyroradii. Subsequently, as long as the gyroradius is smaller than the orthogonal correlation length, those that remain close to the CSC region are accelerated by a (non-magnetic moment conserving) betatron-like mechanism due to the inhomogeneous 𝐮× B_0 ẑ electric field associated to outflows in current sheets. For parameters typical of hot solar coronal loops we have found that the parallel correlation length is longer than the loop length, and our results consider this specific case. Nevertheless they may also apply to the unbounded (e.g., periodic) case, which we plan to investigate thoroughly in upcoming work. In general we expect the current sheet length along z to be strongly correlated with the magnetic field parallel correlation length λ_∥. Therefore we expect that CSC regions connected to any such current sheet of length ∼λ_∥ to have a similar structure to those found here around the current sheet. But as field lines are traced further away at distances larger than λ_∥ we expect the CSC region to fragment and the associated FLRW to lose anisotropy and acquire more homogeneous properties, i.e., the CSC field lines will at that point connect and diffuse isotropically throughout the volume (hence mostly in regions with low current).Therefore in unbounded systems we expect heat and accelerated particles to be initially confined to CSC regions around current sheets (of length ∼λ_∥), but further away heat and particles would distribute more uniformly throughout the plasma. This picture is strongly consistent with recent analyses of solar wind data <cit.>, where temperature is found to peak in regions with high magnetic field gradients, while it rapidly descends to approach the ambient solar wind temperature as distance from those regions increases. 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"authors": [
"A. F. Rappazzo",
"W. H. Matthaeus",
"D. Ruffolo",
"M. Velli",
"S. Servidio"
],
"categories": [
"astro-ph.SR",
"physics.plasm-ph",
"physics.space-ph"
],
"primary_category": "astro-ph.SR",
"published": "20170627180150",
"title": "Coronal Heating Topology: the Interplay of Current Sheets and Magnetic Field Lines"
} |
[email protected] Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France [email protected] Department of Physics, Box 1843, Brown University, Providence, RI 02912-1843 USA Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France The Reynolds stress, or equivalently the average of the momentum flux, is key to understanding the statistical properties of turbulent flows.Both typical and rare fluctuations of the time averaged momentum flux are needed to fully characterize the slow flow evolution. The fluctuations are described by a large deviation rate function that may be calculated either from numerical simulation, or from theory. We show that, for parameter regimes in which a quasilinear approximation is accurate, the rate function can be found by solving a matrix Riccati equation.Using this tool we compute for the first time the large deviation rate function for the Reynolds stress of a turbulent flow. We study a barotropic flow on a rotating sphere, and show that the fluctuations are highly non-Gaussian.This work opens up new perspectives for the study of rare transitions between attractors in turbulent flows. 47.27.eb, 47.27.wg, 05.40.-a, 05.10.GgFluctuations and large deviations of Reynolds stresses in zonal jet dynamics T. Tangarife December 30, 2023 ============================================================================§ TOMÁSRegretfully, Tomás Tangarife suddenly and unexpectedly passed away a few months before completing the research reported in this paper. Most of the science discussed in this paper was developed in patient work by Tomás, and is part of his PhD thesis. F. Bouchet and J. B. Marston pay homage to the unique friendship and passion for science of Tomás, and would like to remember the intense and enriching collaboration that led to these scientific results. Tomás' quiet and constant character, his generosity, and his deep thoughts, were always a source of happiness and joy to his friends and colleagues.§ INTRODUCTION For a wide range of applications, in physics, engineering, and geophysics, the determination of the behavior of the average or typical behavior of a turbulent flow is a key issue.Since the work of Reynolds more than one century ago, the role of momentum fluxes and their divergence, or their averages called Reynolds stresses, have been recognized to play the key role. In order to be more specific, we now consider the very simple case of a two dimensional flow on a plane or in a channel, with an averageflow that is parallel to the 𝐞_x direction, U(y)𝐞_x (where x and y are Cartesian coordinates).We also assume that all averaged quantities do not depend on x. The spatially averaged equation of motion for thefluid reads ∂ U/∂ t=-∂/∂ y𝔼(<uv>)+D[U],where D[U] is the average dissipation operator, 𝔼(<uv>) is the Reynolds stress, and ∂/∂ y𝔼(<uv>) is the momentum flux divergence along the 𝐞_x direction. The symbol 𝔼 is either an ensemble or time average (for a time average ∂ U/∂ t=0), while <.> denotes a spatial average.The spatial average is an average along the 𝐞_x direction.The spatial average can be avoided, but it is often useful to include for practical reasons.Because the Reynolds stress is the key quantity that determines the average flow behavior it has been extensively studied experimentally, numerically and theoretically, for a wide range of turbulent flows (see for instance classical turbulence textbooks <cit.>.Beyond the average value, fluctuations of the momentum flux <uv>, or its divergence ∂/∂ y(<uv>), are very important quantities in a variety of dynamical circumstances. By contrast with the average value, as far as we know, no work has been devoted so far to study such fluctuations, and we undertake this task as the main aim of the paper.An important example of when fluctuations play an important role is in the case of time scale separation between the typical time τ_U for the evolution of the parallel flow (or jet) and the time τ_e for the evolution of the turbulent fluctuations (or eddies): τ_e≪τ_U. Such time scale separation is common when the parallel flow has a very large amplitude; classical examples include some regimes of two dimensional, geostrophic, or plasma turbulence.Then, following the classical results of stochastic averaging for systems with two timescales, a natural generalization of Reynolds average equation is ∂ U/∂ t=-∂/∂ y𝔼_U(<uv>)+∂/∂ yζ_U+D[U],where now 𝔼_U means an average over a time window short compared to the typical time evolution of the parallel flow U, and we still call 𝔼_U(<uv>) the Reynolds stress that now depends on the state of U at time t, and ζ_U(y,t) characterizes the Gaussian typical fluctuations of the momentum flux <uv>. 𝔼_U(<uv>) and ζ_U represent two aspects of the action of the unresolved eddies on the mean flow, the average and typical fluctuations respectively. In such a situation of time scale separation,ζ_U is a white in time Gaussian field whose variance is related through a Kubo formula to the variance of the time average of the momentum fluxr_v=1/T∫dt <uv>,where the time average is over a time window of duration T, which is assumed to be short compared to the time scale for the evolution of U, but large compared with the evolution of the turbulent fluctuations: τ_e≪ T≪τ_U. We call the fluctuation of (<ref>) the Reynolds stress fluctuations (the fluctuation of the time averaged momentum fluxes, over finite but long times T). In many instances, rarer and non Gaussian fluctuations are also important. Then (<ref>) does not contain the relevant information and one wants to go beyond the study of the second moment of(<ref>). In the asymptotic regime τ_e≪ T, the probability distribution function of r_v takes a very simple form P(r_v,T)T→∞≍exp(-TI_v[r_v])), where ≍ is a logarithmic equivalence (the logarithms of the right and left hand sides of the equation are equivalent in the limit T →∞). This relation is called the large deviation principle. (For a review, see Ref. touchette2009large.) The large deviation rate function I_v[r_v]characterizes the fluctuations of the time averaged Reynolds stress, both typical (the second variations of I_v[r_v]gives the statistics of ζ_U), and very rare.In many examples of turbulent flows, it has been observed that the dynamics has several "attractors” (see for instance <cit.> and references therein ; by “attractor” we mean here stationary solutions of the deterministic Reynolds equation ∂ U/∂ t=-∂/∂ y𝔼_U(uv)). Then rare fluctuations of the Reynolds stress characterized by the large deviation rate function I_v, are responsible for rare transitions between attractors. For all these reasons, it is very important to be able able to compute I_v and to be able to study its properties from a fluid mechanics point of view. We develop theoretical and numerical tools to study Reynolds stress fluctuations, and compute the large deviation rate function I_v.First we sample empirically (from time series generated from numerical simulations) the large deviation rate function, using the method developed in reference rohwer2014convergence. In addition to this empirical approach, we determine the Reynolds stress fluctuations and large deviation rate function directly for the case of the quasilinear approximation to the full non-linear dynamics.The quasilinear approximation amounts at neglecting the eddy-eddy interactions (fluctuation + fluctuation → fluctuationtriads) while retaining interactions between the mean flow and the eddies, and may thus be expected to be accurate when the magnitude of the average flow is much larger than the fluctuations. Such a quasilinear approximation, investigated at least as early as 1963 by Herring <cit.>, is believed to be accurate for the 2D Navier-Stokes equation, barotropic flows, or quasigeostrophic models, on either a plane, a torus, or a sphere, for a range of parameters (discussed below).Two dimensional flows are a particularly favorable setting for the quasi-linear approximation because, as Kraichnan showed in his seminal 1967 paper <cit.>, an inverse cascade ofenergy to the largest scales is expected, leading to the formation of coherent structures with non-trivial mean flows <cit.>.For unforced perfect flows, the large scale structurescan be predicted through equilibrium statistical mechanics (see for instance <cit.>).For forced and dissipated flows eddies both sustain, andinteract with, the large-scale flows, and both processes are captured by the quasi-linear approximation. By contrast, the scale-by-scale cascade of energy that plays a central role in Kraichnan's picture <cit.> relies on eddy + eddy → eddy processes that are neglected in the quasi-linear approximation <cit.>.The quasilinear approximation has been shown to be self-consistent <cit.> in the limit when a time scale separation exists between a typical large scale flow inertial time scale τ_i and a flow spin up or spin down time scale τ_s: τ_i≪τ_s (then τ_U≃τ_s and τ_e≃τ_i). This time scale separation condition may however not be necessary. Other factors may favor the validity of the quasilinear approximation, for instance the forcing of the flow through a large number of independent modes, through either a broad band spectrum, or a small scale forcing, keeping the total energy injection rate fixed. The energy transfer is then the same for all forcing spectrums, but with a braod band spectrum each eddy has reduced amplitude, lessening the interaction between eddies. The range of validity of the quasilinear approximation has not been fully understood yet. When the quasilinear approximation is valid, and when one further assumes that the forcing acts on small scales only, one can predict explicitly the averaged Reynolds stress <cit.> and sometimes the averaged velocity profile. The Gaussian fluctuations of the Reynolds stress may be parameterized phenomenologically <cit.>. The spatial structure of the Gaussian fluctuations has also been studied theoretically.It has been proven to have a singular part with white in space correlation function and a smooth part (see <cit.>, section 1.4.3, or <cit.>, see also <cit.>). Within the context of the quasilinear approximation, we show that the Reynolds stress fluctuations and its large deviation rate function can be studied by solving a matrix Riccati equation. The equation can be easily implemented and solved by a generalization of the classical tools used to solve Lyapunov equation for the two-point correlation functions. This mathematical result is the main reason why we study the Reynolds stress fluctuations for the quasilinear dynamics in this first study. Moreover we show that the matrix Riccati equation is a much more computationallyefficient way to study rare fluctuations than through the traditional route of direct numerical simulation.The calculation is illustrated for the case of barotropic flow on the sphere <cit.>, for which the relevance of the quasilinear approximation, over certain parameter ranges, has been recognized for a some time now. For the case of a barotropic flow it is technically more convenient to discuss the dynamics in terms of the equation of motion for the vorticity, so we study the corresponding Reynolds stress that drives the vorticity. Section <ref> introduces the barotropic equation on the sphere and its quasilinear approximation. Section <ref> discusses the fluctuations of the Reynolds stresses, without time average. Section <ref> is an introduction to averaging for stochastic processes. It explains pedagogically how an equation for the slow degrees of freedom, for instance the Reynolds equation (<ref>), can be obtained. The relation between the statistics of the noise term, ζ_U, in equation (<ref>), and the large deviation of the Reynolds stress (<ref>) is explained. A short introduction to the large deviation rate function is also provided. Finally, the matrix Riccati equation that permits direct calculation of the large deviation rate function is derived both in a general framework, and in the case of the quasilinear approximation of the barotropic equation on the sphere. Section <ref> uses the solution of the matrix Riccati equation in order to study numerically the zonal energy balance and the time scale separation in the inertial limit. Section <ref> discusses the computation of the large deviation rate function for the time averaged Reynolds stresses of the barotropic equation on the sphere. Section <ref> discusses the main conclusions and presents some perspectives.§ BAROTROPIC EQUATION AND QUASI–LINEAR APPROXIMATION Here we discuss the barotropic equation and its quasilinear approximation that is expected to be valid when a time scale separation exists between the typical time for the evolution of the zonal flow and that of the evolution of the eddies.We study the dynamics of zonal jets in the quasi-geostrophic one-layer barotropic model on a sphere of radius a, rotating at rate Ω,{ ∂ω/∂ t+J(ψ,ω)+2Ω/a^2∂ψ/∂λ=-κω-ν_n(-Δ)^nω+√(σ)η, u=-1/a∂ψ/∂ϕ, v=1/acosϕ∂ψ/∂λ,ω=Δψ.where ω is the relative vorticity, v=(u,v) is the horizontal velocity field, ψ is the stream function and J(ψ,ω)=1/a^2cosϕ(∂_λψ·∂_ϕω-∂_λω·∂_ϕψ) is the Jacobian operator. The coordinates are denoted (λ,ϕ)∈[0,2π]×[-π/2,π/2], λ is the longitude and ϕ is the latitude. All fields ω,u,v and ψ can be decomposed onto the basis of spherical harmonics Y_ℓ^m(ϕ, λ), for example ψ(ϕ, λ)=∑_ℓ=0^∞∑_m=-ℓ^ℓψ_m,ℓ Y_ℓ^m(ϕ, λ)All fields ω,u,v and ψ are 2π-periodic in the zonal (λ) direction, so we can also define the Fourier coefficients in the zonal direction, ψ_m(ϕ)≡1/2π∫_0^2πψ(ϕ, λ) ^-imλ λ=∑_ℓ=|m|^∞ψ_m,ℓ P_ℓ^m(sinϕ),with the associated Legendre polynomials P_ℓ^m(sinϕ).In (<ref>), κ is a linear friction coefficient, also known as Ekman drag or Rayleigh friction, that models the dissipation of energy at the large scales of the flow <cit.>. Hyper-viscosity ν_n(-Δ)^n accounts for the dissipation of enstrophy at small scales and is used mainly for numerical reasons.Most of the dynamical quantities are independent of the value of ν_n, for small enough ν_n. η is a Gaussian noise with zero mean and correlations 𝔼[η(λ_1,ϕ_1,t_1)η(λ_2,ϕ_2,t_2)]=C(λ_1-λ_2,ϕ_1,ϕ_2)δ(t_1-t_2), where C is a positive-definite function and 𝔼 is the expectation over realizations of the noise η. C is assumed to be normalized such that σ is the average injection of energy per unit of time and per unit of mass by the stochastic force √(σ)η. There is no symmetry reason to enforce homogeneous forcing over a rotating sphere, which only has axial symmetry.Thus it is natural to consider forcing that varies with latitude.The barotropic equation is sometimes used to describe the vertically-averaged atmospheric dynamics. The stochastic forcesmodel the driving influence of the baroclinic instability on the barotropic flow.Baroclinic instabilities are typically strongest at mid-latitude. §.§ Time scale separation between large scale and small scale dynamics §.§.§ Energy balance and non–dimensional equations The inertial barotropic model (eq. (<ref>) with κ=ν_n=σ=0) conserves the energy ℰ[ω]=-1/2∫ωψr (we denote by r=a^2cosϕ ϕλ), the moments of potential vorticity 𝒞_m[ω]=∫(ω+f)^mr with the Coriolis parameter f(ϕ)=2Ωsinϕ, and the angular momentum L[ω]=∫ωcosϕr. The average energy balance for the dissipated and stochastically forced barotropic equation is obtained applying the Ito formula <cit.> to (<ref>). It reads dE/dt=-2κ E-2ν_nZ_n+σ,where E=𝔼[ℰ[ω]] is the total average energy and Z_n=𝔼[-1/2∫ψ(-Δ)^nωr]. The term -2ν_nZ_n in (<ref>) represents the dissipation of energy at the small scales of the flow. In the regime we are interested in, most of the energy is concentrated in the large-scale zonal jet, so the main mechanism of energy dissipation is the linear friction (first term in the right-hand side of (<ref>)). In this turbulent regime, energy dissipated by hyper-viscosity can be neglected. Then, in a statistically stationary state, E_stat≃σ/2κ, expressing the balance between stochastic forces and linear friction in (<ref>).The estimated total energy yields a typical jet velocity of U∼√(σ/2κ). The order of magnitude of the time scale of advection and stirring of turbulent eddies by this jet is τ_eddy∼a/U. We perform a non-dimensionalization of the stochastic barotropic equation (<ref>) using τ_eddy as unit time and a as unit length. The non-dimensionalization may be carried out bysetting a=1 and using the non-dimensionalized variables t'=t/τ_eddy, ω'=ωτ_eddy, ψ'=ψτ_eddy, Ω'=Ωτ_eddy,α=κτ_eddy=√(2κ^3/σ),ν_n'=ν_nτ_eddy, σ'=στ_eddy^3=2α, and a rescaled force η'=η√(τ_eddy) such that 𝔼[η'(λ_1,ϕ_1,t'_1)η'(λ_2,ϕ_2,t'_2)]=C(λ_1-λ_2,ϕ_1,ϕ_2)δ(t'_1-t'_2). In these new units, and dropping the primes for simplicity, the stochastic barotropic equation (<ref>) reads∂ω/∂ t+J(ψ,ω)+2Ω∂ψ/∂λ=-αω-ν_n(-Δ)^nω+√(2α)η.In (<ref>), α is an inverse Reynolds' number based on the linear friction and ν_n is an inverse Reynolds' number based on hyper-viscosity. The turbulent regime mentioned previously corresponds to ν_n≪α≪1. In such regime and in the units of (<ref>), the total average energy in a statistically stationary state is E_stat=1.We are interested in the dynamics of zonal jets in the regime of small forces and dissipation, defined as α≪1.In the next section we show that the dynamics corresponds to a regime in which the zonal jet evolves much more slowly than the surrounding turbulent eddies.§.§.§ Decomposition into zonal and non–zonal components In order to decompose (<ref>) into a zonally averaged flow and perturbations around it, we define the zonal projection of a field ⟨ψ⟩ (ϕ)≡ψ_0(ϕ)=1/2π∫_0^2πψ(λ,ϕ) λ.The zonal jet velocity profile isdefined by U(ϕ)≡⟨ u⟩ (ϕ). In most situations of interest, the stochastic force in (<ref>) does not act direcly on the zonal flow: ⟨η⟩ =0. Then the perturbations of the zonal jet is proportional to the amplitude of the stochastic force in (<ref>). We thus decompose the velocity field as v=U e_x+√(α)δ v and the relative vorticity field as ω=ω_z+√(α)δω with ω_z≡⟨ω⟩, where α is the non-dimensional parameter defined in (<ref>).We call the perturbation velocity δ v and vorticityδω the eddy velocity and eddy vorticity, respectively. With the decomposition of the vorticity field, the barotropic equation (<ref>) reads{ ∂ω_z/∂ t=α R-αω_z-ν_n(-Δ)^nω_z∂δω/∂ t=-L_U[δω]-√(α)NL[δω]+√(2)η, .with R(ϕ)≡-⟨ J(δψ,δω)⟩the zonally averaged advection term, where the linear operator L_U readsL_U[δω]=1/cosϕ(U(ϕ)∂_λδω+γ(ϕ)∂_λδψ)+αδω+ν_n(-Δ)^nδω,with γ(ϕ)=∂_ϕω_z(ϕ)+2Ωcosϕ, and whereNL[δω]=J(δψ,δω)-⟨ J(δψ,δω)⟩is the non-linear eddy-eddy interaction term. Using ω_z(ϕ)=-1/cosϕ∂_ϕ(U(ϕ)cosϕ) and the first equation of (<ref>), we get the evolution equation for the zonal flow velocity U(ϕ)∂ U/∂ t=α f-α U-ν_n(-Δ)^nU ,where f(ϕ) is such that R(ϕ)=-1/cosϕ∂_ϕ(f(ϕ)cosϕ). f is minus the divergence of the Reynolds' stress.§.§.§ Quasi-linear and linear dynamics In this section we discuss the quasilinear approximation to the barotropic equation and the associated linear dynamics.In the limit of small forces and dissipation α≪1, the perturbation flow is expected to be of small amplitude. Then the non-linear term NL[δω] in (<ref>) is negligible compared to the linear term L_U[δω]. Neglecting these non-linear eddy-eddy interaction terms, we obtain the so-called quasi-linear approximation of the barotropic equation <cit.>,{ ∂ω_z/∂ t=α R-αω_z-ν_n(-Δ)^nω_z∂δω/∂ t=-L_U[δω]+√(2)η. .The approximation leading to the quasi-linear dynamics (<ref>) amounts at suppressing some of the triad interactions. Nonetheless, the inertial quasi-linear dynamics has the same quadratic invariants as the initial barotropic equations. The average energy balance for the quasi-linear barotropic dynamics (<ref>) is thus the same as the one for the full barotropic dynamics (<ref>).For many flows of interest, for example Jovian jets, the turbulent eddies δω evolve much faster than the zonal jet velocity profile U <cit.>. In (<ref>) and (<ref>), the natural time scale of evolution of the zonal jet is of order 1/α, while the typical time scale of evolution of the perturbation vorticity δω is of order 1. In the regime α≪1, we thus expect to observe a separation of time scales between the evolution of ω_z and δω, consistent with the definition of α as the ratio of the inertial time scale τ_eddy and of the dissipative time scale 1/κ, see (<ref>). In the regime α≪1, it is natural to consider the linear dynamics of δω with U held fixed,∂δω/∂ t=-L_U[δω]+√(2)η .The relevance of (<ref>) as an effective description of turbulent eddy dynamics is further discussed later. In particular, we show in section <ref> that the correlation time of Reynolds' stresses resulting from the linear dynamics (<ref>) —the most relevant time scale related to the dynamics of eddies and their action on the evolution of the zonal jet— is of the order or smaller than τ_eddy, holding even as α decreases. It means that the time scale separation hypothesis that leads us to consider the linear dynamics (<ref>) is self-consistent in the limit of weak forces and dissipation α≪1.§.§.§ Reynolds averaging for the vorticity equationIn the introduction we discussed Reynolds averaging and Reynolds stresses for the simplest possible case: a two dimensional flow that does not break the symmetry along the direction 𝐞_x. We now adapt the discussion to two dimensional flows on a sphere.As it is much more convenient to work directly with the vorticity equation, we discuss Reynolds averaging for the vorticity equation only. Our aim is to write the counterpart of Eq. (<ref>) and (<ref>), for the vorticity equation. In the cases when there is a time scale separation between the evolution of the slow zonal and the fast non zonal part of the flow, averaging either Eq. (<ref>) or Eq. (<ref>) leads to an effective equation for the low frequency evolution of the zonal vorticity∂ω_z/∂ t=α𝔼(R)-αω_z-ν_n(-Δ)^nω_z + ξ_ω_z,where 𝔼(R) is the average of the vorticity flux R (<ref>), and the white in time Gaussian noise ξ_ω describes the typical fluctuations. We consider time averages of the vorticity fluxr=1/T∫dt R(u).The average of r is the term 𝔼(R) appearing in the Reynolds averaged equation (<ref>). We call this term the vorticity Reynolds stress; however it does not have the same physical dimension as the usual stress. When the time average is over a time window of duration T which is assumed to be short compared to the time scale for the evolution of U, but large compared with the evolution of the turbulent fluctuations: τ_e≪ T≪τ_U,we call the fluctuations of (<ref>) the vorticity Reynolds stress fluctuations (the fluctuation of the time averaged vorticity fluxes, over finite but long times T).In the asymptotic regime τ_e≪ T, the probability distribution function of r takes the simple large deviation form P(r,T)T→∞≍exp(-TI[r])). The variance of ξ_ω is given by a Kubo formula, and is simply related to the second variations of I. We note that there exists a simple relation between the Reynolds stress large deviations rate function I_v, that describes the averages of the actual momentum fluxes that appear in the velocity equation, and the vorticity Reynolds stress large deviation rate function I. In the following we study the vorticity Reynolds stress only. For simplicity, as there is no ambiguity, we call these quantities Reynolds stresses and Reynolds stress large deviation rate functions, omitting the word vorticity.§.§ Numerical implementation Direct numerical simulations (DNS) of the barotropic equation (<ref>), the quasi-linear barotropic equation (<ref>) and the linear equation (<ref>) are performed using a purely spectral code with a fourth-order-accurate Runge-Kutta algorithm and an adaptive time step[A program that implements spectral DNS for the non-linear and quasi-linear equations, solves the non-linear Riccati equation, and includes graphical tools to visualize statistics, is freely available.The application “GCM” is available for OS X 10.9 and higher on the Apple Mac App Store at URL http://appstore.com/mac/gcm]. The spectral cutoffs defined by ℓ≤ L, |m|≤min{ℓ,M} in the spherical harmonics decomposition of the fields are taken to be L=80 and M=20. In all the simulations, the rotation rate of the sphere is Ω=3.7 in the units defined previously.The stochastic noise is implemented using the method proposed in Ref. lilly1969numerical, with a non-zero renewal time scale τ_r larger than the time step of integration. For τ_r much smaller than the typical eddy turnover time scale, the noise can be considered as white in time.Whenever one considers the linear dynamics (<ref>), modes with different values of m decouple, thanks to the zonal symmetry. Then the statistics of the contribution of the Reynolds stress coming from different values of m are independent. The statistics for the total Reynolds stress can be computed from the statistics of the contribution of each independent value of m. It isnatural and simpler to study the contribution from each different value of m independently. For this reason we consider in this study a force that acts on one mode only. However, as explained in the previous section the validity of the quasilinear approximation is favored by the use of a broad band spectrum forcing, or a forcing acting on a large number of small scale modes, or both.Forcing only one mode is the most unfavorable case from the point of view of the accuracy of the quasilinear approximation.Larger time scale separation may be required in this case to ensure the accuracy of the quasilinear approximation.However whenever the quasilinear approximation is accurate, the statistics of the Reynolds stress arising from the forced mode are accurately described by the methods reported here.The forcing only acts on the mode |m|=10, ℓ=10, which is concentrated around the equator (see figure <ref>).With such a forcing spectrum and setting α=0.073, the integration of the quasi-linear barotropic equation (<ref>) leads to a stationary state characterized by a strong zonal jet with velocity U(ϕ), represented in Figure <ref>.We spectrally truncate thejet to its first 25 spherical harmonics to fix the mean flow in the simulation of the linear barotropic equation (<ref>). We use hyper-viscosity of order 4 with coefficient ν_4 such that the damping rate of the smallest mode is 4. To assess that hyper-viscosity is negligible in the large scale statistics, simulations of the linear equation with ν_4=4 and ν_4=2 are compared in sections <ref>, <ref> and <ref>. § EQUAL-TIME STATISTICS OF VORTICITY FLUXESThe aim of this section is to illustrate that fluctuations of equal-time vorticity flux R (<ref>) may be strongly non Gaussian. We prove that vorticity flux fluctuations have exponential tails with a distribution close to that of Gaussian product statistics <cit.>. While equal-time fluctuations of the vorticity flux are important for high frequency jet variability, Reynolds stresses (time average of the vorticity fluxes) are more important for the long term evolution of the jet. Beginning in section IV, we study Reynolds stresses, and their large deviations.The evolution of the mean flow ω_z(ϕ,t) is given by the advection term R(ϕ,t)=-⟨ J(δψ,δω)⟩, through (<ref>) or (<ref>). In most previous statistical approaches to zonal jet dynamics, only the averaged advection term, the Reynolds stress, was considered. This is for instance the case in S3T <cit.> and CE2 <cit.> approaches.Such restriction gives a good approximation of the relaxation of zonal jets towards the attractors of the dynamics, that is expected to be quantitatively accurate in the inertial limit α→0 <cit.>. However, replacing the advection term R by its average does not describe fluctuations of the vorticity fluxes, that may lead to fluctuations of zonal jets. Understanding the statistics of vorticity fluxes beyond their average value is thus a very interesting perspective. In this section, we study the whole distribution function of vorticity fluxes, as computed from direct numerical simulations.The zonally averaged advection term is a function of latitude ϕ and can be decomposed with spherical harmonicsaccording to (<ref>). We denote by R_ℓ(t)≡ R_0,ℓ(t) the ℓ-th component in the spherical harmonics decomposition of R(ϕ,t). All R_l for odd values of l larger than one have non-zero amplitudes (the amplitude of the l=1 mode is zero because total angular momentum about the polar axis remains zero). In the following, for simplicity,we focus our analysis on R_3 only, that has the largest contribution. The probability density functions of R_3, computed either from direct numerical simulations of the barotropic equation (<ref>), or the quasi-linear barotropic equation (<ref>) or the linear equation (<ref>), with the forcing spectrum specified in section <ref> and with α=0.073, are shown in Figure <ref>. Figure <ref> shows that the probability distribution of R_3 is not affected by the choice of small scale dissipation.In the linear dynamics (<ref>), the eddy vorticity evolves according to the linearized barotropic equation close to the fixed base flow U(ϕ) shown in Figure <ref>. In the quasi-linear dynamics (<ref>), the zonal mean flow has the same average velocity profile U(ϕ), but this zonal flow is allowed to fluctuate. This difference in the dynamics of the zonal flow between linear and quasi-linear equations explains the slight difference observed in the corresponding advection term histograms (respectively blue curve and orange curve in Figure <ref>), namely, the probability density function is more spread (the vorticity fluxes fluctuate more) in the quasi-linear dynamics than in the linear dynamics.In contrast, the probability density function of R_3 computed from the non-linear integration (yellow curve in Figure <ref>) is very different from the other ones for two reasons: the average zonal flow is different from the fixed zonal flow used in the linear dynamics, and the dynamics of δω is also different from the quasi-linear dynamics because of the non-linear eddy-eddy interaction terms in (<ref>) (this is expected, as forcing a single mode is the most unfavorable case from the point of view of the validity of the quasilinear approximation, as explained in section <ref>). In all three cases, the probability distribution functions in Figure <ref> show large fluctuations and heavy tails. For instance it is clear that typical fluctuations of the vorticity flux have much larger amplitude than the value of their average (the variance is much larger than the average). While essential for understanding the high frequency and small variability of the jets, on the slow time scale, the jet evolution is described by time averaged vorticity fluxes (the Reynolds stress).In all of the simulations, the distribution of the vorticity flux shows exponential tails. This can be easily understood for the case of the linear equation (<ref>). Indeed, in this case the statistics of the eddy vorticity are exactly Gaussian (δω is an Ornstein-Uhlenbeck process <cit.>). Then, the statistics of R(ϕ) can be calculated explicitly, as we explain now.Using (<ref>) we can write the vorticity flux asR(ϕ)=-1/cosϕ∑_mim(ψ_m·∂_ϕω_-m+∂_ϕψ_m·ω_-m),where ω_m(ϕ) is the m-th Fourier coefficient of δω, and ψ_m(ϕ) is the associated stream function. The Ornstein-Uhlenbeck process ω_m(ϕ) is a Gaussian random variable at each latitude ϕ. The sum of Gaussian random variables is a Gaussian random variable, so ψ_m(ϕ), ∂_ϕψ_m(ϕ) and ∂_ϕω_m(ϕ) are also Gaussian random variables at each latitude ϕ. All these Gaussian random variables have zero mean, and in general they are correlated in a non-trivial way.The vorticity flux (<ref>) is thus of the form R=ξ_1ξ_2+…+ξ_M-1ξ_M where ξ_1,…,ξ_M are M real-valued[We can restrict ourselves to real ξ_m decomposing ω_m and ψ_m into real and imaginary parts.] correlated Gaussian variables with zero mean. We denote by ξ the column vector with components ξ_1,…,ξ_M. By definition, the probability distribution function of ξ is P_ξ(ξ)=1/Zexp(-1/2ξ^TG^-1ξ),where ξ^T denotes the transpose vector of ξ, G is the covariance matrix of ξ, and Z is a normalisation constant. The probability density function of R, denoted P_R, is given by P_R(R) =∫ξ P_ξ(ξ)δ(R-ξ_1ξ_2-…-ξ_M-1ξ_M) =∫ξ_2…ξ_m 1/|ξ_2| P_ξ(R-ξ_3ξ_4-…-ξ_M-1ξ_M/ξ_2,ξ_2,…ξ_M).Using the change of variable ζ_m=ξ_m/√(|R|) for m=2,…,M, the first argument of P_ξ becomes √(|R|)R/|R|-ζ_3ζ_4-…-ζ_M-1ζ_M/ζ_2, so we obtain: P_R(R)=1/Z∫ζ_2…ζ_M |R|^M-2/2/|ζ_2| exp(-|R|Q_±(ζ_2,…,ζ_M)),where Q_± is a function of (ζ_2,…,ζ_M), that depends only on the sign of R, according to R=±|R|. The tails of the distribution P_R correspond to the limits R→±∞. In both limits, |R|→∞ so we can perform a saddle-point approximation in the above integral, and get ln(P_R(R))R→±∞∼-|R|μ_±,where the rates of decay are defined by μ_±=min_ζ_2,…,ζ_M{ Q_±(ζ_2,…,ζ_M)} .The exponential tails of the distribution P_R are direct consequences of the fact that the eddy vorticity δω evolving according to the linear equation (<ref>) is a Gaussian process and of the fact that R is quadratic in δω. This simple argument explains the exponential tails observed in probability density functions of the zonally averaged advection term in simulations of the linear dynamics (<ref>) (blue curve in Figure <ref>), where the vorticity field is exactly an Ornstein-Uhlenbeck process.In the quasi-linear and non-linear dynamics, the zonal flow and eddies evolve at the same time scale. As a consequence, the dynamics of the eddy vorticity is not linear, and its statistics are not Gaussian. However, we observe that the probability density functions of eddy vorticity are nearly Gaussian (skewness -0.0147 and kurtosis 3.8079 in the quasi-linear case, skewness -0.0037 and kurtosis 3.3964 in the non-linear case, compared to skewness 0.0172 and kurtosis 3.0028 in the linear case). The previous argument can thus also be applied empirically to explain the exponential tails observed in the curves corresponding to quasi-linear and non-linear simulations in Figure <ref>. The same analysis has been performed on direct numerical simulations of the deterministic 2-layer quasi-geostrophic baroclinic model <cit.>, see Figure <ref>. In this case, the eddy vorticity statistics are highly non-Gaussian, while statistics of the vorticity flux have exponential tails similar to those found in the one-layer case. The observation indicates that the previous explicit calculation might not be the most general explanation of the exponential distribution of vorticity fluxes. § AVERAGING AND LARGE DEVIATIONS IN SYSTEMS WITH TIME SCALE SEPARATION As explained in section <ref>, we are interested in the regime where zonal jets evolve much slower than the surrounding turbulent eddies. In this section, we present some theoretical tools (stochastic averaging, large deviation principle) that can be applied to study the effective dynamics and statistics of slow dynamical variables coupled to fast stochastic processes. Most of these tools are classical ones <cit.>, except for the explicit results presented in section <ref> <cit.>. Application of these general tools to the quasi-linear barotropic model is considered in sections <ref> and <ref>.Consider the stochastic dynamical system{ dx/dt=α f(x,y)dy/dt=b(x,y)+η. where 0<α≪1, and where η is a Gaussian random column vector with zero mean and correlations 𝔼[η(t_1)η^T(t_2)]=Cδ(t_1-t_2) with the correlation matrix C. In the case we are interested in, the random vector y is actually the eddy vorticity field, and x is the zonal jet vorticity or velocity. For simplicity we use vector notation x=(x_ℓ)_1≤ℓ≤ Lin this section, the formal generalization to the field case is straightforward, see sections <ref> and <ref>.In (<ref>), the variable x typically evolves on a time scale of order 1/α, while y evolves on a time scale of order 1. When there is a time scale separation between zonal jets and eddies, defined by α≪1, the quasi-linear barotropic equation (<ref>) is a particular case of the system (<ref>). Note however that in that case, dissipation terms of order α are present in b(x,y). The general results presented in this section usually do not take into account such terms <cit.>. As a consequence, in sections <ref> and <ref>we make sure that our results do not depend on the dissipative terms in the limit α→0.The goal of stochastic averaging is to give an effective description of the dynamics of x over time scales of order 1/α, where the effect of the fast process y is averaged out. The effective dynamics describes the attractors of x, the relaxation dynamics towards these attractors and the small fluctuations around these attractors, in the regime α≪1.For quasi-geostrophic zonal jets dynamics, stochastic averaging leads to a kinetic description of zonal jets <cit.>, related to statistical closures of the dynamics (S3T <cit.> and CE2 <cit.>).The effective dynamics obtained through stochastic averaging or statistical closures is not able to describe arbitrarily large fluctuations of the slow process x. Such rare events are of major importance in the long-term dynamics of x. For instance in the case where the system (<ref>) has several attractors, transitions between the attractors are governed by large fluctuations of the system. The description of such transitions (transition probability, typical transition path) cannot be done through a stochastic averaging procedure.Large deviation theory is a natural framework to describe large fluctuations of x in the regime α→0. The large deviation principle <cit.> gives the asymptotic form of the probability density of paths { x(t)} _0≤ t≤ T when α≪1, with the effect of the fast process y averaged out.Information about the typical effective dynamics of x as obtained through stochastic averaging is captured,but the principle allows us to go further to describe arbitrarily rare events. In cases of multistability of x, the Large Deviation Principle yields the asymptotic expression of the transition probability from one attractor to another, the average relative residence time in each attractor, and the typical transition path { x(t)} _0≤ t≤ T that links two attractors in a given time T≳1/α, among other relevant statistical quantities. Implementing the large deviation principle in practice for systems like (<ref>) and for the quasilinear dynamics is one of the goals of this work.In the effective descriptions of x provided by stochastic averaging and the Large Deviation Principle, the dynamics of y is approximated by its stationary dynamics with x held fixed, the so-called virtual fast process. The mathematics is described in section <ref>. The effective dynamics of x over time scales t≫1 provided by stochastic averaging ispresented in section <ref>. The Large Deviation Principle for (<ref>) is stated in section <ref>, and in section <ref> we give a method to estimate the quantities involved in the Large Deviation Principle from simulations of the virtual fast process.§.§ The virtual fast process In slow-fast systems like (<ref>), the time scale separation implies that at leading order, the statistics of y are very close to the stationary statistics of the virtual fast process ỹ(u) dỹ/du=b(x,ỹ(u))+η(u),where x is held fixed <cit.>. The time scale separation hypothesis is relevant only when the fast process described by (<ref>) is stable (for instance has an invariant measure and is ergodic). The stationary process (<ref>) depends parametrically on x, and the expectation over the invariant measure of (<ref>) is thus denoted 𝔼_x. The statistics of ỹchange when x evolves adiabatically on longer timescales.Forquasilinear barotropic dynamics (<ref>), the virtual fast process is the linearized barotropic equation close to the fixed stable zonal flow U (<ref>) (the necessity for U to be stable for the quasilinear hypothesis to be correct was emphasized in reference <cit.>.) The process (<ref>) is relevant only if a time scale separation effectively exists between the evolutions of x and y. In practice, the time scale separation hypothesis in (<ref>) can be considered to be self-consistent if the typical time scale of evolution of the virtual fast process (<ref>) is of order one, while the slow variable evolves on a time scale of order 1/α. From the point of view of the interaction with the dynamics of x, the most relevant time scales related to the evolution of ỹ(u) are the correlation times of processes f_ℓ(x,ỹ(u)) and f_ℓ'(x,ỹ(u)), defined as <cit.>τ_ℓ,ℓ'=lim_t→∞1/t∫_0^t∫_0^t𝔼_x[[f_ℓ(x,ỹ(u_1))f_ℓ'(x,ỹ(u_2))]]/2𝔼_x[[f_ℓ(x,ỹ)f_ℓ'(x,ỹ)]] u_1u_2where 𝔼_x[[X_1(u_1)X_2(u_2)]]≡𝔼_x[X_1(u_1)X_2(u_2)]-𝔼_x[X_1(u_1)]𝔼_x[X_2(u_2)] is the covariance of X_1 at time u_1 and X_2 at time u_2. If ℓ=ℓ', τ_ℓ,ℓ is called the auto-correlation time of the process f_ℓ(x,ỹ(u)). In all these expressions, x is fixed and 𝔼_x is the average over realizations of the fast process (<ref>) in its statistically stationary state. The correlation times {τ_ℓ,ℓ'} give an estimate of the time scales of evolution of the terms that force the slow process x in (<ref>). In the regime α≪1,we can consider a time Δ t much larger than the auto-correlation times τ_ℓ,ℓ' but much smaller than the typical time for the evolution of x itself: τ_ℓ,ℓ'≪Δ t≪1/α. Over such time scale, (<ref>) can be integrated to give x(t+Δ t)=x(t)+α∫_t^t+Δ tf(x(u),y(u))u≃ x(t)+α∫_t^t+Δ tf(x(t),ỹ(u))u,where in obtaining the last equality we have used the fact that over time Δ t the process x has almost not evolved. The relation (<ref>) is used in the following to derive equations for the average behaviour, typical fluctuations and large fluctuations of x, in the time scale separation limit α≪1.§.§ Average evolution and energy balance for the slow process We now describe the typical dynamics of x over time scales Δ t such that τ_ℓ,ℓ'≪Δ t≪1/α, recovering classical results from stochastic averaging <cit.>. Because the time Δ t in (<ref>) is much larger than the typical correlation time of the components of f(x,ỹ(u)), by the Law of Large Numbers we can replace the time average by a statistical average: 1/Δ t∫_t^t+Δ tf(x,ỹ(u))u≃ F(x) where F(x)≡𝔼_x[f(x,ỹ(u))] is the average force acting on x, computed in the statistically stationary state of the virtual fast process (<ref>). Then, the average evolution of x at leading order in αΔ t≪1 is Δ x/Δ t≡x(t+Δ t)-x(t)/Δ t≃α F(x(t)).In the case of zonal jet dynamics in barotropic models, x is the zonally averaged vorticity (or velocity) and F(x) is the average advection term R. The effective dynamics (<ref>) isvery close to S3T-CE2 types of closures <cit.> or to kinetic theory <cit.>. This point is further discussed in section <ref>.The effective dynamics (<ref>) is not enough to describe the effective energy balance related to the slow process x. Indeed, replacing the time averaged force in (<ref>) by its statistical average amounts to neglecting fluctuations in the process f(x,ỹ(u)). The fluctuations are however relevant in the evolution of quadratic forms of x. In particular, if we define the energy of the slow degrees of freedom as E=1/2x· x=∑_ℓE_ℓ with E_ℓ=1/2x_ℓ^2, an equation for E_ℓ can be derived using (<ref>),E_ℓ(t+Δ t)≃E_ℓ(t)+α x_ℓ(t)∫_t^t+Δ tf_ℓ(x(t),ỹ(u))u+α^2/2∫_t^t+Δ t∫_t^t+Δ tf_ℓ(x(t),ỹ(u_1))f_ℓ(x(t),ỹ(u_2))u_1u_2.Define Z_ℓ,ℓ'(x)≡lim_Δ t→∞1/Δ t∫_0^Δ t∫_0^Δ t𝔼_x[[f_ℓ(x,ỹ(u_1))f_ℓ'(x,ỹ(u_2))]]u_1u_2 ,then using again that Δ t is much larger than the correlation time of f(x,ỹ(u)) we get Δ E_ℓ/Δ t≃α x_ℓF_ℓ(x)+α^2/2Z_ℓ,ℓ(x).This relation is the energy balance for the slow evolution of x: p_mean,ℓ=α x_ℓF_ℓ(x) is the average energy injection rate by the mean force F(x), and p_fluct,ℓ=α^2/2Z_ℓ,ℓ(x) is the average energy injection rate by the typical fluctuations of the force f, as quantified by Z(x). Neglecting the term p_fluct,ℓ in (<ref>), we recover the energy balance we would have obtained by computing the evolution of E_ℓ from (<ref>). This observation confirms the fact that fluctuations of f, which are not taken into account in (<ref>), are relevant in the effective dynamics of x.§.§ Large Deviation Principle for the slow process §.§.§ Large deviation rate function for the action of the fast variable on the slow variable Equations (<ref>) and (<ref>) give the evolution of x and x· x at leading order in α≪1. Such effective evolution equations can also be found in a more formal way using stochastic averaging <cit.>. The effective equations only describe the low-order statistics of the slow process:The average evolution and typical fluctuations (variance or energy). In contrast, the Large Deviation Principle gives access to the statistics of both typical and rare events, also in the limit α≪1.For system (<ref>), the Large Deviation Principle was first proved by Freidlin (see Ref. freidlin2012random and references therein). It states that the probability density of a path of the slow process x, denoted 𝒫[x], satisfies <cit.>ln𝒫[x]α→0∼-1/α∫ℒ(x(t),ẋ(t))twith ℒ(x,ẋ)≡min_θ{ẋ·θ-H(x,θ)} and where H(x,θ) is the scaled cumulant generating function H(x,θ)≡lim_Δ t→∞1/Δ tln𝔼_x[exp(θ·∫_0^Δ tf(x,ỹ(u))u)],where we recall that 𝔼_x is an average over realisations of the virtual fast process (<ref>) in its statistically stationary state.Quantities H and ℒ are classical definitions from Large Deviation Theory <cit.>. The knowledge of the function H(x,θ) is equivalent to the knowledge of ℒ(x,ẋ), which gives the probability of any path of the slow process x through (<ref>). Computing H(x,θ) is thus a very efficient way to study the effective statistics of x(t), even when extremely rare events that are not described in the effective equations (<ref>) and (<ref>) play an important role.Because the Large Deviation Principle (<ref>) describes both rare events and typical events, information about the effective dynamics (<ref>, <ref>) is encoded in the definition of the scaled cumulant generating function. Indeed, a Taylor expansion in powers of θ in (<ref>) givesH(x,θ)=∑_ℓθ_ℓF_ℓ(x)+1/2∑_ℓ,ℓ'θ_ℓθ_ℓ'Z_ℓ,ℓ'(x)+O(θ^3),with F(x)≡𝔼_x[f(x,ỹ(u))] and Z given by (<ref>). The terms appearing in the leading order evolution of x (<ref>) and of the energy (<ref>) are thus contained in the scaled cumulant generating function, through (<ref>).Higher-order terms in (<ref>) involve cubic and higher-order cumulants of large time averages of the process f(x,ỹ(u)). If this process is a Gaussian process,its statistics are only given by its first and second order cumulants <cit.>. As a consequence, for such process H(x,θ) is quadratic in θ and (<ref>) is exact (corrections of order θ^3 are exactly zero).In practice, the scaled cumulant generating function (<ref>) involves the virtual fast process (<ref>). This stochastic process depends only parametrically on x, which means that we do not have to study the coupled system (<ref>) in order to compute H(x,θ). This result is consistent with the time scale separation property of (<ref>). In quasi-linear systems such as the quasi-linear barotropic dynamics, the virtual fast process is an Ornstein-Uhlenbeck process, which is particularly simple to study. This specific class of systems is considered next in section <ref>.§.§.§ Quasi-linear systems with action of the fast process on the slow one through a quadratic force: the matrix Riccati equation We are particularly interested in the more specific class of systems defined by{ dx/dt=α y^Tℳy+α g(x)dy/dt=-L_x[y]+η. where ℳ is a symmetric matrix, and L_x is a linear operator acting on y that depends parametrically on x. The system (<ref>) is a particular case of (<ref>) with f(x,y)=y^Tℳy+g(x) and b(x,y)=-L_x[y].When x is the zonal flow vorticity profile and y is the eddy vorticity, the quasi-linear barotropic dynamics (<ref>) is an example of such a system, where the quadratic form y^Tℳy defines the zonally averaged advection term R and g(x) contains the dissipative terms acting on the large-scale zonal flow x, and where L_x is the linearized barotropic operator close to the zonal flow x (see also section <ref>).We now describe the effective dynamics and large deviations of x in the system (<ref>), in the limit α→0. In this limit, the statistics of y are very close to the statistics of the virtual fast process (<ref>), which in this case readsdỹ/dt=-L_x[ỹ]+η,where x is frozen. Equation (<ref>) describes an Ornstein-Uhlenbeck process, whose stationary distribution is Gaussian <cit.>. Then, the stationary statistics of (<ref>) are fully determined by the mean and covariance of ỹ. The mean is zero, and the covariance G_ij=𝔼[ỹ_iỹ_j] is given by the Lyapunov equationd G/d t+L_xG+GL_x^T=C.The Lyapunov equation (<ref>) converges to a unique stationary solution whenever (<ref>) has an invariant measure. We recall that such an invariant measure is required for the time scale separation hypothesis to be relevant.The effective dynamics of x over times Δ t≪1/α is given by (<ref>). In the case of (<ref>), it readsΔ x/Δ t≃α[ℳ· G_∞(x)+g(x)]with ℳ· G_∞(x)=∑_i,jℳ_ij(G_∞)_ij(x) where G_∞ is the stationary solution of the Lyapunov equation (<ref>).Simulating the effective slow dynamics (<ref>) can be done by integrating the Lyapunov equation (<ref>), using standard methods[The application “GCM” integrates the equation <ref> and the effective dynamics <ref>.].It provides an effective description of the attractors of x, and of the relaxation dynamics towards the attractors. Examples of such numerical simulations of (<ref>) in the case of zonal jet dynamics in the barotropic model can be found for instance in Refs. bakas2015s3t,Srinivasan-Young-2011-JAS,tobias2011astrophysical,tobias2013direct,marston2014direct.In order to describe large fluctuations of x in (<ref>), we need to use the Large Deviation Principle (<ref>). In practice, we compute the scaled cumulant generating function (<ref>). As proven in Ref. BouchetTangarifeVandenEijnden2015, for the system (<ref>), the scaled cumulant generating function is given byH(x,θ)=θ· g(x)+(CN_∞(x,θ))where C is the covariance matrix of the noise η in (<ref>) and N_∞(x,θ) is a symmetric matrix, stationary solution of d N/d t+NL_x+L_x^TN=2NCN+θℳ.Equation (<ref>) is a particular case of a matrix Ricatti equation, and in the following we refer to (<ref>) as the Ricatti equation.θ is the parameter of the cumulant generating function (<ref>) that defines H. Whenever θ is in the parameter range for which the limit in (<ref>) exists, called the admissible θ range, Eq. (<ref>) has a stationary solution. For the case in this section, with a linear dynamics with a quadratic observable, the admissible θ range is easily studied through the analysis of the positivity of a quadratic form. One can conclude that the admissible θ range is an interval containing 0. All the information regarding the large deviation rate function is contained in the values of H for θ in this range. The Ricatti equation (<ref>) is similar to the Lyapunov equation (<ref>), and it can be solved using similar methods[Note that the ordering of products with L_x and L_x^T differs between (<ref>) and (<ref>).]. Moreover, the numerical implementation of (<ref>, <ref>) can be easily checked using the relation with the Lyapunov equation (<ref>). Namely, (<ref>) implies that.dH/dθ|_θ=0=ℳ· G_∞(x)+g(x).The first term in the right-hand side is computed from the Lyapunov equation (<ref>), while the left-hand side is computed from the Ricatti equation (<ref>) together with (<ref>).In section <ref>, we present a numerical resolution of (<ref>) for the case of the quasi-linear barotropic equation on the sphere, andcompute directly the scaled cumulant generating function using (<ref>). We show that (<ref>) can be very easily solved for a given value of θ. This means that the result (<ref>) permits the study of arbitrarily rare events in zonal jet dynamics extremely easily, through the Large Deviation Principle (<ref>). Such result is in clear contrast with approaches through direct numerical simulations, which require that the total time of integration increases as the probability of the event of interest decreases. This limitation of direct numerical simulations in the study of rare events statistics is made more precise in next section.§.§ Estimation of the large deviation function from time series analysis In this section we present a way to compute the scaled cumulant generating function (<ref>) from a time series of the virtual fast process (<ref>), for instance one obtained from a direct numerical simulation. Many of the technical aspects of this empirical approach follow Ref. rohwer2014convergence.Consider a time series {ỹ(u)} _0≤ u≤ T of the virtual fast process (<ref>), with a given total time window u∈[0,T]. Because the quantities of interest like H(x,θ) involve expectations in the stationary state of the virtual fast process, we assume that the time series {ỹ(u)} _0≤ u≤ T corresponds to this stationary state. We use the continuous time series notation for simplicity. The generalization of the following formulas to the case of discrete time series is straightforward. For simplicity, we also denote by R(u)≡ f(ỹ(u)), the quantity for which the scale cumulant generating function H(θ)=lim_t→∞1/tlog𝔼exp(θ∫_0^tR(u) u) should be estimated.The basic method to estimate the scaled cumulant generating function (<ref>) is to divide the full time series {ỹ(u)} _0≤ u≤ T into blocks of length Δ t, to compute the integrals ∫_t_0^t_0+Δ tR(u) u over those blocks, andto average the quantity exp(θ·∫_t_0^t_0+Δ tR(u) u). For a small block length Δ t, the large-time regime defined by the limit Δ t→∞ in the theoretical expression of H (<ref>) is not attained. On the other hand, too large values of Δ t —typically of the order of the total time T— lead to a low number of blocks, and thus to a very poor estimation of the empirical mean. Estimating H thus requires finding an intermediate regime for Δ t. More precisely, we expect this regime to be attained for Δ t equal to a few times the correlation time of the process R(u), defined by <cit.>τ≡lim_Δ t→∞∫_0^Δ t∫_0^Δ t𝔼_z[[ R(u_1) R(u_2) ] ]d u_1d u_2/2Δ t 𝔼_z[[ R^2 ] ]= ∫_0^∞𝔼_z[[ R(u) R(0) ] ]d u/𝔼_z[[ R^2 ] ] ,where 𝔼_z[[R(u_1) R(u_2)]] is the covariance of R at time u_1 and at time u_2. The second equality is easily obtained assuming that the process R(u) is stationary. Because of the infinite-time limit in (<ref>), the estimation of τ suffers from the same finite sampling problem as the estimation of H.Finding a block length Δ t such that the estimation of H and τ is accurate is thus a tricky problem. In the following, we propose a method to find the optimal Δ t and estimate the quantities we are interested in. The proposed method is close to the “data bunching” method used to estimate errors in Monte Carlo simulations <cit.>.§.§.§ Estimation of the correlation time We first consider the problem of the estimation of τ in a simple solvable case, so the numerical results can be compared directly to explicit formulas. Consider the stochastic process R=w^2 where w is the one-dimensional Ornstein-Uhlenbeck process dw/dt=-w+η,where η is a Gaussian white noise with correlation 𝔼(η(t)η(t'))=δ(t-t'). A direct calculation gives the correlation time of R, τ=1/2. Using (<ref>) and (<ref>), the scaled cumulant generating function can also be computed explicitly (see for instance Ref. BouchetTangarifeVandenEijnden2015). We obtainH(θ) = 1/2 - 1/2√(1-2θ),defined for θ≤ 1/2.For a time series { R(u)} _0≤ u≤ T, we denote by R̅_T=1/T∫_0^TR(u) u and by _T(R)=1/T∫_0^T(R(u)-R̅_T)^2u respectively the empirical mean and variance of R over the full time series. Weestimate the correlation time τ defined in (<ref>) using an average over blocks of length Δ t, τ_Δ t=1/2Δ t _T(R)𝔼_T/Δ t[(∫_t_0^t_0+Δ t(R(u)-R̅_T) u)^2],where 𝔼_T/Δ t[h_t_0] is the empirical average over realisations of the quantity h_t_0 inside the brackets[Explicitely, 𝔼_T/Δ t[(∫_t_0^t_0+Δ t(R(s)-R̅_T) s)^2]=Δ t/2T∑_k=0^2T/Δ t-2(∫_kΔ t/2^kΔ t/2+Δ t(R(u)-R̅_T) u)^2 ,assuming for simplicity that T/Δ t is an integer. Generalisations to any T,Δ t is straightforward, replacing 2T/Δ t by its floor value. The 50% overlap is suggested by Welch's estimator of the power spectrum of a random process <cit.>.].To find the optimal value of Δ t, we plot τ_Δ t as a function of Δ t in figure <ref>. For small values of Δ t, the large-time limit in (<ref>) is not achieved, which explains the low values of τ_Δ t. For too large values of Δ t, the empirical average 𝔼_T/Δ t in (<ref>) is not accurate due to the small value of T/Δ t (small number of blocks), which explains the increasing fluctuations in τ_Δ t as Δ t increases. The optimal value of Δ t —denoted Δ t^⋆ in the following— is between the values giving these artificial behaviours. It should satisfy T≫Δ t^⋆≫τ_Δ t^⋆. Here, one can read Δ t^⋆≃10 and τ_Δ t^⋆≃0.5, so this optimal Δ t^⋆ satifies the aforementioned condition. The estimated value τ_Δ t^⋆ is in agreement with the theoretical value τ=1/2.The error bars for τ_Δ t are given by Δτ_Δ t=√((τ_Δ t)/N_terms), where (τ_Δ t) is the empirical variance associated with the average 𝔼_T/Δ t defined in (<ref>), and N_terms is the number of terms in this sum (roughly N_terms≃2T/Δ t).§.§.§ Estimation of the scaled cumulant generating function The self-consistent estimation of the correlation time τ presented in the previous section gives the optimal value Δ t^⋆ of the block length. Then, the scaled cumulant generating function is computed for a given value of θ as H_T(θ)≡1/Δ t^⋆ln𝔼_T/Δ t^⋆[exp(θ∫_t_0^t_0+Δ t^⋆R(u) u)],where 𝔼_T/Δ t is the empirical average over the blocks, as defined in (<ref>). However, the knowledge of H(x,θ) for an arbitrarily large value of |θ| leads to the probability of an arbitrarily rare event for the slow process x through the Large Deviation Principle (<ref>). This is in contradiction with the fact that the available time series { R(u)} _0≤ u≤ T is finite. In other words, the range of values of θ for which the scaled cumulant generating function H_T(θ) can be computed with accuracy depends on T.Indeed, for large positive values of θ, the sum 𝔼_T/Δ t^⋆ in (<ref>) is dominated by the largest term exp(θI_max) where I_max=max_t_0{∫_t_0^t_0+Δ tR(u) u} is the largest value of ∫_t_0^t_0+Δ tR(u) u over the finite sample { R(u)} _0≤ u≤ T. Then H_T(θ)∼1/Δ t^⋆I_maxθ for θ≫1. This phenomenon is known as linearization <cit.>, and is clearly an artifact of the finite sample size. We denote by θ_max the value of θ such that linearization occurs for θ>θ_max . Typically, we expect θ_max to be a positive increasing function of T. The same way, H_T(θ)∼-1/Δ t^⋆I_minθ for θ<0 and |θ|≫1, with I_min=min_t_0{∫_t_0^t_0+Δ tR(u) u}. In a similar way, we define θ_min as the minimum value of θ for which linearization occurs. Typically, we expect θ_min to be a negative decreasing function of T.The convergence of estimators like (<ref>) is studied in Ref. rohwer2014convergence, in particular it is shown that error bars can be computed in the range [θ_min/2,θ_max/2] for a given time series { R(u)} _0≤ u≤ T. An example of a computation of H_T(θ) is shown in Figure <ref> for the one-dimensional Ornstein-Uhlenbeck process, and compared to the explicit solution. The full error bars in Figure <ref> are given by the error from the estimation of τ and the statistical error described in Ref. rohwer2014convergence. The method shows excellent agreement with theory, and exposes non-Gaussian behavior. In sections <ref> and <ref>, we apply the tools (estimation of the correlation time and of the scaled cumulant generating function) to study the statistics of Reynolds' stresses in zonal jet dynamics.§ ZONAL ENERGY BALANCE AND TIME SCALE SEPARATION IN THE INERTIAL LIMIT In this section we discuss the effective evolution and effective energy balance for zonal flows in the inertial regime ν_n≪α≪1, using the general results of section <ref> and numerical simulations. §.§ Effective dynamics and energy balance for the zonal flow Using (<ref>) and (<ref>), the effective evolution of the zonal jet velocity profile U(ϕ,t) in the regime ν_n≪α≪1 reads∂ U/∂ t≃α F[U] - α U - ν_n (-Δ)^n U,with F[U]≡𝔼_U[f] where f is minus the Reynolds' stress divergence and 𝔼_U is the average in the statistically stationary state of the linear barotropic dynamics (<ref>), with U held fixed.Equation (<ref>) describes the effective slow dynamics of zonal jets in the regime ν_n≪α≪1, it is the analogous of the kinetic equation proposed in Ref. Bouchet_Nardini_Tangarife_2013_Kinetic. In particular, the attractors of (<ref>) are the same as the attractors of a second order closure of the barotropic dynamics <cit.>. As explained in a general setting in section <ref>, equation(<ref>) only takes into account the average Reynolds' stresses (through the term F[U]).As a consequence it does not describe accurately the effective zonal energy balance. Quantifying the influence of fluctuations of Reynolds' stresses on the zonal energy balance is one of the goals of this study. We now derive the effective zonal energy balance, anddescribe the relative influence of average and fluctuations of Reynolds' stresses using numerical simulations. First note that the hyperviscous terms in (<ref>) essentially dissipate energy at the smallest scales of the flow. In the turbulent regime we are interested in, such small-scale dissipation is negligible in the global energy balance. For this reason, the viscous terms can be neglected in (<ref>) and in the zonal energy balance. Note however that some hyper-viscosity is still present in the numerical simulations of the linear barotropic equation (<ref>), in order to ensure numerical stability. For consistency, we make sure that the hyper-viscous terms do not influence the numerical results, see Figure <ref>. The kinetic energy contained in zonal degrees of freedom reads E_z=∫ϕ E(ϕ) with E(ϕ)=πcosϕ U^2(ϕ). Using (<ref>) we get the equation for the effective evolution of E(ϕ): 1/αdE/dt=p_mean(ϕ)-2E+α p_fluct(ϕ) .The left hand side is the instantaneous energy injection rates into the zonal mean flow. It is equal to the sum of the average Reynolds' stresses p_mean(ϕ)≡2πcosϕ F[U](ϕ)U(ϕ), -2E, and the fluctuations of Reynolds' stresses α p_fluct(ϕ)≡απcosϕ Z[U](ϕ), whereZ[U](ϕ)≡lim_Δ t→∞1/Δ t∫_0^Δ t∫_0^Δ t𝔼_U[[f(ϕ,u_1)f(ϕ,u_2)]]u_1u_2 . Integrating (<ref>) over latitudes, we obtain the total zonal energy balance1/αdE_z/dt=P_mean-2E_z+α P_fluct,with P_mean≡∫ϕ p_mean(ϕ) and α P_fluct≡∫ϕ α p_fluct(ϕ).All the terms appearing in (<ref>) and (<ref>) can be easily estimated using data from a direct numerical simulation of the linearized barotropic equation (<ref>). Indeed, F[U](ϕ) can be computed as the empirical average of f(ϕ) in the stationary state of (<ref>), and Z[U](ϕ) can be computed using the method described in section <ref> to estimate correlation times[The statistical error bars for p_fluct are computed from the error in the estimation of Z[U](ϕ), which is similar to the estimation of the correlation time τ described in section <ref>. The statistical error bars for p_mean are computed from the error in the estimation of the average F, given by (δ F)^2=1+2τ/Δ t/N(F) where τ is the autocorrelation time of F, Δ t the time step between measurements of the Reynolds' stress and N the total number of data points <cit.>.].The functions F[U] and Z[U](ϕ) may be computed directly from the scaled cumulant generating function H, using (<ref>). Computing H using the Ricatti equation (<ref>, <ref>) andusing (<ref>), we have a very easy way to compute the terms appearing in the effective slow dynamics (<ref>) or in the zonal energy balance equations (<ref>) and (<ref>), without having to simulate directly the fast process (<ref>).We now describe the results obtained by solving numerically the linearized barotropic equation (<ref>), where the mean flow velocity, U, is obtained from a quasilinear simulation as described in the end of section <ref>, and represented in Figure <ref>. The energy injection rates P_mean and α P_fluct, computed using both of the methods explained above, with different values of the non-dimensional damping rate α are represented in Figure <ref>. The first term P_mean (solid curve) is roughly of the order of magnitude of the dissipation term in (<ref>) (recall we use units such that E_z≃1). The second term α P_fluct is about an order of magnitude smaller than P_mean. In this case, the energy balance (<ref>) implies that the zonal velocity is actually slowly decelerating. Here, neglecting α P_fluct in (<ref>) leads to an error in the zonal energy budget of about 5–10%. This confirms the fact that fluctuations of Reynolds' stresses are only negligible in a first approximation, and that they should be taken into account in order to obtain a quantitative description of zonal jet evolution. However, we emphasize that only one mode is stochastically forced in this case (see section <ref> for details). When several modes are forced independently, the Reynolds' stress divergence f(ϕ) is computed as the sum of independent contributions from each mode. If the number K of forced modes becomes large, then the Central Limit Theorem implies that the typical fluctuations of f(ϕ) (and thus α P_fluct) roughly scale as 1/K. In Figure <ref>, K=1 so we are basically considering the case where fluctuations of Reynolds' stresses are the most important in the zonal energy balance. In other words, this is the worst case test for CE2 types of closures. In most previous studies of second order closures like CE2, a large number of modes is forced <cit.>, so in these cases p_fluct(ϕ) and α P_fluct are most likely to be negligible in the zonal energy balance.We also observe that P_mean increases up to a finite value as α≪1, while α P_fluct is nearly constant over the range of values of α considered. We further comment the behavior in the following. The spatial distribution of the energy injection rates p_mean(ϕ) and p_fluct(ϕ) are represented in Figures <ref> and <ref>, <ref>. Both p_mean(ϕ) and p_fluct(ϕ) are concentrated in the jet region ϕ∈[-π/4,π/4], which is also the region where the stochastic forces act (see Figure <ref>). In Figure <ref>, we observe that p_mean is always positive. This means that the turbulent perturbations are everywhere injecting energy into the zonal degrees of freedom, i.e. the average Reynolds' stresses are intensifying the zonal flow U(ϕ) at each latitude. This effect is predominant at the jet maximum and around the jet minima (around ϕ=±π/8). We also observe that p_mean (and thus F[U]) converges to a finite value as α decreases. A similar result has been obtained for the two dimensional Navier–Stokes equation under the assumption that the linearized equation close to the base flow has no normal mode, using theoretical arguments <cit.>. Those assumptions are not satisfied here, thus indicating that the finite limit of F[U] as α vanishes is a more general result. This result is extremely important, indeed it implies that the effective dynamics (<ref>) is actually well-posed in the limit α→0.By definition, p_fluct(ϕ) is necessarily positive. In Figure <ref>, we see that p_fluct(ϕ) keeps increasing as α decreases in the region away from the jet maximum (roughly for |ϕ|∈[π/16,π/4]). This is in contrast with the behaviour of p_mean(ϕ) (fig. <ref>). We note that such a behaviour for p_fluct(ϕ) has been obtained recently for the two-dimensional Navier-Stokes equation under the assumption that the base flow has no mode <cit.>. However, the range of values of α considered here is not wide enough to check precisely those theoretical results. We also observe in Figure <ref> that p_fluct(ϕ) is relatively small in the region of jet maximum ϕ≃ 0. This means that Reynolds' stresses tend to fluctuate less in this area. In the context of the deterministic two-dimensional Euler equation linearized around a background shear flow, it is known that extrema of the background flow lead to a decay of the perturbation vorticity (depletion of the vorticity at the stationary streamline <cit.>). In a stochastic context, this implies that the perturbation vorticity δω is expected to fluctuate less in the region of jet extrema, in qualitative agreement with what is observed in Figure <ref>. §.§ Empirical validation of the time scale separation hypothesis In this paper we assumed a large separation in time scales: the eddies δω evolves much faster than the zonal flow U, permitting the quasilinear approximation.It has been shown in Ref. Bouchet_Nardini_Tangarife_2013_Kinetic,tangarife-these that for the linearized dynamics close to a zonal jet U, the autocorrelation function of both the eddy velocity and the Reynolds stresses are finite in the limit α→0, even if the dissipation vanishes in this limit. An effective dissipationtakes place, thanks to the Orr mechanism (see Refs. Bouchet_Nardini_Tangarife_2013_Kinetic,tangarife-these). This result ensures that time scale separation assumption is valid for small enough α (the eddies δω evolve on a time scale of order one, and the zonal flow U evolves on a time scale of order 1/α).The consistency of this assumption for any value of α can also be tested numerically. For this purpose, we compute the maximum correlation time of the Reynolds' stress divergence f(ϕ), defined as[In this spherical geometry the maximum is taken over the inner jet region ϕ∈[-π/7,π/7].] τ_max^α≡max_ϕlim_t→∞1/t∫_0^t∫_0^t𝔼_U^α[[f(ϕ,s_1)f(ϕ,s_2)]]/2𝔼_U^α[[f^2(ϕ)]] s_1s_2. Wecheck whether or not τ_max^α≪ 1/α, where 1/α isthe dissipative time scale. The results are summarized in Figure <ref>. We observe that τ_max^α converges to a finite value as α decreases, as expected from the theoretical analysis <cit.>, and this value is smaller than the inertial time scale (equal to one by definition of the time units). This means that the typical time scale of evolution of the Reynolds' stress divergence is much smaller than the dissipative time scale 1/α as soon as 1/α is much larger than one, justifying the time scale separation hypothesis.§ LARGE DEVIATIONS OF REYNOLDS STRESSESIn section <ref>, we studied the effective energy balance for the zonal flow U(ϕ) using numerical simulations of the linearized barotropic dynamics (<ref>). This effective description of zonal jet dynamics takes into account the low-order statistics of Reynolds' stresses: average and covariance.In order to study rare events in zonal jet dynamics, we must employ the large deviation principle. The goal of this section is to apply the theoretical tools presented in sections <ref> and <ref> to the study of rare events statistics in zonal jet dynamics. §.§ Large Deviation Principle for the time-averaged Reynolds' stresses We first formulate the Large Deviation Principle for the quasi-linear barotropic equations (<ref>) in the regime α≪1, and present some properties of the large deviations functions. The numerical results are presented in section <ref>. The Large Deviation Principle presented here is equivalent to the one presented in a more general setting in section <ref>. Consider the evolution of ω_z from the first equation of (<ref>). Over a time scale Δ t much smaller than 1/α but much larger than the correlation time τ we can writeΔω_z/Δ t≡1/αω_z(t+Δ t) - ω_z(t)/Δ t≃1/Δ t∫_t^t+Δ t R(u) d u - ω_z(t) ,where we have used the fact that ω_z has not evolved much from t and t+Δ t (because Δ t≪1/α), while R(u) has evolved according to (<ref>) with a fixed ω_z (or equivalently a fixed U). We also neglect hyper-viscosity in the evolution of ω_z, which is natural in the turbulent regime we are interested in. Note however that some hyper-viscosity is still present in the numerical simulations of (<ref>), in order to ensure numerical stability. For consistency, we make sure that the hyper-viscous terms have no influence on the numerical results (see Figure <ref>). We denote by P_Δ t[Δω_z/Δ t] the probability distribution function of Δω_z/Δ t, with a fixed t (and thus a fixed ω_z(t)), but with an increasing Δ t.This regime is consistent with the limit of time scale separation α→0, where ω_z is nearly frozen while δω keeps evolving. From (<ref>), P_Δ t[Δω_z/Δ t] is also the probability density function of the time-averaged advection term 1/Δ t∫_t^t+Δ t R(u) d u. The Large Deviation Principle gives the asymptotic expression of P_Δ t[Δω_z/Δ t] in the regime Δ t ≫τ, namelyln P_Δ t[Δω_z/Δ t]Δ t→∞∼ -Δ t ℒ[Δω_z/Δ t] .The function ℒ is called the large deviation rate function. It characterizes the whole distribution of Δω_z/Δ t in the regime Δ t≫τ, including the most probable value and the typical fluctuations. Our goal in the following is to compute numerically ℒ[Δω_z/Δ t]. This can be done through the scaled cumulant generating function (<ref>). Using (<ref>), the definition (<ref>) can be reformulated asH[θ] = lim_Δ t→∞1/Δ tln∫dω̇_z P_Δ t[ω̇_z] exp(θ·Δ t ω̇_z)Because ω_z is a field, here θ is also a field depending on the latitude ϕ, and H is a functional. For simplicity, we stop denoting the dependency of H in ω_z. In (<ref>), we also have used the notation θ_1·θ_2≡∫ϕ cosϕ θ_1(ϕ)θ_2(ϕ) for the canonical scalar product on the basis of spherical harmonics.Using (<ref>) in (<ref>) and using a saddle-point approximation to evaluate the integral in the limit Δ t→∞, we get H[θ] = sup_ω̇_z{θ·ω̇_z- ℒ[ω̇_z] },i.e. H is the Legendre-Fenschel transform of ℒ. Assuming that H is everywhere differentiable, we can invert this relation asℒ[Δω_z/Δ t] = sup_θ{θ·Δω_z/Δ t- H[θ] } .The scaled cumulant generating function H[θ] can be computed either from a time series of δω (see section <ref>) or solving the Ricatti equation (see section <ref>). Then the large deviation rate function ℒ can be computed using (<ref>), and this gives the whole probability distribution of Δω_z/Δ t (or equivalently of the time-averaged Reynolds' stresses) through the Large Deviation Principle (<ref>).In the following, we implement this program and discuss the physical consequences for zonal jet statistics. We first give a simpler expression of H[θ], that makes its numerical computation easier. §.§ Decomposition of the scaled cumulant generating function Using the Fourier decomposition (<ref>), we can decompose the perturbation vorticity as δω(λ,ϕ) = ∑_m ω_m(ϕ)e^imλ, where ω_m satisfies∂ω_m/∂ u=-L_U,m[ω_m]+√(2)η_m,where the Fourier transform of the linear operator (<ref>) reads L_U,m[ω_m](ϕ)=-im/cosϕ(U(ϕ)ω_m(ϕ)+γ(ϕ)ψ_m(ϕ))-αω_m(ϕ)-ν_n(-Δ_m)^nω_m(ϕ).In (<ref>), η_m(ϕ,t) is a Gaussian white noise such that η_-m=η_m^*, with zero mean and with correlations 𝔼[η_m(ϕ_1,t_1)η_m^*(ϕ_2,t_2)]=c_m(ϕ_1,ϕ_2)δ(t_1-t_2), 𝔼[η_m(ϕ_1,t_1)η_m(ϕ_2,t_2)]=0,where c_m is the m-th coefficient in the Fourier decomposition of C in the zonal direction.Using the Fourier decomposition, the zonally averaged advection term can be written R(ϕ)=∑_mR_m(ϕ) with R_m(ϕ)=-im/cosϕ∂_ϕ(ψ_m·ω_-m). Using this expression and the fact that ω_m_1 and ω_m_2^* are statistically independent for m_1≠ m_2, the scaled cumulant generating function (<ref>) can be decomposed as[The time t in the upper and lower bounds of the integral in (<ref>) are not relevant here, as we are considering the statistically stationary state of (<ref>).]H[θ]≡lim_Δ t→∞1/Δ tln𝔼_U[exp(θ·∫_0^Δ t(R(u)-ω_z) u )]=-θ·ω_z+∑_mH_m[θ],withH_m[θ]=lim_Δ t→∞1/Δ tlog𝔼_Uexp[∫ϕ cosϕ θ(ϕ)∫_0^Δ tR_m(ϕ,u) u].We recall that 𝔼_U is the average in the statistically stationary state of (<ref>).In the following, we consider the case where only one Fourier mode m is forced, for simplicity and to highlight deviations from Gaussian statistics. If several modes are forced, their contributions to the scaled cumulant generating function add up, according to (<ref>).Finally, consider the decomposition of the zonally averaged advection term into spherical harmonics (<ref>), R_m(ϕ)=∑_ℓR_m,ℓ P_ℓ^0(sinϕ). Using θ(ϕ)=θ_ℓ P_ℓ^0(sinϕ) in (<ref>), we investigate the statistics of the ℓ-th coefficient R_m,ℓ. The associated scaled cumulant generating function (<ref>) is denoted H_m,ℓ(θ)≡ H_m[θ P_ℓ^0(sinϕ)], and the large deviation rate function is denotedℒ_m,ℓ(ω̇_ℓ) = sup_θ_ℓ{θ_ℓ ω̇_ℓ- H_m,ℓ(θ_ℓ) } .§.§ Numerical results The function H_m,ℓ defined in previous section can be computed either from a time series of ω_m(ϕ,u) using the method described in section <ref>, or solving the Ricatti equation as described in section <ref>. Then, the large deviation rate funtion is computed using (<ref>). We now show the results of these computations and discuss the physical consequences. We describe the results obtained by solving numerically the linearized barotropic equation (<ref>), where we use the mean flow U the flow obtained from a quasilinear simulation as described in the end of section <ref>, and represented in Figure <ref>.§.§.§ Scaled cumulant generating function An example of computation of H_m,ℓ(θ) is shown in Figure <ref>, with m=10, ℓ=3 and α=0.073. The linearized barotropic equation (<ref>) is integrated over a time T_max=54,500, with fixed mean flow given in Figure <ref>, and the value of R_m,ℓ is recorded every 0.03 time units (the units are defined in section <ref>). The scaled cumulant generating function (<ref>) is estimated following the procedure described in section <ref>(thick black curve in Figure <ref>). Because the time series of R_m,ℓ is finite, H_m,ℓ(θ) can only be computed with accuracy on a restricted range of values of θ (see section <ref> for details), here θ∈[θ_min/2,θ_max/2] = [-0.6,1.1].The scaled cumulant generating function (<ref>) is also computed solving numerically the Ricatti equation (<ref>) and using (<ref>) (yellow curve in Figure <ref>). We observe almost perfect agreement between the direct estimation of H_m,ℓ (black curve in Figure <ref>) and the computation of H_m,ℓ using the Ricatti equation (yellow curve). The integration of the Ricatti equation was done with a finer resolution and a lower hyper-viscosity than in the simulation of the linearized barotropic equation (<ref>), the agreement between both results in Figure <ref> thus shows that the resolution used in the simulation of (<ref>) is high enough, and that the effect of hyper-viscosity is negligible.We stress that the computation of H_m,ℓ(θ) using the Ricatti equation (<ref>) does not require the numerical integration of the linear dynamics (<ref>). Typically, the integration of (<ref>) over a time T_max=54,500 takes about one week, while the resolution of the Ricatti equation (<ref>) for a given value of θ is a matter of a few seconds. This enables the investigation of the statistics of rare events (large values of |θ| in Figure <ref>) extremely easily, as we now explain in more detail.§.§.§ Rate function and departure from Gaussian statistics The main goal of this study is to investigate the statistics of rare events in zonal jet dynamics, that cannot be described by the effective dynamics studied in section <ref>. Using the previous numerical results, we now show how to quantify the departure from the effective description.The large deviation rate function ℒ_m,ℓ entering in the Large Deviation Principle (<ref>) can be computed from H_m,ℓ using (<ref>). The result of this calculation[Here the Legendre-Fenschel transform (<ref>) is estimated as ℒ_m,ℓ(ω̇_z)=θ^⋆·ω̇_z - H_m,ℓ(θ^⋆) where θ^⋆ is the solution of ω̇_z = ∂_θ H_m,ℓ(θ^⋆). Other estimators could be considered <cit.>.] is shown in Figure <ref> (yellow curve).Because of the relation (<ref>), ℒ_m,ℓ can also be interpreted as the large deviation rate function for the time-averaged advection term, denoted R̅_m,ℓ,Δ t≡1/Δ t∫_0^Δ tR_m,ℓ(u) d u. In other words, the probability distribution function of R̅_m,ℓ,Δ t in the regime Δ t≫τ satisfiesln P_m,ℓ,Δ t(R̅) Δ t≫τ∼ -Δ tℒ_m,ℓ(R̅). The Central Limit Theorem states that for large Δ t≫τ, the statistics of R̅_m,ℓ,Δ t around its mean ℛ_m,ℓ≡𝔼_U[R̅_m,ℓ,Δ t]=𝔼_U[R_m,ℓ] are nearly Gaussian. A classical result in Large Deviation Theory is that the Central Limit Theorem can be recovered from the Large Deviation Principle <cit.>. Indeed, using the Taylor expansion of H_m,ℓ in powers of θ (<ref>) and computing the Legendre-Fenschel transform (<ref>), we getℒ_m,ℓ(R̅)= 1/2𝒵_m,ℓ(R̅ - ℛ_m,ℓ)^2 + O((R̅ - ℛ_m,ℓ)^3)with 𝒵_m,ℓ≡lim_Δ t→∞Δ t 𝔼_U[[R̅_m,ℓ,Δ t^2]]. Using the Large Deviation Principle (<ref>), this means that the statistics of R̅_m,ℓ,Δ t for small fluctuations around ℛ_m,ℓ are Gaussian with variance 𝒵_m,ℓ/Δ t, which is exactly the result of the Central Limit Theorem. Then, the difference between the actual rate function ℒ_m,ℓ(R̅) and its quadratic approximation (right-hand side of (<ref>)) gives the departure from the Gaussian behaviour of R̅_m,ℓ,Δ t. From (<ref>), the Gaussian behaviour is expected to apply roughly for |R̅ - ℛ_m,ℓ|≤σ_m,ℓ,Δ t with σ_m,ℓ,Δ t≡√(𝒵_m,ℓ/Δ t). The values of ℛ_m,ℓ±σ_m,ℓ,Δ t are represented by the black vertical lines in Figure <ref>[The value of Δ t used in this estimation is the optimal one Δ t^⋆, defined in section <ref>.]. The quadratic approximation of the rate function is also shown in Figure <ref> (purple curve). As expected, the curves are indistinguishable from each other between the vertical lines (typical fluctuations), and departures from the Gaussian behaviour are observed away from the vertical lines (rare fluctuations). Namely, the probability of a large negative fluctuation is much larger than the probability of an equally large fluctuation for a Gaussian process with same mean and variance as R̅_m,ℓ,Δ t. On the contrary, the probability of a large positive fluctuation is much smaller than the the probability of the same fluctuation for a Gaussian process with same mean and variance as R̅_m,ℓ,Δ t.The kinetic description basically amounts at replacing R̅_m,ℓ,Δ t by a Gaussian process with same mean and variance. From the results summarized in Figure <ref>, we see that such approximation leads to a very inaccurate description of rare events statistics. Understanding the influence of the non-Gaussian behavior of R̅_m,ℓ,Δ t on zonal jet dynamics is naturally a very interesting perspective of this work. § CONCLUSIONS AND PERSPECTIVES In this work we carried out a first study of the typical and large fluctuations of the Reynolds stress in fluid mechanics. Reynolds stress is certainly a key quantity in studying the largest scales of turbulent flows. This is especially true whenever a time scale separation is present, in which case it can be expected that an effective slow equation governs the large scale flow evolution (see equation (<ref>)). Not only the averaged momentum flux (the Reynolds stress) and averaged advection terms are essential, but also their fluctuations (that we call the Reynolds stress fluctuations). We studied the case of a zonal jet for the barotropic equation on a sphere, in a regime for which time scale separation is relevant. For this case, we show that the probability distribution function of the equal-time (without time average) advection term has a distribution with typical fluctuations which are very large compared to the average, and with heavy tails. These probability distribution functions have exponential tails, both for the quasilinear and fully non-linear dynamics cases.For quasilinear dynamics we gave a simple explanation for these exponential tails.When one is interested in the low frequency evolution of the jet, these high frequency fluctuations of the advection term and momentum fluxes are not relevant. We discussed that the natural quantity to study is the large deviation rate function for the time averaged advection term (that we call the Reynolds stress large deviation rate function). We have proposed two methods to compute this rate function. First an empirical method, directly from the time series of the advection term, that could be applied to any dynamics. Second we show that for the quasilinear dynamics, the Reynolds stress large deviation rate function can be computed as the contraction of a solution of a matrix Riccati equation. We demonstrated that such a computation can be performed by generalizing classical algorithms used to compute Lyapunov equations. Solving the matrix Riccati equation is much more computationally efficient, by several orders of magnitude, compared to accumulating statistics by numerical simulation, and gives direct and easy access to the probability of rare events. The approach is however limited to the quasilinear dynamics so far. We discussed the Reynolds stress large deviation rate again for the specific case of a zonal jet that arises in turbulent barotropic flow on the rotating sphere. We illustrated the computation of the Reynolds stress large deviation rate, both using the empirical method and the Riccati equation. These two approaches give a very good agreement. This large deviation rate function clearly illustrate the existence of non-Gaussian fluctuations. The non Gaussian fluctuations are much more rare than Gaussian ones for positive values of the Reynolds stress component and much less rare than Gaussian for negative values. Our work illustrates the possibility to compute Reynolds stress large deviation rate functions. It opens up a number of perspectives. A next step would be to study the spatial structure of the Reynolds stress fluctuation, and describe it from a fluid mechanics perspective.It would help to answer the following questions: What are the dominant spatial pattern for the fluctuations of the Reynolds stresses? What causes them? What is their effect on the low frequency variability of the large scale flow?The most interesting application of the Reynolds stress large deviation rate functions may be the study of rare long term evolutions of the large scale flow. For instance, in many examples, rare transitions between turbulent attractors have been observed, leading to a bistability phenomenology. In order to study quantitatively such a bistability phenomenology, for instance in order to compute transitions rates and transitions paths between attractors, one could consider equation (<ref>) in the framework of Freidlin–Wentzell theory. The large deviation rate function we studied in this work would then be the basic building block, that would allow to define an action that should be minimized to compute transition paths and transition rates. In order to compute the action, the large deviation rate function should then be computed for any flow U along a possible transition path, as described in section <ref> for a single example of a flow U.An essential question, at a more mathematical level, is the validity of the quasilinear approximation as far as rare events are concerned. The self consistency of the quasilinear approach has been discussed theoretically by focusing on the average Reynolds stress <cit.>. This point has also been verified numerically in this work, through the study of properties of the energy balance (see section <ref>) and through the verification of the fact that the linear equation correlation time has a limit when α→ 0 (see section <ref>). However this does not necessarily imply that the quasilinear approximation is self-consistent as far as fluctuations, and more specifically rare fluctuations, are concerned. This could be addressed by studying the properties of solutions to the Ricatti equation in the limit α→0 to assess whether or not the small scale dissipative mechanism (either viscosity or hyperviscosity) affects the statistics of the rare fluctuations.This problem is left as a prospect for future work. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013 Grant Agreement No. 616811) (F. Bouchet and T. Tangarife) and from the US NSF under Grant No. DMR-1306806 (J. B. Marston). J. B. Marston would also like to thank the Laboratoire de Physique de l'ENS de Lyon and CNRS for hosting a visit where some of this work was carried out. We thank the reviewers for their extremely careful reading of our paper and for their useful suggestions. | http://arxiv.org/abs/1706.08810v3 | {
"authors": [
"Freddy Bouchet",
"J. B. Marston",
"T. Tangarife"
],
"categories": [
"physics.flu-dyn",
"physics.class-ph",
"physics.data-an",
"physics.geo-ph"
],
"primary_category": "physics.flu-dyn",
"published": "20170627121516",
"title": "Fluctuations and large deviations of Reynolds stresses in zonal jet dynamics"
} |
^1 Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany ^2 Faculty of Physics, Moscow State University, Moscow, 119991 Russia ^3 Department of Physics, Tbilisi State University, Chavchavadze Avenue 3, 0128 Tbilisi, Georgia [email protected] study demonstrates how the spin quantum dynamics of a single Fe atom adsorbed on Cu(001) can be controlled and manipulated by the vibrations of a nearby copper tip attached to a nano cantilever by virtue of the dynamic magnetic anisotropy. The magnetic properties of the composite system are obtained from ab initio calculations in completely relaxed geometries and turned out to be dependent considerably on the tip-iron distance that changes as the vibrations set in.The level populations, the spin dynamics interrelation with the driving frequency, as well as quantum information related quantities are exposed and analyzed. Functionalizing Fe adatoms onCu(001)as a nanoelectromechanical system Michael Schüler^1, Levan Chotorlishvili^1, Marius Melz^1, Alexander Saletsky^2, Andrey Klavsyuk^2, Zaza Toklikishvili^3, Jamal Berakdar^1 December 30, 2023 =============================================================================================================================================== § INTRODUCTIONMicroelectromechanical (MEMS) or nanoelectromechanical systems (NEMS) are at the verge of the classical-quantum world <cit.> and can thus sense, possibly coupled, quantum-classical properties. For instance, tiny vibrating cantilever were shown to detect a single spin <cit.>.The sensitivity depends on the mean phonon number with the cantilever dynamics turning quantum as the phonons number decreases.Related to these observations, this field promises a new rout to quantum information nanomechanical devices. An example is the setup consisting of a single nitrogen-vacancy (NV) centre in a diamond nanocrystal deposited at the extremity of a SiC nanowire <cit.>. The quantum NV spin dynamics is observed to be coupled to the nanomechanical oscillator by means of the time-resolved nanocrystal fluorescence and photon-correlation measurements. This dynamic can be influenced by external fields such as a non-homogeneous magnetic field.A clear advantage of utilizing the spin-degrees of freedom of the NV is their long decoherence times even at room temperatures <cit.>.Further phenomena emerge when considering strongly coupled nonlinear NEMS in which case phenomena such as nonlinear resonances can be exploited for the control of the energy transfer between the coupled NEMS <cit.>.In the present work we propose a new type of NEMS based on a single magnetic Fe adatom deposited on a Cu(001) substrate. A proper choice of the driving frequency allows controlling the level populations in the system. The proposed (scanning-tunneling microscopy) STM-type or (atomic force microscopy) AFM-type setup is thus a hybrid system utilizing the quantum nature of single adsorbed atoms or molecules on the surface, which were designed as studied in an impressively controllable way experimentally (for example in Refs. <cit.>). The magnetic properties of Fe and Co adatoms on a Cu_2N/Cu(100)-c(2× 2) surface were determined experimentally viax-ray magnetic dichroism measurements <cit.>.The possibly classical cantilever dissipative dynamics is coupled the quantum spin dynamics of the adsorbates since, as demonstrated below, the magnetic anisotropy is affected by the tip-adsorbate distance, and hence by the tip vibrational motion. This coupling might be exploited to access the topology or the local magnetic properties of spin systems <cit.>.§ THEORETICAL FRAMEWORK Specifically, weconsider a single magnetic Fe atom deposited on a Cu(001) surface. A similar setup consisting ofFe or Mn on copper coated by an Cu_2N overlayer, was shown to havea large magnetic anisotropy and relaxation times<cit.>. Our calculations are carried out in the presence of a tip apex as in AFM experiments and reveal a substantial dependence of the magnetic anisotropy on the distance between the tip and the Fe adatom.Thus, in theproposed setup the magnetic properties of the single-atom are coupled to the oscillations of a nano-sized cantilever carrying the tip apex (FIG. <ref>(a)). As the characteristic frequencies of such nano-mechanical oscillators are known to reach the gigahertz regime<cit.>, frequencies in the range of ∼100 GHz become feasible upon a further downscaling of the cantilever and can thus match the typical energy scale of the spin system (which is in the range of few meV).§ AB INITIO CALCULATIONSAb initio density-functional calculations of the ground state and the energy difference upon changing the magnetization axis of the Fe atom were performed using the projector augmented-wave (PAW) technique <cit.>, as implemented in the Vienna ab initio simulation package (VASP) <cit.>.The calculations are based on density-functional theory with the generalized gradient approximation (GGA) <cit.>.We used the same methodology used in previous calculations of the magnetic anisotropy of Co and Fe adatoms on Rh(111), Pt(111) and Cu(100) substrates <cit.>. We note in this context thatSTM experiments were performed after the tip were incontact with the surface, and hence the tip is most likely covered by the surface material <cit.>.In this casea tip with surface atoms is often used in computer calculations <cit.>.Our computational models consists 125 copper atoms representing the surface, the iron atom, and 5 additional atoms for simulating the presence of a tip apex, as depicted in FIG. <ref>(b).The unit cell has a size of 12.87 Å in the x and y directions (parallel to the surface), whereas the extent of the z direction (perpendicular to the surface) amounts to 31.89 Å. At this slab thickness, the interaction between the tip and the repeated image of the surface is negligible. A cutoff energy of 300 eV is used. The calculations including spin-orbit coupling require a fine k-point mesh for the Brillouin-zone integrations. Test calculations were performed for iron atom on a Cu(001) surface for three different k-point grids: 3×3×1 , 3×3×2 , and 5×5×1 generated by the Monkhorst-Pack scheme <cit.>, in conjunction with a modest Gaussian smearing method. A 3×3×1 grid provided the best compromise between accuracy and computational efforts. The calculations were performed in two steps. First the coordinates of the iron atom and the positions of the atoms in the three topmost layers of the substrate (apart from the tip) were optimized using scalar-relativistic calculations until the forces on all unconstrained atoms were converged to less than 0.01 eV/Å.In the second step, the geometry and the electronic and magnetic ground states resulting from the scalar-relativistic calculations were used to initialize the relativistic calculations including spin-orbital coupling.Recent work <cit.> demonstrated that relaxations of Fe and Co adatom on Pt(111) with and without spin-orbit coupling are almost identical.After a geometry optimization of the full cluster (apart from the tip) for every position of the tip, we computed the magnetic anisotropy energy as the difference of the respective ground state energies upon varying the magnetization axis. We found that the dependence on the angle θ measured from the Fe-tip axis is well described by the lowest-order anisotropy term δsin^2 θ, as it is known for similar systems. The dependence on the angle ϕ measured along the plane on the other hand turned out to be rather weak. Furthermore, weanalyzed the spin density n_↑(r⃗) - n_↓(r⃗) to investigate the degree of localization of the magnetization. The result is presented in FIG. <ref> for small values of the density in two characteristic planes along the symmetry directions. We conclude that the Fe atom slightly polarizes the tip and the substrate below. Especially for the latter we observe the typical behavior of a spin density associated with this kind of anisotropy.However, the major contribution to the magnetization is confined within the direct vicinity of the Fe atom, confirming that the effective surface spin can be interpreted as the magnetic moment offew atoms.§ MODELING THE SPIN DYNAMICS In FIG. <ref>(b) we show our results for the dependence of the magnetic anisotropy parameter δ and of the magnetic moments associated with the spin (μ_S), and the angular momentum (μ_L) for four different values of the distance a between the last tip atom and the iron atom.The spin magnetic moment on the iron atom without the tip-adatom interaction is 2.94 μ_B. This result agrees well with previous density functional calculations <cit.>.It should be noted, that the magnetic anisotropy parameter δ for an atom on the surface is very sensitive to the interatomic distances <cit.> and the arrangement of the atom <cit.>. It was demonstrated that the structural relaxation of the adatom and the substrate reduces significantly the magnetic anisotropy energy <cit.>.Therefore, compared to ab initio calculations for the Fe adatom on the ideal Cu(001) surface, our magnetic anisotropy energy obtained in a fully relaxed geometry is several times less than the value presented in Ref. <cit.>. Based on the fitting functions displayed in FIG. <ref>(b) we are now able to formulate the Hamiltonian describing the effective surface angular momentum with the parametric dependence on a asĤ(a) = -[g_S(a) + g_L(a)]μ_B B_0 Ĵ_x - δ(a) Ĵ^2_z.We assume a magnetic field with a strength B_0 is applied along the x axis. Approximating the total angular momentum with 2 turns out to be an adequate description and can be confirmed experimentally by means of inelastic tunneling spectroscopy <cit.>. The distance-dependent gyromagnetic ratios for spin (angular) momentum g_S(a) (g_L(a)) account for the varying magnitude of the total magnetic moment (see Fig. <ref>(b)), as extracted from our ab initio calculations.We fix B_0 to the value of 4 T and takethe ground state as the initial state. As one can readily show for eq. (<ref>), the expectation value with respect to all eigenstates of Ĵ_y and Ĵ_z is exactly zero. Thisholds true even for the case of the time-dependent Schrödinger equation, when replacing a→ a(t).To map out the spin dynamics for a representative case we choose a by a_0=4 Åand B_0=4 T. The energies of the eigenstates |ξ_n⟩ (on the ordinate axis) and the expectation values of J_x (abscissa) are shown in the inset in FIG. <ref>. For a high density of phonons the spin dynamics originates from an oscillation of the tip apex according to a(t)=a_0 + bsin(ω t)for t>0 (we assume a(t)=a_0 for t≤ 0). This corresponds to a setup where the system is initially in its ground state and is driven out of equilibrium by the cantilever oscillations for t>0. The oscillation amplitude is chosen as b=0.9 Å.Before discussing the results, let us elaborate on the qualitative aspects of the dynamics. Since the spin is driven by an effectively time-dependent anisotropy, i. e. the coupling to the operator Ĵ^2_z, the induced transitions allow for changing the spin projection ⟨Ĵ_x ⟩ only. The magnetic moment μ will thus remain parallel to ⟨Ĵ_x ⟩. Therefore, only the longitudinal spin dynamics can be induced, limiting the transitions from the ground state to only the two excited states that match in symmetry.In FIG. <ref> we present the resulting spin dynamics in dependence on ω. The color map plot (lower right figure) shows the population of the ground state (which we have chosen as the initial state). Interestingly, the magnetic moment is hardly affected by the variation of the magnetic anisotropy for the major part of the frequency range. Apart from that, a couple of distinct lines indicate an optimal setting for the parametersto drive the angular momentum to some excited states. A more detailed analysis reveals that the dynamics for the frequencies indicated by the dashed lines (labelled by 1,2,3) exhibits almost Rabi-like transitions from the ground state to a single excited state. This behavior canbe explored further by a Floquet analysis. We therefore expand the angular momentum wave function as|ξ(t)⟩=∑^2J+1_n=1 c_n |ϕ_n(t) ⟩= ∑^2J+1_n=1 c_n e^i ε_n t|f_n(t) ⟩ ,where ε_n are the quasi energies and |ϕ_n(t)⟩ =e^i ε_n t|f_n(t)⟩ the Floquet eigenvectors. Both can be obtained by solving the eigenvalue problem of the time-evolution operator U(t,0) at t=T≡ 2π/ω, sinceÛ(T,0)|f_n(0)⟩ = e^i ε_n T|f_n(0)⟩(note that |f_n(t+T)⟩ = |f_n(t) ⟩). Before discussing the solution of eq. (<ref>), let us briefly revisit how the dynamics shown in FIG. <ref> can be explained within the Floquet theory. Eq. (<ref>) expressesthe expansion of the time-dependent spin state in terms of the orthonormal Floquet states |f_n(t)⟩, with the projection coefficients c_n. For the case c_n=c_n_0δ_n,n_0, the projection ⟨ξ(0)|ξ(t) ⟩ amounts to exp[-i ε_n_0 t] ⟨ f_n_0(0)|f_n_0(t) ⟩, such that the population of the initial state remains one at multiples of T. Assuming on the other hand c_n= (δ_n,n_1±δ_n,n_2)/√(2) yields the stroboscopic time evolution|⟨ξ(0)|ξ(k T) ⟩|^2 = cos^2[(ε_n_1-ε_n_2)k T].These two scenarios explain the dynamics observed in FIG. <ref>, where slow, Rabi-like population transfer (with a frequency corresponding to the difference of two quasienergies) is overlayed with fast oscillations (which originate from the overlaps of the type ⟨ f_n(0)|f_m(t) ⟩ and are thus periodic with the frequency ω). The quasienergies obtained from eq. (<ref>) are presented in FIG. <ref>(a) (upper panel), along with the projection |c_n|^2=|⟨ξ(0)|f_n(0) ⟩|^2 (lower panel). As pointed out, the decisivefactor for the depletion of the ground state is at least two projection coefficients being different from zero. For this reason, we have ordered the quasienergies according to the magnitude of |c_n|^2. As it turned out, only two of the Floquet states display a significant contribution to the initial state. Therefore, only their projection is shown in FIG. <ref>(a). For the complete picture of the behaviour of the quasienergies however, the third state and its eigenvalue are included in the upper panel.The vertical lines in FIG. <ref>(a) demonstrate that the scenario for the dynamics according to eq. (<ref>)occurs only at the crossing points of the quasienergies, where (at least) two branches exchange their character, that is the magnitude of their projection coefficients. A more detailed analysis reveals that all crossings are avoided crossings. The difference of the quasienergiesbecomes thus relatively small, leading to the slow dynamics in FIG. <ref>. For the exemplary values of ω,FIG. <ref>(b)–(c) provides a magnification of the crossing points and gives their width. Converting the quasienergy gap into a time scale results in exactly the characteristic time of the slow dynamics in FIG. <ref>.§ ENTANGLEMENT MEASURES With cooling down the system, oscillations of the nanocantilever become inherently quantum. Thus the nanocantilever can detect inter-level transitions of the single spin. This statement is generic for a broad class of the nanomechanical systems and is valid for our model as well. Our model is exactly solvable (which is in fact its merit) and allowto exploreanalytically the entanglement between the cantilever and the system. In contrast to the above, where we investigated the case of a moderate elongations of the cantilever, leading to a nonlinear coupling, we consider linear coupling only. In addition to the feasibilityofanalytical solutions, the quantum fluctuations and the oscillations of the cantilever occur on a smaller scale, while large elongations are associated with the classical case which is studied above.In order to quantify the entanglement in the system, we explore von Neumann entropy. In the quantum information theorythe von Neumann entropy is known as the "entanglement entropy" of the reduced density matrix. The technical details of the von Neumann entropy and hence of thereduced density matrix in our case are given in Appendix A. To construct a quantized model, the tip-substrate distance is replaced by a → a_0 + Δ a ∑_α(â_α + â^†_α),where â^†_α (â_α) is the creation (annihilation) operator of the cantilever modes. Modern technologies enabled the fabrication of dual mode (α=1,2) cantilevers, for more details see <cit.>.The resulting Hamiltonian, in lowest order in the oscillation amplitude Δ a readsĤ = -[g_S(a_0)+g_L(a_0)]μ_BB_0 Ĵ_x + ∑_αΩ_αâ^†_αâ_α - Δ aδ'(a_0) Ĵ^2_z ∑_α=1,2(â_α + â^†_α).Similar to the classical case analyzed above, the transition operator Ĵ^2_z allows the transition from the ground state to two excited states only. Hence, the Hamiltonian (<ref>) can be reduced to a three-level system in spin space. We assume that the cantilever frequencies Ω_1,2 match the excitation energies ω_1,2=E_1,2-E_0.We solve directly analytically for the Schrödinger equationi ∂ |Ψ⟩/∂ t = Ĥ |Ψ⟩,using the following ansatz|Ψ(t)⟩ = C_1(t)|n_1,n_2,1 ⟩ + C_2(t)|n_1-1,n_2,2 ⟩ + C_3(t)|n_1,n_2-1,3 ⟩ + C_4(t)|n_1,n_2,4 ⟩ + C_5(t)|n_1,n_2,5 ⟩.Here n_1,n_2 quantify the number of phonons in the cavity with the frequencies Ω_1,Ω_2.Taking into account Eq. (<ref>), (<ref>) we consider the resonance condition E_2-E_1≈Ω_1, E_3-E_1≈Ω_2. After standard calculations we obtain C_1(t)= exp(-iΔ_1 t) { C_1(0)cos(γ√(n_1+n_2)t)-iC_2(0) √(n_1)sin(γ√(n_1+n_2) t)/√(n_1 + n_2)-iC_3(0) √(n_2)sin(γ√(n_1 + n_2)t)/√(n_1+n_2)}, C_2(t)= exp(-i(Δ_2-Ω_1) t) { -iC_1(0) √(n_1)sin(γ√(n_1+n_2) t)/√(n_1 + n_2)+C_2(0) ( n_1 cos(γ√(n_1 + n_2)t)+n_2 )/n_1+n_2+ C_3(0)√(n_1n_2)/n_1+n_2( cos(γ√(n_1 + n_2)t)-1 ) }, In Eq. (<ref>) and (<ref>) we introduced the notation Δ_m=E_m+n_1Ω_1+n_2Ω_2. Further simplifications enabling an analytical treatment are ⟨ 2| Ĵ^2_z|1⟩≈⟨ 3| Ĵ^2_z|1⟩≡ g_0giving rise to the effective coupling constant γ = g_0Δ a δ'(a_0). While the solution Eq. (<ref>)-(<ref>) is obtained for a fixed value of the magnetic field B_0=4T, it is valid for an arbitrary field. Changing the magnetic fieldrescales the level spacing, thus leading to a slight rescaling of the Rabi-like transition frequencies.The quantities we are interested in, such as level populations I_n(t)=C_n(t)C_n^*(t) and von Neumann entropy S=-tr(ρ̂lnρ̂) (where ρ̂=∑_m nC_n(t)C_m^*(t)|m⟩⟨ n| is the density matrix of the system) can be calculated directly from (<ref>) and (<ref>).For this purpose we need to consider the averaging of Eq. (<ref>)–(<ref>) over the phonon distribution functions for the coherent states w_n_1,2=√(λ_1,2^n_1,2)/√(n_1,2!)exp(-λ_1,2/2). Here λ_1,2 is the mean phonon number λ_1,2≫ 1 corresponding to the classical limit.For the calculations of the average level populations we perform summation over the phonon numbers n_1 and n_2:I_n(t)=|C_n(t)|^2=∑_n_1,n_2=0^∞w_n_1^2w_n_2^2|C_n(t)|^2 =∑_n_1,n_2=0^∞e^-λ_1e^-λ_2λ_1^n_1λ_2^n_2/n_1!n_2!|C_n(t)|^2 .The coefficients C_n(t) have a sharp maximum near the mean phonon numbers λ_1,2≫ 1, and the width Δ n_1,2 of their distribution is rather small Δ n_1,2≫λ_1,2. This allows performing the summation analytically and obtaining expressions for the level populations and for the von Neumann entropy. The explicit expression is presented in <ref>.With the analytical solutions at hand, we can now study the population dynamics and the von Neumann entropy of the spin system due to the interaction with the cantilever. FIG. <ref> depicts this dynamics with an oscillation amplitude of Δ a = 0.1 Å. Due to the strong dependence of the anisotropy δ(a) on the tip-sample distance, the dynamics at a_0=3.0 Å (FIG. <ref>(a)) and a_0=4.0 Å (FIG. <ref>(b)) occurs on very different time scales.We clearly see that quantum revivals in level populations are synchronized with the sudden death of von Neumann entropy. This behavior is more prominent in the case of a strong coupling (FIG. <ref>(a)). Obviously with the increase of the phonon number λ the period of quantum revivals becomes larger. In the limit of the classical field λ≫ 1 the revival time tends to infinity. Meaning that the classical field like thermal bath thermalizes the system and leads to irreversibility.§ CONCLUSIONSIn summary, we performed ab initio calculations of the magnetic properties of a single Fe atom adsorbed on Cu(001).We demonstrated that the electronic and the magnetic properties of adatoms are strongly affected by the tip-surface distance.Based on these results weproposed a newtype of NEMS consisting ofasingle magnetic Fe adatoms depositedon a Cu(001) substrate and analyzed its fundamental properties and possible operation scheme.§ VON NEUMANN ENTROPY Using Eq. (<ref>)–(<ref>), a straightforward derivation yields the von Neumann entropyS=-η_1lnη_1-η_2lnη_2-η_3lnη_3.λ_1=λ_2=λ,β=γ√(2λ)t,α=γ t/√(2 λ), ⟨ I_1(t)⟩=1/2(1+exp[2λ(cosα-1)]cos(β+2λsinα)); ⟨ I_2(t)⟩=⟨ I_3(t)⟩= =1/4(1-exp[2λ(cosα-1)]cos(β+2λsinα)); ⟨ I_4(t)⟩=⟨ I_5(t)⟩=0.Here, we used the following notation in order to obtain a compact expression: a=1/2(1+exp[2λ(cosα-1)]cos(β+2λsinα)); b=√(2)/4exp[2λ(cosα-1)]sin(β+2λsinα); d=1/4(1-1/4λ)(1+exp[2λ(cosα-1)]cos(β+2λsinα)); η_1=1/16λ(1-exp[2λ(cosα-1)]cos(β+2λsinα)); η_2=1/4(1+2d+a+√(9a^2+32b^2-6a(1+2d)+(1+2d)^2)); η_3=1/4(1+2d+a-√(9a^2+32b^2-6a(1+2d)+(1+2d)^2)); § ACKNOWLEDGMENTS AKacknowledges the financial support by the joint program of MSU-DAAD Vladimir Vernadsky (A/12/89268) and Volnoe Delo foundation . Computational resources were provided by the Supercomputing Center of Lomonosov Moscow State University. The work was partially funded by the German Science Foundation under SFB 762. § REFERENCESiopart-num | http://arxiv.org/abs/1706.08321v1 | {
"authors": [
"Michael Schuler",
"Levan Chotorlishvili",
"Marius Melz",
"Alexander Saletsky",
"Andrey Klavsyuk",
"Zaza Toklikishvili",
"Jamal Berakdar"
],
"categories": [
"quant-ph",
"cond-mat.mes-hall"
],
"primary_category": "quant-ph",
"published": "20170626110634",
"title": "Functionalizing Fe adatoms on Cu(001) as a nanoelectromechanical system"
} |
[ [ ḍrhpRiemann-Hilbert ProblempropositionPropositiontheoremTheoremcorollaryCorollarylemmaLemmadefinitiondefinitionDefinitionremarkRemarkaddtoresetequationsection matrix[1][c] -ifnextchar@ifnextchar *@̧MaxMatrixCols #1Large-degree asymptotics of rational Painlevé-IV functions]Large-degree asymptotics of rational Painlevé-IV functions associated to generalized Hermite polynomials[R. J. Buckingham]Department of Mathematical SciencesUniversity of CincinnatiPO Box 210025Cincinnati, OH [email protected]://homepages.uc.edu/ buckinrt/The Painlevé-IV equation has three families of rational solutions generatedby the generalized Hermite polynomials.Each family is indexed by two positiveintegers m and n.These functions have applications to nonlinear waveequations, random matrices, fluid dynamics, and quantum mechanics.Numericalstudies suggest the zeros and poles form a deformed n× m rectangulargrid.Properly scaled, the zeros and poles appear to densely fillcertain curvilinear rectangles as m,n→∞ with r:=m/n a fixedpositive real number.Generalizing a method of Bertola and Bothner<cit.> used to study rational Painlevé-II functions, weexpress the generalized Hermite rational Painlevé-IV functions in termsof certain orthogonal polynomials on the unit circle.Using the Deift-Zhounonlinear steepest-descent method, we asymptotically analyze the associatedRiemann-Hilbert problem in the limit n→∞ with m=r· n forr fixed.We obtain an explicit characterization of theboundary curve and determine the leading-order asymptotic expansion of thefunctions in the pole-free region. [ Robert J. Buckingham========================§ INTRODUCTION Rational solutions of the Painlevé-IV equationw_yy=(w_y)^2/2w+3/2w^3+4yw^2+2(y^2-α)w+β/w,w:ℂ→ℂ with parameters α,β∈ℂ arise in the study of steady-state distributions of electric charges for atwo-dimensional Coulomb gas in a parabolic potential <cit.>; rational solutions of the defocusing nonlinear Schrödinger equation<cit.>, the Boussinesq equation <cit.>, theclassical Boussinesq system <cit.>, and the point vortexequations with quadrupole background flow <cit.>;rational-logarithmic solutions of the dispersive water wave equation and themodified Boussinesq equation <cit.>; rational extensions of theharmonic oscillator and related exceptional orthogonal polynomials<cit.>;and the recurrence coefficients forpolynomialsorthogonal to the weight e^-x^2|x|^n and Gaussian Unitary Ensemble matriceswith repeated eigenvalues <cit.>.The fact that these functionshave interesting mathematical properties in their own right is suggested byplots of the zeros and poles.Indeed, as α and β vary alongcertain sequences, the zeros and poles (when appropriately scaled) appear toform strikingly regular patterns in the complex plane that densely fill outcurvilinear rectangles (for the rational functions that can be expressed interms of generalized Hermite polynomials;see Figures<ref>–<ref>) and curvilinear rectangleswith equilateral curvilinear triangles attached to the edges (for the rationalsolutions expressed in terms of generalized Okamoto polynomials)<cit.>.In this work we explicitly determine the boundary curvesfor the rational Painlevé-IV functions associated to the generalized Hermitepolynomials, and derive the leading-order asymptotic expansions of theserational functions in the exterior of the zero/pole region. Various other geometric patterns are also seen in theplots of poles and zeros of rational solutions of the Painlevé-II equationand equations in the Painlevé-II hierarchy <cit.>,the Painlevé-III equation <cit.>, systems of the symmetricPainlevé-IV hierarchy <cit.>, and the Painlevé-V equation<cit.>, as well as certain Wronskians of Hermite polynomialsthat are extensions of the generalized Hermite polynomials and have connectionsto Young diagrams <cit.>. Recently, significant progress has been made in understanding the rationalsolutions of the Painlevé-II equation, which can be indexed by a singleinteger m.As m→∞, appropriately scaled zeros and poles ofthese rational functions densely fill a region T bounded by a curvilineartriangle.By analyzing a Riemann-Hilbert problem derived from theGarnier-Jimbo-Miwa Lax pair, the large-m behavior of these functions (andcertain functions arising in the study of critical behavior in thesemiclassical sine-Gordon equation whose logarithmic derivatives are therational Painlevé-II functions <cit.>) was rigorouslycalculated with error terms outside T in terms of elementary functions,inside T in terms of Riemann theta functions, along edges of T in terms oftrigonometric functions, and at corners of T in terms of the tritronquéePainlevé-I solution <cit.>.In a laterwork, Bertola and Bothner <cit.> reproduced part of theseresults, in particular the equation for the boundary of T and informationabout the location of the zeros and poles, by deriving a new determinantalformula for the squares of the associated Yablonskii-Vorob'ev polynomialsand applying Riemann-Hilbert analysis to a related family of orthogonalpolynomials.Joint with Balogh, they also used their method to obtain theboundary of the zero region for the generalized Yablonskii-Vorob'ev polynomialsassociated to the Painlevé-II hierarchy <cit.>. Miller and Sheng <cit.> have recently shown that,for monodromy data corresponding to rational solutions, the Riemann-Hilbertproblem associated to the Flaschka-Newell Painlevé-II Lax pair is equivalentto the Riemann-Hilbert problem for orthogonal polynomials studied byBertola and Bothner.In this work we use the Bertola-Bothner orthogonal polynomial approach toanalyze the rational Painlevé-IV functions associated to the generalizedHermite polynomials.Set [l] α_m,n^(I):=2m+n+1,β_m,n^(I):=-2n^2,𝒫_-1/z^(I) := {(α_m,n^(I),β_m,n^(I)):m≥ 0,n≥ 1},α_m,n^(II):=-(m+2n+1),β_m,n^(II):=-2m^2,𝒫_-1/z^(II) := {(α_m,n^(II),β_m,n^(II)):m≥ 1,n≥ 0},α_m,n^(III):=n-m,β_m,n^(III):=-2(m+n+1)^2,𝒫_-2z^(III) := {(α_m,n^(III),β_m,n^(III)):m,n∈ℕ_0}, α_j,k^(Oka):=j,β_j,k^(Oka):=-2(2k-j+13)^2,𝒫_-2/(3z)^(Oka):={(α_j,k^(Oka),β_j,k^(Oka)):j,k∈ℤ}.where ℕ_0 denotes the nonnegative integers.It is known thatthe Painlevé-IV equation (<ref>) has a rational solution if and only if(α,β)∈𝒫_-1/z^(I)∪𝒫_-1/z^(II)∪𝒫_-2z^(III)∪𝒫_-2/(3z)^(Oka). Furthermore, for fixed (α,β) this rational solution is uniquewhen it exists <cit.>.Thefamilies of rational solutions to (<ref>)corresponding to 𝒫_-1/z^(I)∪𝒫_-1/z^(II),𝒫_-2z^(III), and 𝒫_-2/(3z)^(Oka) arereferred to as the -1/z, -2z, and -2/(3z) hierarchies, respectively. The rational functions corresponding to 𝒫_-2/(3z)^(Oka)can be constructed from the generalized Okamoto polynomials.The rationalsolutions of (<ref>) for(α,β)∈𝒫_-1/z^(I)∪𝒫_-1/z^(II)∪𝒫_-2z^(III) can be contructedfrom generalized Hermite polynomials.We will analyze these rational solutionsin the remainder of this work.The generalized Hermite polynomials H_m,n(y) are defined for m,n∈ℕ_0by the recurrence relations 2mH_m+1,nH_m-1,n= H_m,nH_m,n” - (H_m,n')^2 + 2mH_m,n^2, 2nH_m,n+1H_m,n-1= -H_m,nH_m,n” + (H_m,n')^2 + 2nH_m,n^2and the initial conditionsH_0,0=H_1,0=H_0,1=1,H_1,1=2y. The name arises from the fact that H_m,1(y) = H_m(y) and H_1,n(y) = i^-nH_n(iy),where for m∈ℕ_0, H_m(y) is the standard Hermite polynomialdefined by the generating function e^2sy-s^2 = ∑_n=0^∞H_n(y)s^n/n!.The generalized Hermite polynomials also have the symmetry H_m,n(iy) = i^mnH_n,m(y).While we will not use them, it is interesting to note that their zeros satisfyvarious sum relations that generalize the Stieltjes relations for the zerosof Hermite polynomials <cit.>. The connection to the rational Painlevé-IV functions is that w_m,n^(I)(y) := d/dylog(H_m+1,n(y)/H_m,n(y))solves the Painlevé-IV equation (<ref>) with parameters(α,β)=(α_m,n^(I),β_m,n^(I)),w_m,n^(II)(y) := -d/dylog(H_m,n+1(y)/H_m,n(y))solves (<ref>) with parameters(α,β)=(α_m,n^(II),β_m,n^(II)), andw_m,n^(III)(y) := -2y+d/dylog(H_m,n+1(y)/H_m+1,n(y)) = -2y - w_m,n^(I)(y) - w_m,n^(II)(y)solves (<ref>) for(α,β)=(α_m,n^(III),β_m,n^(III)). §.§ Outline and resultsOur starting point is the known identity (<ref>) expressing thegeneralized Hermite polynomial H_m,n in terms of a Hankel determinant ofHermite polynomials.In Lemma <ref> we rewrite this as aHankel determinant of certain moments (defined in (<ref>)) of a measuresupported on the unit circle.This establishes a connection to the associatedorthogonal polynomials on the unit circle (see (<ref>)), and wewrite the rational Painlevé-IV functions in terms of these orthogonalpolynomials and their normalization constants in (<ref>) and (<ref>) (see also (<ref>)).We write down the standardRiemann-Hilbert problem associated to the orthogonal polynomials, and showhow to directly extract the rational Painlevé-IV functions from theRiemann-Hilbert problem in Lemmas <ref> and<ref>.In <ref> we compute the so-called g-function, astandard tool used to regularize the Riemann-Hilbert problem and turnoscillatory jumps into constants.By studying topological changes in the levellines of the related phase function φ, we derive an explicit form ofthe boundary curve, which we now state.Fix r∈[1,∞).Let x_c(r) bethe unique value of x satisfying r^4x^8-24r^2(r^2+r+1)x^4+32r(2r^3+3r^2-3r-2)x^2 - 48(r^2+r+1)^2 = 0with (x_c)>0 and (x_c)>0.The four points{± x_c,±x_c} will be the four corners of the boundary of theelliptic region as well as the four branch points of a function Q we willdefine shortly.While it ispossible to solve (<ref>) exactly since it is a quartic inx^2, we simply note that for r=1 the corner points are the four xvalues satisfying x^4 = 36-24√(3) (r=1),so that x_c(1)≈ 1.086+1.086i (compare Figure <ref>).Nowdefine Q(x;r) as the unique function satisfying 3(1+r)^2Q^4+8(1+r)r^1/2xQ^3+4(r-1+rx^2)Q^2-4=0such that Q(x;r)=-x+𝒪(x^-2) as x→+∞ and cut as shownin Figure <ref>.Also defineS(x;r) := (1+r)Q(x;r)^3+2r^1/2xQ(x;r)^2.Then let a(x;r) and b(x;r) be the two values of z satisfying z^2-S(x;r)z+Q(x;r)^2 = 0.For definiteness we choose (a)<(b) for(x_c)≤(x) ≤(x_c) and(a)>(b) for (x_c)≤(x) ≤(-x_c). Throughout we restrict our analysis to(x_c) ≤(x) ≤(-x_c), which issufficient due to the symmetry (<ref>). We now specify a contour Σ connecting a and b. Define R(z;x,r):=(z^2-S(x;r)z+Q(x;r)^2)^1/2 with R(z)=z+𝒪(1) as z→∞ and branch cutchosen as the straight line segment between a and b.Now defineφ(z;x,r)≡φ(z) by φ(z) :=R(z)/Qz^2 + (1+r-S/2Q^3)R(z)/z - (1+r)log(2z+2R(z)-S)+ (r-1)log(2QR(z)-Sz+2Q^2/z) + log(S^2-4Q^2) - (1+r)iπ.Here all logarithms are chosen with principal branches (as we will only needthe real part of φ the particular choice is unimportant). There is a levelline of (φ(z)) connecting a and b traveling in theclockwise direction around the origin;we call this bounded contour Σ. Now set R(z;x,r) to be the function satisfying R(z;x,r)^2 = z^2-S(x;r)z+Q(x;r)^2that is analytic for z∉Σ and satisfies R(z)=z+𝒪(1) asz→∞. Note we have the useful relations S=a+b,Q=R(0),Q^2=ab.Also define R_c(x;r) ≡ R_c := -((1+r)^2Q^4+2(1+r)QS+4)^1/2/(1+r)Q,with the choice of branch inherited from R(z).Then we have the followingdefinitions of the elliptic region in which the zeros and poles of therational Painlevé-IV functions lie (at least asymptotically) and thecomplementary genus-zero region.See Figures <ref> and<ref>.Fix r∈[1,∞).Then the elliptic regionE_r is the boundeddomain of the complex plane defined by the curves { (1+r)r^1/2x/2 R_c - (1+r)log(2R_c-4/(1+r)Q-S)+ (r-1)log((1+r)Q^3+(1+r)Q^2R_c+S) + log(S^2-4Q^2) } = 0.The genus-zero region is the complement of the closure of the ellipticregion.In <ref> we carry out the Deift-Zhou nonlinearsteepest-descent analysis <cit.>of the Riemann-Hilbert problemfor the orthogonal polynomials. This consists of several standard steps:*Conjugating the jump matrices by a matrix involving the g-function,which identifies the contours that will contribute to the leading-ordersolution.*Opening lenses so all jumps are constants or decaying tothe identity as n→∞.*Solving the model problem obtained by disregarding jumps closeto the identity.*Controlling the errors and showing that the model solution is a goodapproximation to the exact problem.Following this procedure, we obtain the following asymptotic formulas for therational Painlevé-IV functions valid in the genus-zero region.Fix p,q∈ℕ (the positive integers) with p≥ q and setr:=p/q.Fix x in the genus-zero region as defined in Definition<ref>.Then as m,n→∞ along the sequence{m,n}={jp,jq} for j∈ℕ we have1/n^1/2w_m,n^(I)(m^1/2x) = -1/Q(x,r) - S(x,r)/2Q(x,r)^2 + 𝒪(1/n).Theorem <ref> is illustrated in Figure <ref>.Fix p,q∈ℕ with p≥ q and setr:=p/q.Fix x in the genus-zero region as defined in Definition<ref>.Then as m,n→∞ along the sequence{m,n}={jp,jq} for j∈ℕ we have1/n^1/2w_m,n^(II)(m^1/2x) = 1/Q(x,r) - S(x,r)/2Q(x,r)^2 + 𝒪(1/n).Theorem <ref> is illustrated in Figure <ref>.Finally, combining these two theorems with (<ref>) immediately givesthe following. Fix p,q∈ℕ with p≥ q and setr:=p/q.Fix x in the genus-zero region as defined in Definition<ref>.Then as m,n→∞ along the sequence{m,n}={jp,jq} for j∈ℕ we have1/n^1/2w_m,n^(III)(m^1/2x) = -2r^1/2x + S(x,r)/Q(x,r)^2 + 𝒪(1/n).To understand the behavior of the rational Painlevé-IV functions asm,n→∞ with r=m/n fixed, it is sufficient to consider the caser≥ 1 due to the symmetry (<ref>).In the case 0<r<1 (seeFigure <ref>) the natural variable is χ:=n^-1/2y, since thezeros of H_m,n(n^1/2χ) are bounded in the χ plane asn→∞.§.§ A comment on the literatureBefore beginning our analysis we make a few remarks regarding a recent paperby Novokshenov and Schelkonogov <cit.> thatconcerns some of the same questions we address here.In particular, theyare interested in the distribution of the zeros of w_n,n^(III) forlarge n.The proposed strategy is intriguing:determine a Riemann-Hilbertproblem for w_0,0^(III) and then apply Schlesinger/Bäcklundtransformations to obtain Riemann-Hilbert problems for w_n,n^(III). Unfortunately, <cit.> expressingw_n,n^(III) (or, in their notation, u_n,n) in terms of the solutionof the Riemann-Hilbert problem in <cit.> is notcorrect.This means that the subsequent asymptotic results for the rationalPainlevé-IV functions are also incorrect, including<cit.> and<cit.> describing the asymptotic behavior ofw_n,n^(III) and <cit.> for the locationof the zeros.In fact, it is not possible to extract any information aboutw_n,n^(III) from the Riemann-Hilbert problem in<cit.>.In their notation, thisproblem is to find a matrix Y(ξ) analytic for ξ∉ℝsatisfyingY_+(ξ)=Y_-(ξ) 1 2π i e^-n(ξ^2-x^2)0 1for ξ∈ℝ; Y(ξ)=(𝕀+𝒪(ξ^-1))ξ^2n0 0ξ^-2n as ξ→∞ (here the parameter x is, after scaling, the independent variable for thePainlevé-IV functions and is the same as our x defined in(<ref>) if m=n).Then the function(2π ie^nx^2)^-σ_3/2 Y(ξ) (2π ie^nx^2)^σ_3/2 satisfies a Fokas-Its-Kitaev Riemann-Hilbert problem <cit.>for the (standard) Hermite polynomials.The solution to this problem can bewritten exactly in terms of H_n, H_n-1, and their Cauchy transforms,which is not enough information to construct H_n,n or w_n,n^(III).§.§ NotationWe denote the positive integers by ℕ and the nonnegative integersby ℕ_0. If f is a function defined on a specified oriented contour, then f_+(f_-) denotes the boundary value taken from the left (right).Matrices aredenoted by bold capital letters, with the exception of the 2× 2identity matrix 𝕀 and the Pauli matrix σ_3:= 1 0 0 -1 .The (jk)-entry of a matrix M is denoted by [ M]_jk. Acknowledgements.The author thanks Ferenc Balogh, Thomas Bothner,Walter Van Assche, Peter Miller, and Arno Kuijlaars for helpful discussions,the Charles Phelps Taft Research Center for a Faculty Release Fellowship, andthe National Science Foundation for support via grants DMS-1312458 andDMS-1615718.§ THE ASSOCIATED ORTHOGONAL POLYNOMIALS To analyze the asymptotic behavior of these functions we will use adeterminantal formula. Define τ_m,n(y) by τ_m,0(y):=1 and by the n× n Hankel determinantτ_m,n(y) :=H_m(y) H_m+1(y)⋯H_m+n-1(y) H_m+1(y) H_m+2(y)⋯H_m+n(y)⋮ ⋮ ⋱ ⋮H_m+n-1(y) H_m+n(y)⋯H_m+2n-2(y) _n× n for n≥ 1.Then τ_m,n is related <cit.> to the generalized Hermite polynomial H_m,n byτ_m,n(y) = (-1)^⌈(n-1)/2⌉(∏_k=0^n-1[k!2^k]) H_m,n(y),where ⌈·⌉ denotes the ceiling function.We rewriteτ_m,n in terms of certain moments as follows. Let the contour C be the unit circle with clockwiseorientation.For ζ∈ C, define the measureν̣_m(ζ;y):=exp(2y/ζ-1/ζ^2)ζ^mdζ/2π iζ.Define the momentsμ_k^(m)(y) := -∮_Cζ^kν̣_m(ζ;y).Now, via the generating function (<ref>), theCauchy integral formula for derivatives, and the change of variabless=ζ^-1, we see we can write the standard Hermite polynomials as H_m+j(y) = ^̣m+j/^̣m+js.(e^2sy-s^2)|_s=0 = -(m+j)!/2π i∮_C e^2sy-s^2ṣ/s^m+j+1 = -(m+j)!∮_Cζ^jν̣_m(ζ;y) = (m+j)!μ_j^(m)(y).In particular, this means we can write τ_m,n(y) =m!μ_0^(m)(y) (m+1)!μ_1^(m)(y)⋯(m+n-1)!μ_n-1^(m)(y) (m+1)!μ_1^(m)(y) (m+2)!μ_2^(m)(y)⋯(m+n)!μ_n^(m)(y)⋮ ⋮ ⋱ ⋮(m+n-1)!μ_n-1^(m)(y) (m+n)!μ_n^(m)(y)⋯(m+2n-2)!μ_2n-2^(m)(y) _n× n.Define the related n× n Hankel determinant T_m,n(y) := |μ_j+k-2^(m)(y)|_j,k=1^n = μ_0^(m)(y)μ_1^(m)(y)⋯ μ_n-1^(m)(y)μ_1^(m)(y)μ_2^(m)(y)⋯ μ_n^(m)(y)⋮ ⋮ ⋱ ⋮ μ_n-1^(m)(y)μ_n^(m)(y)⋯ μ_2n-2^(m)(y) _n× n.Certain ratios of these determinants can be expressed in terms of normalizationconstants for a family of orthogonal polynomials (see (<ref>)below).We now show how to relate τ_m,n with T_m,n (with shiftedindices), thus providing a bridge between the rational Painlevé-IV functionsand the orthogonal polynomials. τ_m,n(y) = (∏_k=0^n-1[(m+k)!2^k])· T_m-n+1,n(y).We start by writing the right-hand side of (<ref>) in terms ofHermite polynomials: (∏_k=0^n-1[(m+k)!2^k])· T_m-n+1,n = ∏_k=1^n-12^k m!/(m-n+1)!H_m-n+1 (m+1)!/(m-n+2)!H_m-n+2 ⋯ (m+n-1)!/m!H_mm!/(m-n+2)!H_m-n+2 (m+1)!/(m-n+3)!H_m-n+3 ⋯ (m+n-1)!/(m+1)!H_m+1 ⋮ ⋮ ⋱ ⋮ m H_m-1(m+1)H_m⋯ (m+n-1)H_m+n-2H_m H_m+1 ⋯H_m+n-1.Our goal is to manipulate τ_m,n into this form.We start bycompletely reversing the order of the rows: τ_m,n = ∏_k=1^n-1(-1)^kH_m+n-1H_m+n ⋯H_m+2n-2H_m+n-2H_m+n-1 ⋯H_m+2n-3H_m+n-3H_m+n-2 ⋯H_m+2n-4 ⋮ ⋮ ⋱ ⋮ H_m+1H_m+2 ⋯H_m+n H_m H_m+1 ⋯H_m+n-1.Note that the nth row is in the desired form (up to the overall constant). We now perform a set of operations on the first n-1 rows that will leavethe (n-1)st row in the desired form.Repeating this set of operations onthe first n-2 rows, then the first n-3 rows, and so on, will establishthe identity.The Hermite polynomials satisfy the recursion relationH_m+1(y) = 2yH_m(y) - 2mH_m+1(y).Using this in the top row gives τ_m,n = ∏_k=1^n-1(-1)^k2yH_m+n-2-2(m+n-2)H_m+n-3 ⋯2yH_m+2n-3 - 2(m+2n-3)H_m+2n-4H_m+n-2 ⋯H_m+2n-3H_m+n-3 ⋯H_m+2n-4 ⋮ ⋱ ⋮ H_m+1 ⋯H_m+n H_m⋯H_m+n-1.Note that we can eliminate the terms proportional to y by subtracting amultiple of the second row from the first row.We can then pull out thecommon -2 factor from the first row, and subtract a multiple of the thirdrow from the first row to change the coefficients in front of the Hermitepolynomials.The result isτ_m,n = -2∏_k=1^n-1(-1)^km H_m+n-3 ⋯(m+n-1)H_m+2n-4H_m+n-2 ⋯H_m+2n-3H_m+n-3 ⋯H_m+2n-4 ⋮ ⋱ ⋮ H_m+1 ⋯H_m+n H_m⋯H_m+n-1.We now carry out the same procedure on rows 2,3,…,n-1:apply therecursion relation, use the next row to remove terms proportional to y,and then use the subsequent row to change the coefficient of the firstentry to m.(For row n-1 the leading coefficient in column 1 is alreadym once the y-terms are removed).Once every row has been modified inthis way we obtain τ_m,n = 2^n-1∏_k=1^n-2(-1)^km H_m+n-3 ⋯(m+n-1)H_m+2n-4m H_m+n-4 ⋯(m+n-1)H_m+2n-5m H_m+n-5 ⋯(m+n-1)H_m+2n-6 ⋮ ⋱ ⋮ m H_m-1 ⋯(m+n-1)H_m+n-2 H_m⋯H_m+n-1.This fixes the last two rows.We now repeat this procedure on rows1,...,n-2, the only difference being that we change the leadingcoefficients in column 1 to m(m-1).The result isτ_m,n = ∏_k=n-2^n-12^k∏_j=1^n-3(-1)^jm(m-1) H_m+n-5 ⋯(m+n-1)(m+n-2)H_m+2n-6m(m-1) H_m+n-6 ⋯(m+n-1)(m+n-2)H_m+2n-7 ⋮ ⋱ ⋮ m(m-1) H_m-2 ⋯(m+n-1)(m+n-2)H_m+n-3m H_m-1 ⋯(m+n-1)H_m+n-2 H_m⋯H_m+n-1.Note that now the final three rows have the intended form.Repeating thisprocedure n-3 more times, each time involving one less row than before andmodifying the leading coefficient appropriately (i.e. so the last rowchanged has the correct coefficient), yields the form(<ref>), as desired.We observe that the result of Lemma <ref> can be written interms of Hermite polynomials as |H_m+j+k-2(y)|_j,k=1^n = ∏_k=0^n-12^k·|(m+k-1)!/(m-n+j+k-1)!H_m-n+j+k-1(y)|_j,k=1^n.Hankel determinants of orthogonal polynomials such as the expression on theleft-hand side are known as Turánians.The Hermite Turánian canbe expressed as a Wronskian for general m<cit.> and evaluated in closed form for m=0<cit.>.For more background andreferences on Turánians see <cit.>.For fixed m∈ℕ_0, define the monic orthogonal polynomialsψ_n^(m), n≥ 0, by∮_Cψ_n^(m)(ζ;y)ζ^jν̣_m(ζ;y) = δ_jnh_n^(m)(y),j=0,…,n,where δ_jn is the Kroneker delta function and h_n^(m)(y) is thenormalization constant (that is, constant in ζ but with parametricdependence on y). Then (see, for example, <cit.>) the value of the orthogonal polynomials evaluated at ζ=0 can beexpressed in terms of determinants viaψ_n^(m)(0;y) = (-1)^nT_m+1,n(y)/T_m,n(y),and the normalization constant h_n^(m) can be expressed ash_n^(m)(y) = -T_m,n+1(y)/T_m,n(y).Note that (<ref>) and (<ref>) provide ways toshift the two indices of T_m,n(y).Applying (<ref>),(<ref>), (<ref>), and (<ref>) to(<ref>)–(<ref>) gives w_m,n^(I)(y) = d/dylog(τ_m+1,n(y)/τ_m,n(y)) = d/dylog(T_m-n+2,n/T_m-n+1,n) = ∂/∂ ylog(ψ_n^(m-n+1)(0;y))andw_m,n^(II)(y) = d/dylog(τ_m,n(y)/τ_m,n+1(y)) = d/dylog(T_m-n+1,n(y)/T_m-n,n+1(y)) = ∂/∂ ylog(ψ_n^(m-n)(0;y)/h_n^(m-n)(y)).Note that w_m,n^(III)(y) can also be expressed in terms of theorthogonal polynomials and their normalization constants through the previoustwo equations and (<ref>).We now introduce theFokas-Its-Kitaev Riemann-Hilbert problem <cit.> in order to analyzethe large-degree behavior of the orthogonal polynomials.[Unscaled orthogonal polynomial problem] Fix y∈ℂ and m,n∈ℕ. Seek a 2× 2 matrix M_m,n(ζ;y)with the following properties:Analyticity:M_m,n(ζ;y) is analytic for ζ∈ℂ except on C (the unit circle orientedclockwise) with Hölder-continuous boundary values.Jump condition:The boundary values taken byM_m,n(ζ;y) on C are related by the jump condition M_m,n+(ζ;y)= M_m,n-(ζ;y)11/2π iζexp(2y/ζ-1/ζ^2+mlogζ) 0 1 , ζ∈ C.Normalization:As ζ→∞, the matrixM_m,n(ζ;y) satisfies the conditionM_m,n(ζ;y) = (𝕀+𝒪(ζ^-1))ζ^nσ_3with the limit being uniform with respect to direction.This Riemann-Hilbert problem is solvable exactly when ψ_n^(m) exists,andψ_n^(m)(ζ;y) = [ M_m,n(ζ;y)]_11 (that is, the 11-entry of M) whileh_n^(m)(y) = -2π ilim_ζ→∞ζ[ M_m,n(ζ;y)ζ^-nσ_3-𝕀]_12.Motivated by the exponent in (<ref>), we define rescaled versions ofy and ζ:x:=m^-1/2y,z:=n^1/2ζ.These definitions suggest scaling the orthogonal polynomials as well.Define Ψ_n^(m)(z;x):=n^n/2ψ_n^(m)(z/n^1/2;m^1/2x), ℋ_n^(m)(x):=n^n+m/2h_n^(m)(m^1/2x).These new polynomials satisfy the orthogonality relations∮_CΨ_n^(m)(z;x)z^jdV_m(z;x) = δ_jnℋ_n^(m)(x),j=0,…,n,dV_m:=exp(n[2r^1/2x/z-1/z^2])z^r· ndz/2π iz,where r=m/n.The desired rational functions can be expressed in terms of the scaledorthogonal polynomials asm^1/2w_m,n^(I)(m^1/2x) = ∂/∂ xlog(Ψ_n^(m-n+1)(0;x)) = ∂/∂ xΨ_n^(m-n+1)(0;x)/Ψ_n^(m-n+1)(0;x) and m^1/2w_m,n^(II)(m^1/2x) = ∂/∂ xlog(Ψ_n^(m-n)(0;x)/ℋ_n^(m-n)(x)).We now pose a Riemann-Hilbert problem for the orthogonal polynomialsΨ_n^(m-n+1)(z;x). [Scaled orthogonal polynomial problem] Fix x∈ℂ and m,n∈ℕ with m≥ n and setr=m/n.Find the unique 2× 2 matrix N_m,n(z;x)with the following properties:Analyticity:N_m,n(z;x) is analytic in z except on C (the unit circle oriented clockwise) with Hölder-continuous boundary values.Jump condition:The boundary values taken byN_m,n(z;x) on C are related by the jump condition N_m,n+(z;x)= N_m,n-(z;x)11/2π ie^-nθ(z;x,r)0 1 ,z∈ C, where θ(z;x,r):=(1-r)log z -2r^1/2x/z + 1/z^2.Normalization:As z→∞, the matrixN_m,n(z;x) satisfies the conditionN_m,n(z;x) = (𝕀+𝒪(z^-1))z^nσ_3with the limit being uniform with respect to direction.It is immediate that Ψ_n^(m-n+1)(0;x) = [ N_m,n(0;x)]_11 andℋ_n^(m-n+1)(x) = -2π ilim_z→∞z[ N_m,n(z;x)z^-nσ_3-𝕀]_12.In the next two lemmas we show how to extract w_m,n^(I) andw_m,n^(II) directly from the solution of the Riemann-Hilbert problem. Write the expansion of N_m,n(z;x) about z=0 as N_m,n(z;x) =N_0(x) +N_1(x)z + 𝒪(z^2),where N_0(x) and N_1(x) are independent of z.Then 1/n^1/2w_m,n^(I)(m^1/2x) = ([ N_0(x)]_11[ N_0(x)]_22+[ N_0(x)]_12[ N_0(x)]_21-1)[ N_1(x)]_11/[ N_0(x)]_11 - 2[ N_0(x)]_12[ N_1(x)]_21.From (<ref>), we haveΨ_n^(m-n+1)(0;x)=[ N_0(x)]_11.Thus, from the last expression in (<ref>) we merely need toexpress ∂/∂ xΨ_n^(m-n+1)(0;x) = [∂/∂ x N_m,n(0;x)]_11 in terms of (undifferentiated)entries of N_m,n.Define N_m,n(z;x):= N_m,n(z;x)e^-nθ(z;x,r)σ_3/2.This function is analytic in ℂ\{0∪ C} with a jumpdiscontinuity on C that is independent of x (and z).This means that∂/∂ x N_m,n(z;x) has the sameproperties with the same jump on C.It follows that W_m,n(z;x):=(∂/∂ x N_m,n(z;x)) N_m,n(z;x)^-1 is analytic in ℂ\ 0.Inserting (<ref>) into(<ref>) givesW_m,n(z;x) = (∂/∂ x N_m,n(z;x)) N_m,n(z;x)^-1 + nr^1/2/z N_m,n(z;x)σ_3 N_m,n(z;x)^-1.This shows that W_m,n(z;x) has a simple pole at z=0 and, inparticular, that z W_m,n(z;x) is entire in z.Insertingthe large-z expansion (<ref>) into (<ref>) (using(<ref>)) shows that W_m,n(z;x)=𝒪(z^-1) asz→∞.This demonstrates that z W_m,n(z;x) is bounded asz→∞.Therefore Liouville's theorem tells us thatz W_m,n(z;x) is a constant matrix (i.e. independent of z withparametric dependence on x).This constant can be determined byconsidering (<ref>) and noting that the first summand on theright-hand side is bounded as z→ 0.ThusW_m,n(z;x) = nr^1/2/z N_m,n(0;x)σ_3 N_m,n(0;x)^-1.Combining (<ref>) and (<ref>) gives ∂/∂ x N_m,n(z;x) = nr^1/2/z( N_m,n(0;x)σ_3 N_m,n(0;x)^-1 N_m,n(z;x) -N_m,n(z;x)σ_3).Evaluating both sides at z=0 (using the expansion (<ref>) onthe right-hand side) yields ∂/∂ x N_m,n(0;x) = nr^1/2( N_0(x)σ_3 N_0(x)^-1 N_1(x) -N_1(x)σ_3).Therefore ∂/∂ xΨ_n^(m-n+1)(0;x) = [∂/∂ x N_m,n(0;x)]_11 = nr^1/2[ ( [ N_0(x)]_11[ N_0(x)]_22 + [ N_0(x)]_12[ N_0(x)]_21 - 1)[ N_1(x)]_11 - 2[ N_0(x)]_11[ N_0(x)]_12[ N_1(x)]_21].Combining (<ref>), (<ref>), and(<ref>) finishes the proof.Write the expansion of N_m,n(z;x) as z→∞ as N_m,n(z;x) = (𝕀+ N_-1(x)/z+𝒪(1/z^2))z^nσ_3 and recall the expansion (<ref>) about z=0.Then1/n^1/2w_m+1,n^(II)((m+1)^1/2x) = (1/n^1/2w_m,n^(I)(m^1/2x) + 2[ N_0]_11[ N_0]_12/[ N_-1]_12)(1+𝒪(1/m)).Here w_m,n^(I) can be expressed in terms of N_m,n viaLemma <ref>.Starting from (<ref>), we shift m→ m+1 and use Lemma<ref> to discover1/n^1/2w_m+1,n^(II)((m+1)^1/2x) = (1/n^1/2∂/∂ xlog(Ψ_n^(m-n+1)(0;x)) - 1/n^1/2∂/∂ xlog(ℋ_n^(m-n+1)(x)))1/(m+1)^1/2 = (1/n^1/2w_m,n^(I)(m^1/2x) - 1/n· r^1/2∂/∂ xlog(ℋ_n^(m-n+1)(x)))m^1/2/(m+1)^1/2 = (1/n^1/2w_m,n^(I)(m^1/2x) - 1/n· r^1/2∂/∂ xℋ_n^(m-n+1)(x)/ℋ_n^(m-n+1))(1+𝒪(1/m)).From (<ref>) and (<ref>) we haveℋ_n^(m-n+1)(x) = -2π i[ N_-1(x)]_12.We now express ∂/∂ xℋ_n^(m-n+1)(x) = -2π i∂/∂ x[ N_-1(x)]_12 in terms ofundifferentiated entries of N_m,n.Insert the large-z expansion(<ref>) into the expression (<ref>) forW_m,n:W_m,n(z;x) = 1/z(∂/∂ x N_-1(x) + n· r^1/2σ_3) + 𝒪(1/z^2).Recalling from the proof of Lemma (<ref>) thatz W_m,n is a constant matrix, the 𝒪(z^-2) terms mustbe identically zero.Combining this expression with(<ref>) gives ∂/∂ x N_-1(x) + n· r^1/2σ_3 = nr^1/2 N_0(x)σ_3 N_0(x)^-1.Taking the (12)-entry of both sides generates ∂/∂ x[ N_-1(x)]_12 = -2nr^1/2[ N_0(x)]_11[ N_0(x)]_12.Using (<ref>) and (<ref>) in (<ref>)completes the proof of the lemma. § DETERMINATION OF THE BOUNDARY CURVE We begin the Riemann-Hilbert analysis by finding the g-function andrelated phase function φ.This will be sufficient to specify theboundary of the elliptic region, which will be used in <ref>to compute the asymptotics of the rational Painlevé-IV functions in thegenus-zero region. §.§ Construction of the g-functionSuppose two complex numbers a=a(x,r) and b=b(x,r) are given, along with anoriented contour Σ=Σ(x,r) from a to b (specifying thesequantities is part of the process of defining the g-function).Thegenus-zero g-function is determined via the following Riemann-Hilbert problem. [The g-function] Fix x∈ℂ and r∈[1,∞) andfind g(z)=g(z;x,r) such that Analyticity:e^g(z;x,r) is analytic for z∈ℂexcept on Σ, where it attains Hölder-continuous boundary values at allinterior points.The function g(z;x,r) also has a logarithmic branch cutthat will play no role since g only appears exponentiated.Jump condition: g_+(z)+g_-(z) = θ(z)+ℓ,z∈Σfor some constant ℓ=ℓ(x;r).Normalization: g(z)=log z + 𝒪(1/z),z→∞.There are some values of x for which it is not possible to pick a singleconnected contourΣ such that this Riemann-Hilbert problem is solvable.When it ispossible, then the resulting outer model problem (see Riemann-Hilbert Problem<ref> below) has jumps on a single band and the associated Riemannsurface is genus zero.As a result, we dub the region where theRiemann-Hilbert problem where g is solvable the genus-zero region(see Definition <ref>).We then show that the Painlevé-IVfunctions are (asymptotically) free of zeros and poles in this region.Given g(z) and ℓ, we could define a function φ byφ(z;x,r)=θ(z;x,r) - 2g(z;x,r) + ℓ. In actuality, we will work in the opposite order, first determiningφ'(z), integrating to find φ(z), and thenusing (<ref>) to find the explicit formula for g(z). Note φ'(z) is specified by the following Riemann-Hilbert problem. [The phase function φ] Fix x∈ℂ andr∈[1,∞) and find φ'(z)≡φ'(z;x,r) such that Analyticity:φ'(z;x,r) is analytic for z∈ℂ except at z=0 and on Σ, where itattains Hölder-continuous boundary values at all interior points.Jump condition:φ'_+(z)+φ'_-(z) = 0,z∈Σ.Pole at z=0:φ'(z) = θ'(z) + 𝒪(1) = -2/z^3+2r^1/2x/z^2+1-r/z + 𝒪(1),z→ 0.Normalization:φ'(z)=-1+r/z + 𝒪(1/z^2),z→∞.We now see how the defining relations (<ref>) and(<ref>) for Q and S arise.If we momentarily assume a(x;r),b(x;r), and Σ are known, then we can define R(z;x,r) by(<ref>).Furthermore, writing a+b as S and R(0) as Q, then wecan see that in order to satisfy the analyticity, jump, and normalizationconditions in Rieman-Hilbert Problem <ref>, we can chooseφ'(z) to have the formφ'(z) = -((1+r)z+2/Q)R(z)/z^3.Now for φ'(z) to satisfy the pole condition (<ref>) atz=0, S and Q must satisfy the moment conditions(1+r)Q^3-S/2Q^2 = -r^1/2x, 4Q^2-2(1+r)SQ^3-S^2/8Q^4 = r-1/2.Solving the first equation for S yields the relation (<ref>). Plugging that into the second yields the quartic equation(<ref>) for Q.The specific sheet so thatQ(x;r)=-x+𝒪(x^-2) as x→∞ is chosen so the signaturecharts in Figures <ref> and <ref> hold. Furthermore, we have assumedR^2 = z^2-(a+b)z+ab = z^2-Sz+Q^2, so we must therefore specify a and bby (<ref>).We pause to indicate how the branch points of Q(x) can be identified.Forany branch point x_b, the pair {x_b,Q(x_b)} must satisify(<ref>) as well as its derivative with respect to Q,12(1+r)^2Q^3 + 24(1+r)r^1/2xQ^2 + 8(r-1+rx^2)Q = 0,since the implicit function theorem must fail at a branch point.Multiplying(<ref>) by Q gives an equation with a term proportional toQ^4.This can be used to remove the term proportional to Q^4 in(<ref>), yielding 8(1+r)r^1/2xQ^3 + 8(r-1+rx^2)Q^2 - 16 = 0.Now (<ref>) can be used again to remove the term proportional toQ^3, giving 3(1+r)(rx^2+1-r)Q^2 + 2r^1/2x(rx^2+r-1)Q + 6(1+r) = 0.Now dividing (<ref>) gives an equation that can be used toeliminate the term proportional to Q^2, yielding a linear equation for Q that givesQ = -r^2x^4+4r^2+4r+4/2r^1/2(1+r)(rx^3+2(1-r)x).Plugging this into (<ref>) yields the octic equation(<ref>) for x that the branch points must satisfy.Thisequation is actually quartic in x^2, and so the roots can be determinedexactly.For r∈[1,∞), two of the roots are on the real axis, two areon the imaginary axis, and one is in each open quadrant.A series expansion ofQ about the points on the axes shows that Q is actually analytic there, andthe four branch points are the ones off the axes (recall that Q is also thesolution of a quartic (<ref>), and so can be written downexplicitly to perform the series expansions).We return to the process of determining φ.Now Q, S, a, and bare well defined by (<ref>), (<ref>), and(<ref>).So far we have seen that, for any choice of Σ, if wedefine R by (<ref>) then φ'(z) must be given by(<ref>).The time has come to specify Σ. Recall the definition of R in (<ref>). Then the function φ(z;x,r)≡φ(z) asdefined in (<ref>) is an antiderivative of (<ref>)(with R replaced with R). The integrationconstant is chosen so φ(a)=0.Now(φ(a))=(φ(b)), and for |x|sufficiently large there are twocontours connecting a and b that do not pass through z=0 (in fact, theexistence of both of these contours is equivalent to being in the genus-zeroregion – see Lemma <ref>).We choose Σ to be thecontour connecting a to b when traveling clockwise around the origin.Nowthat Σ isdefined, we can define R(z) by (<ref>) (which amounts to a deformationof the branch cut for R(z)), and defineφ(z;x,r)≡φ(z) viaφ(z) :=R(z)/Qz^2 + (1+r-S/2Q^3)R(z)/z - (1+r)log(2z+2R(z)-S)+ (r-1)log(2QR(z)-Sz+2Q^2/z) + log(S^2-4Q^2) - (1+r)iπ.Here the branches of the logarithms are chosen so φ_+(z)+φ_-(z)=0for z∈Σ, a choice that depends on both x and r. The behavior of the Riemann-Hilbert problem is controlled by (φ(z)) (see Figures <ref> and <ref>).Now we can set g(z;x,r) := 1/2θ(z;x,r) - 1/2φ(z;x,r) + ℓ(x;r)/2,where only ℓ remains unspecified.The role of ℓ is to ensure thenormalization (<ref>) for g(z), so we choose ℓ(x;r) := 2lim_z→∞(log z - 1/2θ(z;x,r) + 1/2φ(z;x,r)).While ℓ(x;r) can be computed in terms of elementary functions, we willnot need its explicit form. §.§ The boundary and corners of the elliptic regionFor generic values of x and r the function φ'(z) (recall(<ref>)) has three distinct zeros at a(x;r),b(x;r), and c(x;r):=-2/(1+r)Q(x;r).The transition from the genus-zero region to the elliptic region occurs whenone of the zero-level lines of (φ) crosses c, i.e.(φ(c))=0.See the plots with x≈ 1.0253 andx≈ 1.0253i in Figure <ref> and the plot withx≈ 1.2953 in Figure <ref>.This condition can be writtenin the more explicit form (<ref>), where R_c=R(c).It isimportant to note that the boundary of the curvilinear rectangles illustratedin Figures <ref>–<ref> are not the only curves alongwhich (φ(c))=0.There are four additional curves that start at thefour corners and tend to infinity (see Figure <ref>).Thesignature chart of (φ(z)) along one of these lines is illustratedin the plot with x=1.2+1.2i in Figure <ref>.Nevertheless,the genus-zero Riemann-Hilbert analysis in <ref> will gothrough without change along these curves, so they are part of the genus-zeroregion.As illustrated in Figure <ref>, the breaking mechanism at theboundary of the elliptic region depends on whether(x_c)<(x)<(x_c) or(x_c)<(x)<(-x_c).In the first case, a regionin which (φ(z))>0 is pinched off, as in the plot withx≈ 1.0253 in Figure <ref>.Looking ahead to Figure<ref>, this means it is no longer possible to passthe gap contour Γ through this region in which its jump is exponentiallyclose to the identity, and it is necessary to open a second band to control theRiemann-Hilbert problem once x has moved into the elliptic region.On theother hand, for (x_c)<(x)<(-x_c) (see the plot withx≈ 1.0253i in Figure <ref>), it is a region in which(φ(z))<0 that is pinched off.In this case the gap Γremains controlled, and the necessary modification occurs on the bandΣ.We conjecture that, as x enters the elliptic region from the topboundary, a second band opens up directly on Σ and then moves closer tothe origin as (x) decreases.This gives a consistent picture in which,just inside the boundary, there is one small and one large band.As xmoves clockwise, the larger band rotates clockwise in the z-plane while thesmall band rotates counterclockwise.The small band is near an endpointof the large band exactly when x is near a corner of the boundary region. We emphasize the Riemann-Hilbert analysis in <ref> goesthrough uniformly for all x in the genus-zero region as long as x staysbounded away from the boundary curve. We now identify the corner points.These are the values of x for whichc(x)=a(x) or c(x)=b(x) (see the plot with x=x_c in Figure<ref>, as well as <cit.> for a similaranalysis for the Painlevé-II equation).In either case we havec^2-Sc+Q^2=0 from(<ref>).Using (<ref>) and (<ref>) toexpress S and c in terms of Q, x, and r yields 3(1+r)^2Q^4 + 4(1+r)r^1/2xQ^3 + 4 = 0.Adding this to (<ref>) gives6(1+r)^2Q^4 + 12(1+r)r^1/2xQ^3 + 4(r-1+rx^2)Q^2 = 0,which is equivalent to (<ref>), the derivative of(<ref>) with respect to Q.Once (<ref>) holds,the analysis following that equation used to determine the branch points ofQ also holds, and so the corner points must satisfy (<ref>). While there are eight solutions to that equation, only four of them are offthe coordinate axes, and so the geometry of the boundary shows that thecorners are {± x_c,±x_c}.§ ASYMPTOTIC EXPANSION OF THE RATIONAL PAINLEVÉ-IV FUNCTIONS We now apply the Deift-Zhou nonlinear steepest-descent method to obtain anapproximation of N_m,n(z;x).We perform a series oftransformationsN_m,n(z;x) → O_m,n(z;x) → P_m,n(z;x) → Q_m,n(z;x) ≈ R_m,n(z;x).The first transformation (to O_m,n) deforms the jump contours awayfrom the unit circle and onto Σ∪Γ, where Γ lies in aregion where (φ)>0.The second transformation (to P_m,n)introduces the g-function to regularize the jump matrices.In the thirdtransformation (to Q_m,n) we open lenses, which replacesrapidly oscillating jump matrices with ones that are approximately constant. The associated Riemann-Hilbert problem is then replaced with a constant-jumpproblem that can be solved exactly for R_m,n.A key point is thatthe error in approximating Q_m,n with R_m,n can becontrolled, as we will show in Lemma <ref>. §.§ Initial deformation of the contours (N_m,n→ O_m,n)The first step is to deform the jump contours away from the unit circle C. Define a smooth, non-self-intersecting contour Γ starting at b andending at a whose interior is entirely in the region in which(φ(z))>0 (see Figure <ref>).The existenceof Γ in the genus-zero region is shown below in Lemma<ref>.Then Σ∪Γ is a topological deformationof C, as shown in Figure <ref>.Define D_into be the region in the interior of the unit circle but the exterior ofΣ∪Γ, and D_out to be the region in the exterior of theunit circle but the interior of Σ∪Γ (again see Figure<ref>).It is possible one of these regions may beempty.Then defineO_m,n(z;x) :=N_m,n(z;x) 11/2π ie^-nθ(z;x,r) 0 1 , z∈ D_in, N_m,n(z;x) 1-1/2π ie^-nθ(z;x,r)0 1 , z∈ D_out, N_m,n(z;x), z∈ℂ\{D_in∪ D_out}. Now O_m,n(z;x) satisfies exactly the same Riemann-Hilbert problem asN_m,n(z;x) (i.e. Riemann-Hilbert Problem <ref>) with Creplaced by Σ∪Γ.§.§ Introduction of the g-function (O_m,n→ P_m,n)DefineP_m,n(z;x):=e^-nℓσ_3/2 O_m,n(z;x)e^-n(g(z;x,r)-ℓ/2)σ_3.The jump for z∈ C is V_m,n^( P) =P_m,n-^-1 P_m,n+ =e^-n(g_+-g_-) 1/2π ie^n(g_++g_–θ-ℓ)0 e^n(g_+-g_-).Recall that φ(z;x,r) is defined in (<ref>). Note from (<ref>) that g_+(z)-g_-(z) = -φ_+(z) = φ_-(z)for z∈Σ.Also taking into account the asymptotic behavior(<ref>), we are led to the following Riemann-Hilbertproblem. [Introduction of φ] Fix a complex number x in the genus-zero region and m,n∈ℕ withm≥ n and set r=m/n. Determine the unique 2× 2 matrix P_m,n(z;x) with thefollowing properties:Analyticity:P_m,n(z;x) is analytic for z∈ℂ except on Σ∪Γ where itachieves Hölder-continuous boundary values. See Figure<ref>.Jump condition:The boundary values taken byP_m,n(z;x) are related by the jump conditions P_m,n+(z;x)= P_m,n-(z;x) V_m,n^( P)(z;x),whereV_m,n+^( P)(z;x) =e^nφ_+(z;x,r) 1/2π i0 e^nφ_-(z;x,r), z∈Σ,11/2π ie^-nφ(z;x,r)0 1 , z∈Γ.Normalization:As z→∞, the matrixP_m,n(z;x) satisfies the conditionP_m,n(z;x) = 𝕀+𝒪(z^-1)with the limit being uniform with respect to direction.§.§ Opening of the lenses (P_m,n→ Q_m,n)On Σ, the jump matrix V_m,n^( P) has thefactorizatione^nφ_+ 1/2π i0 e^nφ_- =1 0 2π i e^nφ_-101/2π i-2π i 01 0 2π i e^nφ_+1 .We introduce the lens regions Ω_± and the lens boundaries L_± asshown in Figure <ref>.The boundaries L_± are taken to lie inside the regions in whichφ(z)<0 and be such that 0∉(Ω_+∪Ω_-).Make thechange of variablesQ_m,n(z;x):= P_m,n(z;x) 1 0 -2π ie^nφ(z;x,r)1 , z∈Ω_+,P_m,n(z;x) 1 0 2π ie^nφ(z;x,r)1 , z∈Ω_-, P_m,n(z;x),otherwise.We have the following Riemann-Hilbert problem. [Lens-opened problem] Fix a complex number x in the genus-zero region and m,n∈ℕ withm≥ n, and set r=m/n.Determine the unique 2× 2 matrixQ_m,n(z;x) with the following properties:Analyticity:Q_m,n(z;x) is analytic forz∈ℂ\{Σ∪Γ∪ L_+∪ L_-} withHölder-continuous boundary values.See Figure <ref>.Jump condition:The boundary values taken byQ_m,n(z;x) are related by the jump conditionQ_m,n+(z;x)= Q_m,n-(z;x) V_m,n^( Q)(z;x),whereV_m,n^( Q)(z;x) =01/2π i-2π i 0 , z∈Σ,1 0 2π i e^nφ(z;x,r)1 , z∈ L_±,11/2π ie^-nφ(z;x,r)0 1 , z∈Γ.Normalization:As z→∞, the matrixQ_m,n(z;x) satisfies the conditionQ_m,n(z;x) = 𝕀+𝒪(z^-1)with the limit being uniform with respect to direction.§.§ The model and error problemsThe jumps for Q_m,n(z) decay to the identity matrix except for z∈Σ (although this decay is not uniform near a and b).We nowdefine a model solution R_m,n(z) that is a goodapproximation for Q_m,n(z) (up to 𝒪(n^-1)) everywherein the complex plane.We begin by defining the outer model Riemann-Hilbertproblem, which is obtained by neglecting all decaying jumps. [The outer model problem] Fix a complex number x in the genus-zero region and m,n∈ℕ withm≥ n and set r=m/n.Determine the unique 2× 2 matrixR_m,n^ (out)(z;x) with the following properties:Analyticity:R_m,n^ (out)(z;x) is analytic in z except on Σ with Hölder-continuous boundaryvalues in the interior of Σ and at worst quarter-root singularitiesat the endpoints.Jump condition:The boundary values taken byR_m,n^ (out)(z;x) on Σ are related by the jump condition R_m,n+^ (out)(z;x)= R_m,n-^ (out)(z;x)01/2π i-2π i 0 .Normalization:As z→∞, the matrixR_m,n^ (out)(z;x) satisfies the conditionR_m,n^ (out)(z;x) = 𝕀+𝒪(z^-1)with the limit being uniform with respect to direction.This constant-jump problem can be solved in a standard way by diagonalizingthe matrix (thereby reducing the problem to two scalar problems) and thenusing the Plemelj formula.Alternately, it is straightforward to check thatRiemann-Hilbert Problem <ref> is satisfied by R_m,n^ (out)(z;x) := γ(z;x,r)+γ(z;x,r)^-1/2 γ(z;x,r)-γ(z;x,r)^-1/4ππ(γ(z;x,r)-γ(z;x,r)^-1)γ(z;x,r)+γ(z;x,r)^-1/2,where γ(z;x,r):=(z-a/z-b)^1/4 is analytic for z∉Σ and satisfies lim_z→∞γ(z)=1.The outer model solution R_m,n^(out)(z) is a goodapproximation of Q_m,n(z) for all z except in smalln-independent neighborhoods 𝔻_a and 𝔻_b of the bandendpoints a and b, respectively.Here the decay of the jumps on L_±and Γ to the identity is not uniform.However, it is possible toconstruct functions R_m,n^(a)(z) and R_m,n^(b)(z) interms of Airy functions thatsolve the Riemann-Hilbert problem exactly in their respective neighborhood andclosely match the outer parametrix R_m,n^(out)(z) on theboundaries.The construction of Airy parametrices is standard (see, forexample, <cit.>).Here we follow<cit.>.First, we have the local expansionsφ(z) = C_a(z-a)^3/2 + 𝒪((z-a)^5/2),z∈𝔻_a,φ(z) = 2π i + C_b(z-b)^3/2 + 𝒪((z-a)^5/2),z∈𝔻_b,(for appropriate choices of the square roots) where C_a and C_b are nonzeroand independent of z.Then define two local coordinates s_a(z) := e^iπ(3n/4)^2/3ϕ(z)^2/3 forz∈𝔻_a;s_b(z) := (3n/4)^2/3(ϕ(z)-2π i)^2/3 forz∈𝔻_bsuch that if z∈𝔻_a then Γ is mapped to the negative realaxis, whileif z∈𝔻_b then Γ is mapped to the positve realaxis.Set V:=1/√(2) 1 -i -i 1and define the analytic prefactorsB_a(z) :=R_m,n^(out)(z)(2π i)^-σ_3/2 -i -i 1 -1 (e^-iπs_a(z))^σ_3/4, B_b(z) : =R_m,n^(out)(z)(2π i)^-σ_3/2 -i i 1 1s_b(z)^-σ_3/4.Let A(s) be the function defined in<cit.> and built out of Airy functions withjumps on (s)∈{0,±2π/3,π} as given in<cit.> and satisfying A(s) = s^σ_3/4/2√(π) -1 i 1 i (𝕀+ 1/48s^3/2 1 6i 6i -1+ 𝒪(s^-3))e^-2s^3/2σ_3/3,s→∞.Also let A(s) be the function defined in<cit.> and built out of Airy functions withjumps on (s)∈{0,±π/3,π} as given in<cit.> and satisfying A(s) = (e^-iπs)^-σ_3/4/2√(π) 1 -i 1 i (𝕀+ i/48s^3/2 -1 6i 6i 1+ 𝒪(s^-3))e^-2is^3/2σ_3/3,s→∞.Then the Airy parametrices areR_m,n^(a)(z) := i√(π) B_a(z) A(s_a(z))e^2is_a(z)^3/2σ_3/3(2π i)^σ_3/2,z∈𝔻_a, R_m,n^(b)(z) := -i√(π) B_b(z) A(s_b(z))e^2s_b(z)^3/2σ_3/3(2π i)^σ_3/2,z∈𝔻_b.The explicit form of the parametrix is only necessary to recover the𝒪(n^-1) terms in the solution of the Riemann-Hilbert problem.For us it suffices to know that R_m,n^(a)(z) satisfies the samejump conditions as Q_m,n(z) for z∈𝔻_a,R_m,n^(b)(z) satisfies the same jump conditions asQ_m,n(z) for z∈𝔻_b, and R_m,n^(a)(z)= R_m,n^(out)(z)(𝕀+𝒪(n^-1))forz∈∂𝔻_a;R_m,n^(b)(z)= R_m,n^(out)(z)(𝕀+𝒪(n^-1))forz∈∂𝔻_buniformly for x in the genus-zero region bounded away from the corners of theelliptic region.At the corners one of the band endpoints collides with thethird critical point c and a different parametrix is required (see<cit.> for a related analysis for the rational Painlevé-IIfunctions).The global model solution is now defined as R_m,n(z;x):= R_m,n^(out)(z;x), z∈ℂ\{𝔻_a∪𝔻_b}, R_m,n^(a)(z;x), z∈𝔻_a, R_m,n^(b)(z;x), z∈𝔻_b.The error or ratio function is S_m,n(z;x):= Q_m,n(z;x) R_m,n(z;x)^-1.It satisfies the following Riemann-Hilbert problem.Note in particular thatS_m,n(z) has no jump across Σ or inside 𝔻_a or𝔻_b, but does have jumps across ∂𝔻_a and∂𝔻_b. [The error problem] Fix a complex number x in the genus-zero region and m,n∈ℕ withm≥ n and set r=m/n.Determine the unique 2× 2 matrixS_m,n(z;x) with the following properties:Analyticity:S_m,n(z;x) is analytic in z except onJ^(S):=∂𝔻_a∪∂𝔻_b∪((L_+∪ L_-∪Γ)∩(𝔻_a∪𝔻_b)^c) with Hölder-continuous boundary values.Weorient ∂𝔻_a and ∂𝔻_b clockwise.SeeFigure <ref>.Jump condition:The boundary values taken byS_m,n(z;x) are related by the jump conditionsS_m,n+(z;x)= S_m,n-(z;x) V_m,n^( S)(z;x),whereV_m,n^( S)(z;x) =R_m,n-(z;x) V_m,n^( Q)(z;x) R_m,n+(z;x)^-1 = R_m,n^ (out)(z;x)1 0 2π i e^nφ(z;x,r)1R^ (out)_m,n(z;x)^-1, z∈ L_±∩(𝔻_a∪𝔻_b)^c,R_m,n^ (out)(z;x) 11/2π ie^-nφ(z;x,r)0 1R^ (out)_m,n(z;x)^-1, z∈Γ∩(𝔻_a∪𝔻_b)^c, R^ (a)_m,n(z;x) R^ (out)_m,n(z;x)^-1, z∈∂𝔻_a, R^ (b)_m,n(z;x) R^ (out)_m,n(z;x)^-1, z∈∂𝔻_b.Normalization:As z→∞, the matrixS_m,n(z;x) satisfies S_m,n(z;x) = 𝕀+𝒪(z^-1)with the limit being uniform with respect to direction.We now show that the jump matrices for the error solution S_m,n aresmall as n→∞.Fix δ>0.Then for z∈ J^(S) V_m,n^( S)(z;x) = 𝕀 + 𝕆(1/n)with the error term uniform in x if dist(x,E_r)>δ.For z∈∂𝔻_a∪∂𝔻_b, the necessaryestimate is given by (<ref>).What remains is to show that, inthe genus-zero region, the signature chart of (φ(z)) has thetopology shown in the genus-zero plots in Figures <ref> and<ref>.More specifically, weneed to show that (except for the endpoints a and b), L_± can be placedentirely in a region in which (φ(z))<0, and Γ entirely in aregion in which (φ(z))>0.If so, then with these choices we findthat V_m,n^( S)(z;x) is exponentially close to theidentity on the relevant parts of L_± and Γ, and so(<ref>) holds.As a level set of a function that is harmonic except on Σ and at z=0,{z:(φ(z))=0} consists of a finite number of smooth arcs.Localanalysis at infinity shows there are no zero-level lines of (φ(z))there.The only points at which two or more zero-level lines can intersectare the critical points a, b, and c (see (<ref>)) or theorigin.A direct calculation shows that a and b are distinct and nonzero. We also saw in <ref> that c can coincide with a or b,but only at the corners of the elliptic region, which we avoid.Furthermore,c cannot be zero since Q has no finite singularities.Therefore, we canassume all four points a, b, c, and 0 are distinct.By construction,(φ(a))=(φ(b))=0, and local analysis shows there are threezero-level lines of (φ(z)) emanating from both a and b. Similarly, local analysis at the pole z=0 shows there are four zero-levellines of (φ(z)) intersecting at the origin.As we have seen in<ref>, (φ(c)) is generically nonzero, but thereare four semi-infinite arcs in the complex x-plane along which(φ(c))=0, in which case four zero-level lines of (φ(z))intersect at c.First, assume x is such that (φ(c))≠ 0.In this case we havethree arcs each emerging from a and b and four from 0.Therefore not allthe arcs from a and b can connect to the origin, and at least one mustjoin a and b.Closed contours that are level lines of harmonic functionsmust enclose singularities, and so there are two options:either a secondarc connects a to b and passes around the opposite side of the origin fromthe first such arc, or the other four arcs from a and b all connect to theorigin.We are in the first situation for x sufficiently large and eitherx purely real or purely imaginary (for illustration see the plots withx=1.2 and x=1.2i in Figure <ref>).In this case the signaturechart necessarily has the form show in those plots since (φ(z))<0for z sufficiently large.The only allowable mechanism for the contourtopology to change as x varies is for c to intersect a zero-level lineof (φ(z)), which we have seen only occurs on the semi-infinite arcs.Therefore, off these four arcs we see that the contours L_± and Γcan be chosen appropriately in the exterior of the elliptic region.We now consider x such that (φ(c))=0.The signature chart at acorner point can be seen to have the form shown in the plot with x=x_c inFigure <ref>.It is possible to continuously vary x to thevalue that interests us keeping c on a zero-level curve of (φ(z)).Therefore the signature chart of (φ(z)) must (topologically) havethe form illustrated in the plot with x=1.2+1.2i in Figure<ref>, from which it is clear the contours L_± and Γcan be chosen as needed. We have finally arrived at a small-norm Riemann-Hilbert problem forS_m,n(z;x), that is, one with jumps close to the identity.Thefollowing analysis is standard (see, for example, <cit.> or<cit.>).Recursively define the functions U_0(z):=𝕀,U_k(z):=-1/2π i∫_J_-^(S) U_k-1(u)( V_m,n^( S)(u)-𝕀)/z-udu,in which J_-^(S) means the integration is performed along the minus-side ofJ^(S).Then S_m,n(z) is the sum of an infinite Neumann series:S_m,n(z) = 𝕀 - 1/2π i∑_k=1^∞∫_J^(S) U_k-1(u)( V_m,n^( S)(u)-𝕀)/z-udu.This gives us the boundS_m,n(z) = (𝕀+𝒪(1/(|z|+1)n)),n→∞ that holds uniformly for z∈ℂ\ J^(S) and for x a fixeddistance away from the elliptic region. §.§ The asymptotic expansionWe now prove the main theorems.Retracing the various transformations givesN_m,n(z;x) = e^nℓσ_3/2 S_m,n(z;x) R_m,n(z;x)e^n(g(z;x,r)-ℓ/2)σ_3,z∈ℂ\{Ω_+∪Ω_-∪ D_in∪ D_out}.We therefore have N_m,n(z) = (𝕀+𝒪(1/(|z|+1)n)) γ(z)+γ(z)^-1/2e^ng(z) γ(z)-γ(z)^-1/4πe^-n(g(z)-ℓ)π(γ(z)-γ(z)^-1)e^n(g(z)-ℓ) γ(z)+γ(z)^-1/2e^-ng(z), z∈ℂ\{Ω_+∪Ω_-∪ D_in∪ D_out∪𝔻_a ∪𝔻_b ∪ J^(S)}.In particular, this expression holds for z=0 and for |z| sufficientlylarge.We expand g(z) and γ(z) about z=0:g(z;x,r) = g_0(x,r) + g_1(x,r)z + 𝒪(z^2), γ(z;x,r) = γ_0(x,r) + γ_1(x,r)z + 𝒪(z^2),wherein γ_0 = (a/b)^1/4, γ_1 = a-b/4ab(a/b)^1/4 (interestingly, it will turn out that we will not need the explicit form ofg_0 or g_1).Thus, recalling the expansion (<ref>) forN_m,n, we compute [] [ N_0]_11 = γ_0+γ_0^-1/2e^ng_0(1+𝒪(n^-1)),[ N_0]_12 = γ_0-γ_0^-1/4πe^-n(g_0-ℓ)(1+𝒪(n^-1)), [ N_0]_21 = π(γ_0-γ_0^-1)e^n(g_0-ℓ)(1+𝒪(n^-1)),[ N_0]_22 = γ_0+γ_0^-1/2e^-ng_0(1+𝒪(n^-1)), [ N_1]_11 = 1/2[(γ_0+γ_0^-1)ng_1 + γ_1 - γ_1/γ_0^2]e^ng_0(1+𝒪(n^-1)), [ N_1]_21 = π[(γ_0-γ_0^-1)ng_1 + γ_1 + γ_1/γ_0^2]e^n(g_0-ℓ)(1+𝒪(n^-1)).Inserting these into (<ref>) gives 1/n^1/2w_m,n^(I)(m^1/2x) = 2γ_1(x,r)/γ_0(x,r)(1-γ_0(x,r)^2)/(1+γ_0(x,r)^2) + 𝒪(n^-1).Using the expressions (<ref>) for γ_0 and γ_1produces 1/n^1/2w_m,n^(I)(m^1/2x) = 1/(a(x,r)b(x,r))^1/2 - a(x,r)+b(x,r)/2a(x,r)b(x,r) + 𝒪(n^-1).Finally, using the identities S=a+b and Q=-(ab)^1/2 gives(<ref>) in the genus-zero region.This completes the proof ofTheorem <ref>.See also Figure <ref>.Next, we compute the asymptotic expansion of w_m,n^(II), startingfrom Lemma <ref>.From (<ref>) we have [ N_m,n(z)]_12 = γ(z)-γ(z)^-1/4πe^-n(g(z)-ℓ)(1+𝒪(n^-1))for x in the genus-zero region.We expand γ(z) at infinity asγ(z) = 1 + b-a/4z + 𝒪(1/z^2).Using the last two equations along with g(z)=log(z)+𝒪(z^-1)and the expansion (<ref>) shows [ N_-1]_12 = (b-a)e^nℓ/8π(1+𝒪(n^-1)).Taking this along with (<ref>) and then (<ref>)shows [ N_0]_11[ N_0]_12/[ N_-1]_12 = γ_0^2-γ_0^-2/b-a = -1/(ab)^1/2 = 1/Q.We now plug this and (<ref>) into the result of Lemma<ref> to see1/n^1/2w_m+1,n^(II)((m+1)^1/2x) = ( 1/Q(x,r) - S(x,r)/2Q(x,r)^2 + 𝒪(n^-1) )(1+𝒪(m^-1)).As long as we agree r=m/n is fixed, we can replace𝒪(m^-1) with 𝒪(n^-1).Therefore, we have1/n^1/2w_m,n^(II)(m^1/2x) = 1/Q(x,m-1/n) - S(x,m-1/n)/2Q(x,m-1/n)^2 + 𝒪(n^-1).From the dependence of Q(x,r) and S(x,r) on r, we can replaceQ(x,m-1/n) and S(x,m-1/n) with Q(x,r) and S(x,r),respectively, at the price of an 𝒪(n^-1) error, so we obtainour final result (<ref>).This completes the proofs ofTheorems <ref> and <ref> (as Theorem <ref> followsimmediately from Theorems <ref> and <ref>).See Figure<ref> for plots demonstrating the convergence forw_m,n^(II). 99BaloghBB:2016 F. 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Johnson, “String theory without branes,”,2006.Kametaka:1983 Y. Kametaka, “On poles of the rational solution of the Toda equation of Painlevé-II type,”Proc. Japan Acad. Ser. A Math. Sci.59,358–360, 1983.Kapaev:1997 A. Kapaev, “Scaling limits in the second Painlevé transcendent,”J. Math. Sci.83, 38–61, 1997. Translated fromZap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)209, 60–101, 1994.Muskhelishvili:1992 N. Muskhelishvili, Singular Integral Equations, Second Edition, Dover, 1992.Murata:1985 Y. Murata, “Rational solutions of the second and fourth Painlevé equations,”Funkcialaj Ekvacioj28, 1–32, 1985.DLMF NIST Digital Library of Mathematical Functions., Release 1.0.6 of 2013-05-06.Online companion to <cit.>.OLBC10 F. W. J. Olver, D. W. Lozier. R. F. Boisvert, and C. W. Clark, editors.NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010.Print companion to <cit.>.Roffelson:2010 P. Roffelson, “Irrationality of the roots of the Yablonskii-Vorob'ev polynomials and relations between them,” SIGMA Symmetry Integrability Geom. Methods Appl.6, 1–11, 2010.Roffelson:2012 P. Roffelson, “On the number of real roots of the Yablonskii-Vorob'ev polynomials,” SIGMA Symmetry Integrability Geom. Methods Appl.8, 1–9, 2012. ] ] | http://arxiv.org/abs/1706.09005v1 | {
"authors": [
"Robert Buckingham"
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"categories": [
"math-ph",
"math.CA",
"math.MP",
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"primary_category": "math-ph",
"published": "20170627183952",
"title": "Large-degree asymptotics of rational Painleve-IV functions associated to generalized Hermite polynomials"
} |
Relational Algebra for In-Database Process MiningRemco Dijkman1 Juntao Gao2 Paul Grefen1 Arthur ter Hofstede3,1 ====================================================================§ INTRODUCTION In its simplest incarnation <cit.>, the weak gravity conjecture states that a consistent, quantized theory of gravity coupled to an Abelian gauge theory must contain at least one charged, massive particle satisfyingm ≤ q ,where m is the particle mass and q the particle charge. Because Newton's constant G_N = 1/^2, the bound implies gravity is the weakest force. All known string compactifications with Abelian gauge forces satisfy the conjecture. Moreover, it reconciles the absence of global symmetries in string theory with the q→ 0 limit of Abelian gauge theories. Within the context of perturbative string theory, the authors of <cit.> demonstrate that modular invariance of effective worldsheet theories evidently implies a version of the conjecture. Extensions of the weak gravity conjecture apply to p-form gauge fields of any p ≥ 0 in arbitrary spacetime dimensions D ≥ 3 <cit.>. In this paper, we focus on p = 1, D = 4.Although string theory automatically satisfies the weak gravity conjecture, the authors of <cit.> use black holes to argue that all healthy effective field theories should obey a weak gravity conjecture. Suppose a black hole has charge Q and mass M. Assuming cosmic censorship, M ≥ Q. The black hole may decay via Hawking radiation or Schwinger pair production. For black holes far from extremality, Hawking radiation dominates. If the black hole only emits charged particles with charge q, and mass m, then conservation of charge implies that Q/q particles are produced. The black hole evolves to a state with mass m Q/q, which is less than M by conservation of energy. Through this process, the black hole approaches extremality, Q/M = 1.[We work in Planck units, i.e. = 1.] At extremality, the black hole's temperature is zero, and Hawking radiation ceases. Such a black hole is stable unless there is a charged particle with q/m > 1, in which case particle-antiparticle pairs are produced via Schwinger pair production. Pair production emits charged matter from the black hole; the black hole is no longer extremal. On the other hand, if the weak gravity conjecture is violated, a large number of stable extremal black hole states exist in the full quantum theory.[Previously, it was believed that the presence of a large number of stable, Planck sized extremal black hole states would violate known entropy bounds <cit.>. However, Casini <cit.> casts doubt on this assertion by carefully examining properties of relative entropy, showing that entropy bounds may not necessarily rule out remnants.] While a proliferation of stable quantum states does not itself signal a sickness from the effective field theory's perspective, it does appear physically undesirable.Recent research directions have focused on sharpening and defining the weak gravity conjecture using effective field theory. The authors of <cit.> propose a stronger form of the weak gravity conjecture by studying matter gauged under a U(1)^N symmetry group. They claim that the convex hull of the charge-to-mass vectors z_i for each species i of particles gauged under the U(1)^N group must contain the unit ball |z_i| ≤ 1. The same authors also attempt to frame the conjecture in terms of unitarity and causality of infrared scattering amplitudes <cit.>, but <cit.> discusses counterexamples to their original argument. A series of papers <cit.> combine intuition from black hole physics with considerations from effective field theory to sharpen the conjecture and to cast doubt on the consistency of field theories that violate it, such as large field axion inflationary models.Nonetheless, an inherent sickness in effective field theories violating the weak gravity conjecture has eluded discovery. Proving the conjecture from a “bottom-up” perspective within the realm of flat space effective field theory may prove too difficult, or impossible. Consequently, effective field theories on large black hole backgrounds provide an ideal setting to test the conjecture without needing to invoke assumptions or intuition from some unknown UV theory. Presumably, we should be able to treat the near horizon physics of large black holes semi-classically due to the smallness of the Ricci curvature. One expects that entanglement of macroscopic fields across the horizon should tell us something about the underlying gravitational theory, even in a semi-classical setup.Let us suppose a proliferation of stable black hole states is a property of sick effective field theories. It is plausible that the sickness would manifest itself by violating known properties of semi-classical entropy. The past decade has seen immense progress in unravelling entropy inequalities that encode deep connections between field theory and semi-classical gravity <cit.>. It is natural to speculate that macroscopic entropy might be powerful enough to discriminate between effective field theories that live in the landscape or swampland.Sen et al. laid the foundation to study black hole entropy in effective field theory <cit.>. They calculate logarithmic corrections to black hole entropy from the Euclidean path integral over the near horizon black hole geometry. One may work with the near horizon geometry directly because of the attractor mechanism, which Sen et al. also show applies to non-BPS black holes in the near-extremal limit. They further justify their methodology by matching the macroscopic entropy results with microscopic state counting using the 21 duality. In low energy effective theories descending from string theory, the results match on both sides of the duality.These papers do not address the macroscopic entropy due to fields interacting with the background field strength. The presence of a background electric flux modifies the effective masses of the matter fields near the horizon. The flux depends on the radius of the black hole. If the fields have sufficiently small mass relative to their charge, the coupling to the flux renders the near-horizon geometry unstable. It decays rapidly due to Schwinger pair production of particle-antiparticle pairs, which precludes us from calculating the macroscopic entropy with Sen's formalism. On the contrary, whenever the weak gravity conjecture is violated or saturated, the geometry is stable. No symmetry protects the stability of the extremal black hole in the non-supersymmetric theories we consider. We expect that perturbations of the extremal geometry may alter the black hole entropy in a way incompatible with known entropy inequalities after we account for quantum effects.The purpose of this paper is to confirm this hypothesis. To our knowledge, this is the first concrete demonstration that entropy inequalities may discriminate between effective theories that live in the swampland or landscape in a controlled, semi-classical environment. We consider D=4 scalar matter gauged under a U(1)^N gauge group in a large, extremal black hole background. The scalar matter violates the weak gravity conjecture. The scalar is minimally coupled to the gravitational and gauge fields. We do not include any non-renormalizable interactions or scalar-scalar interactions. We compute the exact, non-perturbative macroscopic contribution of the gauged scalar to the entropy of the black hole.[We hold the external gauge and gravitational fields fixed. Determining the full macroscopic entropy requires the gauge and gravitational sectors as well. Note however that the quantum corrections of fields neutral under the gauge symmetry are generically subleading. The calculation is exact in the semiclassical limit because the action is quadratic in the gauged scalar field.] We choose a renormalization condition that sets an extremal black hole solution with large charge |Q⃗| to its classical value. We consider a perturbation to the black hole whereby a neutral particle with energy E crosses the black hole horizon. We demonstrate that any small perturbation violates the second law for a sufficiently large initial black hole solution.[What we refer to as the second law is typically referred to as the generalized second law in the literature. We omit the word “generalized” because the generalized second law is the second law once one accounts for all sources of entropy.] Consequently, we prove the weak gravity conjecture for a single scalar. §.§ Related work Qualitatively similar results to our entropy calculation appear in <cit.>. However, not all of their quantitative results match ours exactly. We believe that this results from the formalism they use to calculate the entropy of the black hole, which is not exactly equivalent to ours. We also believe that their conclusions and interpretation of results differ significantly enough from our own. Moreover, they do not attempt to prove the weak gravity conjecture using entropy inequalities, although they allude to this possibility.A separate application of the second law towards understanding the weak gravity conjecture appears in <cit.>, which appeared during the preparation of this manuscript. However, their calculation is orthogonal to ours. Their paper argues for the weak gravity conjecture using a bound on relaxation rates of quasinormal modes of near-extremal black holes. Although related to the second law, the connection is indirect: the second law implies the relaxation rate bound, which in turn implies the weak gravity conjecture. In this paper, we present a a more direct link between the second law and the weak gravity conjecture.§ SETUP Consider a charged, non-rotating black hole. The metric isds^2 = - (r - r_+)(r - r_-)/r^2 dt^2 +r^2/(r - r_+)(r - r_-) dr^2 + r^2 dΩ_S^D-2^2,wherer_± =M ±√(M^2 - |Q⃗|^2)are the outer and inner horizons of the black hole in units where = 1. M is the ADM mass of the black hole spacetime. The black hole is a solution of Einstein's equations, where the stress-energy tensor descends from a U(1)^N gauge theory action. The classical action isS_0 = 1/16π∫ d^D x √(detg)( ^2 R - ∑_n = 1^N F_μν^(n)F^(n)μν ).where g is the determinant of the spacetime metric, R is the Ricci scalar, and F^(n) is the field strength for the n^th gauge field. The background gauge fields A_μ^(n) are a Coulomb potential in the appropriate gauge:A_μ^(n) = (Q^(n)/r, 0, …, 0).In the extremal limit, M → |Q⃗|, the coordinates of the horizons degenerate tor_E^2 = |Q⃗|^2.We may compute the macroscopic entropy of the classical geometry and quantum fluctuations about it using the near-horizon geometry <cit.>.[This is computationally beneficial because there are no conifold singularities in the near-horizon geometry.] After an appropriate choice of coordinates and Wick rotation to Euclidean signature, the near-horizon geometry in D = 4 spacetime dimensions is described by[Roughly, coshη corresponds to the proper distance from the outer horizon in the near-horizon geometry. Details on deriving this form of the metric by taking the near-horizon and extremal limits may be found in <cit.>. The utility of working with this form of the metric is that there are no conical singularities.]ds^2 = r_E^2( dη^2 + sinh^2ηdθ^2 + dψ^2 + sin^2ψdφ^2),where θ is 2π-periodic.[The coordinate θ is related to Euclidean time by a rescaling. The Euclidean time coordinate has infinite periodicity for extremal black holes. The normalization of Euclidean time such that it has period 2π permits us to find a finite result for the macroscopic entropy.] The near-horizon extremal metric factorizes as .The macroscopic entropy of the black hole may be calculated by calculating the effective action for the quantum fluctuations about the classical background. We work with the normalization of the Euclidean action in <cit.>. The effective action splits into a classical (S_0) and quantum (Δ W_eff) component:W_eff = S_0+ Δ W_eff.UsingF^(n)_ηθ = Q^(n)sinhηandR = 2/r_E^2,we obtainS_0 = - 2 β r_E -4π r_E^2,where β = 4π r_E coshη_0 is the inverse temperature of the near-extremal black hole induced by theboundary cutoff.[The cutoff is implicitly taken to infinity, indicating that the black hole has a temperature that limits to zero, as expected for near-extremal black holes.] The first term in the classical part of the effective action is the classical entropy. The second is the classical black hole energy multiplied by the inverse temperature of the black hole.Quantum corrections to the effective action may be calculated by splitting each field Φ into their classical background value Φ_cl and fluctuations about the background Φ_q:Φ(x) = Φ_c(x) + Φ_q(x).If we truncate the action for the fluctuations about the background at quadratic order, we may calculate the one-loop contribution to the effective action. This classical action changes by Δ W_eff <cit.>:Δ W_eff = ∫ d^4 x√(det g) Δℒ_eff = 1/2π r_E^4( coshη_0 - 1 ) Δℒ_eff,where Δℒ_eff is the effective Lagrangian. The first term corrects the ground state energy, regularized by an infrared cutoff η_0.[This IR cutoff renders the volume of E finite.] The second term corrects the macroscopic entropy <cit.>:S_quant = -1/2π r_E^4 Δℒ_eff.From this expression, it is explicitly clear that in the near-extremal limit, where we can take β→∞, that the difference in entropies between two near-extremal geometries automatically satisfies the first law of thermodynamics.Calculating the quantum correction to the macroscopic entropy reduces to calculating Δℒ_eff.[Some places in the literature refers to the quantum correction we compute as S_out, and the macroscopic entropy as S_gen.] The evolution operator along Euclidean worldline time for a particle with worldline Hamiltonian Ĥ is the heat kernel <cit.>K(x,x';s) = ⟨x'|e^-sĤ|x ⟩.To derive Ĥ for fluctuations of a scalar field about a classical background, consider the minimally gauged scalar field action:S_ϕ = ∫ d^4 x √(detg)( -g^μνϕ ( ∇_μ + q A_μ )( ∇_ν + q A_ν ) ϕ + m^2 ϕϕ ),where ∇_μ is the covariant derivative compatible with the metric g_μν. The worldline Hamiltonian for the ϕ field isĤ = -g^μν ( ∇_μ + q A_μ )( ∇_ν + q A_ν ) + m^2.Inserting Ĥ into the heat kernel, we obtain the quantum correction to the effective action:Δℒ_eff = 1/2∫_ε^∞ds/s∫ d^4 x √(det g)K(s),where K(s) ≡ K(x,x;s).[K(s) is independent of x by translational symmetry.] A small distance cutoff ε[With dimensions length squared.] must be imposed due to divergences at the lower bound of the s integral.We may calculate the heat kernel in two ways. Perturbatively, we may perform an expansion of the heat kernel for small s <cit.>. We express the heat kernel in powers of the Riemann curvature, field strengths, and their contractions, multiplied by the appropriate power of s. The geometric expansion yields the perturbative, one-loop contribution to the effective action. This is the familiar small s expansion of the heat kernel. For an arbitrary scalar field, this expansion readsTo find an exact solution, we decompose the heat kernel as a sum of the eigenfunctions f_n(x) and eigenvalues κ_n of Ĥ <cit.>:K(x,x';s) = ∑_n f_n(x) f_n^*(x) e^-κ_n s.By performing the sum, we obtain the resummed one-loop contribution to the effective action. If the action is quadratic in the field Φ, then the resummed one-loop correction is the exact correction to the effective action for the Φ field in the presence of fixed, external A_μ^(n) and g_μν. Although the heat kernel only resums one-loop diagrams, the effects of higher loop processes from internal gravitons and gauge particles are encoded in effective vertices, which may be verified in a Feynman diagrammatic expansion.[The same phenomenon occurs in the Euler-Heisenberg Lagrangian, cf. <cit.>.]Armed with the exact effective action, we extract its logarithmic corrections in the limit where |Q⃗| and |q⃗·Q⃗| are large, but |q⃗| is small. After choosing a renormalization scheme or redefining couplings by appropriately absorbing the effective field theory cutoff, we obtain the macroscopic entropy due to the Φ field. Note that because A_μ and g_μν are held fixed, their contribution to the entropy must be estimated from their separate one-loop contribution to the effective action. Additionally, one must characterize the backreaction on the gauge and gravitational fields induced by the scalar fluctuations.[We may calculate the semiclassical backreaction by solving Einstein's equations with the stress-tensor replaced by its one-loop corrected expectation value. We later show backreaction effects to be negligible for the perturbations of the renormalized effective action for the specific geometry we study.]§ MACROSCOPIC ENTROPY§.§ Contribution to Entanglement Entropy from Neutral Scalars We want to compute the quantum correction to the macroscopic entropy due to a gauged scalar. Let us review the calculation for a neutral, massless scalar. For each field, there are four contributions to the entropy:S = S_0 + S_div + S_CT + S_finwhere S_0 is the classical contribution to the entropy, S_div is the UV divergent quantum correction, S_CT is the entropy from counterterms that regulate UV divergences, and S_fin is from finite quantum corrections to the entropy. Because the heat kernels of the individual fields add at one-loop, the total entropy is the sum of the individual fields' contributions to the entropy. Beyond one-loop, we must estimate the magnitude of entropic contributions from quantum fluctuations of the background geometry backreacting on one-another.To compute the heat kernel of the scalar field in thegeometry, we express Ĥ as the sum of the scalar Laplacian operator on AdS_2 and the scalar Laplacian on S^2. The heat kernel factorizes asK(s) = K_AdS_2(s) K_S^2(s).The eigenfunctions ofare the spherical harmonics Y_ℓ m(ψ,φ)/r_E^2. Only the m = 0 eigenfunctions contribute to K(s). At ψ = 0,Y_ℓ 0(0) = √(2ℓ + 1/4π),and Y_ℓ 0 has eigenvalues ℓ (ℓ +1)/r_E^2. Therefore,K_(s) = 1/4π r_E^2∑_ℓ = 0^∞ (2ℓ + 1) e^-sℓ (ℓ +1)/r_E^2.The eigenvalues and eigenfunctions of the S^2 Laplacian are unaffected by the gauge covariant coupling of the ϕ field to the background gauge field.The eigenfunctions of the neutral, massless scalar Laplacian onare given in <cit.>. The full expression simplifies significantly at the origin of thecoordinate system. There, the eigenfunctions aref(λ) = √(λtanh(λ)/2π r_E^2),where λ is a positive real number. The eigenvalues areκ(λ) = λ^2 + 1/4/r_E^2.Therefore, the heat kernel isK_(s) = 1/2π r_E^2∫_0^∞ dλ λtanh(πλ) e^-(λ^2 + 1/4)s/r_E^2.We interpret λtanh(πλ) as the density of states for the neutral scalar in thebackground geometry.Combining these results, we obtain the heat kernel for the neutral, massless scalar on the near-horizon background geometry K(s) = 1/16π^2r_E^4 s^2 ( 1 + s^2/45 ).Consequently, the divergent contribution to the entropy in the large |Q⃗| limit in Planck units isS_div =+ r_E^2/4ε^2 + 1/180log(ε/r_E^2).This is the exact divergent correction to the macroscopic entropy of the black hole due to the quantum fluctuations of a neutral scalar, previously derived in <cit.>.[Up to exponentially suppressed terms and backreaction of the background fields.] The result is exact because the action is quadratic in the scalar field, and we formally solved for the heat kernel using equation (<ref>) without a perturbative expansion.The result matches the familiar small s expansion of the heat kernel in powers and contractions of curvature invariants. The coefficients of the heat kernel expanded in s are related to local quantities computed in the background geometry,K(s) = ∑_n=0^∞ a_2n(R_μνρσ, F_μν)s^n-2e^-sm^2,where, for a massive scalar field in an arbitrary background geometry the coefficients area_0= 1/8π^2∫ d^4x √(detg)a_2= 1/8π^2∫ d^4x √(detg) 1/6 Ra_4= 1/8π^2∫ d^4x √(detg) ( 12 ∇_μ∇^μ + 5 R^2 - 2 R_μνR^μν + 2 R_μνρσR^μνρσ - 30 q^2 F_μνF^μν ). For the near-horizon geometry, the constant part of K(s), which is a_4(s) in four-dimensions, may be reduced toa_4 = 1/720π^2 R_μνR^μν = 1/720π^2 r_E^2,as expected. We have set m^2=0 for the massless field considered in this section. For a massive field, the logarithmic divergence is damped:S_div,log = 1/180log ( ε/m^2 ).If the mass is smaller the inverse radius of the extremal black hole, it is appropriate to expand the exponential for small s. The logarithmic divergence is a modification of the massless scalar's logarithmic divergence:S_div,log =(1/180+ 1/8 m^2 r_E^2) log ( ε/r_E^2 ).It may be checked <cit.> that this extra term contributes to the renormalization of the cosmological constant. When we study the gauged scalar, it is important to note that the extra divergence present in that answer takes the form of a divergent cosmological constant contribution without any expansion of the exponential.Because the expression for the entropy is UV divergent, we must append counterterms to the effective action to cancel the divergences. Schematically denote each counterterm by δ_𝒪𝒪, where 𝒪 is the operator which receives a divergent correction, and δ_𝒪 is the counterterm. The heat kernel in the small effective mass limit has no exponential suppression. Therefore, the counterterm δ_𝒪 introduces an arbitrary length scale ℓ satisfying ε < ℓ^2 < r_E^2 to cancel the divergence in the logarithmic term that occurs when we take ε→ 0. Schematically, each counterterm takes the formδ_𝒪𝒪 = -∑_n = 1^d/2 c^(n)_𝒪ε^-2n - c^(0)_𝒪log(ℓ^2/ε) = -∑_n = 1^d/2 c^(n)_𝒪ε^-2n - c^(0)_𝒪[ log(ℓ_0^2/ε) + log(ℓ^2/ℓ_0^2) ].The c^(n)_𝒪 coefficients represent the coefficients of the divergent parts of the ε^-2n portions of the effective action in the ε→ 0 limit. We introduce two arbitrary length scales ℓ and ℓ_0. The length scale ℓ_0 does not contribute to the entropy of the initial extremal black hole solution we consider, as it cancels out. However, to simplify calculations, we fix the last term in the above expression for all black hole solutions. When we renormalize both black hole solutions, this fixes both ℓ and ℓ_0. Because ℓ_0 does not appear in the entropy for the extremal black hole, choosing a renormalization condition for the initial extremal black hole fixes ℓ. When we apply a linearized perturbation to the extremal black hole, we have chosen a convention where all terms in the entropy above change except for the last, finite counterterm. We then renormalize this black hole solution, which fixes ℓ_0. All other black hole solutions obtained from further perturbations of the renormalized solution run with changes in the black hole parameters (charge, gauge coupling, radius) as dictated by our initially chosen renormalization conditions.We implicitly choose a renormalization condition that exactly cancels any non-logarithmic divergences. We only discuss the logarithmically divergent counterterms in what follows, unless otherwise specifed. For the massless scalar, we must add a counterterm for the R_μνR^μν operator. Its contribution to the expression for the entropy isS_CT,log = -1/180log(ε/ℓ^2),where ℓ is the arbitrary renormalization scale, in units of length. The renormalized quantum contribution to the entropy isS_qu = 1/180log(ℓ^2/r_E^2),at extremality. If we can trust the extremal approximation near-extremality, we may simply replace the extremal radius with the outer radius of the black hole, r_E → r_+. We do this when we consider small, linear perturbations to the near horizon geometry. Because ℓ is an ambiguous scale, we fix it by specifying our renormalization condition. For example, we may choose a condition that for a black hole of charge Q⃗_0 at extremality, the quantum corretion to the black hole entropy vanishes exactly. Because the entropy depends on the radius of the black hole, the quantum entropy of another extremal black hole of charge Q⃗_0' Q⃗_0 or of a near-extremal black hole of charge Q⃗_0 is non-zero. In other words, the entropy runs with the radius of the black hole.The case of a massive scalar is different. For a massive scalar with m > 1/r_E, we may not expand the exponential term that suppresses the heat kernel. The logarithmic contribution to the entropy is, therefore,S_div,log = 1/180log(ε/m^2).Up to a finite term that is independent of the black hole radius, we may choose a logarithmically divergent counterterm for R_μνR^μν whose contribution to the entropy isS_CT,log = 1/180log(m^2/ε),which cancels the divergence exactly. There is no ambiguous renormalization scale that must be specified. This is in line with the reasoning that only massless neutral particles contribute to the entropy of large black holes. The exception is for particles with very small mass, i.e. m < r_E^-1. In that case, the renormalization to the R_μνR^μν operator proceeds in the same way. An extra operator must be renormalized to absorb the extra divergent contributions to the heat kernel. The structure of the divergent terms exactly matches the contribution to the cosmological constant. We renormalize the cosmological constant to absorb its divergence <cit.>. Its counterterm contributes a logarithmically divergent term to the entropyS_CT,log = 1/360m^4r_E^4log(ℓ^2/ε).The renormalized correction to the entropy isS_qu =( 1/180 + 1/360m^4r_E^4)log(ℓ^2/r_E^2).§.§ Entropy of Gauged Scalars The coupling of the gauged scalar to the background field modifies the eigenvalues and eigenfunctions of the scalarLaplacian <cit.>. In the near-horizon geometry, the background field strength for the n^th gauge field in the Wick rotated spacetime isF^(n)_ηθ = i Q^(n)sinhη.Suppose instead that the scalar is coupled to a constant background magnetic monopole field B⃗ = q⃗·Q⃗sin(ψ)ψ̂×φ̂/r_E^2. There is a continuous and discrete delta-function normalizable spectrum. The continuous eigenvalues areκ(λ)_B = (λ - q⃗·Q⃗)^2 + (q⃗·Q⃗)^2 + 1/4/r_E^2.The density of continuous states becomesλtanh(πλ) →λsinh(2πλ)/cosh(2πλ) + cos(2πq⃗·Q⃗).Wick rotating q⃗·Q⃗→ i q⃗·Q⃗, where q⃗ is the elementary charge vector of the ϕ field, we obtain the density of states for the ϕ field in the constant background electric field:λtanh(πλ) →λsinh(2πλ)/cosh(2πλ) + cosh(2πq⃗·Q⃗).The scalar heat kernel for the near-horizon geometry isK(s) = 1/8π^2 r_E^4∑_ℓ = 0^∞ (2ℓ + 1) ∫_0^∞ dλλsinh(2πλ)/cosh(2πλ) + cosh(2πq⃗·Q⃗) e^-s(λ^2 + ℓ(ℓ+1) + 1/4 + r_E^2 m^2 - (q⃗·Q⃗)^2)/r_E^2.Because the coupling q⃗ appears in the argument of a hyperbolic cosine function in the denominator of the density of states, we conclude that the resummed heat kernel represents the non-perturbative scalar field contribution to the effective action in a fixed, constant, external electric field. The result is not, however, the full quantum correction to the heat kernel. The gauge fields and gravitational field themselves contribute to the entropy. Furthermore, allowing the external gauge and gravitational fields to vary induces backreaction effects on the scalar's effective action.[The entropy due to the gauge and gravitational fields has already been tabulated in equation (<ref>).]Let us compute the divergent contributions to the effective action. Logarithmic divergences are universal and may be found in the region of integration ε≪ s ≪ r_E^2. Therefore, we expand the resummed heat kernels for small s≡ s/r_E^2. The total heat kernel is the product of theandheat kernels, weighted by a factor of e^-s(r_E^2 m^2 - (q⃗·Q⃗)^2). The expansion of theheat kernel is <cit.>:K_(s) = 1/4π r_E^2 s e^s/4 ( 1 + 1/12s + 7/480s^2 + 𝒪(s^3)).We perform the small s expansion of theheat kernel in its resummed form. The denominator of thedensity of states has an asymptotic expansion1/cosh(2πλ) + cosh(2πq⃗·Q⃗) = 1 + ∑_n = 1^∞ ( U_n(-cosh(2πq⃗·Q⃗)) - U_n-2(-cosh(2πq⃗·Q⃗))) e^-2π n λ,where U_n(x) is the n^th Chebyshev polynomial of the second kind. The expansion converges when 0 ≤λ < |q⃗·Q⃗|, regardless of the size of |q⃗·Q⃗|. This may be checked readily by using the ratio test. The first term in the series may be evaluated directly. To evaluate the subsequent terms, we expand e^-λ^2 s for small s. Denoteℱ_n(x) = Li_n(x + √(x^2 - 1)) + Li_n(x - √(x^2 - 1)),where Li_n is the n^th polylogarithm. We integrate over λ to find theheat kernel:K_(s) = -e^-s/4/4π r_E^2 s [1 + s/2π^2ℱ_2(-cosh(2πq⃗·Q⃗))+ s^2( 7/480 + 1/24π^2ℱ_2(-cosh(2πq⃗·Q⃗)) -3/4π^2ℱ_4(-cosh(2πq⃗·Q⃗)))]. Through the last step, we have not made any assumptions concerning the size of |q⃗·Q⃗|. All expansions performed have been independent of it. Now, let us take the large |q⃗·Q⃗| limit. We find thatLi_2(-cosh(2πq⃗·Q⃗))= -π^2/6 - 2π^2 (q⃗·Q⃗)^2 + 𝒪 (sech^2(q⃗·Q⃗)) Li_4(-cosh(2πq⃗·Q⃗))= -7π^4/360 - π^4/3(q⃗·Q⃗)^2 - 2π^4/3(q⃗·Q⃗)^4 + 𝒪 ( sech^2(q⃗·Q⃗)).All together, the unrenormalized, large |q⃗·Q⃗| heat kernel isK(s) = e^-s(r_E^2 m^2 - (q⃗·Q⃗)^2)/16π^2 r_E^4 s^2 [ 1 - s ( q⃗·Q⃗ )^2 + s^2( 1/45 + 1/6 (q⃗·Q⃗)^2 + 1/2(q⃗·Q⃗)^4) + 𝒪(s^3)].Note that our result reduces to the heat kernel of a single neutral scalar in the extremal black hole near-horizon geometry when q⃗ = 0 <cit.>. Higher order terms in s contribute to finite portions of the effective action, which contribute negligibly to differences in the entropy.[One may check that the finite contributions to the entropy scale as |q⃗|^2n|Q⃗|^4 for n >2. However, differences between the near-extremal and extremal black hole entropies scale as |q⃗|^2n |Q⃗|^2. Because we are interested in the |q⃗| → 0 limit, the finite contributions to the entropy are suppressed, as expected. Our logarithmic result, however, does not rely on the smallness of |q⃗·Q⃗|, as shown explicitly in the work outlined above.] A similar, yet quantitatively different, result appears in <cit.>.Using the heat kernel, we may determine the logarithmic correction to the effective action, and thereby the logarithmic correction to the entropy. To connect with the weak gravity conjecture, we want to know the entropy for the resummed heat kernel, which has the |Q⃗|^4 dependence. The resummed, unrenormalized logarithmic correction to the macroscopic entropy from a single gauged scalar of mass m and charge q⃗ in the large |q⃗·Q⃗| limit for fixed A_μ^(n), g_μν isS_div, log = 1/4 ( 1/45 + 1/6 (q⃗·Q⃗)^2 + 1/2 (q⃗·Q⃗)^4) log(ε/(r_E^4 m^2 - r_E^2 (q⃗·Q⃗)^2)).When r_E^2 m^2 = (q⃗·Q⃗)^2, there is no exponential suppression, and the logarithmically divergent contribution to the entropy isS_div, log = 1/4 ( 1/45 + 1/6 ( q⃗·Q⃗ )^2 + 1/2 (q⃗·Q⃗)^4) log ( ε/r_E^2 ). Suppose that weak gravity conjecture is satisfied but not saturated. The exponent in the resummed heat kernel before integration over λExponent = e^-s(λ^2 + r_+^2 m^2 - (q⃗·Q⃗)^2)grows with increasing s for sufficiently small λ. We interpret this as an IR instability in the spectrum. The IR instability yields an imaginary contribution to the effective action <cit.>. The magnitude of the imaginary contribution corresponds to the amount of pair production that occurs at the near-horizon geometry. We expect that one must resort to a computation of the macroscopic entropy using the Euclidean action defined on the global black hole geometry due to the instability. Additionally, we expect that it is no longer justified to work with the classical black hole background without considering how the instability backreacts on the geometry. We leave this topic for future work. §.§ Renormalization of Gauged Scalar Entropy Let us specify renormalization conditions for the initial extremal black hole solution. The black hole we consider has charge Q⃗. We assume that |q⃗| is small, |Q⃗| is large, and |q⃗·Q⃗| is large. We choose counterterms that cancel ε^-n divergences for n ≥ 1. The coefficient of the logarithmically divergent term is much larger than the classical contribution to the entropy. However, this does not imply that the correction for this black hole solution is large. We choose a renormalization condition that allows us to still work in the semi-classical regime. For the perturbation we consider, we choose a renormalization condition that sets ℓ_0 to the inverse Planck mass. Note that for large perturbations, the quantum corrections to the perturbed black hole become non-negligible.The entanglement entropy calculated with all loop orders is given by equation (<ref>). There are two important pieces of this result. First, we have the divergent term of the form S_div, log = 1/4 ( 1/45 + 1/6 (q⃗·Q⃗)^2 + 1/2 (q⃗·Q⃗)^4) log(ε/r_E^2). Comparing this to equation (<ref>), we see that this logarithmically divergent contribution resembles the contribution to the entropy from a neutral scalar field with a small mass. There are two important differences. First, the places where the small quantity m/r_E appear in the expansion of the heat kernel are exactly replaced by factors of (q⃗·Q⃗)^2. This indicates that unlike the one-loop approximation to the gauged scalar heat kernel (cf appendix), there is an extra divergent contribution to the exact heat kernel from a cosmological constant term. As with the massless neutral scalar, we may cancel the divergence from the other two terms by inserting counterterms for R_μνR^μν and F_μνF^μν. As may be confirmed in <cit.>, the (q⃗·Q⃗)^4 requires renormalization of the cosmological constant.The second difference is the argument of the logarithm: it depends on (q⃗·Q⃗)^2. It becomes clear what to do with the logarithmic divergence if we rewrite its contribution to the entropy in the following way:S_div, log = 1/4 ( 1/45 + 1/6 (q⃗·Q⃗)^2 + 1/2 (q⃗·Q⃗)^4)( log(ε/r_E^2) - log (r_E^2 m^2) - log ( 1 - (q⃗·Q⃗)^2/r_E^2 m^2 )).Surprisingly, the divergent terms for a massive gauged scalar look like the divergent terms for a neutral scalar in the small mass limit, with the r_E^2 m^2 coefficient swapped for (q⃗·Q⃗)^2. The other terms are resummed, finite corrections to the entropy. Their contributions come from an infinite sum of (F_μνF^μν)^n-type operators. They do not require counterterms because of the lack of dependence on ε. Because they depend on the radius of the black hole, their contribution can only be cancelled for a specific black hole solution. In general, they contribute a non-zero, finite correction to the entropy at arbitrary black hole mass and charge.We renormalize the entropy as we did for the neutral scalar in the small mass limit, with the only new feature being a F_μνF^μν counterterm. The renormalized entropy isS_div, log = 1/4 ( 1/45 + 1/6 (q⃗·Q⃗)^2 + 1/2 (q⃗·Q⃗)^4)( log(ℓ^2/r_E^2) - log (r_E^2 m^2) - log ( 1 - (q⃗·Q⃗)^2/r_E^2 m^2 )).We specify a renormalization condition that sets the finite contributions to the entropy from resummation as well as the divergent terms equal to zero for a black hole of fixed charge Q⃗ exactly at extremality. This removes the ambiguity for the renormalization scale ℓ and removes all divergences.§ VIOLATING THE SECOND LAW§.§ Setup The second law states that entropy increases under any physical process:dS ≥ 0.For healthy semi-classically treated effective field theories in curved space, the second law has been proven within various settings, e.g. <cit.>. The entropy S has contributions from the classical and quantum parts of the effective action: the Bekenstein-Hawking area term as well as quantum corrections from the macroscopic fields:𝒮 = - W_eff =-(S_0 + S_quant), S_quant = S_div + S_CT,where we have neglected subleading finite corrections in S_quant. In our physical scenario, the entropy changes when the black hole consumes a neutral particle because the black hole's radius increases.Let subscript f denote final quantities, subscript i initial quantities, A ≡ 4 π r_+^2 the area, and S_quant the quantum entropy correction. ThenS_0,f - S_0,i≥ S_quant,i - S_quant,ffollows from the second law.In our thought experiment, we let a single neutral particle crosses the black hole horizon with energy E. This induces a linearized perturbation of the extremal black hole geometry. The black hole charge remains fixed. The initial black hole entropy has been set to its classical entropy S_0 and energy E_0 values by choosing the appropriate renormalization condition:Δ W_eff,i = 0, S_0,i = π r_E^2 = π |Q⃗|^2, E_0,i = r_E = M = |Q⃗|.By conservation of energy,the black hole mass shifts toM_f = M + δ M.ThenM_f > |Q⃗|. The perturbed black hole receives a quantum contribution to its entropy because we have already specified fixed counterterms for the effective action of the black hole and the divergent contributions to the entropy depend on the radius of the black hole. The quantum contribution to the entropy may be mathematically traced to the fact that it runs with the radius of the horizon of the black hole. Because the quantum contribution to the exact contribution of the gauged scalar to the black hole entropy modulo backreaction outscales the classical contribution, we expect large perturbations to the classical geometry may induce large quantum backreaction. We therefore consider small perturbations to the geometry and write the near-extremal radius r_+ of the perturbed black hole asr_+^2 = r_E^2 + δ r^2.Note that in what follows we only consider the gauged scalar matter sector and small perturbations to the geometry in our second law analysis. We justify our result in the next section by demonstrating that effects from all other fields are subleading at one-loop and suppressed at higher loop orders and that quantum backreaction may be neglected for small perturbations of the geometry.The only modification to the entropy at the level of linearized backreaction arises from the change in the near-horizon electric field, which shifts from 2|q⃗|^2cos^2(ϑ) ≡ (q⃗·Q⃗)^2/r_E^2 to (q⃗·Q⃗)^2/r_+^2. The logarithmic correction to the classical entropy of the new black hole is[In Planck units = 1.]S_quant = 1/4 [ 1/45 + 1/6 (q⃗·Q⃗)^2 + 1/2 (q⃗·Q⃗)^4 + 𝒪 (sech^2[(q⃗·Q⃗)^2])] log ( m^2 r_+^4 - (q⃗·Q⃗)^2 r_+^2/m^2 r_E^4 - (q⃗·Q⃗)^2 r_E^2 ).Ignoring exponentially suppressed contributions and keeping only the 𝒪(|Q⃗|^2) or higher terms, the bound equation (<ref>) becomes| Q⃗|^2 ≤32π/|q⃗|^4 cos^4(ϑ)m^2 - |q⃗|^2 cos^2(ϑ)/2 m^2 - |q⃗|^2 cos^2(ϑ) - 1/31/|q⃗|^2 cos^2(ϑ).,where ϑ is the angle between Q⃗ and q⃗. The bound applies to particles violating or saturating (m^2 = |q⃗|^2cos^2(ϑ)) the conjecture. The dependence on δ r^2 cancels on both sides of the bound for small δ r^2. We may always choose an initially large, extremal black hole such that we violate the bound. A conservative interpretation of the result is that there is a maximum charge allowed in the macroscopic theories considered. This would require the appearance of some instability for large black holes. There is no evidence that this is the case, however, as we discuss in the next section.It is natural to wonder if our result is nullified when instanton tunnelling, quantum backreaction, and effects from other fields are accounted for. The answer is no. Because the differences in quantum contributions to the entropy dominate differences in classical contributions to the entropy, large black holes are stable against splitting into multiple black holes whose charge adds up to the charge of the large black hole. This is more general than the statement that no Schwinger pair production occurs for extremal black holes formed in theories violating the weak gravity conjecture. Quantum effects dominate differences in entropy, but do not dominate the classical expressions for the entropy themselves for a suitable renormalization condition. One may worry that changes in energy, related to the backreaction of the quantum fields on the classically perturbed geometry, are important too. In fact, quantum backreaction on the black hole mass only appears at 𝒪((δ r^2)^2). Moreover, as aforementioned and cited, contributions at one-loop from other massless fields, such as massless matter, other gauge fields, and the gravitational field, are always subleading with respect to the classical entropy of the black hole. For large black holes, only the one-loop answer contributes in the large |Q⃗| limit: higher loop contributions are suppressed by factors of inverse radii of the black hole. We leave these results to the appendix.We cannot emphasize enough that the answer we have obtained is an exact answer that extends beyond the one-loop approximation: higher loop factors have in effect been resummed because we computed the full partition function for the scalar field. The reason we could do this is because the action is quadratic in the scalar field, so the Euclidean path integral reduces to a Gaussian integral. Because of the special geometry of the near horizon region, we were able to compute this result analytically. Any error in our result is of order 𝒪(sech(|q⃗·Q⃗|)), which is suppressed in the large |q⃗·Q⃗| limit. As shown in the appendix, quantum backreaction does not affect the classical geometry at order δ r^2 after the neutral particle crosses the black hole horizon. This is the only effect that is not explicitly captured by our exact computation.§ CONSISTENCY CHECKS§.§ Subleading Contributions from Neutral Matter, Gauge Fields, and Gravitational Field at All Loop Orders Our expression for the exact heat kernel of the scalar field indicates a second law violation. We have not included effects from the two other fields present: the U(1)^N gauge field and the gravitational field. This is because these contributions are subleading. The reason that the scalar had such a large contribution to its entropy is because it couples to the background gauge field. Therefore, the action for the ϕ field includes contributions from positive powers of the background black hole charge. This is not the case for the gravitational and gauge actions.Let us first consider the one-loop contributions to the heat kernel from the gauge and gravitational fields. The total heat kernel for the full theory is the sum of the individual heat kernels, so we can consider each field separately. At one-loop, we only need to consider the quadratic action for each field. The one-loop expression for N U(1) vector fields and the gravitational field has already been known for some time, calculated by Sen in <cit.>:S = A/4 - 1/180 ( 964 + 62 N) log(A).Note that the quantum correction is subleading. Moreover, Sen et al. demonstrate in <cit.> that higher loop contributions are suppressed in the large black hole mass limit. Therefore, the one-loop result for the macroscopic answer suffices. §.§ Suppression of Quantum Backreaction for Small Classical Perturbations In our analysis, we choose renormalization conditions such that the quantum correction to the extremal black hole entropy with charge Q⃗ is absorbed into the tree-level, classical value for the entropy. The linear perturbation to the black hole induced by a neutral particle crossing the horizon causes the quantum entropy to run, because the entropy depends on the radius of the black hole. The quantum correction to the entropy of the perturbed black hole solution is smaller than the classical entropy of the perturbed black hole. However, the difference between the classical entropies of the initial and final black holes is smaller than the difference in quantum corretions to the entropy. It is for this reason that the second law is violated. We use the exact expression for the scalar field effective action for fixed, external classical backgrond fields, accounting only for classical backreaction. Here we provide a back of the envelope argument that quantum backreaction does not modify our result at 𝒪(δ r^2).The mass of the black hole may be expressed via the first law asM = T S + |Q⃗|,where T is the temperature, S is the entropy, Q⃗ is the charge, and we have set the chemical potential to one. We assume that quantum corrections to all quantities written above are factored into this formula. For a stationary, charged black hole, these are the only sources that can contribute to the black hole mass.Let us consider backreaction on the charge of the black hole. The charge of the perturbed black hole receives quantum corrections that are of orderQ⃗_qu∝ -|q⃗|^4 |Q⃗|^0 δ r^2.In the small |q⃗| limit, we may assume that these are subleading and ignore these corrections only if the quantum corrections to the mass do not dominate.The perturbed black hole has a classical correction to its mass proportional to the temperature of the black hole. The classical temperature isT = 1/2π ( 1/r_+ - |Q⃗|^2/r_+^3 ),which evalutes toT = δ r^2/|Q⃗|^3 + 𝒪((δ r^2)^2).It may be checked that quantum corrections do not modify this order of magnitude estimate. The thermal contribution to the mass of the black hole has a classical and quantum component. The classical contribution arises from the classical entropy:M_cl = 4π T r_+^2 = 4πδ r^2/|Q⃗| + 𝒪((δ r^2)^2).The quantum correction to the entropy is proportional to 𝒪(δ r^2)|q⃗|^4|Q⃗|^4. Therefore, the quantum correction to M is proportional to (δ r^2)^2:M_qu∝ - |q⃗|^4|Q⃗| 𝒪((δ r^2)^2).In the small δ r^2 expansion, this is smaller than the classical backreaction near-extremality. We conclude that we may ignore quantum backreaction effects in our thought experiment. A full analysis should utilize the semi-classical Einstein equations. We leave this to future work. §.§ Stability of the Near Horizon Geometry A black hole with charge Q⃗ is not the only classical geometry asymptotic to thein the near-horizon limit. Other geometries that contribute to the path integral are multi-black hole solutions, where the total charge of the black holes equals Q⃗. When the weak gravity conjecture is satisfied, tunneling processes may occur in which the initialnear-horizon geometry fragments into multiplegeometries. The simplest example is the Brill instanton, wherein one initialspace tunnels into two disconnected spaces. Let us review the calculation using the classical piece of the effective action first, following <cit.>. For simplicity of presentation, we work with a U(1) gauge group in the remainder of this section. The background gauge field in the two black hole solution isA_t(x⃗) = Q_1/|x⃗ - x⃗_1| + Q_2/|x⃗ - x⃗_2|.Further details may be found in <cit.>. The instanton action is half the negative difference of the initial and final black hole entropies,[The factor of 1/2 appears because the transition probability between solutions is proportional to e^-Δ S.]S_inst = -1/2Δ S. Consider the Bekenstein-Hawking term without quantum corrections. The Brill instanton action isS_inst = π Q_1 Q_2.The transition amplitude from the charge Q black hole to the Q_1 and Q_2 charged black holes isA_Q → Q_1 + Q_2∝ e^1/2Δ S,up to normalization. Consequently, the transition probability isP_Q → Q_1 + Q_2∝ e^-πQ_1 Q_2.The probability is less than one for non-zero Q_1 and Q_2, as expected.[It is implicit in what follows that Q_1 and Q_2 have the same sign.] When Q_1→ q, this answer represents the probability amplitude for brane-antibrane production, i.e. Schwinger pair production. Now consider quantum corrections to the macroscopic entropy from matter neutral under the U(1) gauge symmetry. The logarithmic terms are subleading. Therefore, the instanton action is still positive, because S(Q) ≥ S(Q_1) + S(Q_2). We interpret this to mean that large black holes dominate the Euclidean path integral, with an exponentially suppressed probability that the black holes fragment into multi-black hole solutions. Note that fragmentation and Schwinger pair production would preclude us from maintaining a sufficient level of control over the process we consider.Now consider the quantum corrected black hole entropy when the weak gravity conjecture is violated or saturated for a non-supersymmetric gauged scalar. In the large charge limit, one may verify thatS(Q) < S(Q_1) + S(Q_2).Consequently, the physical instanton process is not fragmentation; rather, it is black hole growth from an initial two black hole state to a single black hole final state. This is consistent with the Q_1→ q limit: there is no pair production. Similarly, there is no black hole fragmentation. Instead, the correct instanton action corresponds to two initialstates transitioning into one finalstates. In the Q_1→ q limit, this is a process akin to the thought experiment in the previous section. The combined state of the black hole and a particle eventually transitions to a final state where the black hole consumes the charged particle. In conclusion, the large-charged black hole in our setup does not fragment into a multitude of black holes or charged particles. This is in accord with the kinematics arguments presented in the introduction. The theory contains only subextremal objects in its spectrum, so the extremal black hole has no decay channels. Moreover, the instanton analysis implies that the Euclidean path integral is dominated by small black hole classical saddle points, i.e. remnants. We claim that the absence of a decay channel affords us sufficient control over the process we consider.It is clear now that black hole growth is the favored physical process in theories violating the weak gravity conjecture, at least in situations where backreaction can be neglected. We speculate that the reversal of the Brill instanton violates unitarity. Renormalize the large extremal black hole effective action. One may tune the effective action such that the entropy is positive, despite the seemingly large quantum correction. However, no mechanism exists within the IR theory that prevents the black hole from continuing to grow unbounded. When the black hole grows, the decreasing quantum correction outcompetes the increasing Bekenstein-Hawking term. Counterintuitively, larger black holes hide fewer microscopic states behind the horizon than smaller black holes. Because growth may occur without bound, the entropy eventually becomes negative, indicating that the black hole contains less than one microscopic state. We expect that this behavior is forbidden in a unitary theory. Therefore, we speculate that the scalar violating the weak gravity conjecture is secretly non-unitarity, even at the level of effective field theory. § CONCLUSION AND FUTURE WORK This paper comprises a proof of the weak gravity conjecture, obtained from studying the macroscopic entropy of gauged scalars on a semiclassical near-extremal black hole background. Our choice of renormalization conditions allows us to safely neglect non-linear metric backreaction. The quantum corrected entropy violates the second law if the conjecture is not satisfied. When the conjecture is satisfied, the black hole near extremality decays rapidly due to Schwinger pair production, which allows the theory to evade the troubling thermodynamic violation. Therefore, we establish that it is necessary that a weak gravity conjecture is obeyed.[We leave it to future work to determine if it is sufficient.]. Our calculation demonstrates that entropy inequalities may discriminate between effective field theories that live in the landscape versus the swampland. Although effective field theories that violate the weak gravity conjecture do not obviously violate unitarity, positivity, or causality, the violation of the second law indicates that some sickness lurks within them. In conclusion, we propose that a violation of the second law modulo backreaction indicates an IR obstruction to a UV completion in a unified theory <cit.>.Our analysis does not truly address weak gravity in effective field theory or on arbitrary perturbations of the black hole background. We only consider the minimally gauged, minimally coupled quadratic action of the D = 4 gauged scalar. A follow-up paper <cit.> bridges the gap: we address the conjecture in arbitrary dimensions and non-minimal interactions, including non-renormalizable terms. We limited our analysis here to the minimal quadratic action for ease of presentation and because we could obtain an exact result. We extend our result to actions with higher dimension operators and to actions with multiple scalars in <cit.>. In particular, we prove the generalized electric weak gravity conjecture of <cit.> in our follow-up paper.It would be worthwhile to extend our methodology to arbitrary p-form gauge fields. For example, while it is expected that there is a weak gravity conjecture for p > 1, it is unclear if p = 0 axions are subject to a weak gravity conjecture. If they are, then there are direct implications for inflationary model building. In particular, large field axion inflation would violate the p = 0 weak gravity conjecture <cit.>.Although our results directly apply to the weak gravity conjecture, they might also apply to the Ooguri-Vafa conjecture <cit.>.[See also <cit.>.] Ooguri and Vafa claim that there are no stable non-supersymmetric AdS vacuua whose cosmological constant is supported by a flux. If true, then the conjecture has serious implications for non-supersymmetric AdS/CFT. Large-N brane constructions and Kaluza-Klein compactifications include extremal particles in the bulk spectrum. Our result demonstrates a conflict between thermodynamics and non-supersymmetric, gauged extremal particles, suggesting a route to proving the conjecture.The extensions aforementioned do not capture the full potential of our methodology. We propose that the armamentarium of entropy technology at our disposal may define new, undiscovered constraints on effective field theories compatible with quantum gravity. Our follow-up paper provides minor evidence in favor of the proposal <cit.>. The power of the methodology lies within the relative ease of calculating macroscopic entropy of IR field content in semi-classical gravitational backgrounds. One may remain agnostic as to the full UV completion of the effective theory. Nonetheless, if the effective theory violates known entropy inequalities in the IR, then there exists some obstruction to a UV completion.§ ACKNOWLEDGEMENTS The authors are especially grateful to Matt Reece for thoroughly reviewing a late-stage draft of the manuscript and Aron Wall for extensive, thoughtful discussion and debate. We also thank Raphael Bousso, Venkatesa Chandrasekar, Netta Engelhardt, Illan Halpern, Petr Hořava, Juan Maldacena, Arvin Moghaddam, Fabio Sanches, and Ziqi Yan for conversations during various stages of preparation of this manuscript. The authors are also very grateful to John Brown and Xiaobei Wei for their hospitality at our second home, Sophie's Cuppa Tea.C.M. is supported by a National Science Foundation Graduate Research Fellowship. The work of Z.F. is supported in part by the Berkeley Center for Theoretical Physics, by the National Science Foundation (award numbers 1214644, 1316783, and 1521446), by fqxi grant RFP3-1323, and by the US Department of Energy under Contract DE-AC02-05CH11231. This work was completed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293.§ ONE-LOOP CALCULATION The exact heat kernel for the minimally gauged scalar in the presence of fixed external background fields does not match the expected one-loop result. In the one-loop heat kernel Suppose that m ≫ q. Then we may expand the s(q⃗·Q⃗)^2 part of the argument of the exponent:K(s) ≈1/16π^2 r_E^4 s^2 [ 1 + s^2( 1/45 + 1/6 (q⃗·Q⃗)^2) + 𝒪(s^4)] e^-sr_E^2 m^2.This is exactly what one would obtain in the geometric expansion of the heat kernel in the large |Q⃗| limit:K(s)≈1/16π^2 s^2( 1 + s/6 R + s^2/45 R_μνR^μν + 1/6 q^2 F_μνF^μν) e^-sr_E^2 m^2.The (q⃗·Q⃗)^4 term is cancelled by the background gauge field term when we expand the exponential. Therefore, in the large mass limit, one may verify that only a q^2 F_μνF^μν counterterm is required to cancel the divergence due to powers of q⃗·Q⃗ that appear in the final result, which can be seen by performing the small s expansion or, likewise, expanding the exponent in our exact result.[It can be verified that it is only in this limit that the approximation of the integrand made in <cit.> is justified. Because this is not a focus of our paper, we refrain from providing further commentary in this paper on this detail.] utcaps 10ArkMot06 N. 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Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,” http://dx.doi.org/10.1088/1126-6708/2006/10/014JHEP 10 (2006) 014, http://arxiv.org/abs/hep-th/0602178arXiv:hep-th/0602178 [hep-th]. OogVaf16 H. Ooguri and C. Vafa, “Non-supersymmetric AdS and the Swampland,” http://arxiv.org/abs/1610.01533arXiv:1610.01533 [hep-th]. FreKle16 B. Freivogel and M. Kleban, “Vacua Morghulis,” http://arxiv.org/abs/1610.04564arXiv:1610.04564 [hep-th]. MalMic98 J. M. Maldacena, J. Michelson, and A. Strominger, “Anti-de Sitter fragmentation,” JHEP 02 (1999) 011, http://arXiv.org/abs/hep-th/9812073hep-th/9812073. | http://arxiv.org/abs/1706.08257v3 | {
"authors": [
"Zachary Fisher",
"Christopher J. Mogni"
],
"categories": [
"hep-th",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170626072056",
"title": "A Semiclassical, Entropic Proof of a Weak Gravity Conjecture"
} |
Managing a Fleet of Autonomous Mobile Robots (AMR) using Cloud Robotics Platform Aniruddha Singhal, Nishant Kejriwal, Prasun Pallav, Soumyadeep Choudhury, Rajesh Sinha and Swagat KumarEmail IDs:The authors are with TCS Research, Tata Consultancy Services, New Delhi, India 201309. Accepted 24/05/2017 ================================================================================================================================================================================================================ In this paper, we provide details of implementing a system for managing a fleet of autonomous mobile robots (AMR) operating in a factory or a warehouse premise. While the robots are themselves autonomous in its motion and obstacle avoidance capability, the target destination for each robot is provided by a global planner. The global planner and the ground vehicles (robots) constitute a multi agent system (MAS) which communicate with each other over a wireless network. Three different approaches are explored for implementation. The first two approaches make use of the distributed computing based Networked Robotics architecture and communication framework of Robot Operating System (ROS) itself while the third approach uses Rapyuta Cloud Robotics framework for this implementation. The comparative performance of these approaches are analyzed through simulation as well as real world experiment with actual robots. These analyses provide an in-depth understanding of the inner working of the Cloud Robotics Platform in contrast to the usual ROS framework. The insight gained through this exercise will be valuable for students as well as practicing engineers interested in implementing similar systems else where.In the process, we also identify few critical limitations of the current Rapyuta platform and provide suggestions to overcome them. Fleet Management System, Multi-AMR control, Rapyuta, Cloud Robotics Platform, Robot Operating System, MAS, Gazebo, Gzweb § INTRODUCTION The last couple of decades have witnessed a steady rise in robot-based industrial automation. These industrial robots are comparatively inexpensive and are capable of carrying out repeated tasks at high speed and great accuracy and hence, are widely deployed in the industries of mass production. In spite of this, the robotic automation has remained confined only to big industries who can pay for elaborate assembly lines built around these robots to compensate for their lack of intelligence. In addition, this involves writing and testing extensive programs to take into account all possible cases that a robot might encounter during its operation. In short, the current robot-based industrial automationrequires huge investment both in terms of capital and time, making it unaffordable to small and medium enterprises. This scenario is poised to change with the rise of service robots <cit.> <cit.>, which unlike their industrial counterparts, can work in unstructured environments while learning and adapting tochanges around them. These robots are designed to be safe and can work collaboratively with humans in close proximity without any protective fencing. These robots could be programmed very easily and intuitively through demonstrations by operators themselves. This gives rise to a field known as teaching by demonstration paradigm <cit.> which can be used for changing the robot behavior on the fly. Similarly, these robots will exhibit higher level of intelligence in taking autonomous decisions based on sensory perception.According to International Federation of Robotics (IFR) <cit.>, service robotics is going to drive the growth in robotic industry in the coming decade. This growth will be partly due to the increased adoption of robots in industries as well as domestic environments.Cloud Robotics <cit.> <cit.> will play a significant role in the growth of service robotics by augmenting the robot capabilities while reducing the per unit costs of each robot.This will become possible as the robots can off-load computationally intensive tasks on to the cloud for processing, can collaborate with other robots and humans over network, can learn new skills instantly from internet. Cloud Robotics can be used for providing “Robotics-as-a-Service” based solutions where robots could be dynamically combined to give support to specific applications <cit.>. One such application that is being considered in this paper is a vehicle fleet management system for warehouse and factory shop floors. A vehicle fleet management system comprises various software and hardware components which facilitates optimum utilization of vehicles in meeting pre-defined goals. One such example is the use of Kiva mobile robots <cit.> for moving goods within Amazon fulfillment centers <cit.>. These autonomous ground vehicles (AGVs) are programmed to move autonomously along predefined tracks.However, the schedule and routes are provided by a centralized planner which also carries out resource allocation and manages job assignment to individual robots. Such a system also includes effective modules that facilitates efficient collaboration between machines and robots <cit.>. In this paper, we are looking into a simpler version of this fleet management system where a group of autonomous vehicles are required to follow desired paths provided by a global path planner as shown in Figure <ref>. This figure shows the essential components required for implementing such a fleet management system. The current location of robots as well as new obstacles detected on the way are used to update the environment map which, in turn, is used by the global planner to create new paths for the robots.The user or the operator provides the goals or destinations for each robot in this case. However, such goals may also come from an ERP (Enterprise Resource Planning) system in an industrial setting. The autonomy of each robot is governed by the navigation module that implements SLAM (Simultaneous localization and mapping) <cit.> as well as obstacle avoidance capabilities. Unlike the existing systems that focus on system integration involving various software and hardware components <cit.> <cit.> <cit.>, we are particularly interested in exploring various software frameworks like ROS <cit.> and Rapyuta <cit.> for implementing such systems.To be specific, we provide details of three implementation in this paper. First two make use of the distributed control and communication framework of Robot Operating System (ROS) <cit.> and the last implementation uses Rapyuta cloud robotics engine <cit.>. A comparative analysis of these approaches are carried out which provides an understanding of underlying challenges, which if addressed, may increase the usability of the platform. The working of these implementations are demonstrated through several simulation as well as real world experiments. In short, the contributions made in this paper could be summarized as follows: (1) We provide three different implementations of a fleet management system for autonomous ground vehicles using ROS and Rapyuta platforms. This includes single-master based ROS system, multi-master based ROS system and Rapyuta-based cloud robotics system. (2) The working details of these implementations are provided for both simulation as well as actual experiments which could serve as operation manual for students, researchers and practicing engineers who would like to implement similar systems in other domains. (3) Through rigorous comparative performance analysis, we identify the critical limitations of existing cloud robotics platform which, if solved, will improve the usability of these platforms.The rest of this paper is organized as follows. An overview of related work is provided in the next section. The three approaches of implementing fleet management system is described in Section <ref>. The comparative performance analysis of these systems for simulation and actual experiments are provided in Sections <ref> and <ref> respectively. The limitations of the current implementation which provides direction for future work is discussed in Section <ref> followed by conclusion in Section <ref>.§ RELATED WORKIn this section we provide a brief overview of several related work. This will also serve as a background material for various core concepts that will be repeatedly referred in the rest of this paper. §.§ Robot Operating SystemRobot Operating System (ROS) <cit.> is a software framework for managing and controlling multiple robots. It uses a peer-to-peer topology for communication between robot processes, supports multiple programming languages and provides tools for robot software development.Readers can refer to online wiki <cit.> to know about ROS in detail. For the sake of completion, some of the common concepts which will be used frequently are listed below for the sake of completeness. (1) Nodes areprocesses that perform computation. They can communicate with each other by passing messages. (2) Topics are medium over which nodes exchange messages. They provide a link between two nodes. A topic is channel for anonymous communication. Multiple nodes can publish/subscribe to a given topic.(3) Subscriber is a node which listens to the messages that are published to a topic. (4) Publisher is a node which writes to a topic from which other nodes can subscribe. (5) roscore is a set of nodes which are necessary for ROS environment to work. roscore starts a ROS master node, ROS permanent server and a node where logs are published.(6) AMCL(Adaptive Monte Carlo Localization)<cit.> is an inbuilt package in ROS that is used by the robots to localizes themselves in the map. (7) TF is a package that lets the user keep track of multiple coordinate frames over time. TF maintains the relationship between coordinate frames in a tree structure buffered in time, and lets the user transform points, vectors, etc., between any two coordinate frames at any desired point in time. (8) GMapping <cit.> package provides laser-based SLAM (Simultaneous Localization and Mapping) capability. It runs as a ROS node called . This node can be used for creating 2-D occupancy grid map of the environment from laser and pose data collected by a mobile robot.§.§ Cloud Robotics Platform: RapyutaRapyuta <cit.> is an open-source cloud robotics framework. It provides an elastic computing model which dynamically allocates secure computing environment for robots. In this way it helps in solving the problem of unavailability of high computing power on robots. The Rapyuta framework is based on clone-based model <cit.> where each robot connected to the cloud has a system level clone on the cloud itself which allows them to offload heavy computation into the cloud.These clones are tightly interconnected with high bandwidth making it suitable for multi-robot deployment. In addition, Rapyuta provides access to libraries of images, maps otherwise known as RoboEarth knowledge repository <cit.> <cit.> and, provides framework that facilitates collaborative robot learning and human computation <cit.>. A number of applications have been reported in literature that demonstrate the applicability and usefulness of the platform. This includes collaborative mapping <cit.> <cit.>, robot-grasping <cit.>, tele-presence <cit.> and ubiquitous manufacturing <cit.>.Readers are also referred to <cit.> <cit.> <cit.> for a comparative study on several other cloud robotics platforms reported in the literature. While a cloud-based system offers several advantages, it also poses several challenges which if solved can greatly enhance the usability of such platforms. Some of these challenges include network latency, data interaction and security <cit.>. Also a slightly related work is done by Turnbull et al in which they have made a system to detect position of robots through a camera placed on ceiling and control their motion so that they don't collide. They have exploited the large computation power provided by the cloud.<cit.>. A collision avoidance and path planning system which works on individual robots also exist <cit.>. They have used common ROS topic for inter robot communication and AMCL for localization.§.§ Fleet Management SystemA fleet management system <cit.> <cit.> <cit.> <cit.> primarily concerns itself with managing a group of vehicles to meet the goals and objectives obtained from an enterprise computer system. While most of the existing system focus on integrating various software and hardware components to ensure efficient utilization of resources, there has been very few efforts at generalizing the underlying architecture to make it more flexible and generic.Authors in <cit.> do propose to use a cloud infrastructure to implement formation control of a multi-robot system by using an external camera system for detecting and tracking individual robots. While a cloud infrastructure is used for image processing, it does not use a generic framework like Rapyuta.In this paper, we primarily implement a simplified fleet management system using Rapyuta cloud robotics engine. The implementation is carried out through simulation as well as physical experiments using actual robots. The purpose of this work is to provide an insight into the working of the cloud robotics framework as well as identifying the limitations of current architecture. We also attempt to offer suggestions for overcoming these limitations and thereby improving the usability of the Rapyuta cloud robotics framework. The details of implementations for fleet management system is described next in this paper. § THE METHODS In this section, we provide details of our implementation of a simplified fleet management system as shown in Figure <ref>. It primarily consists of four modules: (1) a user, an operator or an ERP system that provides goals or target destination for each robot, (2) a global planner that computes the path to be taken by each robot based on the current state of the environment (3) Autonomous Mobile Robots (AMR) having capability for autonomous navigation and obstacle avoidance; and (4) an environment map which could be updated with the information of new obstacles detected by the robots. The user is also free to update the availability of routes for any robot by creating obstacles in the environment map. The above fleet management system is implemented using three methods: (1) single-master system, (2) multi-master system and (3) Cloud Robotics platform. The first two methods make use of the distributed computing and communication architecture of Robot Operating System (ROS) <cit.> while the last methods uses Rapyuta cloud robotics framework <cit.>. The details of each implementation and their respective pros and cons are presented next in this section. §.§ Single Master System In a single master system,runs on one machine which is called the master. Other nodes work in a distributed fashion on different machines. The nodes can run anywhere on the network except the driver nodes, which runs on the system that is directly connected to the hardware.All the nodes need to connect tothe master. They connect viawhich can be set infile of the respective machines as shown below. All the machines in the network have a bidirectional connection with each other. Also, the host IP and the master IP will be same in case of the master machine.0.5pt0.5ptSome of the common tasks like localization, mapping etc. runs on every robot resulting in nodes with same name under . A single launch file cannot be used to launch the nodes as it will create a conflict and the previous running node will be overridden with the new instance of the same name. This problem is resolved by introducing namespace andtags in the launch file as shown below.0.5pt 0.5ptThe single master system can be set up by following the steps given below: * Setupin each robot as shown above.* Append suitable namespace andto the nodes corresponding to each robot.* Runon the master.* Launch each individual robot. A single master system is handy for quick testing of algorithms on a single robot because of its simple setup process. Its simplicity, however, does not provide much advantage as the number of robots increase in the environment.A schematic diagram of a working instance of single master system is shown in Figure <ref>. It shows one master runningand two client robots connected to the master over LAN. As one can see, all the topics from one robot is available for subscription by the all other robots as well. These topics are shown as dotted ellipse. The topics generated by the robot is shown as solid ellipses.Making topics available to everyone all the time may lead to some security concern as one would like to have some control over who can access which topics. In other words, this would require additional overhead to restrict access to the topics of a given robot by the other. Secondly, the bandwidth requirement for a single master system with multiple robots is comparatively higher as all the topics are available over the network for subscription. Moreover, having a single master makes the whole system vulnerable because ifdies, service based communication between the nodes get stopped. Topic based communication can still work because once a connection between nodes is established via topics,is no longer needed, but new topics cannot be created withoutrunning.Also, as the number of robots increase, it becomes increasingly cumbersome to deal with conflict among similar topics and namespace resolution. §.§ Multi Master SystemMany of the limitations of a single master system can be overcome by having multiple masters running their own independentas shown in Figure <ref>. This makes the system robust as the failure of one will not lead to the failure of the complete system. Since the visibility of topics is limited to the scope of eachenvironment, there are no namespace conflict with topics in a multi-master system. All the nodes and services are local to that robot. However, it is possible to share a minimum number of topics with other robots through remapping as and when required. Since only a limited number of topics are shared, the bandwidth required in a multi-master system is less compared to that in a single master system for the same task. To implement a multi-master System, a package calledis needed <cit.> and can be easily installed as shown below. This allows two important processes,andto run simultaneously. The function ofisto send multicast messages to the network so that all environments become aware of each other. It also monitors the changes in the network and intimates all ROS masters about these changes. The other process calledenables us to select which topics can shared between different . Withoutnode, no information can be accessed by other s. The following commands are required to be executed to install and activate multi-master mode in each machine:0.5pt0.5ptIt is to be noted that the host and master IPs are same on each machine. This is unlike the single-master case where these two IPs could be different for a given machine. The namespace conflict in multi-master system can be avoided using a relay node. The use of relay node can be understood in the context shown in Figure <ref>. The global planner needs to access pose data from Robot 1 and 2 for carrying out path planning. Each of these two robots publish pose data to a topic calledunder their respective s. To avoid conflict, one has to relay theof Robot 1 to the topicand that of Robot 2 torespectively. This can be done by executing the following command on each of the robots:0.5pt0.5ptAs shown in the above figure, the global planner can now accessthese new topics calledandfor obtaining their respective pose data.Even though multi-master system saves us from several problems encountered in a single master system, it still does not provide solution to some other problems such as scalability, load balancing and lower computation power. As number of robots increase, one needs to reconfigure system files manually for each robot to enable multi-casting. It does not make efficient use of the processing power available because, by default, the processes are not distributed such that load on each machine is balanced. Bandwidth usage in multi-master system is still high compared to a cloud-based system due to the difference in network protocols <cit.> used by different machines. In a multi-master system, each machine has a limited on-board computational hardware which can not be augmented to accommodate for higher demand in the run time. This limits the usability of multi-machine system.§.§ Cloud Robotics System Many of the limitations of a multi-master system can be solved by having a cloud infrastructure to which the robots can offload computationally heavy tasks. In this paper, Rapyuta cloud robotics engine <cit.> <cit.> is used for implementing the fleet management system. As discussed earlier, it is a Platform as a Service (PaaS) framework suitable for developing robotic applications. The schematic of such an implementation is shown in Figure <ref>. It shows four main components: (1) a cloud server which includes both software as well as hardware infrastructure; (2) Physical or simulated Robots and their working environment. (3) an user interface for interacting with the system and (4) an operator or an ERP system to provide goals for the system. The inner working of this cloud-based implemented could be better understood by studying the Figure <ref> that provides a process level overview of the system showing nodes, topics and interconnection pathways among various modules of the fleet management system. The figure shows a five agent system implemented using four physical machines (three robots and a server).Each robot runs processes for localization and autonomous navigation through nodesandrespectively. The processes related to Rapyuta cloud robotics engine runs on the server machine. It also runs processes for global planner which generates paths for the robots. In a general scenario, the global planner and all related optimization algorithms can run on a separate physically machine on the network. Hence, it is shown as a separate block in the Figure <ref> similar to the blocks corresponding to robots. As shown in this figure, the global planner publishes data into two types of topics. The first topic iswhich provides paths generated by the planner in the form of an array of grid block numbers. Each robot subscribes to its correspondingto know the cell locations that it needs to traverse. The second topic, called , is a binary number which indicates whether the current goal locations received from the global planner is to be discarded by the robot or not. The binary value for the topicfor a given robot is set if a cell on its path is blocked either by an user or by an obstacle detected by the robot sensors. The grid cells could also be blocked by an ERP (Enterprise Resource Planning) system indicating non-traversable regions in the environment.Whenever the value foris set, the robot discards previously received goal locations and uses new values available at the correspondingtopic.These topics are subscribed by the respectivenodes on the cloud which, in turn, publish necessary topics for use subscription by the physical robots. Before going further, a brief understanding of Rapyuta organization will be useful for understanding the configuration steps described later. Rapyuta has the following four main components <cit.>. (1) Computing environments are the Linux containers <cit.> <cit.> used for running various ROS based robot applications; (2) Communication protocols: are the standard protocols used for internal and external communication between cloud, container and robot processes. (3) Core Task Set: for managing all process and tasks. They are further divided into three groups, namely, robot task set, environment task set and container task set. (4) Command Data Structures: are the necessary formats used for various system administration activities. The setup process for the cloud robotics based fleet management system involves two main step:* Create configuration files providing details of interaction between cloud and robots. * Launch these files using system commands on server as well as robot clients. In the remaining part of this section, we provide the details of configuration on server as well as the clients.§.§.§ Configuration of Cloud ServerThe configuration for the cloud-based fleet management system is shown in Figure <ref>. The dotted box shows the activities within the cloud server.The first process which needs to be started on the server is the Master Task Set which controls and manages all other processes on the cloud. It takes up an IP calledand listens on port 8080. This process is started by executing following Linux command:0.5pt0.5ptThe next process which needs to be started on the server is the Robot Task Set which is responsible for managing communicationwith physical robots. It can be started using the following command: 0.5pt0.5ptThe third task which needs to be started is the Container Task Set responsible for managing containers which are the basic computing environment on the cloud. The corresponding command is: 0.5pt0.5ptEach Linux container (LXC) takes up its own IP and port to communicate with master. Linux containers need not be collocated with the Rapyuta server () and can run on any other machine on the network. It is also possible to have multiple containers. The linux containers are capable of running standard available ROS nodes or user-created nodes to perform a specific task.Inside each Linux container, lies the fourth and final core task set known as Environment Task Set. This task set allows the ROS nodes running within the container to communicate with other nodes running on other Linux containers and robots on the network. The configuration for these environment tasks for containers are provided in the configuration files used by the individual clients as will be explained in the next section.The Figure <ref> also shows two main types of connection for communication among various processes. One is for internal communication within different Rapyuta processes and, the other one is for external communication between Rapyuta processes and robots. Internally, Rapyuta communicates over UNIX Sockets. For instance, the master task set uses portfor communication and is referred to as an . The processes within the Linux container communicate with robot end points through communication ports or . The corresponding port number isand is represented by the letter `P' (stands for ports) in the above figure. The robot endpoints provide interfaces for converting external format (e.g. JASON) into internal format of robots (e.g. ROS messages). On the other hand, ports are used for internal communication between endpoint processes.The external communication between Rapyuta processes and robots uses web-socket protocol. This communication is overport which is also knows asor .Readers can refer to <cit.> for more details. The figure also shows the process IDs (PID) for all related topics and nodes.§.§.§ Configuration for RobotsIn order to demonstrate the working of the system, Turtlebots <cit.> are used as autonomous mobile robot (AMR) platforms for our fleet management system. After setting up the cloud, robot processes are required to be started on each robot. Each robot is made alive within ROS environment by usingcommand. Other functionalities of the robot (autonomous navigation, obstacle avoidance, localization etc.) are activated through a standard ROS launch file. The connection between robot and Rapyuta is established usingcommand with a local configuration file available on each robot. The basic commands for setting up robots are as follows:0.5pt0.5ptThe configuration files are written in JSON and are used for sending request instructions to master task set for establishing connection with the cloud. The configuration file for each robot has the following four main components: (1) Containers, (2) Nodes, (3) Interfaces and (4) Connections. Other than this, the first part of the configuration file is used to send HTTP request to the cloud. This part appears as shown below: 0.5pt0.5ptAs shown above, the request is sent on portand in response, Rapyuta sends the endpoint's URL to the robot as a JSON encoded response. This received URL is used by the robot to connect with the cloud through port . These ports are configured at the time of installation. Upon establishing the connection the robot requests for container creation and it is done by the following block in the configuration file:0.5pt0.5ptThis creates a container inside Rapyuta having a unique tag provided by the key . Each container starts with the necessary processes or daemons like , , etc. and looks for the nodes which needs to be run inside the container. This information is provided in the `node' block in the configuration file as shown below: 0.5pt0.5ptThe keyrefers to the name of the container where these nodes are to be created,specifies the name for the node,tells the master task set about the needed packages. The keytells the name of the executable,contains the arguments to be passed andsegregates the processes inside the container giving us the flexibility to run multiple copies of the same executable independently inside a container. Once the nodes are up, it is necessary to define interfaces for each robot. Interfaces primarily refer to various kinds of sensor data that will be shared with the cloud or other robots in the network. This is specified by the following block in the configuration file: 0.5pt0.5ptThe keyrefers to the endpoint tag which is either a robot end or a container end and accordingly, a robot ID or a container tag can be mentioned as its value. The keyis the interface tag and is unique in the scope of an endpoint tag.defines the type of the interface tag which can be subscriber, publisher, service client or service provider as defined by Rapyuta <cit.>. refers to the class name and it defines the message type for publisher or subscriber andis the address of ROS topic. After defining the interfaces, it is necessary to specify the connections between various endpoints as shown in the following block:0.5pt0.5ptThis part establishes the connection between interfaces. The points to be connected are defined asand .§ SIMULATIONS & EXPERIMENTSIn this section we will provide details of how different components of fleet management system work. The modules that are being discussed here include global planner, gazebo simulation model and the web-based user interface.§.§ Global PlannerAs discussed earlier, the global planner is responsible for generating paths for robots between their current locations and the target destinations provided by the operator. It receives the location information from each of the robots, the destination information for these robots from the operator and, uses the latest map to generate necessary paths for the robots. In its simplified form, it implements a Dijkstra algorithm <cit.><cit.> <cit.> on a grid map to find shortest path between two cells as shown in Figure <ref>. In this figure, the robots are represented by filled circles. The start and end destinations of these robots are represented by the symbol pair {S_i, E_i},i=1,2,…,N where N=3 in this case. The Figure <ref> shows the case when no obstacles are present in the map. As soon as the path information is transmitted to the robots, they start following their respective paths as shown by the trail of circular dots on their paths. The Figure <ref> shows the case when an obstacle is created (or detected) in the cell number 26 at any time during this motion. This results in generation of new paths by the global planner. In a simulated environment, the robots can react instantaneously to this change. However, the robots may take some in a real world scenario due to factors like communication delay and inertia of motion as shown in this figure. The global planner may also include several other factors such as, battery life of robots, additional on-board sensor or actuator on robots (in case of a heterogeneous scenario) and other environmental conditions to solve a multi-objective optimization problem to generate these paths. Our purpose in this paper has been to demonstrate the working of a complete fleet management system which invariably requires such a centralized planner for task allocation and towards this end, we pick up the simplest path planner as an example. Readers are free to explore other planners in the same context.§.§ Simulation Environment The simulated environment for the fleet management system is created using Gazebo <cit.> <cit.>, which is an open-source software well integrated with ROS. The steps required for this on a Ubuntu Linux environment are as follows: * Create "" and "" file and place them in a folder preferably in folder .* Create a launch file similar toand set "" argument as the address of your newly created world by using the following command:0.5pt0.5ptThe resulting simulated environment is shown in Figure <ref>. It also shows three obstacles (cuboidal blocks) and three robots which are spawned in the environment. The grid cells on the floor correspond to the grid map used by the global planner shown in Figure <ref>. Whenever an user blocks a cell in the grid map, a cuboidal block is spawned in the Gazebo environment. The slow performance of Gmapping algorithm in Gazebo simulation might be overcome by tweaking some scan matching parameters as shown below: 0.5pt0.5pt §.§ Web User Interface A web interface is also built to interact with the robots and view the robot movement. This is shown in Figure <ref>. The figure shows two windows - one for visualizing the robot motion in a simulated environment and an interactive grid map for user interaction. The user can block out cells to spawn obstacles in the simulated environment and select starting position for robots. The interface is created using [http://gazebosim.org/gzweb] which is a web graphics library (WebGL) for Gazebo simulator. Like , it is a front-end interface toand provides visualization of the simulation. It is a comparatively thin and light weight client that can be accessed through a web browser. The organization of this interface is shown in Figure <ref>. usesfor visualization and interacts withthrough .which forms the core of the Gazebo simulator can interact with user programmes written with ROS APIs. This web-based interface makes the whole system platform-independent where an user can access the system over internet without having to worry about installing the pre-requisite software on his/her system. §.§ The Experimental SetupIn this section, we provide details of our real world experiment with physical robots. Three Turtlebots <cit.> are used as autonomous mobile robots (AMR) in a lab environment as shown in Figure <ref>. The map of the environment is created by using Gmapping SLAM algorithm available with ROS <cit.>. The map generated is shown in Figure <ref>.Each of the robots run AMCL <cit.> <cit.> to localize itself in the map. It also runs an obstacle avoidance algorithm that uses on-board Kinect depth range information to locate obstacles on the path. These programs are run on a low power Intel Atom processor based netbook with 2 GB RAM that comes with these robots. The map is divided into equispaced 8× 8 grid to match with the grid up used by the global planner shown in Figure <ref>. The server is a 12 CPU machine with Intel Xeon processor with 48 GB of RAM and 2 TB of storage space. The robots and server communicate over a local wireless LAN. The complete video of the experiment <cit.> as well as the source codes <cit.> are made available online for the convenience of users. § PERFORMANCE ANALYSISThe performance of each of the three modes of operation is analyzed by performing two different experiments.The details of the experiment and the resulting analysis is provided in this section.§.§ Experiment 1The schematic of machine configuration used for this experiment is shown in Figure <ref>. It shows two physical machines in the network connected to each other through Wireless LAN.The figure <ref>(a) shows the single-master mode where Machine 1 acts as the master running . Machine 2 runs Gazebo simulation environment as explained in Section <ref> and spawns five Turtlebots in it.Machine 1 apart from runningsubscribes to the Kinect scan data from these robots and prints them on a terminal console. The Figure <ref>(b) shows the multi-master mode of operation where both machines run their ownprocesses. As before, the machine runs its own Gazebo simulation environment and spawns a set of five Turtlebots. Each of the machines runnode to detect other masters in the network. The machine 1 runs thenode to subscribe to the scan data from all robots running on the other machine. The Figure <ref>(c) shows the cloud-based mode of operation where Machine 1 acts as the cloud server running Rapyuta nodes such as ,and . Similar to the previous case, the other machine runs its own Gazebo simulation environment and spawns its own set of five Turtlebots. In addition these machines also runnodes for each of the robot in order to establish connection with the cloud. In this case as well, the server subscribes to the scan data from all robots from both the machines through a container process. The relative performance of each of the modes of implementation can be analyzed by studying the two parameters, namely, network usage and CPU usage of the machines as explained below. The network usage for Machine 1 for all the three configurations is shown in Figure <ref>. It shows that the single master system generates maximum traffic while cloud robotics system generates least network traffic for the same operation. The corresponding CPU usage for the server as well as the clients in each of these three configurations is shown in Figure <ref>. It also shows the default publishing rate of messages at the topics for three configurations. As one can see, a client in the single master system publishes at higher rate (7.5 Hz) compared to that in the multi-master system (4.5 Hz) or the cloud robotics system (4 Hz). This could be linked to the fact that the CPU usage of the client for a single master system (SMS-C) is lowest giving rise to higher publishing rate. A client in multi-master system (MMS-C) and cloud robotics system (CRS-C) is required to run additional processes to establish communication with the server which leads to higher CPU usage and hence, lower publishing rate.This, however, causes more CPU usage and network usage for the server in the single master system (SMS-S).Overall, it appears that it is advantageous to go for a multi-master system or cloud robotics systems compared to a single master system as the former systems lead to lower network traffic at a comparable CPU usage compared to the later.We also plot the Round Trip Time (RTT) for the three modes of implementation. It is the time taken by a packet to go from a sender to a receiver and come back to the sender.In this paper, RTT is computed as follows. A message is published at a node on one machine. This node is subscribed by another machine, which in, turns publishes it on another node. This new node is then subscribed by the first time. The time difference between publishing the message on node and receiving it at another on machine 1 is considered as the round trip time. These two machines are located in the same place communicating over wireless LAN. The resulting RTT for all the three configurations is shown in Figure <ref>. As expected, the round-trip time increases monotonically with increasing data size and it's behaviour is more or less same for all the three configurations.Usually, the round trip time (RTT) is computed for machines which are physically separated by several kilometers <cit.>. Nevertheless, the RTT behaviour will remain more or less same as shown in Figure <ref> as the network delays between the machines will dominate the minor differences arising out of internal processes of each configuration. §.§ Experiment 2In this experiment as well, two physical machines are connected to each other through a Wireless LAN.The experiment is further simplified by removing the Gazebo simulator which has a high computational as well as memory footprint. One of these machines publish images onto a topic which is subscribed by the other machine. The other machine simply echoes this data on a console. The second machine subscribing to the image publishing topic is considered as the server as it either runs aprocess in the single master mode or a Rapyuta engine in the cloud robotics mode of operation. The relative performance of the machines is analyzed and compared in terms of CPU usage and network bandwidth usage as shown in Figure <ref>.The network usage is almost same in all the three cases as all of them use the same publishing rate and there are no other processes / nodes that generate additional network traffic.However, there is a difference in the CPU usage in these implementations. It is highest in Cloud Robotics mode of operation both on client as well as server side. This could be attributed to the additional computational overhead needed for running cloud processes. The multi-master system has the second highest CPU usage owing to the additional computation needed for runningprocesses andprocesses. Since none of these additional processes are there in the single master mode, the CPU usage is least in this case. These observations are in sync with our understanding of the systems as explained in the previous sections. § LIMITATIONS AND FUTURE WORK As summarized above, the single master or multi-master ROS systems are not suitable for deployment of Fleet Management services as a PaaS environment. Both these architectures implement Networked Robotics model based on Robot-to-Robot (R2R) communication. While enabling the familiar ROS based PaaS environment and transparent availability of sensor data across multiple robots, the following key shortcomings or constraints on an individual robot or a fleet of robots has to be noted: (1) Resource Constraints - There are resource constraints on each robot in terms of onboard compute, memory and robot's power supply, motion mode and working environment. Once deployed they cannot be easily upgraded. Algorithms which require access to high dimensional data from multiple robots requiring larger compute infrastructure will remain constrained by the overall network of robots' compute capacities.(2) Communication Constraints - higher bandwidth usage within the R2R network of mobile robots will lead to higher network latencies thereby deteriorating the quality of service.(3) Scalability constraints - on the overall solution as number of robots in a mobile fleet increases.For cloud-based PaaS systems such as Rapyuta, which implements Robot-to-Cloud (R2C) model, the following limitations are identified which need remediation: * In its current form, it does not offer high availability <cit.> for Rapyuta Master taskset and its failure leads to collapse of the whole system. This needs remediation by infrastructural mechanisms in combination with checkpoint-restart utilities <cit.> <cit.>. * Of the five key characteristics of Cloud Services, the current implementation of Rapyuta PaaS lacks one, namely, the elasticity. It has a cannibalized approach for all containers on a host to access compute, storage and network resources on the host machine and does not offer ability to allocate and resize these containers in the run-time as the workload changes over time. The utilities for monitoring the resource consumption are rudimentary and do not offer advice for migration of containers from one host to another or resizing. * In the current implementation of the cloud platform, there are no provisions for managing communication bandwidth to cater to different traffic situations. In practical scenarios for fleet management, having a logical segregation of communication bandwidth between control and data signals will improve the responsiveness of the R2C system. This is a concern when a remote tele-operation is required for an impaired mobile robot in a data centric network environment. Ability to leverage Multi-Path TCP <cit.> <cit.> can also improve the transfer rates with R2C communication as it can make use of multiple interfaces to compensate for congestion in one of the channels.* In a large warehouse of several thousand square feet area, it is possible that all mobile robots may not always have access to Cloud through the Cloud Access point. But with alternate communication modalities like Bluetooth, Zigbee or Wifi Direct - they may have connectivity to nearby robots which, in turn, may have access to the Cloud infrastructure. In such a scenario, a proxy-based <cit.> compute topology will be useful where one robot functions as a group leader to bridge the interaction between the set of nearby out-of-coverage robots and the cloud. The current Rapyuta implementation does not provide this topology and would require extensive changes to enable this. However, the other topologies such as clone-based or peer-based models are easier to implement with the current implementation and may be used along the ROS single-master or multi-master mode to simulate proxy-based systems. * In the current implementation of Rapyuta framework - the partitioning of data and compute across three options -onboard compute on robot itself or robotic R2R network and/or Cloud execution has to be decided upfront and is usually static. Depending on the task with deadline, whether it is a SLAM, Navigation or Grasping task in warehouse, it would be useful to have a framework that can allocate these tasks to suitable compute resources (on edge / fog / cloud) in the run-time. Use of energy-efficient optimization algorithms <cit.> <cit.> for task allocation and subsequent path planning and coordination have to be added on the top of Rapyuta platform for warehouse fleet management. The directions for future work therefore include remediation of the limitations of the Rapyuta Cloud framework and engineering the algorithm layer for task allocation, task planning, path planning and coordination, Grasping, Tele-operations and Collaborative SLAM in context of Picker-to-Parts Warehouse robotics. Future work needs to add the tier of R2R layer with adhoc network (using Multi-Master ROS) with suitable elastic compute topology (Peer, Proxy or Clone) with R2C Rapyuta framework. § CONCLUSIONThis paper presents the details of implementation of a fleet management system for a group of autonomous mobile robots (AMR) using three configurations: single-master, multi-master and cloud robotics platform. The mobile robots are completely autonomous as far as their navigation capabilities are concerned. These robots are required to traverse the paths provided by a global planner. The globalplanner implements a basic path planning algorithm to generate paths between the current robot locations and the desired goal locations set by the operator, taking into account the obstacles which could be created dynamically in run time. The whole system can be controlled or monitored through a web-based user interface. The details of implementation for both simulation as well as actual experiment is provided which will be useful for students and practicing engineers alike. These details provide an insight into the working of each of the these modes of operation allowing us to identify the strengths and weaknesses of each one of them. These insights are further corroborated by analyzing parameters such as, network usage, CPU load and round trip time.We also identify the critical limitations of current cloud robotics platform and provide suggestions for improving them which forms the future direction for our work.IEEEtran[m] l0.2< g r a p h i c s > Aniruddha Singhal is working as a Researcher at Innovation Labs in Tata Consultancy Services. He received his Bachelor's degree in computer science from Madhav Institute of Technology, Gwalior in the year 2014 and Master's degree in System Science from Indian Institute of Technology Jodhpur in 2016. His current research interests include Machine Learning, Computer Vision and Robotics[m] l0.2< g r a p h i c s > Nishant Kejriwal obtained his Bachelor's degree in Computer Science from Indian Institute of Technology Jodhpur in 2012. Since then, he is working as a researcher at Innovation Labs in Tata Consultancy Services. His research interests include Machine Learning, Robotics and Computer Vision.[m] l0.2< g r a p h i c s > Prasun Pallav obtained his Bachelor's degree in computer science engineering from West Bengal University of Technology in the year 2014. Since then he is working as a system engineer at Tata Consultancy Services, New Delhi, India. His research interest includes Linux System Programming, Robotics and Computer Vision.[m] l0.2< g r a p h i c s > Soumyadeep Choudhury obtained his Bachelor's degree in Electronics and Communication Engineering in the year 2015 from Academy Of Technology, West Bengal University Of Technology. Since then, he is working as a researcher at Innovation Labs, Tata Consultancy Services, New Delhi, India. His research interests include Linux System Programming, Robotics and Computer Vision. [m] l0.3< g r a p h i c s > Rajesh Sinha holds a Bachelor's degree in Electrical and Electronics Engineering from BITS Pilani and a Masters in Comparative Religion from Dayalbagh University. He has over 20 years experience of building engineered software and hardware solutions for Transportation, Logistics, Government and Retail Industries and startups. He is currently heading the Smart Machines Programme at Tata Consultancy Services' research and innovation division. [m] l0.2< g r a p h i c s > Swagat Kumar (S'08-M'13) obtained his Bachelor's degree in Electrical Engineering from North Orissa University in the year 2001. He obtained his Master's and PhD degree in Electrical Engineering from IIT Kanpur in 2004 and 2009 respectively.He was a post doctoral fellow at Kyushu University in Japan for about a year. Then he worked as an assistant professor at IIT Jodhpur for about 2 years before joining TCS Research in 2012. He currently heads the robotics research group at Tata Consultancy Services, New Delhi, India. His research interests are in Machine Learning, Robotics and Computer Vision. He is a member of IEEE Robotics and Automation Society. | http://arxiv.org/abs/1706.08931v1 | {
"authors": [
"Aniruddha Singhal",
"Nishant Kejriwal",
"Prasun Pallav",
"Soumyadeep Choudhury",
"Rajesh Sinha",
"Swagat Kumar"
],
"categories": [
"cs.RO"
],
"primary_category": "cs.RO",
"published": "20170627163259",
"title": "Managing a Fleet of Autonomous Mobile Robots (AMR) using Cloud Robotics Platform"
} |
First-principles Equation of State and Shock Compression Predictions of Warm Dense Hydrocarbons Burkhard Militzer December 30, 2023 =============================================================================================== We study the following problem: for given integers d, k and graph G, can we reduce some fixed graph parameter π of G by at least d via at most k graph operations from some fixed set S? As parameters we take the chromatic number χ, clique number ω and independence number α, and as operations we choose the edge contraction and vertex deletion .We determine the complexity of this problem for S={} and S={} and π∈{χ,ω,α} for a number of subclasses of perfect graphs. We use these results to determine the complexity of the problem for S={} and S={} and π∈{χ,ω,α}restricted to H-free graphs. § INTRODUCTIONA typical graph modification problem aims to modify a graph G, via a small number of operations from a specified set S, into some other graph H that has a certain desired property, which usually describes a certain graph class G to which H must belong.In this way a variety of classical graph-theoretic problems is captured. For instance, if only k vertex deletions are allowed and H must be an independent set or a clique, we obtain the Independent Set or Clique problem, respectively. Now, instead of fixing a particular graph class G, we fix a certain graph parameter π. That is, for a fixed set S of graph operations,we ask, given a graph G, integers k and d, whether G can be transformed into a graph G' by using at most k operations from S, such that π(G')≤π(G)-d.The integer d is called the threshold. Such problems are called blocker problems, as the set of vertices or edges involved “block”some desirable graph property, such as being colourable with only a few colours. Identifying the part of the graph responsible for a significant decrease of the parameter under consideration gives crucial information on the graph. Blocker problems have been given much attention over thelast few years, see for instance <cit.>. Graph parameters considered were the chromatic number, the independence number, the clique number, the matching number, the weight of a minimum dominating setand the vertex cover number. So far, the set S always consisted of a single graph operation, which was a vertex deletion, edge deletion or an edge addition.In this paper, we keep the restriction on the size of S by letting S consist of either a single vertex deletion or, for the first time, a single edge contraction.As graph parameters we consider the independence number α, the clique number ω and the chromatic number χ.Before we can define our problems formally, we first need to give some terminology. The contraction of an edge uv of a graph G removes the vertices u and v from G, and replaces them by a new vertex made adjacent to precisely those vertices that were adjacent to u or v in G (neither introducing self-loops nor multiple edges). We say that G can be k-contracted or k-vertex-deleted into a graph G', if G can be modified into G' by a sequence of at most k edge contractions or vertex deletions, respectively. We let π denote the (fixed) graph parameter; as mentioned, in this paper π belongs to {α,ω,χ}. We are now ready to define our decision problems in a general way: .99 Contraction Blocker(π) Instance:a graph G and two integers d,k≥ 0Question:can G be k-contracted into a graph G' such that π(G')≤π(G)-d? .99 Deletion Blocker(π) Instance:a graph G and two integers d,k≥ 0Question:can G be k-vertex-deleted into a graph G' such that π(G')≤π(G)-d?If we remove d from the input and fix it instead, then we call the resulting problems d-Contraction Blocker(π) andd-Deletion Blocker(π), respectively. .99 d-Contraction Blocker(π) Instance:a graph G and an integer k≥ 0Question:can G be k-contracted into a graph G' such that π(G')≤π(G)-d? .99 d-Deletion Blocker(π) Instance:a graph G and an integer k≥ 0Question:can G be k-vertex-deleted into a graph G' such that π(G')≤π(G)-d?The goal of our paper is to increase our understanding of the complexities of Contraction Blocker(π) and Deletion Blocker(π) fo π∈{ω,χ,α}. In order to do so, we will also consider the problems d-Contraction Blocker(π) and d-deletion blocker(π). §.§ Known Results and Relations to Other Problems It is known that Deletion Blocker(α) is polynomial-time solvable for bipartite graphs, as proven both by Bazgan, Toubaline and Tuza <cit.> and Costa, de Werra and Picouleau <cit.>. The former authors also proved thatDeletion Blocker(α) is polynomial-time solvable for cographs and graphs of bounded treewidth. The latter authors also proved that for π∈{ω,χ}, Deletion Blocker(π) is polynomial-time solvable for cobipartite graphs. Moreover, they showed that for π∈{ω,χ,α}, Deletion Blocker(π) is -complete for the class of split graphs, but becomes polynomial-time solvable for this graph class if d is fixed.By using a number of example problems we will now illustrate how the blocker problems studied in this paper relate to a number of other problems known in the literature. As we will see, this immediately leads to new complexity results for the blocker problems. 1. Hadwidger Number and Club Contraction. The Contraction Blocker(α) problem generalizes the well-known Hadwiger Number problem, which is that of testing whether a graph can be contracted into the complete graph K_r on r verticesfor some given integer r. Indeed, we obtain the latter problem from the first by restricting instances to instances (G,d,k) where d=α(G)-1 and k=|V(G)|-r. Note that the diameter and independence number of K_r are both equal to 1. Hence, one can also generalize Hadwiger Number in another way: the Club Contraction problem (see e.g. <cit.>) is that of testing whether a graph G can be k-contracted into a graph with diameter at most s for some given integers k and s.As such, Contraction Blocker(α) can be seen as a natural counterpart of Club Contraction.2. Graph transversals. Blocker problems generalize so-called graph transversal problems. To explain the latter type of problems, for a family of graphs H, the H-transversal problem is to test if a graph G can be k-vertex-deleted, for some integer k, into a graph G' that has no induced subgraph isomorphic to a graph in H. For instance, the problem {K_2}-transversal is the same as Vertex Cover. Here are some examples of specific connections between graph transversals and blocker problems.* Let H be the family {K_p|p≥ 2} of all complete graphs on at least two vertices. Then H-transversal is equivalent to Deletion Blocker(ω) restricted to instances (G,d,k) with d=ω(G)-1.* In our paper we will prove that for a graph G with at least one edge and an integer k ≥ 1, the instance (G,ω(G)-1,k) is a yes-instance of Deletion Blocker(ω) if and only if (G,k) is a yes-instance of Vertex Cover.* The Odd Cycle Transversal problem is to test whether a given graph can be made bipartite by removing at most k vertices for some given integer k≥ 0. This problem is -complete <cit.>, and it is equivalent to Deletion Blocker(χ) for instances (G,d,k) where d=χ(G)-2.* The d-Transversal or d-Cover problem <cit.> is to decide whether a graph G=(V,E) contains a set V' that intersects eachmaximum set satisfying some specified property π by at least d vertices. For instance, if the property is being an independent set, 1-Transversal is equivalent to 1-Deletion Blocker(α).3. Bipartite Contraction. The problem Bipartite Contraction is to test whether a graph can be made bipartite by at most k edge contractions. Heggernes et al. <cit.> proved that this problem is -complete. It is readily seen that 1-Contraction Blocker(χ) and Bipartite Contraction are equivalent for graphs of chromatic number 3.4. Maximum induced bipartite subgraphs. The Maximum Induced Bipartite Subgraph problem is to decide if a given graph contains an induced bipartite subgraph with at least k vertices for some integer k. Addario-Berry et al.<cit.> proved that this problem is-complete for the class of 3-colourable perfect graphs.We observe that, for 3-colourable graphs, 1-Deletion Blocker(χ) is equivalent to Maximum Induced Bipartite Subgraph.5. Cores. The two problems 1-Deletion Blocker(α)and 1-Deletion Blocker(ω) are equivalent to testing whether the input graph contains a set of S of size at most k that intersects every maximum independent set orevery maximum clique, respectively. If k=1, these two problems becomeequivalent to testing whether the input graph contains a vertex that is in every maximum independent set, or in every maximum clique, respectively. In particular, the intersection of all maximum independent sets is known as thecore of a graph.Properties of the core have been well studied (see, for example, <cit.>). In particular, Boros, Golumbic and Levit <cit.> proved that computing if the core of a graph has size at least ℓ is co--hard for every fixed ℓ≥ 1. Taking ℓ=1 gives co--hardness of1-Deletion Blocker(α).6. Critical vertices and edges. The restriction d=k=1 has also been studied when π=χ. A vertex of a graph G is critical if its deletion reduces the chromatic number of G by 1. An edge of a graph is critical or contraction-critical if its deletion or contraction, respectively, reduces the chromatic number of G by 1. The problems Critical Vertex, Critical Edge and Contraction-Critical are to test if a graph has a critical vertex, critical edge or contraction-critical edge, respectively. We note that the first two problems are the restrictions of Contraction Blocker(χ) and Deletion Blocker(χ) to instances (G,d,k) where d=k=1. Complexity dichotomies exist for each of the three problems on H-free graphs, and moreover the latter two problems are shown to be equivalent <cit.>. Graphs with a critical (or equivalently contraction-critical) edge are also called colour-critical (see, for instance, <cit.>).Due to links to problems as the ones above, it is of no surprise that many results for blocker problems are known implicitly in the literature already in various settings.For example, Belmonte et al. <cit.> proved that1-Contraction Blocker(Δ), where Δ denotes the maximum vertex-degree, is -complete even for split graphs. We make use of several known complexity results for some of the related problems stated above for proving our results. §.§ Our Results In Section <ref> we mentioned that Deletion Blocker(π) is known to be -complete for π∈{α,ω,χ} even when restricted to special graph classes. Non-surprisingly, Contraction Blocker(π) is -complete for π∈{α,ω,χ} as well (this follows from our results in Section <ref>, but it is also easy to show this directly).Due to the above, it is natural to restrict inputs to some special graph classes in order to obtain tractable results and to increase our understanding of the computational hardness of the problems.Note that it is not always clear whether Contraction Blocker(π) and Deletion Blocker(π) belong towhen restricted to a graph class G. However, when G is closed under edge contraction or vertex deletion, respectively, and π can be verified in polynomial time, then membership ofholds: we can take as certificate the sequence of edge contractions or vertex deletions, respectively. We present our results in two parts.Part I. In the first part of our paperwe focus on the class of perfect graphs and a number of well-known subclasses of perfect graphs. Most of these classes are not only closed under vertex deletion but also under edge contraction. This enables us to get unified results for the cases π=ω and π=χ (note that ω=χ holds by definition of a perfect graph). Another reason for considering subclasses of perfect graphs is that α, ω, χ can be computed in polynomial time for perfect graphs; Grötschel, Lovász, and Schrijver <cit.> proved this for χ and thus for ω, whereas the result for α follows from combining this result with the fact that perfect graphs are closed under complementation. This helps us with finding tractable results or at least with obtaining membership of(if in addition the subclass under consideration is closed under edge contraction or vertex deletion). Table <ref> gives an overview of the known results and our new results for the classes of perfect graphs we consider. We have unified results for the cases π=ω and π=χ even for the perfect graph classes in this table that are not closed under edge contraction, namely the classes of bipartite graphs;C_4-free perfect graphs with clique number 3; and the class of perfect graphs itself. As the class of perfect graphs is not closed under edge contraction we could for perfect graphs only deduce that the three contraction blocker problems are -hard (even if d=1). As the class of cographs coincides with the class of P_4-free graphs (where P_r denotes the r-vertex path) and split graphs are P_5-free, the corresponding rows in Table <ref> show a complexity jump of all our problems for P_t-free graphs from t=4 to t=5.Recall also from Section <ref> that the Hadwiger Number problem is a special case of Contraction Blocker(α)) As such, our polynomial-time result in Table <ref> for Contraction Blocker(α) restricted to cographs generalizes a result of Golovach et al. <cit.>, who proved that the Hadwiger Number problem is polynomial time solvable on cographs. Part II. In the second part of our paper we give several dichotomy results. First we give, for π∈{α,ω,χ}, complete classifications of Deletion Blocker(π) and Contraction Blocker(π) depending on the size of π, that is, we prove the following theorem. The following six dichotomies hold: (i) Contraction Blocker(α)is polynomial-time solvable for graphs with α=1 and1-Contraction Blocker(α) is -complete for graphs with α=2; (ii) Contraction Blocker(χ)is polynomial-time solvable for graphs with χ=2 and 1-Contraction Blocker(χ) is -complete for graphs with χ=3; (iii) Contraction Blocker(ω) is polynomial-time solvable for graphs with ω=2 and 1-Contraction Blocker(ω) is -complete for graphs with ω=3; (iv) Deletion Blocker(α) is polynomial-time solvable for graphs with α=1 and 1-Deletion Blocker(α) is -complete for graphs with α=2; (v) Deletion Blocker(χ) is polynomial-time solvable for graphs with χ=2 and1-Deletion Blocker(χ) is -complete for graphs with χ=3; (vi) Deletion Blocker(ω) is polynomial-time solvable for graphs with ω=1 and1-Deletion Blocker(ω)is -complete for graphs with ω=2; In particular we extend the hardness proof of Theorem <ref> (iii) in order to obtain the hardness resultfor C_4-free perfect graphs with ω=3 in Table <ref>. We note that some of the results in Table <ref>, such as this result, may at first sight seem somewhat arbitrary. However, we need the result for C_4-free perfect graphs with ω=3 and other results of Table <ref> to prove our other resultsofthe second part of our paper. Namely, by combining the results for subclasses of perfect graphs with other results, we obtain complexity dichotomies for our six blockers problems restricted to H-free graphs, that is, graphs that do not contain some (fixed) graph H as an induced subgraph. These dichotomies are stated in the following summary; here, P_r is the r-vertex path, C_3 is the triangle, and the paw is the triangle with an extra vertex adjacent to exactly one vertex of the triangle, whereasdenotes the induced subgraph relation and ⊕ denotes the disjoint union of two vertex disjoint graphs.Let H be a graph. Then the following holds: (i)If H⊆_i P_4, thenDeletion Blocker(α) is polynomial-time solvable for H-free graphs, otherwise it is-hard or co--hard for H-free graphs. (ii)If H⊆_i P_4, then Contraction Blocker(α) is polynomial-time solvable for H-free graphs, otherwise it is -hard for H-free graphs. (iii) f H⊆_i P_4, then Deletion Blocker(ω) is polynomial-time solvable for H-free graphs; otherwise it is -hard or co--hard for H-free graphs. (iv) Let H≠ C_3⊕ P_1. If H P_4 or H, then Contraction Blocker(ω) is polynomial-time solvable for H-free graphs, otherwise it is -hard or co--hard for H-free graphs. (v) If H P_1⊕ P_3 or H P_4, thenDeletion Blocker(χ) is polynomial-time solvable for H-free graphs, otherwise it is -hard or co--hard forH-free graphs. (vi) If H P_4, thenContraction Blocker(χ) is polynomial-time solvable for H-free graphs, otherwise it is -hardor co--hard forH-free graphs. Statements (i), (ii), (iii), (v), (vi) of Theorem <ref> correspond to complete complexity dichotomies, whereas there is one missing case in statement (iv). In particular we note that statements (v) and (vi) do not coincide for disconnected graphs H. We also observe from Theorem <ref> (i) that Deletion Blocker(α) is computationally hard for triangle-free graphs; in fact we will show co--hardness even if d=k=1. This in contrast to the problem being polynomial-time solvable for bipartite graphs, as shown in <cit.> (see also Table <ref>). §.§ Paper OrganizationSection <ref> contains notation and terminology. Sections <ref>–<ref> contain the results mentioned in Part I. To be more precise, Section <ref> contains our results for cobipartite graphs, bipartite graphs and trees.In Sections <ref> and <ref>, we prove our results for cographs and split graphs, respectively. In Section <ref> we also show that our -hardness reduction for split graphs can be used to prove that the three contraction blockers problems, restricted to split graphs, are [1]-hard when parameterized by d. The latter result means that for split graphs these problems are unlikely to be fixed-parameter tractable with parameter d. In Sections <ref> and <ref> we prove our results for interval graphs and chordal graphs, respectively.Sections <ref> and <ref> contain the results mentioned in Part II. In Section <ref> we first prove dichotomies for the three contraction blocker and three vertex blocker problems when we classify on basis of the size of π∈{α,χ,ω}. In the same section, we modify the hardness construction for 1-Contraction Blocker(ω) to prove that 1-Contraction Blocker(ω) is -complete even forC_4-free perfect graphs with ω=3. In Section <ref> we prove Theorem <ref>.Section <ref> contains a number of open problems and directions for future research. § PRELIMINARIESWe only consider finite, undirected graphs that have no self-loops and no multiple edges; we recall that when we contract an edge no self-loops or multiple edges are created. We refer to <cit.> or <cit.> for undefined terminology and to <cit.> for more on parameterized complexity.Let G=(V,E) be a graph.For a subset S⊆ V, we let G[S] denote the subgraph of G induced by S, which has vertex set S and edge set {uv∈ E|u,v∈ S}. We write H G if a graph H is an induced subgraph of G. Moreover, for a vertex v∈ V, we write G-v=G[V∖{v}] and for a subset V'⊆ V we write G-V'=G[V∖ V']. For a set {H_1,…,H_p} of graphs, a graph G is (H_1,…,H_p)-free if G has no induced subgraph isomorphic to a graph in {H_1,…,H_p}; if p=1 we may write H_1-free instead of (H_1)-free. The complement of Gis the graph G=(V,E) with vertex set V and an edge between two vertices u and v if and only if uv∉ E. Recall that the contraction of an edge uv∈ E removes the vertices u and v from a graph G and replaces them by a new vertex that is made adjacent to precisely those vertices that were adjacent to u or v in G. This new graph will be denoted by G| uv. In that case we may also say that u is contracted onto v, and we use v to denote the new vertex resulting from the edge contraction. The subdivision of an edge uv∈ E removes the edge uv from G and replaces it by a new vertex w and two edges uw and wv.Let G and H be two vertex-disjoint graphs. The join operation ⊗ adds an edge between every vertex of G and every vertex of H. The union operation ⊕ takes the disjoint union of G and H, that is,G⊕ H=(V(G)∪ V(H),E(G)∪ E(H)). We denote the disjoint union of p copies of G by pG. For n≥ 1, the graph P_n denotes the path on n vertices, that is, V(P_n)={u_1,…,u_n} and E(P_n)={u_iu_i+1|1≤ i≤ n-1}. For n≥ 3, the graph C_n denotes the cycle on n vertices, that is,V(C_n)={u_1,…,u_n} and E(C_n)={u_iu_i+1|1≤ i≤ n-1}∪{u_nu_1}. The graph C_3 is also called the triangle. The claw K_1,3 is the 4-vertex star, that is, the graph with vertices u, v_1, v_2, v_3 and edges uv_1, uv_2, uv_3.Let G=(V,E) be a graph. A subset K⊆ V is called a clique of G if any two vertices in K are adjacent to each other.The clique number ω(G) is the number of vertices in a maximum clique of G. A subset I⊆ V is called an independent set of G if any two vertices in I are non-adjacent to each other.The independence number α(G) is the number of vertices in a maximum independent set of G. For a positive integer k, a k-colouring of G is a mapping c: V→{1,2,…,k} such that c(u)≠ c(v) whenever uv∈ E.The chromatic number χ(G) is the smallest integer k for which G has a k-colouring. A subset of edges M⊆ E is called a matching if no two edges of M share a common end-vertex. The matching number μ(G) is the number of edges in a maximum matching of a graph G. A vertex v such that M contains an edge incident with v is saturated by M; otherwise v is unsaturated by M. A subset S⊆ V is a vertex cover of G if every edge of G is incident with at least one vertex of S.The Coloring problem is that of testing if a graph has a k-colouring for some given integer k. The problems Clique and Independent Set are those of testing if a graph has a clique or independent set, respectively, of size at least k.The Vertex Cover problem is that of testing if a graph has a vertex cover of size at most k. We need the following lemma at several places in our paper. Vertex Cover is -complete for C_3-free graphs. An interval graph is a graph such that one can associate an interval of the real line with every vertex such that two vertices are adjacent if and only if the corresponding intervals intersect. A graph is cobipartite if it is the complement of a bipartite (2-colourable) graph. A graph is chordal if it contains no induced cycle on more than three vertices. A graph is a split graph if it has a split partition, which is a partition of its vertex set into a clique K and an independent set I. Split graphs coincide with(2P_2,C_4,C_5)-free graphs <cit.>. A P_4-free graph is also called a cograph. A graph is perfect if the chromatic number of every induced subgraph equals the size of a largest clique in that subgraph.Ahole is an induced cycle on at least five vertices and an antihole is the complement of a hole. A hole or antihole is odd if it contains an odd number of vertices. We need the following well-known theorem of Chudnovsky, Robertson, Seymour, and Thomas. This theorem can also be used to verify that the other graph classesin Table <ref> are indeed subclasses of perfect graphs. A graph is perfect if and only if it contains no odd hole and no odd antihole. § COBIPARTITE GRAPHS, BIPARTITE GRAPHS AND TREES We first consider the contraction blocker problems and then the deletion blocker problems. §.§ Contraction Blockers Our first result is a hardness result for cobipartite graphs that follows directly from a known result. 1-Contraction Blocker(α) is -complete for cobipartite graphs.Golovach, Heggernes, van 't Hof and Paul <cit.> considered the s-Club Contraction problem. Recall that this problem takes as input a graph G and an integer k and asks whether G can be k-contracted into a graph with diameter at most s for some fixed integer s. They showed that 1-Club Contraction is -complete even for cobipartite graphs. Graphs of diameter 1 are complete graphs, that is, graphs with independence number 1, whereas cobipartite graphs that are not complete have independence number 2. We now focus on π=χ and π=ω. For our next result (Theorem <ref>) we need some additional terminology. A biclique is a complete bipartite graph,which is nontrivial if it has at least one edge.A biclique vertex-partition of a graph G=(V,E) is a set 𝒮 of mutually vertex-disjoint bicliques in G such thatevery vertex of G is contained in oneof the bicliques of S. The Biclique Vertex-Partition problem consists in testing whether a given graph G has a biclique vertex-partition of size at most k, for some positive integer k. Fleischner et al. <cit.> showed that this problem is -complete even for bipartite graphs and k=3.We are now ready to prove Theorem <ref>. For π∈{χ,ω}, Contraction Blocker(π) is -complete for cobipartite graphs.Since cobipartite graphs are perfect and closed under edge contractions, we may assume without loss of generality that π=χ. The problem is in , as Coloring is polynomial-time solvable on cobipartite graphs and then we can take the sequence of edge contractions as certificate. We reduce fromBiclique Vertex-Partition. Recall that this problem is -complete even for bipartite graphs and k=3 <cit.>.As the problem is polynomial-time solvable for bipartite graphs and k=2 (see <cit.>), we may ask for a biclique vertex-partition of size exactly 3, in which each biclique is nontrivial.Let (G,3) be an instance of Biclique Vertex-Partition, where G is a connected bipartite graph on n vertices that has partition classes X and Y. We claim that G has a biclique vertex-partition consisting of three non-trivial bicliques if and only if G can be (n-6)-contracted into a graph G' with χ(G')≤ 3 (so d=χ(G)-3).First suppose that G has a biclique vertex-partition S of size 3. Let S_1,S_2,S_3 be the three (nontrivial) bicliques in S. Let A_i,B_i be the two bipartition classes of S_i for i=1,2,3. So, in G, we have that A_1,A_2,A_3, B_1,B_2,B_3 are six cliques that partition the vertices of G, and moreover, there is no edge between a vertex of A_i anda vertex of B_i, for i=1,2,3. In G we contract each clique A_i to a single vertex that we give colour i, and we contract each clique B_i to a single vertex that we give colour i as well. In this way we have obtained a 6-vertex graph G' (so the number of contractions is n-6) with a 3-colouring. Thus, χ(G')≤ 3.Now suppose that G can be (n-6)-contracted into a graph G' with χ(G')≤ 3.We first observe that the class of cobipartite graphs is closed under taking edge contractions; indeed, if e is an edge connecting two vertices of the same partition class, then contracting e results in a smaller clique, and if e is an edge connecting two vertices of two different partition classes, then contracting e is equivalent to removing one of its end-vertices and making its other end-vertex adjacent to every other vertex in the resulting graph. As the class of cobipartite graphs is closed under taking contractions, G' is cobipartite. As cobipartite graphs have independence number at most 2, each colour class in a colouring of G' must have size at most 2. Consequently, G' must have exactly six vertices a_1,a_2,a_3, b_1,b_2,b_3 such that a_1,a_2,a_3 form a clique, b_1,b_2,b_3 form a clique, and moreover,a_i and b_i are not adjacent, for i=1,2,3. This means that we did not contract an edge uv with u∈ X and v∈ Y (as the resulting vertex would be adjacent to all other vertices).Hence, we may assume without loss of generality that for i=1,2,3, each a_i corresponds to a set of vertices A_i⊂ X (that we contracted into the single vertex a_i) and thateach b_i corresponds to a set of vertices B_i⊂ Y (that we contracted into the single vertex b_i). As each pair a_i,b_i is non-adjacent,no vertex of A_i is adjacent to a vertex of B_i. Consequently, in G, we find that each set A_i∪ B_i induces a biclique. Hence, the sets A_1∪ B_1, A_2∪ B_2 and A_3∪ B_3 form a biclique vertex-partition of G that has size 3.We now assume that d is fixed. We show thatd-Contraction Blocker(π) becomes polynomial-time solvable on cobipartite graphs for π∈{χ,ω}. For π=χ, we can prove this even for the class of graphs with independence number at most 2, or equivalently, the class of 3P_1-free graphs, which properly contains the class of cobipartite graphs. For any fixed d≥ 0, the d-Contraction Blocker(χ) problem can be solved in polynomial time for 3P_1-free graphs.Let G be a graph with α(G)≤ 2. Consider a colouring with χ(G) colours. The size of every colour class is at most 2. Hence every subgraph of G induced by two colour classes has at most 4 vertices, and as such has a spanning forest with in total at most 3 edges. This means that we can contract two colour classes to an independent set (that is, to a new colour class) by using at most 3 contractions. This observation gives us the following algorithm. We guess a set of at most 3 contractions. Afterward we decrease d by 1 and repeat this procedure until d=0. For each resulting graph G' we check whether χ(G')≤χ(G)-d. If so, then the algorithm returns a yes-answer and otherwise a no-answer.Let m be the number of edges of G. Then the total number of guesses is at most m^3d, which is polynomial as d is fixed. Because Coloring is polynomial-time solvable on graphs with independence number at most 2and this class is closed under edge contractions, our algorithm runs in polynomial time.Let π∈{χ,ω}. For any fixed d≥ 0, the d-Contraction Blocker(π) problem can be solved in polynomial time on cobipartite graphs.For π=χ this follows immediately fromTheorem <ref>.As cobipartite graphs are perfect and closed under edge contraction, we obtain the same result for π=ω. We now consider the class of bipartite graphs. If π∈{χ,ω}, then Contraction Blocker(π) is trivial for bipartite graphs (and thus also for trees). To the contrary, for π=α, we will showthat Contraction Blocker(π) is -hard for bipartite graphs. The complexity of d-Contraction Blocker(α) remains open for bipartite graphs. Bipartite graphs are not closed under edge contraction. Therefore membership tocannot be established by taking a sequence of edge contractions as the certificate, even though due to König's Theorem (see, for example, <cit.>), Independent Set ispolynomial-time solvable for bipartite graphs.Contraction Blocker(α) is -hard on bipartite graphs. We know from Theorem <ref> that 1-Contraction Blocker(α) is -complete on cobipartite graphs. Consider a cobipartite graph G with m edges and an integer k, which together form an instance of 1-Contraction Blocker(α). Subdivide each of the m edges of G in order to obtain a bipartite graph G'. We claim that (G,k) is a yes-instance of 1-Contraction Blocker(α) if and only if (G',α(G')-1,k+m) is a yes-instance of Contraction Blocker(α). First suppose that (G,k) is a yes-instance of 1-Contraction Blocker(α). In G' we first perform m edge contractions to get G back. We then perform k edge contractions to get independence number 1=α(G')-(α(G')-1). Hence, (G',α(G')-1,k+m) is a yes-instance ofContraction Blocker(α). Now suppose that (G',α(G')-1,k+m) is a yes-instance of Contraction Blocker(α). Then there exists a sequence of k+m edge contractions that transform G' into a complete graph K. We may assume thatK has size at least 4 (as we could have added without loss of generality three dominating vertices to G without increasing k). As K has size at least 4, each subdivided edge must be contracted back to the original edge again. This operation costs m edge contractions, so we contract G to K using at most k edge operations. Hence, (G,k) is a yes-instance of 1-Contraction Blocker(α). This proves the claim and hence the theorem.We complement Theorem <ref> by showing that Contraction Blocker(α) is linear-time solvable on trees. In order to prove this result we make a connection to the matching number μ of a graph. Contraction Blocker(α) is linear-time solvable on trees. Let (T,d,k) be an instance of Contraction Blocker(α), where T is a tree on n vertices. We first describe our algorithm and prove its correctness. Afterwards, we analyze its running time. Throughout the proof let M denote a maximum matching of T.As α(T)+μ(T)=n by König's Theorem (see, for example, <cit.>), we find that (T,d,k) is a no-instance if d>n-μ(T).Assume that d≤ n-μ(T). We observe that trees are closed under edge contraction. Hence, contracting an edge of T results in a new tree T'. Moreover, T' has n-1 vertices and the edge contraction neither increased the independence number nor the matching number. As α(T)+μ(T)=n and similarlyα(T')+μ(T')=n-1, this means that either α(T')=α(T)-1 or μ(T')=μ(T)-1. First suppose that d≤ n-2μ(T). There are exactly σ(T)=n-2μ(T) vertices that are unsaturated by M. Let uv be an edge, such that u is unsaturated.As M is maximum, v must be saturated. Then, by contracting uv, we obtain a tree T' such that μ(T')=μ(T).It follows from the above that α(T')=α(T)-1. Say that we contracted u onto v.Then in T' we have that v is saturated by M, which is a maximum matching of T' as well.Thus, if d≤ n-2μ(T), contracting d edges, one of the end-vertices of which is unsaturated by M, yields a tree T' with μ(T')=μ(T) and α(T')=α(T)-d. Since an edge contraction reduces the independence number by at most 1, it follows that this is optimal. Hence, as d≤ n-2μ(T), we find that (G,T,k) is a yes-instance if k≥ d and a no-instance if k<d. Now suppose that d>n-2μ(T). Suppose that we first contract the n-2μ(T) edges that have exactly one end-vertex that is unsaturated by M. It follows from the above that this yields a tree T' with μ(T')=μ(T) and α(T')=α(T)-(n-2μ(T)). Since T' does not contain any unsaturated vertex, M is a perfect matching of T'. Then, contracting any edge in T' results in a tree T” with μ(T”)=μ(T')-1 and thus, α(T”)=α(T'). If we contract an edge uv∈ M, the resulting vertex uv is unsaturated byM'=M∖{uv} in T”. Hence, as explained above, if in addition we contract now an edge (uv)w, we obtain a tree T”' with α(T”')=α(T”)-1 and μ(T”')=μ(T”). Repeating this procedure, we may reduce the independence number of T by d with n-2μ(T)+2(d-n+2μ(T))=2(d+μ(T))-n edge contractions. Below we show that this is optimal.Suppose that we contract p edges inT. Let T' be the resulting tree. We have α(T')+μ(T')=n-p. As μ(T')≤1 2(n-p), this means that α(T')≥1 2(n-p). If p<2(d+μ(T))-n we have -p 2>-(d+μ(T))+n 2,and thus [α(T')≥ 1 2(n-p); > n 2-d-μ(T)+n 2; =α(T)-d. ]So at least 2(d+μ(T))-n edge contractions are necessary to decrease the independence number by d.It remains to check if k is sufficiently high for us to allow this number of edge contractions. As we can find a maximum matching of tree T (and thus compute μ(T)) in O(n) time by using the algorithm of Savage <cit.>, our algorithm runs in O(n) time. Remark 1. By König's Theorem, we have that α(G)+μ(G)=|V(G)|for any bipartite graph G, but we can only use the proof of Theorem <ref> to obtain a result for treesfor the following reason: trees form the largest subclass of (connected) bipartite graphs that are closed under edge contraction, and this property plays a crucial role in our proof.§.§ Deletion Blockers We first show that all three deletion blocker problems are polynomial-time solvable for bipartite graphs (and thus for trees). It is known already that Deletion Blocker(α) is polynomial-time solvable for bipartite graphs <cit.>. Hence it suffices to prove that the same holds for Deletion Blocker(π) when π∈{χ, ω}. In order to do so we need the following relation between 1-Deletion Blocker(ω) and Vertex Cover.Let G be a graph with at least one edge and let k ≥ 1 be an integer. Then (G,ω(G)-1,k) is a yes-instance of Deletion Blocker(ω) if and only if (G,k) is a yes-instance of Vertex Cover.Let G=(V,E) be a graph with |E|≥ 1. Thus, ω(G) ≥ 2. Let k ≥ 1 be an integer. First suppose that (G,k) is a yes-instance of Vertex Cover, that is, G has a vertex cover V' of size at most k. So, every edge of G is incident to at least one vertex of V'.Then, deleting all vertices of V' yields a graph G' with no edges. This means that ω(G') ≤ 1, and thus (G,ω(G)-1,k) is a yes-instance for Deletion Blocker(ω). Now suppose that (G,ω(G)-1,k) is a yes-instance of Deletion Blocker(ω). Then there exists a set V'⊆ V of size |V'|≤ k such that ω(G-V') ≤ 1.This implies that G-V' has no edges. Thus V' is a vertex cover of G of size at most k. So, (G,k) is a yes-instance for Vertex Cover. Proposition <ref> has the following corollary, which we will apply in this section and at some other places in our paper.Let G be a triangle-free graph with at least one edge and let k ≥ 1 be an integer. Then (G,k) is a yes-instance of 1-Deletion Blocker(ω) if and only if (G,k) is a yes-instance of Vertex Cover. We are now ready to prove the following result. For π∈{χ,ω}, Deletion Blocker(π) can be solved in polynomial time on bipartite graphs.As bipartite graphs are perfect and closed under vertex deletion, the problems Deletion Blocker(ω) and Deletion Blocker(χ) are equivalent. Therefore, we only have to consider the case where π=ω. As bipartite graphs have clique number at most 2, Deletion Blocker(ω) and 1-Deletion Blocker(ω) are equivalent.As bipartite graphs are triangle-free, we can apply Corollary <ref>. To solve Vertex Cover on bipartite graphs, König's Theoremtells us that it suffices to find a maximum matching, which takes O(n^2.5) time on n-vertex bipartite graphs <cit.>.We now consider thethe class of cobipartite graphs. It is known that Deletion Blocker(π) is polynomial-time solvable on cobipartite graphs if π∈{ω, χ} <cit.>. Hence we only have to deal with the case π=α. For this case we prove the following result, which follows immediately from Theorem <ref>. Deletion Blocker(α) can be solved in polynomial time on cobipartite graphs. § COGRAPHS It is well known (see for example <cit.>) thata graph G is a cograph if and only if G can be generated from K_1 by a sequence of operations, where each operation is either a join or a union operation.Recall from Section <ref> that we denote these operations by ⊗ and ⊕, respectively. Such a sequence corresponds toa decomposition tree T, which has the following properties: 1.its root r corresponds to the graph G_r=G;2. every leaf x of T corresponds to exactly one vertex of G, and vice versa, implying that x corresponds to a unique single-vertex graph G_x;3.every internal node x of T has at least two children, is either labeled ⊕ or ⊗, and corresponds to an induced subgraph G_x of G defined as follows: * if x is a ⊕-node, then G_x is the disjoint union of all graphs G_y where y is a child of x;* if x is a ⊗-node, then G_x is the join of all graphs G_y where y is a child of x. A cograph G may have more than one such tree but has exactly one unique tree <cit.>, called the cotree T_G of G, if the following additional property is required: 4. Labels of internal nodes on the (unique) path from any leaf to r alternate between ⊕ and ⊗. Note that T_G has O(n) vertices. For our purposes we must modify T_G by applying the following known procedure (see for example <cit.>). Whenever an internal node x of T_G has more than two children y_1 and y_2, we remove the edges xy_1 and xy_2 and add a new vertex x' withedges xx', x'y_1 and x'y_2. If x is a ⊕-node, then x' is a ⊕-node, and if x is a ⊗-node, then x' is a⊗-node. Applying this rule exhaustively yields a tree in which each internal node has exactly two children. We denote this tree by T_G'. Because T_G has O(n) vertices, modifying T_G into T_G' takes linear time.Corneil, Perl and Stewart <cit.> proved that the problem of deciding whether a graph with n vertices and m edges is a cograph can be solved in time O(n+m). They also showed that in the same time it is possible to construct its cotree (if it exists). As modifying T_G into T_G' takes O(n+m) time, we obtain the following lemma. Let G be a graph with n vertices and m edges.Deciding ifG is a cograph and constructing T_G' (if it exists) can be done in time O(n+m). For two integers k and ℓ we say that a graph G can be (k,ℓ)-contracted into a graph H if G can be modified into H by a sequence containing k edge contractions and ℓ vertex deletions. Note that cographs are closed under edge contraction and under vertex deletion.In fact, to prove our results for cographs, we will prove the following more general result. Let π∈{α,χ,ω}. The problem of determining the largest integer d such that a cograph G with n vertices and m edges can be (k,ℓ)-contracted into a cograph H with π(H)≤π(G)-dcan be solved inO(n^2+mn+(k+ℓ)^3n) time. First consider π=α. Let G be a cograph with n vertices and m edges and let k,ℓ be two positive integers. We first construct T_G'. We then consider each node of T_G' by following a bottom-up approach starting at the leaves of T_G' and ending in its root r. Let x be a node of T_G'. Recall that G_x is the subgraph of G induced by all vertices that corresponds to leaves in the subtree of T_G' rooted at x.With node x we associate a tablethat records the following data: for each pair of integers i,j≥ 0 with i+j≤ k+ℓwe compute the largest integer dsuch that G_x can be (i,j)-contracted into a graph H_x with α(H_x)≤α(G_x)-d. We denote this integer d byd(i,j,x). Leti,j≥ 0 with i+j≤ k+ℓ.Case 1. x is a leaf.Then G_x is a 1-vertex graph meaning that d(i,j,x)=0 if j=0, whereas d(i,j,x)=1 if j≥ 1.Case 2. x is a ⊕-node.Let y and z be the two children of x. Then, as G_x is the disjoint union of G_y and G_z, we find that α(G_x)=α(G_y)+α(G_z).Hence, we have [ d(i,j,x)=max {α(G_x)-(α(G_y)-d(a,b,y)+α(G_z)-d(i-a,j-b,z))|; 0≤ a≤ i, 0≤ b≤ j}; =max{d(a,b,y)+d(i-a,j-b,z)|0≤ a≤ i, 0≤ b≤ j}. ]Case 3. x is a ⊗-node. Since x is a ⊗-node, G_x is connected and as such has a spanning tree T. If i+j≥ |V(G_x)| and j≥ 1, then we can contract i edges of T in the graph G_x followed by j vertex deletions. As each operation will reduce G_x by exactly one vertex, this results in the empty graph. Hence, d(i,j,x)=α(G_x). From now on assume that i+j<|V(G_x)| or j=0. As such, any graph we can obtain from G_x by using i edge contractions and j vertex deletions is non-empty and hence has independence number at least 1.Let y and z be the two children of x. Then, as G_x is the join of G_y and G_z,we find that α(G_x)=max{α(G_y),α(G_z)}.In order to determine d(i,j,x) we must do some further analysis. Let S be a sequence that consists of i edge contractions and j vertex deletions of G_x such that applying S on G_x results in a graph H_x with α(H_x)=α(G_x)-d(i,j,x).We partition S into five sets S_y^e, S_z^e, S_yz^e, S_y^v, S_z^v,respectively, as follows. Let S_y^e and S_z^e be the set of contractions of edges with both end-vertices in G_y and with both end-vertices in G_z, respectively. Let S_yz^e be the set of contractions of edges with one end-vertex in G_y and the other one in G_z. Let a_y=|S_y^e| and let a_z=|S_z^e|. Then |S_yz^e|=i-a_y-a_z. Let S_y^v and S_z^v be the set of deletions of vertices in G_y and G_z, respectively.Let b=|S_y^v|. Then |S_z^v|=j-b. We distinguish between two cases. First assume that S^e_yz=∅. Then a_y+a_z=i. Let H_y be the graph obtained from G_y after applying the subsequence of S, consisting of operations in S_y^e∪ S_y^v, on G_y. Let H_z be defined analogously. Then we have[α(H_x) =max{α(H_y),α(H_z)}; =max{α(G_y)-d(a_y,b,y),α(G_z)-d(a_z,j-b,z)}; = max{α(G_y)-d(a_y,b,y),α(G_z)-d(i-a_y,j-b,z)}, ]where the second equality follows from the definition of S.Now assume that S^e_yz≠∅.Recall thati+j<|V(G_x)| or j=0. Hence α(H_x)≥ 1. Our approach is based on the following observations.First, contracting an edge with one end-vertex in G_y and the other one in G_z is equivalent to removing these two end-vertices and introducing a newvertex that is adjacent to all other vertices of G_x (such a vertex is said to be universal).Second, assume that G_y contains two distinct vertices u and u' and that G_z contains two distinct vertices v and v'. Now suppose that we are to contract two edges from {uv,uv',u'v,u'v'}. Contracting two edges of this set that have a common end-vertex, say edges uv and uv', is equivalent to deleting u,v,v' from G_x and introducing a new universal vertex. Contracting two edges with no common end-vertex, say uv and u'v', is equivalent to deleting all four vertices u,u',v,v' from G_x and introducing two new universal vertices. Because the two new universal vertices in the latter choice are adjacent, whereas the vertex u' may not be universal after making the former choice, the latter choice decreases the independence number by the same or a larger value than the former choice. Hence, we may assume without loss of generality that the latter choice happened. More generally, the contracted edgeswith one end-vertex in G_y and the other one in G_z can be assumed to form a matching.We also note that introducing a new universal vertex to a graph does not introduce any new independent set other than the singleton set containing the vertex itself.We conclude that each edge contraction in S^e_yz may be considered to be equivalent to deleting one vertex from G_y and one from G_z and introducing a newuniversal vertex. If one of the two graphs G_y or G_z becomes empty in this way, then an edge contraction in S^e_yz canbe considered to be equivalent to the deletion of a vertex of the other one. Finally, if both sets G_y and G_z become empty, then we can stop as in that case H_x has independence number 1 (which we assumed was the smallest value of α(H_x)).By the above observations and the definition of S we find thatα(H_x) =max{1,α(G_y)-d(a_y,b+i-a_y-a_z,y),α(G_z)-d(a_z,j-b+i-a_y-a_z,z)}. Hence we can do as follows. We consider all tuples (a_y,b) with 0≤ a_y≤ i and 0≤ b≤ j and computemax{α(G_y)-d(a_y,b,y),α(G_z)-d(i-a_y,j-b,z)}. Let α_x' be the minimum value over all values found. We then consider all tuples (a_y,a_z,b) with a_y≥ 0, a_z≥ 0, a_y+a_z≤ i and 0≤ b≤ j and computemax{1,α(G_y)-d(a_y,b+i-a_y-a_z,y),α(G_z)-d(a_z,j-b+i-a_y-a_z,z)}. Let α_x” be the minimum value over all values found. Then d(i,j,x)=α(G_x)-min{α_x',α_x”}.After reaching the root r, we let our algorithm return the integer d(k,ℓ,r). By construction, d(k,ℓ,r) is the largest integer such that G=G_r can be (k,ℓ)-contracted into a graph H with α(H)≤α(G)-d(k,ℓ,r). We are left to analyze the running time.Constructing T_G' can be done in O(n+m) time by Lemma <ref>. We now determine the time it takes to compute one entry d(i,j,x) in the table associated with a node x. It takes linear time to compute the independence number of a cograph[For a cograph G,compute T_G' and use the formula α(G_x)=α(G_y)+α(G_z) if x is a ⊕-node with children y and z andα(G_x)=max{α(G_y),α(G_z)} otherwise. Alternatively, see for example <cit.> for a linear-time algorithm on a superclass of cographs.]. The total number of tuples (a_y,b) and (a_y,a_z,b) that we need to consider is O((k+ℓ)^3). Note that the table associated with a node x has O((k+ℓ)^2) entries but that we only have to compute α(G_x) once. Hence, it takes O(n+m+(k+ℓ)^3) time to construct a table for a node. As T_G' has O(n) vertices, the total running time is O(n+m)+O(n(n+m+(k+ℓ)^3))=O(n^2+mn+(k+ℓ)^3n).Now consider π=χ. Note that we cannot consider the complement of a cograph (which is a cograph) because an edge contraction in a graph does not correspond to an edge contraction in its complement. However, we can re-use the previous proof after making a few modifications. Let G be a cograph with n vertices and m edges and let k,ℓ be two positive integers. We follow the same approach as in the proof for π=α. We only have to swap Cases 2 and 3 after observing that χ(G_x)=max{χ(G_y),χ(G_z)} if x is a ⊕-node with y and z as its two children and χ(G_x)=χ(G_y)+χ(G_z) if x is a ⊗-node. We can use the same arguments as used in the proof for π=α for the running time analysis as well; we only have to observe that it takes O(n+m) time to compute the chromatic number of a cograph (using the same arguments as before or by using another algorithm of <cit.>).Finally consider π=ω. As cographs are perfect and closed under edge contractions, the proof follows immediately from the corresponding result for π=χ. For π∈{α,χ,ω}, both the Contraction Blocker(π) problem and the Deletion Blocker(π) problem can be solved in polynomial time for cographs.We use Theorem <ref> after setting ℓ=0 for Contraction Blocker(π) and k=0 for Deletion Blocker(π). § SPLIT GRAPHS A split partition (K,I) of a split graph is minimal if I∪{v} is not an independent set for all v∈ K, in other words every vertex v∈ K is adjacent to some vertex u∈ I. Note that for a minimal split partition (K,I) we have α(G)=| I|. A split partition (K,I) is maximal if K∪{v} is not a clique for all v∈ I,in other words every vertex v∈ I is non adjacent to at least one vertex u∈ K. Note that for a maximal split partition (K,I) we have ω(G)=χ(G)=| K|. We first show the following result. Let π∈{α,χ,ω}. For anyfixed d≥ 0,the d-Contraction Blocker(π) problem is polynomial-time solvable on split graphs.First consider π=α. Let (G,k) be an instance of d-Contraction Blocker(α) where G=(V,E) is a split graph. Let (K,I) be a minimal split partition of G. Let I' be the set of vertices in I that have at least one neighbour in K, and let I”=I∖ I'. Because G is a split graph, all vertices of I' belong to the same connected component D of G. Moreover, we have α(G)=|I|=|I'|+|I”|=α(D)+|I”|. First suppose that |I'|≤ d.For (G,k) to be a yes-instance, G must be contracted into a graph G' with α(G')≤α(G)-d=|I'|+|I”|-d≤ |I”|. This means that we must contract D into the empty graph, which is not possible. Hence, (G,k) is a no-instance in this case. Hence, we may assume without loss of generality that |I'|≥ d+1.Suppose that k≥ d+1. If k≥ |I'|, then we contract every vertex of I' onto a neighbour in K. In this way we have k-contracted G into a graph G' with α(G')=|I”|+1= |I'|+|I”|-(|I'|-1)≤ |I'|+|I”|-d=α(G)-d. So, (G,k) is a yes-instance in this case. If k≤ |I'|-1, we contract each vertex of an arbitrary subset of k vertices of I' onto a neighbour in K.In this way we have k-contracted G into a graph G' with α(G')≤ |I'|-k+1+|I”|≤ |I'|+|I”|-d=α(G)-d. So, (G,k) is a yes-instance in this case as well.If k≤ d, then we consider all possible sequences of at most k edge contractions. This takes time O(|E(G)|^k), which is polynomial as d, and consequently k, is fixed. For every such sequence we check in polynomial time whether the resulting graph has stability number at most α(G)-d. As split graphs are closed under edge contraction and moreover are chordal graphs, the latter can be verified in linear time (see <cit.>).Now let π=χ. Let (G,k) be an instance of d-Contraction Blocker(χ) where G=(V,E) is a split graph.Case 1. χ(G)≤ d.For (G,k) to be a yes-instance, G must be k-contracted into a graph G' with χ(G')≤χ(G)-d≤ 0. The only graph with chromatic number at most 0, is the empty graph. However, a non-empty graph cannot be contracted to an empty graph. Hence, (G,k) is a no-instance in this case.Case 2. χ(G)=d+1. For (G,k) to be a yes-instance, G must be k-contracted into a graph G' with χ(G')≤χ(G)-d=1. Hence, every connected component of G' must consist of exactly one vertex. If G has no connected components with edges, then (G,k) is a yes-instance. Otherwise, because G is a split graph, G has exactly one connected component D containing one or more edges. In that case, (G,k) is a yes-instance if and only if k≥ |V(D)|-1; this can be checked in constant time.Case 3. χ(G)≥ d+2.First, assume that k<d. Because every edge contraction reduces the chromatic number by at most 1,(G,k) is a no-instance.Second, assume that k=d.We consider all possible sequences of at most k edge contractions. This takes time O(|E(G)|^k), which is polynomial as d, and consequently k, is fixed. For every such sequence we check in polynomial time whether the resulting graph has chromatic number at most χ(G)-d. As split graphs are closed under edge contractions and moreover are chordal graphs, the latter can be verified in polynomial time(see <cit.>).Third, assume that k>d. We claim that (G,k) is a yes-instance. This can be seen as follows. Let (K,I) be a maximal split partition of G. If k<|K|, then we contract k arbitrary edges of K. The resulting graph G' has a split partition (K',I) with |K'|= |K|-k≤ |K|-d-1. Hence χ(G')≤ |K'|+1≤ |K|-d=χ(G)-d. Note that the latter equality follows from our assumption that (K,I) is maximal. Now suppose that k≥ |K|. We contract |K| arbitrary edges of K. The resulting graph G' has chromatic number χ(G')=2≤χ(G)-d.Hence, in both cases, we conclude that (G,k) is a yes-instance.Finally consider π=ω. We use the previous result combined with the fact that split graphs are perfect and closed under edge contractions. In our next theorem we give two hardness results which, as explained in Section <ref>, show that Theorem <ref> can be seen as best possible. In their proofs we will reduce from the Red-Blue Dominating Set problem. This problem takes as input a bipartite graph G=(R∪ B,E) and an integer k, and asks whether there exists a red-blue dominating set of size at most k, that is, a subset D⊆ B of at most k vertices such that every vertex in R has at least one neighbour in D. This problem is -complete, because it is equivalent to the -complete problems Set Cover and Hitting Set <cit.>. TheRed-Blue Dominating Set problem isalso [1]-complete when parameterized by |B|-k <cit.>. Belmonte et al. <cit.> reduced from the same problem for showing that 1-Contraction Blocker(Δ) is -complete and [2]-hard (with parameter k) for split graphs, but the arguments we use to proveour results are quite different from the ones they used.For π∈{α,χ,ω},the Contraction Blocker(π) problem, restricted to split graphs, is -complete as well as [1]-hard when parameterized by d.The problem is infor π∈{α,χ,ω}, as split graphs are closed under edge contraction and the three problems Clique Coloring and Independent Set are readily seen to be polynomial-time solvable onsplit graphs; hence, we can take the sequence of edge contractions asthe certificate. Recall that we reduce from Red Blue Dominating Set in order to show -hardness and [1]-hardness with parameter d.First consider π=α. Let G=(R∪ B,E) be a bipartite graph that together with an integer k forms an instance ofRed-Blue Dominating Set.We may assume without loss of generality that k≤ |B|. Moreover, we may assume that every vertex of R is adjacent to at least one vertex of B. We add all possible edges between vertices in R. This yields a split graph G^* with a split partition (R,B). Because every vertex in R is assumed to be adjacent to at least one vertex of B in G, we find that (R,B) is a minimal split partition of G^*.BecauseRed-Blue Dominating Set problem is -complete <cit.> and [1]-complete when parameterized by |B|-k <cit.>, it suffices to prove that G has a red-blue dominating set of size at most k if and only if (G^*,|B|-k) is a yes-instance of (|B|-k)-Contraction Blocker(α). We prove this claim below. First suppose that G has a red-blue dominating set D of size at most k. Because k≤ |B|, we may assume without loss of generality that |D|=k (otherwise we would just add some vertices from B∖ D to D).In G^* we contract every u∈ B∖ D onto a neighbour in R. In this way we (|B|-k)-contracted G^* into a graph G'.Note that G' is a split graph that has a split partition (R,D). Becauseevery vertex in R is adjacent to at least one vertex of D in G by definition of D, it is adjacent to at least one vertex of D in G^*.The latter statement is still true for G', as contracting an edge incident to a vertex u∈ B is equivalent to deleting u. Hence, (R,D) is a minimal split partition of G', so α(G')=|D|. Because (R,B) is a minimal split partition of G^*, we have α(G^*)=|B|. This means that α(G')=|D|=|B|-(|B|-|D|)=α(G^*)-(|B|-k). We conclude that (G^*,|B|-k) is a yes-instance of (|B|-k)-Contraction Blocker(α).Now suppose that (G^*,|B|-k) is a yes-instance of (|B|-k)-Blocker(α), that is, G^* can be (|B|-k)-contracted into a graph G' such that α(G')≤α(G^*)-(|B|-k). Recall that α(G^*)=|B|. Hence, α(G')≤ k. Let p be the number of contractions of edges with one end-vertex in B. Note that any such contraction decreases the size of the independent set B by exactly one. If p<|B|-k, then G' contains an independent set of size |B|-p>k, which would mean that α(G')>k, a contradiction. Hence, p≥ |B|-k, which implies that p=|B|-k as we performed no more than |B|-k contractions in total.Let D denote the independent set obtained from B after all edge contractions. Then we find thatk= |B|-(|B|-k)=|B|-p=|D|≤α(G') ≤α(G^*)-(|B|-k) =|B|-(|B|-k) =k. Hence, |D|=α(G'), which means that (D,R) is a minimal split partition of G'. This means that every vertex of R is adjacent to at least one vertex of D in G'. Because all our contractions were performed on edges with one end-vertex in B, we have only removed vertices from G^*, that is, G' is an induced subgraph of G^*. Hence, every vertex of R is adjacent to at least one vertex of D in G'. Consequently, D is a red-blue dominating set of G with size |D|=k.Now consider π=χ. Let G=(R∪ B,E) be a bipartite graph that together with an integer k forms an instance ofRed-Blue Dominating Set.We may assume without loss of generality that k≤ |B|. Moreover, we may assume that every vertex of R is adjacent to at least one vertex of B.We take the bipartite complement of G, that is, we construct the bipartite graph with partition classes R and B, and we add an edge between any two vertices u∈ R and v∈ B if and only if uv∉ E. Then, we add all possible edges between vertices in B. Finally we add a new vertex x to the graph. We make xadjacent to all vertices of B∪ R. This yields a split graph G^* with a split partition (B∪{x},R). Because every vertex in R is assumed to be adjacent to at least one vertex of B in G, itis non-adjacent to at least one vertex of B in G^*.Hence, (B∪{x},R) is a maximal split partition of G^* (we will explain the role of vertex x in our construction later). Similarly to the previous case, we claim that G has a red-blue dominating set of size at most k if and only if (G^*,|B|-k) is a yes-instance of (|B|-k)-Contraction Blocker(χ). We prove this claim below.First suppose that G has a red-blue dominating set D of size at most k. Because k≤ |B|, we may assume without loss of generality that |D|=k (otherwise we would just add some vertices from B∖ D to D).In G^* we contract every u∈ B∖ D onto x. In this way we (|B|-k)-contracted G^* into a graph G'. Note that G' is a split graph that has a split partition (D∪{x},R). Because every vertex in R is adjacent to at least one vertex of D in G by definition of D, it is non-adjacent to at least one vertex of D in G^*. The latter statement is still true for G', as no vertex of D∪ R was involved in any of the edge contractions performed. Hence, (D∪{x},R) is a maximal split partition of G', so χ(G')=|D|+1. Because (B∪{x},R) is a maximal split partition of G^*, we have χ(G^*)=|B|+1. This means that χ(G')=|D|+1=k+1=|B|+1+k+1-(|B|+1)=χ(G^*)-(|B|-k). We conclude that (G^*,|B|-k) is a yes-instance of (|B|-k)-Contraction Blocker(χ).Now suppose that (G^*,|B|-k) is a yes-instance of (|B|-k)-Blocker(χ), that is, G^* can be (|B|-k)-contracted to a graph G' such that χ(G')≤χ(G^*)-(|B|-k). Recall that χ(G^*)=|B|+1. Hence, χ(G')≤ k+1. Let p be the number of contractions of edges between two vertices of B∪{x}.Note that any such contraction decreases the size of the clique B∪{x} by exactly one. If p<|B|-k, then G' contains a clique of size |B|+1-p>k+1, which would mean that χ(G')>k+1, a contradiction. Hence, p≥ |B|-k, which implies that p=|B|-k as we performed no more than |B|-k contractions in total.Let B' denote the clique obtained from B∪{x} after all edge contractions. Then we find that [k+1=|B|+1-(|B|-k); =|B|+1-p; = |B'|; ≤χ(G'); ≤ χ(G^*)-(|B|-k); =|B|+1-(|B|-k); = k+1. ]Hence, |B'|=χ(G'), which means that (B',R) is a maximal split partition of G'. This means that no vertex of R is adjacent to all vertices of B' in G'. We may assume without loss of generality that x∈ B', as we can view any edge contraction of an edge between a vertex u∈ B and x as a contraction of u onto x. Furthermore, suppose we performed a contraction of an edge uu' with u,u'∈ B, say we contracted u onto u'. We change this by contracting u onto x instead. Because x is adjacent to all vertices of B∪ R in G, we find that x is adjacent to all vertices (except to itself) of G' and of any intermediate graph that we obtained while contracting G into G'. Hence, contracting u onto x is equivalent to deleting u. As such, contracting u onto x does not lead to a vertex v∈ R becoming adjacent to all vertices of B'.Consequently, the size of a maximum clique in the modified graph is also equal to |B'|=χ(G'). As we can do the same for any other contraction of an edge between two vertices in B, we may assume without loss of generality that every edge contraction is a contraction of a vertex of B onto x. Let D=B'∖{x}⊆ B. As noted, contracting a vertex of B onto x is the same as deleting such a vertex of B from the graph. Hence, every vertex of D has exactly the same neighbours in G' as it has in G^*. Because every vertex in R is adjacent to x but not to all vertices of B'=D∪{x}, we find that every vertex in R is non-adjacent to at least one vertex of D in G', and consequently, in G^*.Because x∈ B' and |B'|=k+1, we find that |D|=k. We conclude that D is ared-blue dominating set of G with size |D|=k.Finally, consider π=ω. As split graphs are perfect and closed under edge contractions, this case follows directly from the previous case where π=χ. Regarding the Deletion Blocker(π) problem, for π∈{α,χ,ω},we know from <cit.> that it is -complete. In the same paper it was shown that if d is fixed, all three problems become polynomially solvable. § INTERVAL GRAPHS Let G=(V,E) be an interval graph with n vertices and m edges that corresponds to a set of intervals ℐ={I_1, I_2, …, I_n} on the real line. Let V={v_1,…,v_n} be such that vertex v_i corresponds to interval I_i for i=1,…,n. Note that the class of interval graphs is closed under edge contraction. Indeed, contracting an edge v_iv_i corresponds to removing the intervals I_i and I_j and adding a new interval I_ij= I_i ∪ I_j. It is well known (see e.g. <cit.>) that G has at most n maximal cliques which can be linearly ordered in O(n+m) time so that the maximal cliques containing a vertex v_i appear consecutively for i=1,…,n.We first prove a useful lemma for the class of C_4-free graphs, which contains the class of interval graphs as a proper subclass.Let G=(V,E) be a C_4-free graph and let v_1v_2∈ E. Let G|v_1v_2 be the graph obtained after the contraction of v_1v_2 and let v_12 be the new vertex replacing v_1 and v_2. Thenevery maximal clique K in G|v_1v_2 containing v_12 corresponds to a maximal clique K' in G and vice versa, such that (a) either |K|=|K'| and K∖{v_12}=K'∖{v_1};(b) or |K|=|K'| and K∖{v_12}=K'∖{v_2};(c) or |K|=|K'|-1 and K∖{v_12}=K'∖{v_1,v_2}.Moreover, every other maximal clique in G|v_1v_2 is a maximal clique in G and vice versa.Let A_1 (resp. A_2) be the set of neighbours of v_1 (resp. v_2) that are nonadjacent to v_2 (resp. v_1). Let A_3 be the set of vertices adjacent to both v_1 and v_2.Now consider a clique K in G|v_1v_2 containing v_12. As G is C_4-free, we find that G, and hence G|v_1v_2, contains no edge between a vertex in A_1 and a vertex in A_2. Therefore we are in exactly one of the following cases:(i) K contains one or more vertices from both A_1 and A_3 but no vertices from A_2; (ii) K contains one or more vertices from both A_2 and A_3 but no vertices from A_1;(iii) K contains one or more vertices from A_1 but no vertices from A_2 and A_3; (iv) K contains one or more vertices from A_2 but no vertices from A_1 and A_3; (v) K contains one or more vertices from A_3 but no vertices from A_1 and A_2.Suppose we are in case (i).Since K is maximal, it follows that (K∖{v_12})∪{v_1} is a maximal clique in G and thus outcome (a) holds.By symmetry, if we are in case (ii), outcome (b) holds. Assume now that case (iii) occurs. Since K is maximal, it follows that (K∖{v_12})∪{v_1} is a maximal clique in G and thus outcome (a) holds. By symmetry, we conclude that if case (iv) occurs, outcome (b) holds. Finally, suppose that we are in case (v). Then (K∖{v_12})∪{v_1,v_2} is a maximal clique in G and thus outcome (c) holds. Lemma <ref> tells us that if we contract an edge e in a C_4-free graph, every maximal clique containing both end-vertices of e will have its size reduced by exactly one in the resulting graph, and moreover, the size of every other maximal clique of the original graph will remain the same and we do not create any new maximal clique.Let G=(V,E) be an interval graph and let d≥ 0 be an integer. LetK^1 be the first maximal clique of size strictly greater than ω(G)-d starting left on the real line, and let I_x,I_y be the intervals with the rightmost right endpoints among all intervals corresponding to the vertices in K^1. Let B⊆ E be a set of edges such that the graph G' obtained from G after having contracted all edges from B satisfies ω(G')≤ω(G)-d. Then there exists a set B'⊆ E such that B'=(B∖{v_1v_2})∪{xy}, where v_1,v_2∈ K^1 and such that the graph G” obtained from G after contracting all edges in B' satisfies ω(G”)≤ω(G)-d. We first note that, by their definition, x and y are contained in all maximal cliques of size strictly greater than ω(G)-d that contain at least two vertices ofK^1. Moreover, contracting the edge xy instead of another edge v_1v_2 of K^1 does not create cliques of larger size, due to Lemma <ref>. Lemma <ref> tells us that if for an interval graph the answer ofthe Contraction Blocker(ω) problem is yes, then there always exists a set B⊆ E with |B|≤ k such that ω(H)≤ω(G)-d, where H is the graph obtained from G by contracting the edges of B, and xy∈ B where x,y belong to the first maximal clique K in G with size strictly greater than ω(G)-d starting left on the real line and such that I_x,I_y have the rightmost right endpoints among all intervals corresponding to vertices in K. Since interval graphs are closed under edge contractions, we can use this property recursively to obtain a polynomial-time algorithm for Contraction Blocker(π), with π∈{χ,ω}, in interval graphs. Let π∈{χ,ω}. Then Contraction Blocker(π) can be solved in polynomial time on interval graphs.Since interval graphs are perfect and closed under edge contractions, we may assume without loss of generality that π=ω. Let G=(V,E) be an interval graph and let d≥ 0 be an integer. Our algorithm goes as follows. Let K^1 be the first maximal clique of size strictly greater than ω(G)-d starting left on the real line. By Lemma <ref>, we know that if there exists a solution, then there exists one in which we contract the edge xy where x,y∈ K^1 are such that the corresponding intervals I_x,I_y have the rightmost right endpoints among all intervals corresponding to vertices in K^1. So we contract the edge xy. Since the resulting graph is still an interval graph, we may repeat our procedure. We consider again the first maximal clique of size strictly greater than ω(G)-d starting left on the real line and contract the edge whose end-vertices correspond to the intervals with the rightmost right endpoints among all intervals corresponding to vertices in that clique. We continue like this until there is no more maximal clique of size strictly greater than ω(G)-d in the graph.The correctness of our algorithm follows from Lemmas <ref> and <ref>. Indeed, by Lemma <ref> we know that our choice of the edges that we contract is such that at each step there is at least one maximal clique of size strictly greater than ω(G)-d whose size is reduced by one and furthermore, we do not create any new maximal clique. Since an interval graph on n vertices contains at most n maximal cliques, it follows that our algorithms stop after at most nd steps.Since all maximal cliques of an interval graph can be found in time O(n+m), where m is the number of edges, we then find that our algorithm runs in time O(nd(n+m)). Finally, Lemma <ref> ensures that the set of edges we choose to contract has minimum size.The proof of Theorem <ref> can be readily adapted to show polynomial-time solvability of theDeletion Blocker(π) problem on interval graphs for π∈{χ,ω}.Let π∈{χ,ω}. Then Deletion Blocker(π) can be solved in polynomial time on interval graphs. We recall that for π=α the complexity of both problems is open for interval graphs.§ CHORDAL GRAPHS The following result shows that Theorem <ref> cannot be generalized to chordal graphs. For π∈{χ,ω},1-Contraction Blocker(π) is -complete for chordal graphs.Since chordal graphs are perfectand closed under taking edge contractions, we may assume without loss of generality that π=ω.As Clique is polynomial-time solvable on chordal graphs, this means that the problem is in(take the sequence of edge contractions as the certificate). We reduce from Vertex Cover, which is well known to be -complete (see <cit.>). Let G=(V,E) be a graph that together with an integer k forms an instance of Vertex Cover.From G we construct a chordal graph G' as follows. We introduce a new vertex y not in G. We represent each edge e of G by a clique K_e in G' of size |V| so that K_e∩ K_f=∅ whenever e≠ f. We represent each vertex v of G by a vertex in G' that we also denote by v. Then we let the vertex set of G' be V∪⋃_e ∈ EK_e ∪{y}. We add an edge between every vertex in K_e and a vertex v∈ V if and only if v is incident with e in G. In G' we let the vertices of V form a clique. Finally, we add all edges between y and any vertex in V∪⋃_e ∈ EK_e. Note that the resulting graph G' is indeed chordal. Note also that ω(G')=|V|+3 (every maximum clique consists of y, the vertices of a clique K_e and their two neighbours in V).We claim that G has a vertex cover of size at most k if and only if G' can be k-contracted to a graph H with ω(H)≤ω(G')-1. First suppose that G has a vertex cover U of size at most k. For each vertex v∈ U, we contract the corresponding vertex v in G' to y. As |U|≤ k, this means that we k-contracted G' into a graph H. Since U is a vertex cover, we obtain ω(H)≤ |V|+2=ω(G')-1.Now suppose that G' can be k-contracted to a graph H with ω(H)≤ω(G')-1. Let S be a corresponding sequence of edge contractions (so |S|≤ k holds). By Lemma <ref> and the fact that chordal graphs are closed under taking edge contractions,we find that no contraction in S results in a new maximum clique. Hence, as we need to reduce the size of each maximum clique K_uv∪{u,v,y} by at least 1, we may assume without loss of generality that each contraction in S concerns an edge with both its end-vertices in V∪{y}. We construct a set U as follows. If S contains the contraction of an edge uy we select u. If S contains the contraction of an edge uv, we select one of u,v arbitrarily. Because each maximum clique K_uv∪{u,v,y} must be reduced, we find that U⊆ V is a vertex cover. By construction, |U|≤ k. This completes the proof.Similar arguments as in the above proof can be readily used to prove the following result, which shows that Theorem <ref> cannot be generalized to chordal graphs. For π∈{χ,ω},1-Deletion Blocker(π) is -complete for chordal graphs. § SIX DICHOTOMY RESULTS AND C_4-FREE PERFECT GRAPHS WITH Ω=3 In this section we first prove that for π∈{α,χ, ω} the contraction and deletion blocker problems become very quickly -hard when we increase π, that is, we prove Theorem <ref>.Theorem <ref> (restated). The following six dichotomies hold: (i) Contraction Blocker(α)is polynomial-time solvable for graphs with α=1 and1-Contraction Blocker(α) is -complete for graphs with α=2; (ii) Contraction Blocker(χ)is polynomial-time solvable for graphs with χ=2 and 1-Contraction Blocker(χ) is -complete for graphs with χ=3; (iii) Contraction Blocker(ω) is polynomial-time solvable for graphs with ω=2 and 1-Contraction Blocker(ω) is -complete for graphs with ω=3; (iv) Deletion Blocker(α) is polynomial-time solvable for graphs with α=1 and 1-Deletion Blocker(α) is -complete for graphs with α=2; (v) Deletion Blocker(χ) is polynomial-time solvable for graphs with χ=2 and1-Deletion Blocker(χ) is -complete for graphs with χ=3; (vi) Deletion Blocker(ω) is polynomial-time solvable for graphs with ω=1 and1-Deletion Blocker(ω)is -complete for graphs with ω=2; All six problems are readily seen to be in for the above graph classes (it suffices to take the sequence of edge contractions or vertex deletions as a certificate). We prove each of the six statements separately.(i)The problem is trivial if α=1. As cobipartite graphs have independence number at most 2, we can apply Theorem <ref> to obtain -completeness if α=2.(ii) The problem is trivial if χ≤ 2. We now consider the class of graphs with χ=3. Recall that the problem Bipartite Contraction is to test whether a graph can be made bipartite by at most k edge contractions. It is readily seen that 1-Contraction Blocker(χ) and Bipartite Contraction are equivalent for graphs of chromatic number 3. Heggernes, van 't Hof, Lokshtanov and Paul <cit.> observed that Bipartite Contraction is -complete by reducing from the -complete problem Edge Bipartization, which is that of testing whether a graph can be made bipartite by deleting at most k edges. Given an instance (G,k) of Edge Bipartization, they obtain an instance (G',k') of Bipartite Contraction by replacing every edge in G by a path of sufficiently large odd length.Note that the resulting graph G' has chromatic number 3 (assign colour 1 to the vertices of G and give the new vertices colours 2 and 3).(iii) The problem is trivial if ω≤ 2. We now consider the class of graphs with ω=3. We use a polynomial reduction from the problem ONE-IN-3-SAT, which is well known to be -complete (see <cit.>). This problem has as input a set X={x_1,…, x_n} of n boolean variables and a collection C={c_1,…,c_m} of clauses over X∪X̅ such that | c_i|=3 for i=1,…,m.The question is whether there a truth assignment for X such that each clause of C contains exactly one true literal.Let I=(X,C) be an instance of ONE-IN-3-SAT. We construct an instance (G,n+m) of 1-Contraction Blocker(ω), where G is constructed as follows(see Fig. <ref> for an example): * For each variable x∈ X, introduce five vertices forming a triangle and a square sharing exactly one edge. This yields the gadget for the variable x, where the two edges that do not belong to the square correspond to the two literals x and x̅.* For each clause c_i∈ C, introduce three vertices forming a triangle T_i. This yields the gadget for the clause c_i, where each edge corresponds to one of the three literals forming c_i.* For every edge of a triangle T_i corresponding to a literal λ, link its two end-vertices by a matching to the two end-vertices of the edge corresponding to λ in the variable gadget. Observe that (G,n+m) can be obtained in polynomial time. Moreover, ω(G)=3 and G contains exactly n+m disjoint triangles. Thus, in order to obtain a graph G' from G withω(G')=2, we need to contract at least one edge from each of these triangles. We claim that I is a yes-instance of ONE-IN-3-SAT if and only if (G,n+m) is a yes-instance of 1-Contraction Blocker(ω).First suppose that I is a yes-instance. For each variable x which is true (resp. false), we contract the edge corresponding to the literal x (resp. the literal x̅) in the triangle of the variable gadget; for each clause c_i, we contract the unique edge of the clause gadget corresponding to the literal which is set to true (see Fig. <ref>). Thus we contract exactly n+m edges, one in each of the n+m disjoint triangles. For each clause gadget in G, the unique contracted edge is linked to the unique contracted edge in the variable gadget corresponding to the true literal. Hence the four original vertices are transformed into two adjacent vertices. We claim that no new triangles are created by performing the n+m edge contractions. Indeed, when contracting an edge from a clause gadget, we do create a triangle T one edge of which belongs to a variable gadget. But by construction, this edge will necessarily be contracted as well. Thus this triangle T is transformed into a single edge. Hence ω(G')=2, which means that (G,n+m) is a yes-instance.Suppose now that (G,n+m) is a yes-instance. This means that we can obtain a graph G' with ω(G')=2 by contracting n+m edges of G.Since G contains exactly k disjoint triangles, we must, as already mentioned before, contract exactly one edge in each of these triangles. Furthermore, in a variable gadget we must contract an edge not belonging to the square, as otherwise a new triangle is created and hence we would need more than n+m contractions, a contradiction. Let e be an edge in a variable gadget that is contracted. Suppose that e corresponds to a literal λ. In G, e is contained in some squares containing edges of clause gadgets which correspond to λ. Thus, after this contraction, we create new triangles each containing an edge of a clause gadget corresponding to λ. It follows that we must contract the edges in the clause gadgets corresponding to the literal λ, otherwise triangles will remain in G'. Since we use n+m edge contractions, exactly one edge in each clause gadget is contracted. Hence, by assigning the value true to the literal corresponding to the edge contracted in each variable gadget, one literal has value true and the other two have value false in each clause. This yields a positive answer for I, so I is a yes-instance.(iv) & (vi) Both problems are trivial if π∈{α,ω} has value 1. Now consider the class of graphs with ω=2, or equivalently the class of triangle-free graphs. Since Vertex Cover is -complete for triangle-free graphsby Lemma <ref>, we conclude from Corollary <ref> that 1-Deletion Blocker(ω) is -complete for triangle-free graphs. The remainder of statement (iv) follows immediately after recalling that 1-Deletion Blocker(α) can be solved by taking the complement of the input graph and solving 1-Deletion Blocker(ω) instead.(v) First consider the class of graphs with χ=2, which coincides with the class of bipartite graphs. Then the problem becomes equivalent to Independent Set, which ispolynomial-time solvable for bipartite graphs (due to König's Theorem; see, for example, <cit.>). Now consider the class of graphs with χ=3. Recall that the Maximum Induced Bipartite Subgraph problem is to test if a given graph contains an induced bipartite subgraph with at least k vertices for some integer k and that this problem is -complete even for the class of 3-colourable perfect graphs <cit.>. As for 3-colourable graphs 1-Deletion Blocker(χ) is equivalent to Maximum Induced Bipartite Subgraph, we find that 1-Deletion Blocker(χ) is -complete for graphs with chromatic number 3. We have proven each of the six claims and thus have proven the theorem.We note that the graph G in the proof of Theorem <ref> (iii) contains no induced diamond (the complete graph K_4 on four vertices minus an edge) and no induced butterfly (the graph with vertices a,b,c,d,e and edges ab,bc,ca,cd,de,ec). As a graph G is K_4-free if and only if ω(G)≤ 3, we have in fact proven the following. The 1-Contraction Blocker(ω) problem is -complete for the class of (,K_4)-free graphs. We use Theorem <ref> (iii) to prove the following hardness result. For π∈{χ, ω},1-Contraction Blocker(π)is -complete for the class of C_4-free perfect graphs with clique number 3.As before, the problem is readily seen to be in . Let π=ω, or equivalently, π=χ. We adapt the construction used in the proof of Theorem <ref> (iii) by doing as follows for each edge e of the graph G in this proof. First we subdivide e. This gives us two new edges e_1 and e_2. We introduce two new non-adjacent vertices u_e and v_e and make them adjacent to both end-vertices of e_1.Denote the resulting graph by G^*. Note that we got rid of all the induced C_4s while not creating any new induced C_4 in this way. Hence G^* is C_4-free. Moreover, we did not introduce any clique on four vertices. Hence, as ω(G)=3, we also have ω(G^*)=3. The vertices of the original graph together with the subdivision vertices form a bipartite graph on top of which we placed a number of triangles. Hence, G^* contains no odd hole and no odd antihole. By Theorem <ref>, G^* is perfect.We increase the allowed number of edge contractions accordingly and observe that, because of the presence of the vertices u_e and v_e for each edge e, we are always forced to contract the edge e_1, which gives us back the original construction extended with a number of pendant edges (which do not play a role). Note that we have left the class of C_4-free perfect graphs after contracting away the triangles, but this is allowed. We recall that Contraction Blocker(α) is still open for the class of C_4-free perfect graphs as well as Deletion Blocker(π) for π∈{α,χ, ω}, even if d is fixed.§ H-FREE GRAPHS In this section we prove our complexity results for the six blocker problems restricted to H-free graphs, that is, we prove Theorem <ref>. To summarize, for π∈{α,ω,χ} we are able to give a dichotomy both for the contraction and deletion blocker problem except for one open case for the contraction blocker problem when π=ω.We first consider π=α, then π=ω and then π=χ. §.§ When π=α We call a vertex forced if it is in every maximum independent set of a graph <cit.>.Recall that the set of all forced vertices is called the core of a graph and that Boros, Golumbic and Levit <cit.> proved that computing whether the core of a graph has size at least k is co--hard for every fixed k≥ 1.As a special case of their result, the problem of testing the existence of a forced vertex is co--hard.We prove that the latter problem, or equivalently, Deletion Blocker(α) with d=k=1, stays co--hard even forgraphs of girth p+1, or equivalently,(C_3,…,C_p)-free graphs, for any constant p≥ 3 ((the girth of a graph is the length of a shortest cycle in it). Deletion Blocker(α) is co--hard for (C_3,…,C_p)-free graphs for any constant p≥ 3 even if d=k=1. Let G be a graph. We pick one of its edges uv and subdivide uv twice, that is, we replace the edge uv by two new verticesx and y and edges ux, xy, yv. We let G' denote the resulting graph. Note that α(G')=α(G)+1 (see also <cit.>). We claim that G has a forced vertex if and only if G' has a forced vertex.First suppose that G has a forced vertex s. Then s is also a forced vertex of G'. In order to see this consider a maximum independent set I' of G'. For contradiction, suppose that I' does not contain s. Recall that I has size α(G)+1. If x is in I', then its neighbour y is not in I', and thus I'∖{x} is a maximum independent set of G that does not contain s, a contradiction. Hence x is not in I', and for the same reason y is not in I' either. Then u is in I', as otherwise we could putx in I' to get a larger independent set than I'. However, we now find that I'∖{u} is a maximum independent set of G that does not contain s, a contradiction. Hence s belongs I'. We conclude that s is a forced vertex of G' as well.Now suppose that G' has a forced vertex s. First suppose s∈{x,y}, say s=x. Then v is a forced vertex of G. In order to see this consider a maximum independent set I of G. For contradiction, suppose that Idoes not contain v. ThenI∪{y} is a maximum independentset of G' not containing s=x, a contradiction. Hence s does not belong to {x,y}, so s must be in G. Then s is also a forced vertex of G. In order to see this consider a maximum independent set I of G. For contradiction, suppose that I does not contain s.As u and v are adjacentin G, not both of them are in I. Assume without loss of generality thatuis not in I. Then I∪{x} is a maximum independent set of G' that does not contains,a contradiction. We conclude that s is a forced vertex of G. We now subdivide each edge of G a sufficiently number of times (say p times) so that the resulting graph G” is (C_3,…,C_p)-free.By repeatedly applying the above claim, we find that G has a forced vertex if and only if G” has a forced vertex. As deciding whether a graph has a forced vertex is co--hard <cit.>, the result follows.Before we present our two complexity dichotomies for π=α we need one additionalobservation. If H is a (3P_1,2P_2)-free forest, then H P_4.As H is 3P_1-free, H contains at most two connected components. Suppose H contains exactly two connected components. Then, as H is 2P_2-free, at least one of these components must be a P_1. As H is 3P_1-free, this means that H is an induced subgraph of P_1⊕ P_2, so H P_4. Suppose H is connected. As H is 3P_1-free, H contains no claw and no path on more than five vertices. Hence, H P_4.We are now ready to present our first dichotomy. Let H be a graph.If H⊆_i P_4, then Deletion Blocker(α) is polynomial-time solvable for H-free graphs, otherwise it is -hard or co--hard for H-free graphs.Let H be a graph. If H⊆_i P_4, then we use Corollary <ref> to obtain polynomial-time solvability. Suppose H is not an induced subgraph of P_4.If H contains an induced cycle C_r for some r≥ 3, then we pick p=r+1 and apply Theorem <ref> to obtain co--hardness even if d=k=1. Note that for r=5, we could have applied Theorem <ref> to obtain -hardness, as split graphs are C_5-free. Similarly, if r≥ 6, then H contains an induced 2P_2 and we could have applied Theorem <ref> (as split graphs are 2P_2-free) to obtain -hardness as well.Now assume that H is forest. As H is not an induced subgraph of P_4, by Lemma <ref> either 2P_2 H or 3P_1 H. If 2P_2 H, then we apply Theorem <ref> again to obtain -hardness. If 3P_1 H, then we use Theorem <ref> (iv) to obtain -hardness even if d=1, after observing that a graph G is 3P_1-free if and only if α(G)=2. Remark 2. Recall that H-free graphs are closed under vertex deletion. Hence, Deletion Blocker(α) for H-free graphs will be in if we can solve Independent Set for H-free graphs in polynomial time; in that case we can take a sequence of vertex deletions as certificate. To give an example, Independent Set is polynomial-time solvable for P_5-free graphs <cit.>. Hence, for P_5-free graphs, Deletion Blocker(α) is not only -hard (which, as argued in the proof of Theorem <ref>, follows from Theorem <ref>) but even -complete. We now consider the edge contraction variant and present our second dichotomy. Let H be a graph.If H⊆_i P_4, then Contraction Blocker(α) is polynomial-time solvable for H-free graphs, otherwise it is-hard for H-free graphs.Let H be a graph. If H is an induced subgraph of P_4, then we use Corollary <ref> to obtain polynomial-time solvability.Now suppose that H is not an induced subgraph of P_4. If H contains an induced cycle that is odd, then we use Theorem <ref> to obtain -hardness.If H contains an induced cycle that is even, then H either contains an induced C_4 or, if the even cycle has at least six vertices, an induced 2P_2. This means that we can use Theorem <ref> to obtain -hardness after recalling that split graphs are (2P_2,C_4)-free. Assume H contains no cycle. Then H is a forest. If H contains an induced 3P_1, then we use Theorem <ref> (i) to obtain -hardness even if d=1, after observing that a graph G is 3P_1-free if and only if α(G)=2. Assume H is 3P_1-free.Then 2P_2 H by Lemma <ref>, which means we can useTheorem <ref> again to obtain -hardness.§.§ When π=ω The complexity dichotomy for Deletion Blocker(ω) follows immediately from Theorem <ref> after making two observations. First, Deletion Blocker(ω) for H-free graphs is equivalent to Deletion Blocker(α) for H-free graphs. Second, the graph P_4 is self-complementary, that is, P_4=P_4.Let H be a graph. If H⊆_i P_4, then Deletion Blocker(ω) is polynomial-time solvable for H-free graphs; otherwise it is co--hard or -hard for H-free graphs.We now consider the Contraction Blocker(ω) problem for H-free graphs. We start by giving a sufficient condition for computational hardness.Let G be a graph class with the following property: if G∈ G, then so are 2G and G⊕ K_r for any r≥ 1. We call such a graph class clique-proof. If Cliqueis -complete for a clique-proof graph class G, thenContraction Blocker(ω) is co--hard for G, even if d=k=1.Let G be a graph class that is clique-proof. From a given graph G∈ G and given integer ℓ≥ 1 we construct the graph G'=2G⊕ K_ℓ+1. Note that G'∈ G by definition and that ω(G')=max{ω(G),ℓ+1}.It suffice to prove that ω(G)≤ℓ if and only if G' can be 1-contracted into a graph G^* with ω(G^*)≤ω(G')-1. First suppose thatω(G)≤ℓ. Then ω(G')=ω(K_ℓ+1)=ℓ+1. In G' we contract an edge of the K_ℓ+1. This yields the graph G^*=2G⊕ K_ℓ, which has clique number ω(G^*)=ℓ, as ω(K_ℓ)=ℓ and ω(G)≤ℓ. As ω(G')=ℓ+1, this means thatω(G^*)≤ω(G')-1. Now suppose that G' can be 1-contracted into a graph G^* with ω(G^*)≤ω(G')-1.As contracting an edge in one of the two copies of G in G' does not lower the clique number of G', the contracted edge must be in the K_ℓ+1, that is, G^*=2G⊕ K_ℓ. As this did result in a lower clique number, we conclude that ω(G')=ω(K_ℓ+1)=ℓ+1 and ω(G^*)=ω(2G⊕ K_ℓ)= max{ω(G),ℓ}=ℓ. The latter equality implies that ω(G)≤ℓ.We need a number of special graphs, namely the cobanner, bull, the aforementioned butterfly and thepaw (the graph P_1⊕ P_3), which are all displayed in Figure <ref>. We also need the following lemma from Poljak. The Clique problem is -complete for the following classes: (C_5,P_5)-free graphs, K_1,3-free graphs, -free graphs and (,P_5)-free graphs. We use Lemma <ref> in the proof of our next lemma. Let H be a connected graph. If H is neither an induced subgraph of P_4 nor ofthe paw, then 1-Contraction Blocker(ω) is -hard or co--hard for H-free graphs.Let H be a connected graph that is neither an induced subgraph of P_4 nor of the paw. If H contains an induced C_4, use Theorem <ref>. If H contains an induced K_4, diamond or butterfly, useCorollary <ref>. If H contains an induced K_1,3, C_5, P_5, bull or cobanner, use Lemma <ref> with Theorem <ref>. So from now on we may assume that H is (C_4,C_5,P_5,K_1,3,K_4,,,, )-free. Below we show that this leads to a contradiction.First suppose that H contains no cycle. Then, as H is connected, H is a tree. Because H is K_1,3-free, H is a path.Our assumption that H is neither an induced subgraph of P_4 nor of the paw implies that H contains an induced P_5, which is not possible as H is P_5-free.Now suppose that H contains a cycle C. Then C must have exactly three vertices, because H is (C_4,C_5,P_5)-free. As H is not an induced subgraph of the paw, we find that H contains at least one vertex x not on C. As H is connected, we may assume that x has a neighbour on C. Because H is (,K_4)-free, x has exactly one neighbour on C. Let v be this neighbour. Hence, H contains an induced paw (consisting of x, v and the other two vertices of C).As H is not an induced subgraph of the pawand H is connected, it follows that H contains a vertex y∉ V(C)∪{x} that is adjacent to a vertex on C or to x. Suppose that y is adjacent to a vertex of C. Then, as H is (,K_4)-free, y has exactly one neighbour u in C. If u=v then H either contains an induced claw (if x and y are non-adjacent) or an induced butterfly (if x and y are adjacent). Since, by our assumption, this is not possible, it follows that u≠ v. Then, because H is bull-free, we deduce that x and y are adjacent. However, then the vertices, u,v,x,y form an induced C_4, which is not possible as H is C_4-free. We conclude that y is not adjacent to a vertex of C, so y must be adjacent to x only. However, then H contains an induced cobanner, a contradiction. This completes the proof of Lemma <ref>. A graph G is complete multipartite if V(G) can be partitioned into k independent sets V_1,…,V_k for some integer k, such that two vertices are adjacent if and only if they belong to two different sets V_i and V_j. We need a result of Olariu on paw-free graphs. Every connected paw-free graph is either triangle-free or complete multipartite. We are ready to present our result for Contraction Blocker(ω) restricted to H-free graphs. This is the only result where we do not have a dichotomy due to one missing case. Let H≠ C_3⊕ P_1 be a graph. If H P_4 or H, then Contraction Blocker(ω) is polynomial-time solvable for H-free graphs, otherwise it is -hard or co--hard for H-free graphs.First assume that H is connected. If H is an induced subgraph of P_4 then we use Corollary <ref>. If H is an induced subgraph of the paw, then we know from Lemma <ref> that G is either C_3-free or complete multipartite. In the first case one must contract all the edges of an H-free graph in order to decrease its clique number. Hence Contraction Blocker(ω) is polynomial-time solvable for C_3-free graphs. In the second case H is P_4-free, so we can use Corollary <ref> again. If H isneither an induced subgraph of P_4 nor of the paw, then we useLemma <ref>.Now assume that H is not connected. If H contains a connected component that is not an induced subgraph of P_4 or the pawthen we use Lemma <ref> again. Assume that each connected component of H is an induced subgraph of P_4 or the paw. If 3P_1 H or 2P_2 H then we use Theorem <ref> or Theorem <ref>, respectively. Hence, H∈{2P_1,P_2⊕ P_1,C_3⊕ P_1}. In the first two cases H P_4 and thus we can use Corollary <ref>, whereas we excluded the last case.§.§ When π=χ Recall that Deletion Blocker(χ) and Contraction Blocker(χ) are called Critical Vertex andContraction-Critical Edge, respectively, if d=k=1. We need the following result announced in <cit.>; see <cit.> for its proof. If a graph HP_4 or of H P_1⊕ P_3, then Critical Vertex and Contraction-Critical Edge restricted to H-free graphs are polynomial-time solvable, otherwise they are -hard or co--hard. We also need the following result of Král', Kratochvíl, Tuza, and Woeginger. Let H be a graph. If HP_4 or of H P_1⊕ P_3, then Coloring is polynomial-time solvable for H-free graphs, otherwise it is -complete for H-free graphs. We also need the following lemma.Deletion Blocker(χ) is polynomial-time solvable for 3P_1-free graphs.Let G=(V,E) be a 3P_1-free graph with |V|=n and let k ≥ 1 be an integer. Consider an instance (G,k,d) of Deletion Blocker(χ). We proceed as follows. Consider an optimal colouring of G.Since G is 3P_1-free, the size of each colour class is at most 2. Moreover, the number of colour classes of size 1 is the same for every optimal colouring of G. Let ℓ be this number. Hence, there are n-ℓ/2 colour classes of size 2 and χ(G)=ℓ+n-ℓ/2. Now (G,k,d) is a yes-instance if and only if we can obtain a graph G' from G by deleting at most k vertices such that χ(G') ≤χ(G)-d=ℓ+n-ℓ/2 -d. Since G' is also 3P_1-free, the colour classes in any optimal colouring of G' have size at most 2 and thus, G' contains at most 2(ℓ+n-ℓ/2 -d)=n+ℓ-2d vertices. In other words, we need to delete at least 2d-ℓ vertices from G in order to get such a graph G'. As such, (G,k,d) is a no-instance if k < 2d-ℓ. Next we will show that if k ≥ 2d-ℓ, then (G,k,d) is a yes-instance and this will complete the proof. If d ≤ℓ, we delete d vertices representing colour classes of size 1. If d > ℓ, we delete the ℓ vertices representing the colour classes of size 1 and 2(d-ℓ) vertices of d-ℓ colour classes of size 2. In this way we obtain a graph G' whose chromatic number is exactly χ(G)-d.Due to the above, all we need to do is check if k≥ 2d-ℓ. This can be done in polynomial time, since we can compute ℓ in polynomial time due to Theorem <ref>.Two disjoint subsets of vertices in a graph are complete if there is an edge between every vertex of A and every vertex of B. Lemma <ref> implies the following lemma, which we use together with Corollary <ref> and Lemma <ref> to prove Lemma <ref>.The vertex set of every (P_1⊕ P_3)-free graph G can be decomposed into two disjoint sets A and B such that G[A] is 3P_1-free, G[B] is P_4-free and A and B are complete to each other.Let G=(V,E) be a (P_1⊕ P_3)-free graph. Then G is P_1⊕ P_3-free. By Lemma <ref> every connected component of G is triangle-free or complete multipartite. Let A be the union of the verticesof all triangle-free components. Then G[A]=K_3-free, so G[A] is 3P_1-free. Let B=V∖ A. As every component of G[B] is complete multipartite, G[B] is P_4-free. AsP_4=P_4, this means that G[B] is P_4-free. Moreover, A and B are complete to each other in G. Deletion Blocker(χ)is polynomial-time solvable for (P_1⊕ P_3)-free graphs.Let (G,d,k) be an instance of Vertex Deletion Blocker(χ), where G=(V,E) is(P_1⊕ P_3)-free. By Lemma <ref>, the vertex set of G can be decomposed into two disjoint sets A and B such that G_1=G[A] is 3P_1-free, G_2=G[B] is P_4-free and A and B are complete to each other. The latter impliesthat χ(G)=χ(G_1)+χ(G_2). Moreover, this property is maintained when deleting vertices from V.For each pair (k_1,k_2) with k_1+k_2=k we check by how much we can decrease χ(G_1) using at most k_1 vertex deletions and by how much we can decrease χ(G_2) using at most k_2 vertex deletions. We can do this in polynomial timeby Corollary <ref> and Lemma <ref>, respectively. We keep track of the maximum sum of these values. In the end, we are left to check if the value found is at least d or not. Since the number of pairs (k_1,k_2) is at most k, the total running time is polynomial.We can now state and prove the following two dichotomies.Let H be a graph. Then the following holds: * If H P_1⊕ P_3 or P_4, then Deletion Blocker(χ) for H-free graphs is polynomial-time solvable, and it is -hard or co--hard otherwise. * If H P_4, thenContraction Blocker(χ) for H-free graphs is polynomial-time solvable for H-free graphs, and it is -hard or co--hard otherwise. Let H be a graph. If H is neither an induced subgraph of P_4 nor of P_1⊕ P_3, then for both problems we can applyTheorem <ref>. If H P_4, then for both problems we apply Corollary <ref>. In the remaining case H=3P_1 or H=P_1⊕ P_3. Then applying Lemma <ref> gives us the desired dichotomy for DeletionBlocker(χ). And applying Theorem <ref> gives us the desired dichotomy forContraction Blocker(χ) after recalling that cobipartite graphs are 3P_1-free. After proving Theorem <ref>we have shown all six cases of Theorem <ref>.Note that, unlike the case d=k=1 (see Theorem <ref>), the complexity dichotomies of the problems Contraction Blocker(χ) and Deletion Blocker(χ) restricted to H-free graphs are different when H is disconnected. § FUTURE WORK We aim to solve the blank entries in Table <ref>. In particular, we pose the following open problems:Q1. Determine the complexity of Contraction Blocker(α) for interval graphs. Q2. Determine the complexity of Deletion Blocker(α) for interval graphs.We observe that the complexity of the two problems in Q1 and Q2 is unknown for interval graphs even if d is fixed.We also aim to research the complexity of 1-Contraction Blocker(α) for bipartite graphs and chordal graphs, and the complexity of1-Deletion Blocker(α) for perfect graphs and chordal graphs.In addition to the above it would be interesting to generalize our results for the blocker problems restricted to H-free graphs in Section <ref> to families of more than one forbidden induced subgraph H. However, we still need to complete one stubborn remaining case for one problem:Q3. Determine the complexity of Contraction Blocker(ω) for (C_3⊕ P_1)-free graphs.We observe that it is not difficult to construct graph classes for which a blocker problem is tractable, but the original problem is -complete. However, we do not know of such examples of hereditary graph classes. Hence it would be interesting to solve the following question.Q4. For π∈{α,ω,χ}, are Contraction Blocker(π) and Deletion Blocker(π)computationally hard on every hereditary graph class G, for which Independent Set, Clique or Coloring, respectively, is -complete? Several computationally hard cases of our dichotomies for H-free graphs in Theorem <ref> hold in fact even when d=1 or d=k=1. In particular, from Theorems <ref> and <ref> we immediately deduce that if H P_1⊕ P_3 or P_4, then 1-Deletion Blocker(χ) for H-free graphs is polynomial-time solvable, and -hard or co--hard otherwise. However, for the other five variants we still have a number of missing cases to solve.Finally, we aim to determine a dichotomy with respect to H-free graphs for the variant (π∈{α,ω,χ}), where S consists of other graph operations, for instance when S consists of an edge deletion. This variant has been less studied than the vertex deletion and edge contraction variant. 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"authors": [
"Öznur Yaşar Diner",
"Daniël Paulusma",
"Christophe Picouleau",
"Bernard Ries"
],
"categories": [
"cs.DS",
"cs.CC",
"cs.DM",
"math.CO"
],
"primary_category": "cs.DS",
"published": "20170627211625",
"title": "Contraction and Deletion Blockers for Perfect Graphs and $H$-free Graphs"
} |
Multi-agent MILPs: finite-time feasibility and performance guarantees Research was supported by the European Commission under the project UnCoVerCPS, grant number 643921. [footnoteinfo]Corresponding author A. Falsone. Tel. +39-02-23994028. Fax +39-02-23993412. PoliMi]Alessandro [email protected], Oxford]Kostas [email protected], PoliMi]Maria [email protected] [PoliMi]Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy [Oxford]Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom MILP, decentralized optimization, multi-agent networks, electric vehicles. We address the optimal design of a large scale multi-agent system where each agent has discrete and/or continuous decision variables that need to be set so as to optimize the sum of linear local cost functions, in presence of linear local and global constraints. The problem reduces to a Mixed Integer Linear Program (MILP) that is here addressed according to a decentralized iterative scheme based on dual decomposition, where each agent determines its decision vector by solving a smaller MILP involving its local cost function and constraint given some dual variable, whereas a central unit enforces the global coupling constraint by updating the dual variable based on the tentative primal solutions of all agents. An appropriate tightening of the coupling constraint through iterations allows to obtain a solution that is feasible for the original MILP. The proposed approach is inspired by a recent method to the MILP approximate solution via dual decomposition and constraint tightening, and presents the advantage of guaranteeing feasibility in finite-time and providing better performance guarantees. The two approaches are compared on a numerical example on plug-in electric vehicles optimal charging. § INTRODUCTIONIn this paper we are concerned with the optimal design of a large-scale system composed of multiple agents, each one characterized by its set of design parameters that should be chosen so as to solve a constrained optimization problem where the agents' decisions are coupled by some global constraint. More specifically, the goal is to minimize the sum of local linear cost functions, subject to local polyhedral constraints and a global linear constraint. A key feature of our framework is that design parameters can have both continuous and discrete components.Let m denote the number of agents. Then, the optimal design problem takes the form of the following Mixed Integer Linear Program (MILP):min_x_1,…,x_m ∑_i=1^m c_i x_isubject to: ∑_i=1^m A_i x_i ≤ b x_i ∈ X_i,i=1,…,mwhere, for all i=1,…,m, x_i∈^n_i is the decision vector of agent i, c_i x_i its local cost, and X_i = { x_i∈^n_c,i×^n_d,i: D_i x_i ≤ d_i } its local constraint set defined by a matrix D_i and a vector d_i of appropriate dimensions, n_c,i being the number of continuous decision variables and n_d,i the number of discrete ones, with n_c,i+n_d,i=n_i. The coupling constraint ∑_i=1^m A_i x_i ≤ b is defined by matrices A_i∈^p×^n_i, i=1,…,m, and a p-dimensional vector b∈^p. Despite the advances in numerical methods for integer optimization, when the number of agents is large, the presence of discrete decision variables makes the optimization problem hard to solve, and calls for some decomposition into lower scale MILPs, as suggested in <cit.>.A common practice to handle problems of the form of <ref> consists in first dualizing the coupling constraint introducing a vector λ∈^p of p Lagrange multipliers and solving the dual programmax_λ≥ 0-λ b + ∑_i=1^m min_x_i∈ X_i (c_i + λ A_i)x_i,to obtain , and then constructing a primal solution x() = [x_1() ⋯ x_m()] by solving m MILPs given by:x_i(λ) ∈min_x_i∈|(X_i) (c_i + λ A_i)x_i,where the search within the closed constraint polyhedral set X_i can be confined to its set of vertices |(X_i) since the cost function is linear.Unfortunately, while this procedure guarantees x() to satisfy the local constraints since x_i()∈ X_i for all i=1,…,m, it does not guarantees the satisfaction of the coupling constraint.A way to enforce the satisfaction of the coupling constraint is to solve <ref> via a subgradient method, and then use a recovery procedure for the primal variables, <cit.>. Albeit this method is very useful in applications since it allows for a distributed implementation, see e.g. <cit.>, it provides a feasible solution only when there are no discrete decision variables. As a matter of fact, if we let (X_i) denote the convex hull of all points inside X_i, then, the primal solution recovered using <cit.> is the optimal solution _ of the following Linear Program (LP):min_x_1,…,x_m ∑_i=1^m c_i x_i _ subject to: ∑_i=1^m A_i x_i ≤ b x_i ∈(X_i),i=1,…,m.The dual of the convexified problem <ref> coincides with the one of <ref> and is given by <ref> (see <cit.> for a proof). Clearly _∈(X_1)×…×(X_m) does not necessarily imply that _∈ X_1× X_2 ×…× X_m. Therefore the solution _ recovered using <cit.> satisfies the coupling constraint but not necessarily the local constraints.For these reasons recovery procedures for MILPs are usually composed of two steps: a tentative solution that is not feasible for either the joint constraint or the local ones is first obtained through duality, and then a heuristic is applied to recover feasibility starting from this tentative solution, see, e.g., <cit.>. Problems in the form of <ref> arise in different contexts like power plants generation scheduling <cit.> where the agents are the generation units with their on/off state modeled with binary variables and the joint constraint consists in energy balance equations, or buildings energy management <cit.>, where the cost function is a cost related to power consumption and constraints are related to capacity, comfort, and actuation limits of each building. Other problems that fits the structure of <ref> are supply chain management <cit.>, portfolio optimization for small investors <cit.>, and plug-in electric vehicles <cit.>. In all these cases it is of major interest to guarantee that the derived (primal) solution is implementable in practice, which means that it must be feasible for <ref>.Interestingly, a large class of dynamical systems involving both continuous and logic components can be modeled as a Mixed Logical Dynamical (MLD) system, <cit.>, which are described by linear equations and inequalities involving both discrete and continuous inputs and state variables. Model predictive control problems for MLD systems involving the optimization of a linear finite-horizon cost function then also fit the MILP description in <ref>.§.§ BackgroundProblems in the form of <ref> have been investigated in <cit.>, where the authors studied the behavior of the duality gap (i.e. the difference between the optimal value of <ref> and <ref>) showing that it decreases relatively to the optimal value of <ref> as the number of agents grows. The same behavior has been observed in <cit.>. In the recent paper <cit.>, the authors explored the connection between the solutions _ to the linear program <ref> and x() recovered via (<ref>) from the solutionto the dual program <ref>. They proposed a method to recover a primal solution which is feasible for <ref> by using the dual optimal solution of a modified primal problem, obtained by tightening the coupling constraint by an appropriate amount.Let ρ∈^p with ρ≥0 and consider the following pair of primal-dual problems:min_x_1,…,x_m ∑_i=1^m c_i x_i _,ρ subject to: ∑_i=1^m A_i x_i ≤ b-ρx_i ∈(X_i),i=1,…,mandmax_λ≥ 0-λ (b-ρ) + ∑_i=1^m min_x_i∈ X_i (c_i + λ A_i)x_i. _ρ<ref> constitutes a tightened version of <ref>, whereas <ref> is the corresponding dual. For all j=1,…,p, let ∈^p be defined as follows:[]_j = pmax_i∈{ 1,…, m} {max_x_i∈ X_i [A_i]_j x_i - min_x_i∈ X_i [A_i]_j x_i },where [A_i]_j denotes the j-th row of A_i and []_j the j-th entry of .[Uniqueness] Problems <ref> and <ref> with ρ set equal todefined in (<ref>) have unique solutions _, and _, respectively.From <cit.>, we have the following result: Let _ be the solution to <ref> with ρ= given in (<ref>). Under Assumption <ref>, we have that any x(_) satisfying (<ref>), is feasible for <ref>. Let us define= p max_i∈{ 1,…, m}{max_x_i∈ X_i c_i x_i - min_x_i∈ X_i c_i x_i }.Consider the following assumption: [Slater] There exist a scalar ζ > 0 and x̂_i∈(X_i) for all i=1,…,m, such that ∑_i=1^m A_i x̂_i ≤ b -- mζ, where ∈^p is a vector whose elements are equal to one.Then, the sub-optimality level of the approximate solution x(_) to <ref> can be quantified as follows: Let _ be the solution to <ref> with ρ=. Under Assumptions <ref> and <ref>, we have that x(_) derived from (<ref>) with λ=_ satisfies ∑_i=1^m c_i x_i(_) - ≤ + _∞/pζ,whereis the optimal cost of <ref>. Note that both Proposition <ref> on feasibility and Proposition <ref> on optimality require the knowledge of the dual solution _. This may pose some issues if _ cannot be computed centrally, which is the case, e.g., when the agents are not willing to share with some central entity their private information coded in their local cost and constraint set. In those cases, the value of _ can only be achieved asymptotically using a decentralized/distributed scheme to solve <ref> with ρ=.§.§ Contribution of this paper In this paper we propose a decentralized iterative procedure which provides in a finite number of iterations a solution that is feasible for the optimal design problem <ref>, thus overcoming the issues regarding the finite-time computability of a decentralized solution in <cit.>. Furthermore, the performance guarantees quantifying the sub-optimality level of our solution with respect to the optimal one of <ref> are less conservative than those derived in <cit.>.As in the inspiring work in <cit.>, we still exploit some tightening of the coupling constraint to enforce feasibility. However, the amount of tightening is decided through the iterations, based on the explored candidate solutions x_i∈ X_i, i=1,…,m, and not using the overly conservative worst-case tightening (<ref>) as in <cit.> where for all i=1,…,m, the max and min of [A_i]_j x_i are computed letting x_i vary over the whole set X_i. The amount of tightening plays a crucial role in the applicability of Proposition <ref>. In fact, a too large value ofmay prevent <ref> to be feasible when ρ is set equal to , thus violating Assumption <ref>. A less conservative way to select an appropriate amount of tightening can extend the applicability of the approach to a larger class of problems. According to a similar reasoning, we are able to improve the bound on the performance degradation of our solution with respect to the optimal one of <ref> by taking a less conservative value for the quantityin (<ref>) that is used in the performance bound (<ref>).Notably, the proposed decentralized scheme allows agents to preserve the privacy on their local information, since they do not have to send to the central unit either their cost coefficients or their local constraints. § PROPOSED APPROACHWe next introduce Algorithm <ref> for the decentralized computation in a finite number of iterations of an approximate solution to <ref> that is feasible and improves over the solution in <cit.> both in terms of amount of tightening and performance guarantees.Algorithm <ref> is a variant of the dual subgradient algorithm. As the standard dual subgradient method, it includes two main steps: step <ref> in which a subgradient of the dual objective function is computed by fixing the dual variables and minimizing the Lagrangian with respect to the primal variables, and step <ref> which involves a dual update step with step size equal to α(k), and a projection onto the non-negative orthant (in Algorithm <ref> [ · ]_+ denotes the projection operator onto the p-dimensional non-negative orthant ^p_+). The operators max and min appearing in steps <ref>, <ref>, and <ref> of Algorithm <ref> with arguments in ^p are meant to be applied component-wise. The sequence {α(k)}_k≥ 0 is chosen so as to satisfy lim_k→∞α(k) = 0 and ∑_k=0^∞α(k) = ∞, as requested in the standard dual subgradient method to achieve asymptotic convergence. Furthermore, in order to guarantee that the solution to step <ref> in Algorithm <ref> is well-defined, we impose the following assumption on <ref>: [Boundedness] The polyhedral sets X_i, i=1,…,m, in problem <ref> are bounded. If min_x_i∈|(X_i) (c_i + λ(k) A_i)x_i in step <ref> is a set of cardinality larger than 1, then, a deterministic tie-break rule is applied to choose a value for x_i(k+1).Algorithm <ref> is conceived to be implemented in a decentralized scheme where, at each iteration k, every agent i updates its local tentative solution x_i(k+1) and communicates A_ix_i(k+1) to some central unit that is in charge of the update of the dual variable. The tentative value λ(k+1) for the dual variable is then broadcast to all the agents. Note that the agents do not need to communicate to the central unit their private information regarding their local constraint set and cost but only their tentative solution x_i(k).The tentative primal solutions x_i(k+1), i=1,…,m, computed at step <ref> are used in Algorithm <ref> by the central unit to determine the amount of tightening ρ(k+1) entering step <ref>. The value of ρ(k+1) is progressively refined through iterations based only on those values of x_i∈ X_i, i=1, …,m, that are actually considered as candidate primal solutions, and not based on the whole sets X_i, i=1, …,m. This reduces conservativeness in the amount of tightening and also in the performance bound of the feasible, yet suboptimal, primal solution.Algorithm <ref> terminates after a given stopping criteria is met at the level of the central unit, e.g., if for a given number of subsequent iterations x(k) = [x_1(k) ⋯ x_m(k)] satisfies the coupling constraint. As shown in the numerical study in Section <ref>, variants of Algorithm <ref> can be conceived to get an improved solution in the same number of iterations of Algorithm <ref>. The agents should however share with the central entity additional information on their local cost, thus partly compromising privacy preservation.As for the initialization of Algorithm <ref>, λ(0) is set equal to 0 so that at iteration k=0 each agent i computes its locally optimal solutionx_i(1) min_x_i∈|(X_i) c_i x_i.Since ρ(1) = 0, if the local solutions x_i(1), i=1, …,m, satisfy the coupling constraint (and they hence are optimal for the original problem <ref>), then, Algorithm <ref> will terminate since λ will remain 0, and the agents will stick to their locally optimal solutions. Before stating the feasibility and performance guarantees of the solution computed by Algorithm <ref>, we need to introduce some further quantities and assumptions.Let us define for any k≥ 1γ(k) = p max_i∈{1,…,m}{max_r≤ k c_i x_i(r) - min_r≤ k c_i x_i(r) },where {x_i(r)}_r≥ 1, i=1,…,m, are the tentative primal solutions computed at step <ref>.Due to Assumption <ref>, for any i=1,…,m, (X_i) is a bounded polyhedron. If it is also non-empty, then |(X_i) is a non-empty finite set (see Corollaries 2.1 and 2.2 together with Theorem 2.3 in <cit.>). As a consequence, the sequence {γ(k)}_k≥ 1 takes values in a finite set. Since this is a monotonically non-decreasing sequence, it converges in finite-time to some value . The same reasoning can be applied to show that the sequence {ρ(k)}_k≥ 1, iteratively computed in Algorithm <ref> (see step <ref>), and given by[ρ(k)]_j = p max_i∈{1,…,m}{max_r≤ k [A_i]_j x_i(r) - min_r≤ k [A_i]_j x_i(r) },for j=1,…,p, converges in finite-time to somesince it takes values in a finite set and is (component-wise) monotonically non-decreasing. Note that the limiting valuesandfor {ρ(k)}_k≥ 1 and {γ(k)}_k≥ 1 satisfy ≤ and ≤ whereandare defined in (<ref>) and (<ref>). Similarly to <cit.>, defineandas the primal-dual pair of optimization problems that are given by setting ρ equal toin <ref> and <ref>.[Uniqueness] Problemsandhave unique solutions _ and .[Slater] There exists a scalar ζ > 0 and x̂_i∈(X_i) for all i=1,…,m, such that ∑_i=1^m A_i x̂_i ≤ b -- mζ.Note that, since ≤, if Assumption <ref> is satisfied, then Assumption <ref> is automatically satisfied.We are now in a position to state the two main results of the paper. Under Assumptions <ref> and <ref>, there exists a finite iteration index K such that, for all k≥ K, x(k)=[x_1(k) ⋯ x_m(k)], where x_i(k), i=1,…,m, are computed by Algorithm <ref>, is a feasible solution for problem <ref>, i.e., ∑_i=1^m A_i x_i(k) ≤ b, k≥ K and x_i(k)∈ X_i, i=1,…,m. Under Assumptions <ref>-<ref>, there exists a finite iteration index K such that, for all k≥ K, x(k)=[x_1(k) ⋯ x_m(k)], where x_i(k), i=1,…,m, are computed by Algorithm <ref>, is a feasible solution for problem <ref> that satisfies the following performance bound: ∑_i=1^m c_i x_i(k) - ≤ + _∞/pζ. By a direct comparison of (<ref>) and (<ref>) we can see that the bound in (<ref>) is no worse than (<ref>) due to the fact that ≤ and ≤. § PROOF OF THE MAIN RESULTS §.§ Preliminary results Under Assumptions <ref> and <ref>, the Lagrange multiplier sequence {λ(k)}_k≥ 0 generated by Algorithm <ref> converges to an optimal solution of . As discussed after equation (<ref>), there exists a K∈ such that for all k≥ K we have that the tightening coefficient ρ(k) computed in Algorithm <ref> becomes constant and equal to . Therefore, for any k≥ K, Algorithm <ref> reduces to the following two steps x_i(k+1) ∈min_x_i∈|(X_i) (c_i + λ(k) A_i)x_i λ(k+1) = [ λ(k) + α(k) (∑_i=1^m A_i x_i(k+1) - b + ) ]_+ which constitute a gradient ascent iteration for . According to <cit.>, the sequence {λ(k)}_k≥ 0 generated by the iterative procedure (<ref>)-(<ref>) is guaranteed to converge to the (unique under Assumption <ref>) optimal solution of . Let P be a non-empty bounded polyhedron. Consider the linear program min_x∈ P (c+δ)x, where δ is a perturbation in the cost coefficients. Define the set of optimal solutions as (δ). There always exists an > 0 such that for all δ satisfying δ <, we have (δ) ⊆(0). Let u(δ) = min_x∈ P (c+δ)x. Since P is a bounded polyhedron, the minimum is always attained and u(δ) is finite for any value of δ. The set (δ) can be defined as (δ) = {x∈ P: (c+δ)x ≤ u(δ)}, which is a non-empty polyhedron. As such, it can be described as the convex hull of its vertices (see Theorem 2.9 in <cit.>), which are also vertices of P (Theorem 2.7 in <cit.>). Let V = |(P) and V_δ = |((δ))⊆ V. Consider δ = 0. If V_0=V, then, given the fact that, for any δ, (δ) is the convex hull of V_δ and V_δ⊆ V=V_0, we have trivially that (δ) ⊆(0), for any δ. Suppose now that V_0⊂ V. For any choice of ∈ V_0 and x∈ V∖ V_0, we have that c < c x, or equivalently c (-x) < 0. Pick = min_∈ V_0 x ∈ V∖ V_0 -c (-x)/-x and let (x̅^⋆,x̅) be the corresponding minimizer. By construction, (<ref>) is well defined since x̅^⋆ is different from x̅. Since c (-x) < 0 for any ∈ V_0 and x∈ V∖ V_0, we have that > 0. Moreover, for any ∈ V_0 and x∈ V∖ V_0, if δ satisfies δ <, then (c+δ) (-x) = c(-x) + δ(-x) ≤ c(-x) + δ-x< c(-x) + -x ≤ c(-x) + ( -c(-x)/-x) -x= c(-x) -c(-x) = 0, where the first inequality is given by the fact that u v ≤ |u v| together with the Cauchy–Schwarz inequality |u v| ≤uv, the second inequality is due to δ satisfying δ <, and the third inequality is given by the definition ofin (<ref>). By (<ref>) and the definition of u(δ), for any point x_δ in the set V_δ, we have that (c+δ)x_δ≤ (c+δ)x, for all x∈ V, and therefore (c+δ)x_δ≤ (c+δ) for any ∈ V_0⊂ V. By (<ref>), whenever δ<, we have that (c+δ) < (c+δ)x for any choice of ∈ V_0 and x∈ V∖ V_0, therefore (c+δ)x_δ < (c+δ)x for any x∈ V∖ V_0. Since the inequality is strict, we have that x_δ∉V∖ V_0, which implies x_δ∈ V_0. Since this holds for any x_δ∈ V_δ, we have that V_δ⊆ V_0. Finally, given the fact that, for any δ, (δ) is the convex hull of V_δ and V_δ⊆ V_0, we have (δ) ⊆(0), thus concluding the proof. Exploiting Lemma <ref>, we shall show next that each {x_i(k)}_k≥ 1 sequence, i=1,…,m, converges in finite-time to some set. Under Assumptions <ref> and <ref>, there exists a finite K such that for all i=1,…,m the tentative primal solution x_i(k) generated by Algorithm <ref> satisfies x_i(k) ∈min_x_i∈|(X_i) (c_i +A_i)x_i, k≥ K, whereis the limit value of the Lagrange multiplier sequence {λ(k)}_k≥ 0. Consider agent i, with i ∈{1,…,m}. We can characterize the solution x_i(k) in step <ref> of Algorithm <ref> by performing the minimization over (X_i) instead of |(X_i) since the problem is linear and by enlarging the set |(X_i) to (X_i) we still obtain all minimizers that belong to |(X_i). Adding and subtracting A_i x_i to the cost, we then obtain x_i(k) ∈min_x_i∈(X_i) (c_i +A_i + (λ(k-1)-) A_i)x_i. Set δ_i(k-1) = (λ(k-1)-) A_i, and let _i(δ_i(k-1)) be the set of minimizers of (<ref>) as a function of δ_i(k-1). By Lemma <ref>, we know that there exists an _i > 0 such that if δ_i(k-1) < _i, then _i(δ_i(k-1)) ⊆_i(0). Since, by Proposition <ref>, the sequence {λ(k)}_k≥ 0 generated by Algorithm <ref> converges to , by definition of limit, we know that there exists a K_i such that δ_i(k-1) = (λ(k-1)-) A_i < _i for all k≥ K_i. Therefore, for every k≥ K=max{K_1, …, K_m}, we have that x_i(k) ∈_i(0)=min_x_i∈(X_i) (c_i +A_i)x_i, i=1,…,m. This property jointly with the fact that x_i(k) ∈|(X_i), i=1,…,m, leads to (<ref>), thus concluding the proof. §.§ Proof of Theorems <ref> and <ref>Theorem <ref> Theorem 2.5 of <cit.> establishes a relation between the solution _ ofand the one recovered in (<ref>) from the optimal solutionof the dual optimization problem . Specifically, it states that there exists a set of indices I⊆{1,…,m} of cardinality at least m-p, such that [_]^(i) = x_i() for all i∈ I, where [_]^(i) is the subvector of _ corresponding to the i-th agent. Therefore, following the proof of Theorem 3.1 in <cit.>, we have that ∑_i=1^mA_i x_i() = ∑_i∈ I A_i x_i() + ∑_i∈ I^c A_i x_i() = ∑_i∈ I A_i [_]^(i) + ∑_i∈ I^c A_i x_i() = ∑_i=1^m A_i [_]^(i) + ∑_i∈ I^c A_i ( x_i() - [_]^(i)) ≤ b -+ p max_i=1,…,m{ A_i x_i() - A_i [_]^(i)}, where I^c = {1,…,m}∖ I, and b- constitutes an upper bound for ∑_i=1^m A_i [_]^(i) given that _ is feasible for . According to <cit.>, the component [_]^(i) of the (unique, under Assumption <ref>) solution _ tois the limit point of the sequence {x̃_i(k)}_k≥ 1, defined as x̃_i(k) = ∑_r=1^k-1α(r)x_i(r+1)/∑_r=1^k-1α(r). By linearity, for all k≥ 0, we have that A_i x̃_i(k)= ∑_r=1^k-1α(r) A_i x_i(r+1)/∑_r=1^k-1α(r)≥min_r≤ k A_i x_i(r) = _i(k) ≥_i, where the first inequality is due to the fact that all α(k) are positive and the second equality follows from step <ref> of Algorithm <ref>. In the final inequality, _i(k) is lower bounded by _i, that denotes the limiting value of the non-increasing finite-valued sequence {_i(k)}_k≥ 0. Note that all inequalities have to be intended component-wise. By taking the limit for k→∞, we also have that A_i [_]^(i)≥_i. By Proposition <ref>, there exists a finite iteration index K such that x_i(k) satisfies (<ref>). Since (<ref>) holds for any choice of x_i() which minimizes (c_i +A_i)x_i over |(X_i), if k≥ K, then we can choose x_i() = x_i(k). Therefore, for all k≥ K, (<ref>) becomes ∑_i=1^mA_i x_i(k) ≤ b -+ p max_i=1,…,m{ A_i x_i(k) - A_i [_]^(i)}≤ b -+ p max_i=1,…,m{max_r≤ k A_i x_i(r) - A_i [_]^(i)}= b -+ p max_i=1,…,m{_i(k) - A_i [_]^(i)}≤ b -+ p max_i=1,…,m{_i - _i }= b, where the second inequality is obtained by taking the maximum up to k, the first equality is due to step <ref> of Algorithm <ref>, the third inequality is due to the fact that _i is the limiting value of the non-decreasing finite-valued sequence {_i(k)}_k≥ 1 together with (<ref>), and the last equality comes from the definition of ρ(k)=p max{ρ_1(k), …, ρ_m(k) } where ρ_i(k)=_i(k)-_i(k). From (<ref>) we have that, for any k≥ K, the iterates x_i(k), i=1,…,m, generated by Algorithm <ref> provide a feasible solution for <ref>, thus concluding the proof. Theorem <ref> Denote as , , andthe optimal cost of <ref>, , and <ref>, respectively. From Assumption <ref> it follows that , , andare finite. Consider the quantity ∑_i=1^m c_i x_i(k) -. As in the proof of Theorem 3.3 in <cit.>, we add and subtractandto obtain ∑_i=1^m c_i x_i(k) - = ( ∑_i=1^m c_i x_i(k) - ) + ( - ) + (-). We shall next derive a bound for each term in (<ref>). Bound on ∑_i=1^m c_i x_i(k) -: Similarly to the proof of Theorem <ref> for feasibility, due to Theorem 2.5 in <cit.>, have that there exists a set I of cardinality at least m-p such that x_i() = [_]^(i), for all i∈ I. Therefore, ∑_i=1^m c_i x_i () - = ∑_i=1^m c_i x_i() - ∑_i=1^m c_i [_]^(i)= ∑_i∈ I^c c_i x_i() - c_i [_]^(i)≤ p max_i=1,…,m{ c_i x_i() - c_i [_]^(i)}, where I^c = {1,…,m}∖ I. According to <cit.>, the components [_]^(i) of the (unique, under Assumption <ref>) solution _ tois the limit point of the sequence {x̃_i(k)}_k≥ 1, defined as x̃_i(k) = ∑_r=1^k-1α(r)x_i(r+1)/∑_r=1^k-1α(r). By linearity, for all k≥ 1, we have that c_ix̃_i(k)= ∑_r=1^k-1α(r) c_i x_i(r+1)/∑_r=1^k-1α(r)≥min_r≤ k c_i x_i(r) ≥_i, where the first inequality is due to the fact that all α(k) are positive and the last one derives from the fact {min_r≤ k c_i x_i(r)}_k≥1 is a non-increasing sequence that takes values in a finite set, and hence is lower bounded by its limiting value _i. Therefore, by taking the limit for k→∞, we also have that c_i [_]^(i)≥_i. Since (<ref>) holds for any choice of x_i() which minimize (c_i +A_i)x_i over |(X_i), by Proposition <ref> it follows that, for k≥K̅, x_i() = x_i(k) and, as a result ∑_i=1^m c_i x_i(k) - ≤ p max_i=1,…,m{ c_i x_i(k) - c_i [_]^(i)}≤ p max_i=1,…,m{max_r≤ k c_i x_i(r) - c_i [_]^(i)}≤ p max_i=1,…,m{max_r≤ k c_i x_i(r) - _i }, where the second inequality is obtained by taking the maximum up to iteration k and the third inequality is due to (<ref>). Now if we recall the definition of γ(k) in (<ref>) and its finite-time convergence to , jointly with the fact that _i is the limiting value of {min_r≤ k c_i x_i(r)}_k≥1, we finally get that there exists K ≥K̅, such that for k≥ K p max_i=1,…,m{max_r≤ k c_i x_i(r) - _i }= , thus leading to ∑_i=1^m c_i x_i(k) - ≤, k≥ K. Bound on -: Problem <ref> can be considered as a perturbed version of , since the coupling constraint ofis given by ∑_i=1^m A_i x_i ≤ b- and that of <ref> can be obtained by addingto its right-hand-side. From perturbation theory (see <cit.>) it then follows that the optimal costis related toby: - ≤. From Assumption <ref>, by applying <cit.> we have that for all λ≥0 _1≤1/mζ( ∑_i=1^m c_ix̂_i + λ b - ∑_i=1^m min_x_i∈ X_i (c_i + λ A_i)x_i ) ≤1/mζ( ∑_i=1^m c_ix̂_i - ∑_i=1^m min_x_i∈ X_i c_i x_i ) ≤1/ζmax_i=1,…,m{max_x_i∈ X_i c_i x_i - min_x_i∈ X_i c_i x_i }= /pζ, where the second inequality is obtained setting λ = 0, the third inequality comes from the fact that c_ix̂_i ≤max_x_i∈ X_i c_i x_i and that ∑_i=1^m β_i ≤ mmax_i β_i, and the third equality is due to (<ref>). Using (<ref>) in (<ref>) we have -≤≤_1 _∞≤_∞/pζ, where the second inequality is due to the Hölder's inequality. Bound on -: Since <ref> is a relaxed version of <ref>, then -≤ 0. The proof is concluded considering (<ref>) and inserting the bounds obtained for the three terms.§ APPLICATION TO OPTIMAL PEVS CHARGINGIn this section we show the efficacy of the proposed approach in comparison to the one described in <cit.> on the Plug-in Electric Vehicles (PEVs) charging problem described in <cit.>. This problem consists in finding an optimal overnight charging schedule for a fleet of m vehicles, which has to satisfy both local requirements and limitations (e.g., maximum charging power and desired final state of charge for each vehicle), and some network-wide constraints (i.e., maximum power that the network can deliver at each time slot). We consider both version of the PEVs charging problem, namely, the “charge only” setup in which all vehicles can only draw energy from the network, and the “vehicle to grid” setup where the vehicles are also allowed to inject energy in the network.The improvement of our approach with respect to that in <cit.> is measured in terms of the following two relative indices: the reduction in the level of conservativenessΔρ_% = _∞ - _∞/_∞· 100and the improvement in performance achieved by the primal solutionΔ J_% = J_-J_/J_· 100,where J_ = ∑_i=1^m c_i x_i(_) and J_ = ∑_i=1^m c_i x_i(). A positive value for these indices indicates that our approach is less conservative.For a thorough comparison we determined the two indices while varying: i) the number of vehicles in the network, ii) the realizations of the random parameters entering the system description (cost of the electrical energy and local constraints), and iii) the right hand side of the joint constraints. All parameters and their probability distributions were taken from <cit.>.In Table <ref> we report the conservativeness reduction and the cost improvement for the “vehicle to grid” setup. As it can be seen from the table, the level of conservativeness is reduced by 50% while the improvement in performance (witnessed by positive values of Δ J_%) drops as the number of agents grows. This is due to the fact that the relative gap between J_ andtends to zero as m→∞, thus reducing the margin for performance improvement.We do not report the results for the “charge only” setup since the two methods lead to the same level of conservativeness and performance of the primal solution. We also tested the proposed approach against changes of the random parameters defining the problem. We fixed m = 250 and performed 1000 tests running Algorithm <ref> and the approach in <cit.> with different realization for all parameters, extracted independently. Figure <ref> plots an histogram of the values obtained for Δ J_% in the 1000 tests. Note that the cost improvement ranges from 3% to 15% and, accordingly to the theory, is always non-negative. The reduction in the level of conservativeness is also in this case 50%, suggesting that the proposed iterative scheme exploits some structure in the PEVs charging problem that the approach in <cit.> oversees. Also in this case, in the “charge only” setup the two methods lead to the same level of conservativeness and performance.Finally, we compared the two approaches in the “vehicle to grid” setup against changes in the joint constraints. If the number of electric vehicles is m = 250 and we decrease the maximum power that the network can deliver by 37%, then thethat results from applying the approach in <cit.> makes <ref> with ρ = infeasible, thus violating Assumption <ref>. Whereas with our approach <ref> with ρ = remains feasible,being the limiting value for {ρ(k)}_k≥1 in Algorithm <ref>. §.§ Performance-oriented variant of Algorithm <ref>While Algorithm <ref> is able to find a feasible solution to <ref>, it does not directly consider the performance of the solution, whereas the user is concerned with both feasibility and performance with higher priority given to feasibility. This calls for a modification to Algorithm <ref> which also takes into account the performance achieved.Theorem <ref> guarantees that there exists an iteration index K after which the iterates stay feasible for <ref> for all k≥ K. Now, suppose that the agents, together with the A_i x_i(k) also transmit c_i x_i(k) to the central unit, then the central unit can construct the cost of x(k) = [x_1(k),⋯,x_m(k)] at each iteration. When a feasible solution is found, its cost may be compared with that of a previously stored solution, and the central unit can decide to keep the new tentative solution or discard it. This way we are able to track the best feasible solution across iterations.The modified procedure is summarized in Algorithm <ref>. Note that, compared to Algorithm <ref>, each agent is required to transmit also the cost of its tentative solution.To show the benefits of Algorithm <ref> in terms of performance, we run 1000 test with m = 250 vehicles in the “charge only” setup, where we are also able to compute the optimal solution of <ref>, and compare the performance of Algorithm <ref> and <ref> in terms of relative distance from the optimal costof <ref>.Figure <ref> shows the distribution of (J_-)/·100 obtained with Algorithm <ref> (blue) and (J̌-)/·100 obtained with Algorithm <ref> (orange) for the 1000 runs. As can be seen from the picture, most runs of Algorithm <ref> result in a performance very close to the optimal one, while the runs from Algorithm <ref> exhibit lower performance. § CONCLUDING REMARKSWe proposed a new method for computing a feasible solution to a large-scale mixed integer linear program via a decentralized iterative scheme that decomposes the program in smaller ones and has the additional beneficial side-effect of preserving privacy of the local information if the problem originates from a multi-agent system.This work improves over existing state-of-the-art results in that feasibility is achieved in a finite number of iterations and the decentralized solution is accompanied by a less conservative performance certificate. The application to a plug-in electric vehicles optimal charging problem verifies the improvement gained in terms of performance.Future research directions include the development a distributed algorithm, which does not require any central authority but only communications between neighboring agents, and allows for time-varying communications among agents.Moreover, we aim at exploiting the analysis of <cit.> to generalize our results to problems with nonconvex objective functions. abbrv 10aubin1976estimates J.-P. Aubin and I. Ekeland. Estimates of the duality gap in nonconvex optimization. Mathematics of Operations Research, 1(3):225–245, 1976.baumann2013portfolio P. Baumann and N. Trautmann. Portfolio-optimization models for small investors. Mathematical Methods of Operations Research, 77(3):345–356, 2013.Bemporad:1999 A. Bemporad and M. Morari. Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3):407–427, 1999.bertsekas1983optimal D. Bertsekas, G. Lauer, N. Sandell, and T. Posbergh. Optimal short-term scheduling of large-scale power systems. IEEE Transactions on Automatic Control, 28(1):1–11, 1983.bertsekas1999nonlinear D. P. Bertsekas. Nonlinear programming. Athena scientific Belmont, 1999.bertsimas1997introduction D. Bertsimas and J. N. Tsitsiklis. Introduction to linear optimization, volume 6. Athena Scientific Belmont, MA, 1997.boyd2004convex S. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004.dawande2006effective M. Dawande, S. Gavirneni, and S. Tayur. Effective heuristics for multiproduct partial shipment models. Operations research, 54(2):337–352, 2006.falsone2017automatica A. Falsone, K. Margellos, S. Garatti, and M. Prandini. Dual decomposition for multi-agent distributed optimization with coupling constraints. Automatica, 2017. To appear. Online preprint arXiv:1607.00600.geoffrion1974lagrangean A. M. Geoffrion. Lagrangean relaxation for integer programming. Mathematical Programming Study 2, pages 82–114, 1974.ioli2015iterative D. Ioli, A. Falsone, and M. Prandini. An iterative scheme to hierarchically structured optimal energy management of a microgrid. In 54th Conference on Decision and Control (CDC2015), pages 5227–5232, 2015.nedic2009approximate A. Nedić and A. Ozdaglar. Approximate primal solutions and rate analysis for dual subgradient methods. SIAM Journal on Optimization, 19(4):1757–1780, 2009.redondo1999short N. J. Redondo and A. Conejo. Short-term hydro-thermal coordination by lagrangian relaxation: solution of the dual problem. IEEE Transactions on Power Systems, 14(1):89–95, 1999.shor1985minimization N. Shor. Minimization Methods for Non-Differentiable Functions. Springer, 1985.simonetto2016primal A. Simonetto and H. Jamali-Rad. Primal recovery from consensus-based dual decomposition for distributed convex optimization. Journal of Optimization Theory and Applications, 168(1):172–197, 2016.udell2016bounding M. Udell and S. Boyd. Bounding duality gap for separable problems with linear constraints. Computational Optimization and Applications, 64(2):355–378, 2016.vujanic2016decomposition R. Vujanic, P. M. Esfahani, P. J. Goulart, S. Mariéthoz, and M. Morari. A decomposition method for large scale MILPs, with performance guarantees and a power system application. Automatica, 67:144–156, 2016.yamin2004review H. Y. Yamin. Review on methods of generation scheduling in electric power systems. Electric Power Systems Research, 69(2):227–248, 2004. | http://arxiv.org/abs/1706.08788v1 | {
"authors": [
"Alessandro Falsone",
"Kostas Margellos",
"Maria Prandini"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170627112809",
"title": "A decentralized approach to multi-agent MILPs: finite-time feasibility and performance guarantees"
} |
automata,arrows,positioning,calc arrows,matrix,positioning | http://arxiv.org/abs/1706.08590v1 | {
"authors": [
"John McKay",
"Vishal Monga",
"Raghu G. Raj"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170626205332",
"title": "Robust Sonar ATR Through Bayesian Pose Corrected Sparse Classification"
} |
2017/6/28 2017/12/28Herschel Spitzer WISE AKARIμmX_ CO cm^-2 K^-1 km^-1 s cm^-2 cm^-3 km s^-1 v_ LSR ^∘ L_ M_ M_ yr^-1 N(H_2)H II O III S II(l, b) (l, b)= (l, b)∼ (α_ J2000, δ_ J2000) (α_ J2000, δ_ J2000)= (α_ J2000, δ_ J2000)∼C12O C13O CO18J= (J=1–0) (J=2–1) (J=3–2)1Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan 2Institute for Advanced Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan 3Nobeyama Radio Observatory, 462-2 Nobeyama, Minamimaki Minamisaku, Nagano, 384-1305, Japan [email protected]: clouds— Stars:formation — ISM:indivisual objects : S116, S117, S118 Molecular clouds toward three Spitzer bubbles S116, S117 and S118: Evidence for a cloud-cloud collision which formed the threeregions and a 10-pc scale molecular cavity Kengo tachihara1 Received: date / Accepted: date ========================================================================================================================================================================== We carried out a molecular line study toward the three Spitzer bubbles S116, S117 and S118 which show active formation of high-mass stars.We found molecular gas consisting of two components with velocity difference of ∼ 5 .One of them, the small cloud, has typical velocity of -63 and the other, the large cloud, has that of -58 .The large cloud has a nearly circular intensity depression whose size is similar to the small cloud.We present an interpretation that the cavity was created by a collision between the two clouds and the collision compressed the gas into a dense layer elongated along the western rim of the small cloud. In this scenario, the O stars including those in the three Spitzer bubbles were formed in the interface layer compressed by the collision.By assuming that the relative motion of the clouds has a tilt of 45D to the line of sight, we estimate that the collision continued over the last 1 Myr at relative velocity of ∼10 .In the S116–117–118 system theregions are located outside of the cavity. This morphology is ascribed to the density-bound distribution of the large cloud which made theregions more easily expand toward the outer part of the large cloud than inside of the cavity.The present case proves that a cloud-cloud collision creates a cavity without an action of O star feedback, and suggests that the collision-compressed layer is highly filamentary. § INTRODUCTIONO stars have a mass range from 20 to 300in the Local Group <cit.>.They influence the interstellar space considerably by dynamically disturbing the interstellar medium (ISM) over their whole lifetime and by enriching the heavy elements at the end of their lives via supernova explosions. These actions have a significant influence on the galactic evolution.It is therefore of crucial importance to better understand the formation mechanism of O stars. The mass accretion rate required for O star formation is as high as 10^-4 ; otherwise the high stellar radiation pressure halts the mass accretion and higher-mass stars do not grow in mass to a mature O star (e.g., <cit.>). It is often discussed that dense and massive clouds like infrared dark clouds are a plausible site of high-mass star formation.This seems to be a natural scenario, whereas we have often noregions associated with infrared dark clouds; regions where O stars are formed unambiguously areregions which are ionized by ultraviolet photons emitted by O or early B stars.regions in the Galaxy show a variety of morphology and evolutionary stages, from ultra-compactregions to extendedregions (e.g., for review <cit.>), and more careful studies ofregions have a potential to shed a new light on O star formation. In particular, a detailed study of molecular gas, the raw material to form stars, is still not thoroughly made inregions due to a limited angular resolution or a low dynamic range in molecular line observations in spite of the previous efforts on molecular observations towardregions (e.g., Andreson et al. 2009).Spitzer bubbles are seen at 8with a ring-like shape and harbor usually an O star and anregion in the ring.In total 322 bubbles are cataloged in Galactic longitude from -65Dto 65D and in Galactic latitude from -1.5D to 1.5D except the Galactic center (l< ± 10D) <cit.>. It is usually discussed that the ultraviolet photons and stellar winds accelerate and compress the surrounding gas to form an expanding bubble and trigger second-generation star formation (the collect-collapse scenario) <cit.>. A CO survey of Spitzer bubbles was made by <cit.> who observed 43 Spitzer bubbles with JCMT in the CO 3–2 transition. Their results indicate that the gas is ring-like and flattened with no significant sign of expansion as opposed to the wind-blown scenario, raising a puzzle which is not immediately explained by the conventional picture.<cit.>, recently, presented a novel scenario that the bubble in a most typical Spitzer bubble RCW120 <cit.> was formed by cloud-cloud collision via employing the <cit.> model, which numerical simulated head-on collision between small and large spherical clouds. These simulations showed that such uneven collision can create a cavity in the large cloud with a size of the small cloud and the compressed layer between them become dense and self-gravitating to trigger star formation.If the formed star is able to ionize the surroundings, i.e., the inside of the cavity, the model provides an alternative to explain the ring-like shape of RCW120 inside of which is ionized <cit.>.<cit.> did not find an observational signature of expansion of the bubble in the molecular gas, which lead to the cloud-cloud collision scenario. RCW120 has two velocity molecular components, one of which shows a good correspondence with the bubble, and the other associated with the opening of the bubble.These properties are consistent with the results by <cit.>. The two clouds show enhanced temperature toward the ring in spite of their large velocity separation ∼20 , which provides robust verification of their physical association with RCW120.Since the velocity separation is too large to be bound by the cloud gravity, <cit.>concluded that RCW120 is a case of cloud-cloud collision which triggered formation of a single O7 star.It is noteworthy that the new model offers an explanation of the elongated horseshoe morphology of the bubble along with the off-center position of the exciting star close to the bubble bottom; the simple stellar-wind bubble does not offer a natural explanation for such asymmetry. Figure <ref> shows the region of three Spitzer bubbles S116, S117, and S118. Distance of S116 is estimated kinematically to be 5.9 ± 0.9 kpc <cit.>. The green color and red color in Figure <ref> show 8and 24images, respectively. The 8radiation traces the PAH emission and the 24radiation the heated dust by the ultraviolet radiation of the O stars. The bubbles are not perfectly circular and are elongated significantly. The Spitzer bubbles are often isolated, and such a case with three lined-up bubbles are rare.In order to have a better understanding of formation of multiple Spitzer bubbles, we undertook new CO observations of molecular gas toward S116, S117, and S118 by using three mm/sub-mm telescopes NANTEN 2, ASTE, and the Mopra 22-m telescope in the CO rotational transitions. Section 2 gives description of observations and Section 3 presents observational results.Section 4 gives discussion of the results and tests a cloud-cloud collision scenario, and Section 5 concludes the paper. § OBSERVATIONS §.§1–0 with NANTEN2We performed1–0 observations with the NANTEN2 4-m millimeter sub-millimeter radio telescope <cit.>. The observations were carried out in October 2011. The front end was a 4 K cooled SIS mixer receiver. The typical system noise temperature including the atmosphere was ∼250 K in the double-side band (DSB) during observations.The backend was a digital-Fourier transform spectrometer (DFS) with 16384 channels of 1 GHz bandwidth, which corresponds to velocity coverage and resolution of ∼2600and 0.16 , respectively.We used the on-the-fly (OTF) mapping mode to cover an area of 1D× 1D. The pointing accuracy was confirmed to be better than 15” with daily observations toward the Sun. The absolute intensity was calibrated by observing Orion KL everyday. The final half-power beam width (HPBW) is 180”.The final root-mean-square (rms) noise fluctuation of the data was ∼ 1.0 K/ch at a velocity resolution of 0.16 km s^-1.§.§1–0 and1–0 with MopraDetailed1–0 and1–0 distributions around the Spitzer bubbles S116, S117, and S118 were obtained by using the 22-m ATNF (Australia Telescope National Facility) Mopra mm telescope at a high angular resolution of 33” in two periods June-October 2013 and July-August 2014.For the field shown in Figure 1 by a dashed box we simultaneously observed1–0 and1–0 in the OTF mode with a unit field of 4' × 4'. The typical system noise temperature including the atmosphere was between 400 K and 600 K in the single-side band (SSB) during observations.The Mopra backend system “MOPS" which provides 4096 channels across 137.5 MHz in each of the two orthogonal polarizations was used in the observations.The velocity resolution was 0.088and the velocity coverage was 360at 115 GHz.The pointing accuracy was checked every 90 minutes to keep within 4” with observations of 86 GHz SiO masers. The absolute intensity was calibrated by comparing with CO 1–0 data observed with NANTEN2. The obtained data were smoothed to a HPBW of 40” with a 2D Gaussian function and to a 0.6velocity resolution.The final data cube had a rms noise fluctuation of ∼ 0.6 K/ch at a velocity resolution of 0.088 km s^-1.§.§3–2with ASTEObservations of the3–2 emission were performed by using the ASTE 10-m telescope located in Chile in three periods September 2013, June 2014, and November 2015. The waveguide-type sideband-separating SIS mixer receiver for the single sideband (SSB) "CAT345” having system noise temperature of 300 K and the digital spectrometer "MAC” with the narrow-band mode providing 128 MHz bandwidth and 0.125 MHz resolution, which corresponds to 450velocity coverage and 0.43velocity resolution at 345 GHz, were used.The observations were made with the OTF mode at a grid spacing of 7.5”, and the HPBW was 24” at the3–2 frequency. The Observed area is the same as that of the Mopra observations. The pointing accuracy was checked every 90 minutes to keep within 2” with observations of RAFGL 4202 (14h52m23.82s, -62D04'19.2"). The absolute intensity was calibrated by observing W44 and IRC+10216 every 90 minutes. The obtained data was smoothed to a HPBW of 22” with a 2D Gaussian function and to a 1velocity resolution.The final data cube had a rms noise fluctuation of ∼ 0.38 K/ch at a velocity resolution of 0.43 km s^-1. § RESULTS §.§ CO distributionsFigure <ref> shows an infrared image of the present region taken with Spitzer <cit.>. The three Spitzer bubbles S116, S117 and S118 are distributed over 20 pc in the north-south direction. We find nearly ten smallerregions, which are bright warm dust traced by the Spitzer 24 μm emission.The area indicated by the dashed-line box was observed with Mopra and ASTE, while the whole area was mapped with NANTEN2.Figures <ref>a and <ref>b show a large-scale view of the1–0 emission observed with NANTEN2. We find two clouds whose distributions are distinctly different. The -63 cloud has a peak at (314.22D, 0.33D). The -58 cloud is extended along the plane and has a peak at (314.21D, 0.25D) and an intensity depression at (314.3D, 0.36D) in addition to several intensity peaks surrounding.We shall hereafter call the -63 cloud and the -58 cloud the small cloud and the large cloud, respectively, because of their sizes.The small cloud has a sharp intensity decrease to every direction. The large cloud shows a sharp intensity gradient toward the west.Figure <ref> shows velocity channel distributions every 1.3in the1–0 emission taken with the Mopra telescope, which indicates that the small cloud becomes large in size with velocity increase from -69.3to -62.7 . We also find that the CO distribution is extremely filamentary in a velocity range from -66.6to -61.3particularly toward the small cloud. The filamentary structure is described into more detail in section <ref>.Figure <ref> shows detailed distribution of1–0,1-0 and3–2 images of the small cloud (Figure <ref>a, 4b, and 4c) and the large cloud (Figure <ref>d, 4e, and 4f) overlaid with infrared contours.The bubbles S116 and S118 delineate the northern and southern boundary of the small cloud, and S117 is located toward the small cloud. In the large cloud, the cavity is clearly seen with a sharp nearly circular boundary.We find another intensity depression in the north at (314.27D –314.32D, 0.42D –0.44D). The western edge of the large cloud shows a distribution similar to the small cloud at b = 0.3D –0.45D, and S116 and S118 are located also along the edge of the large cloud.In Figures <ref>b and <ref>e the distribution of the ^13CO J=1–0 emission, which is likely optically thin, shows correspondence with the intense part of the ^12CO emission.We find that the effect of self-absorption is small in this region from the similarity between the ^12CO and ^13CO distributions. The physical parameters of the two clouds are as follows.The size of the cavity hole is ∼5 pc in radius for an assumed distance of 5.9 kpc. The mass and peak column density of the small cloud and the large cloud in the area shown in Figure <ref> are (2.0 × 10^4, 0.7 × 10^5 ) and (1 × 10^22 cm^-2, 2 × 10^22 cm^-2), respectively, where an X(CO) factor of 1.0 × 10^20<cit.> was assumed. §.§ Radio continuum distribution and the properties of the O starsIn Figure <ref> the ASTE ^12CO J=3–2 images of the two components are superposed on the SUMMS 843MHz radio continuum distribution <cit.> (Figures <ref>a and <ref>b) and the Spitzer image (Figures <ref>c and <ref>d). We find that the radio continuum distribution coincides well with the Spitzer bubbles. S117 is located toward the center of the small cloud, and it fits the intensity depression in the large cloud (Figure <ref>c and <ref>d). The heavy obscuration does not allow us to directly observe the exciting stars, and instead we used the radio continuum radiation as a measure of the stellar ultraviolet radiation. We estimate the spectral types of the exciting O stars in eachregion from the radio flux by using the relationship given by <cit.> and <cit.> as shown in Table 1, where the exciting star is assumed to be a single star in eachregion.The ultraviolet photon flux was used to estimate stellar spectral types and corresponding stellar mass for ZAMS (zero-age main sequence). As a result, the spectral types are estimated as follows; O6–06.5 for S116, O9 for S117, O7.5–O8 for S118, and the mass of luminosity class V stars are inferred to be 30 , 20 , and 23 , respectively <cit.>.§.§ The intensity ratio of the3–2 to the1–0Figures <ref>a and <ref>d show the intensity ratio of the3–2 to the1–0 (R_3-2/1-0) for the two clouds. R_3-2/1-0 is mainly affected by the3–2 distribution.The typical R_3-2/1-0 of molecular clouds in our galaxy without an extra heat source is R_3-2/1-0 ∼ 0.4 <cit.>. R_3-2/1-0 is enhanced to 1.0–1.4 outside of the cavity toward S116 and S118, while R_3-2/1-0 is 0.6–1.0 around the cavity. The enhanced ratio toward S116 and S118 suggests that the stellar radiation is heatig the gas.Red and blue lines in Figures <ref>b and <ref>e represent the filamentary structures of R_3-2/1-0, where red and blue colors show that the filament is overlapping with theregions (red), or not (blue).These filamentary structures were identified by eye with a criteria of R_3-2/1-0>0.9 and length >∼2 pc.The filamentary structures have a typical size of ∼1 × 3 pc.Multiple filamentary structures are seen not only toward theregion but also in the cavity withoutregions. § DISCUSSIONWe have carried out large scale CO observations with NANTEN2 in the region of the three Spitzer bubbles S116–S117–S118, which includes at least several O stars in addition to a few tens of low mass stars as Akari point sources (Figure <ref>). We made follow up high resolution observations with ASTE and Mopra telescopes. We found that the molecular gas comprises at least two velocity components of different morphology; the -63 cloud (the small cloud) and the -58 cloud (the large cloud), which are extended along the plane over 40 pc if the weak extended CO emission is included in Figure <ref>. The large cloud has an intensity peak and an intensity depression of ∼ 10 pc in size.Higher resolution observations with Mopra and ASTE show that the small cloud grows in size with the increase of velocity (Figure <ref>). §.§ A cloud-cloud collision scenario in S116–S117–S118In order to explain the velocity distribution we propose a hypothesis that cloud-cloud collision took place between the two clouds and that the small cloud pushed the surface of the large cloud to produce the cavity in the large cloud. Figure <ref>a prsents a schematic view of the collision seen from a direction perpendicular to the cloud relative motion. In the plane of Figure <ref>a, θ is an angle of the line of sight to the relative motion of the cloud.The small cloud was separated in the upper left corner of Figure <ref>a prior to collision, and moved along a straight line toward the large cloud.They collided with each other and the small cloud created a cavity in the large cloud.The layer between the two clouds are compressed to form an interface layer, which is denoted by dark blue in front of the small cloud in Figure <ref>a. The two clouds observed on the sky are divided into three sections A, B, and C as shown in Figure <ref>a.A shows the large cloud and the cavity, B the interface layer, the small cloud, and the large cloud with the cavity, and C shows part of the large cloud prior to collision.In order to gain an insight into observed cloud properties, we describe the physical states of colliding clouds by using the hydrodynamical numerical simulations of <cit.>. The simulations deal with head-on collision between a small cloud and a large cloud, which are idealized to be spherically symmetric. We adopt a model listed in Table 2 for discussion; the radius of the small cloud is 3.5 pc and that of the large cloud 7.2 pc. The two are currently colliding at 7and have internal turbulence in the order of 1–2with highly inhomogeneous density distribution. For more details see <cit.>. The cloud parameters do not correspond exactly to the present cloud although the difference does not critically affect a qualitative comparison below. We assume θ = 45^∘ in the following and an epoch of 1.6 Myr after the onset of the collision, where cloud signatures typical to collision are seen. The assumption on θ is not so critical as long as θ is not very close to 0^∘ or 90^∘. The small cloud is producing a cavity in the large cloud by the collisional interaction. The interface layer of the two clouds has enhanced density by collision, where the internal turbulence and the momentum exchange between the clouds mix the gas distribution in space and velocity. The gas in the two clouds is continuously merging into the layer during the collision. Figures <ref>b–<ref>i show the velocity channel distributions every 0.5 . In cloud-cloud collision it might be expected to see two distinct clouds of a narrow line width, whereas the actual distribution in the simulations present a merged single cloud which is continuous in velocity and space. The small cloud is seen at a velocity range from -5.1 to -2.2 , and the large cloud from –1.2 to 1.7 . The velocity range from -2.2 to -1.2corresponds to the interface velocity layer which was created by merging. Note that the velocity ranges of each panel in Figure <ref> do not exactly fit the velocity ranges of the two clouds due to the mixing in velocity. We find that the small cloud becomes large with the increase of velocity by merging of the large cloud as is consistent with the observations. The cavity is produced in the large cloud by the small cloud, while the boundary of the cavity is less clear in the model than the observations, reflecting that turbulence is more enhanced in the model than in the observed cloud.In S116-117-118, there is a displacement of ∼10 pc in the sky between the small cloud and the cavity (Figure <ref>). We interpret that the displacement is caused by projection of a tilt of the cloud relative motion to the line of sight. The collision velocity is estimated to be ∼7if we assume tentatively θ = 45 deg, an angle between the relative motion to the line of sight.Figure <ref> shows a comparison in a position-velocity diagram taken in the direction of the relative motion of the two clouds. Figure <ref>a is an overlay of the small and large clouds taken from Figure <ref>e and Figure <ref>h.Figure <ref>a shows distributions of the two clouds.Figure <ref>b shows a synthetic position-velocity-diagram produced from the same numerical simulations in the plane of the two cloud centers. The clouds as a whole show a “V-shape” as traced by the solid lines, and the main peak is found at X=5–7 pc and V=-4–-1 in Figure <ref>b. This peak is formed by a combination of the small cloud and the interface layer which merged together. There is an intensity depression at X=4–6 pc and V=-2–0 , which correspond to the blue-shifted part of the cavity in the large cloud.Figure <ref>c is the observed two velocity components and Figure <ref>d the observed position-velocity diagram. The ”V-shape” typical to collision is recognized in the Figure <ref>d. Correspondence is seen between the model and the observations qualitatively;the synthetic observations reproduce the cavity atl=314.3Dand =-58and the small cloud at l=314.2Dand =-64 , except for the sharp cut of the large cloud atl=314.2Dwhich is not taken into account in the model. This correspondence lends support for applying the cloud-cloud collision model to the S116-S117-S118 system.The interface layer is strongly compressed by collision, and the O stars ionizing the three Spitzer bubbles S116-117-118 were formed in the layer due to gravitational instability. This explains the distribution of the bubbles along the western surface of the small cloud which is interacting with the large cloud.The timescale of the collision is roughly estimated to be ∼ 1.3 Myr (=10 pc / 7 ). An O stars of 30is formed in 10^5 yr within the timescale for a mass accretion rate 3 × 10^-4 , which is adopted from a typical value in the compressed layer of cloud-cloud collision in magnetohydrodynamical numerical simulations <cit.>. This satisfies the mass accretion rate required to form O stars by overcoming the stellar radiation feedback <cit.>.The collisional compression is possibly extended over 40 pc beyond the size of the cavity in the north-south direction vertical to the projected motion of the two clouds. The layer is observed as a north-south elongated molecular ridge at l=314.2 from 0.1 to 0.6 in b, and we find a possible sign of further triggered formation of lower mass stars along the compressed layer found as at least several compact infrared sources in Figure <ref>. So, it is possible that triggering is extensive in space, while O star formation is probably limited to the region of high molecular column density 1–2× 10^22cm^-2.§.§ Comparison with RCW120In S116-117-118, theregions are located outside the collision-created molecular cavity, although in RCW120 theregion is located inside the cavity.In the both cases the collision formed the interface layer where O star(s) form, and the morphological difference between the two cases are to be explained. The schematic drawings of the two colliding clouds in position-velocity diagram of the two cases, RCW 120 and S116-117-118, are shown in Figure <ref>.RCW120 <cit.>presented a cloud-cloud collision model in order to explain the formation of the O star within the bubble. The diameter of the RCW120 bubble is ∼3 pc, whereas that of the S116-S117-S118 cavity is ∼10 pc.Therefore, the volume of the S116-S117-S118 cavity is larger than the RCW120 cavity by a factor of ∼30 if a spherical cavity is assumed, and the molecular mass inside the cavity is significantly larger in S116-S117-S118 than in RCW120.This offers a possible explanation for that the larger number of O stars in the present case.Figure <ref> shows the schematic images of cloud-cloud collision in RCW 120 and S116–S117–S118.According to the <cit.> model of cloud-cloud collision, we expect that two clouds, one of which is delineating a Spitzer bubble and the other localized inside the bubble, exhibit complementary distribution in the early phase after the collision (Phase 1 in Figure <ref>). Later, the small cloud is destructed by ionization due to the formed star and by collisional merging into the interface layer in RCW120.The difference between the two cases is the location of the O stars. In RCW120 the O star is inside the cavity (Figure <ref>a Phase 3), whereas in S116–S117–S118 the O stars are located outside of the cavity (Figure <ref>b Phase 3).In S116-117-118, we suggest that the collision happened by chance close to the edge of the large cloud.This situation made the shocked layer exposed to the outside of the large cloud where density drops suddenly.We infer that this ad foc geometry in S116-117-118 caused the O star formation outside the collision-created molecular cavity.§.§ The physical properties of the collisional interface layerThe S116–S117–S118 cavity demonstrates clearly the role of cloud-cloud collision in creating a cavity, which shows a well-ordered nearly circular boundary.Because no O star exists inside the cavity, there is no room for the cavity to be produced by an O star. InRCW120 the inside wall of the cavity is partially ionized and the physical conditions are affected by the ionization, implying that the collisional interface do not keep the conditions just after the collision.The present case is different because it provides physical conditions in the shock-compressed layer unaffected by the ionization.Although S117 seems to lie toward the peak of the small cloud, S117 is not affecting the small cloud and the inside of the cavity as shown by no enhanced R_3-2/1-0 ratio toward S117 in Figure <ref>d. In this context, we pay attention to that the interface layer toward the small cloud and the cavity exhibit highly filamentary distribution which is obvious in the R_3-2/1-0 distribution in Figure <ref>a.<cit.> showed that molecular filaments are formed in the shocked interface in a cloud-cloud collision, and similar filament formation is seen in the simulations by <cit.> and <cit.>.The width and length of the filaments in these simulations deserve a further detailed comparison with observations in order to better understand the physical processes in collision.Figures <ref>c and <ref>f show the distribution of1-0 images of the small cloud and the large cloud overlaid on the filamentary distribution of R_3-2/1-0.The high1-0 intensity part and the R_3-2/1-0 filamentary structures in the collision interface are not corresponding with each other, but the high R_3-2/1-0 regions rather correspond to the edge of the1-0 intensity peaks.Since there is noregion toward the cavity, the high R_3-2/1-0 is possibly due to heating by the collisional shock but not by the enhancement of density or irradiation by ultraviolet photons of O stars.In the density regime concerned molecular cooling time scale is ∼ 10^4 yr andcollisional shock heating may by responsible for the high line intensity ratio in such a small timescale.S116–S117–S118 may be a rare case where heating by ultraviolet radiation is not important in a collision-produced cavity, allowing us to test isolate contribution of the shock heating only. § CONCLUSIONSWe carried out a molecular line study toward the three Spitzer bubbles S116, S117 and S118.The region is associated with nearly ten smallerregions, indicating active formation of high-mass stars over a length of 50 pc although the region did not attract much attention until now. The detailed molecular data in the present work lead to the following conclusions which offer a novel insight into the formation of the Spitzer bubbles; * The molecular clouds in the region of the three Spitzer bubbles S116–S117–S118 include two velocity components; one of them, the small cloud, has-63and the other, the large cloud, has -58 , while the two clouds appear to be continuously distributed in a position-velocity diagram.The large cloud has a cavity, an intensity depression, which is apparently correlated with the morphology of the small cloud.The two of the Spitzer bubbles S116 and S118 and additional smallregions are distributed along the northwestern and southwestern edges of the small cloud, and the other S117 toward the peak of the small cloud. * We present an interpretation that the cavity was created by a collision between the two clouds∼1 Myr ago and the collision compressed the gas into a dense layer elongated in the north-south direction over an extent of ∼20 pc at a kinematic distance of 5.9 kpc.In the compressed layer produced by the collision the O stars exciting theregions including the three Spitzer bubbles were formed.We show that a position-velocity diagram including the small cloud and the cavity exhibits a pattern characteristic to cloud-cloud collision simulated numerically by <cit.>.By assuming that the relative motion of the clouds has 45^∘ to the line of sight, we estimate that the collision velocity and the collision timescale to be ∼ 7and ∼ 1 Myr, respectively.* The morphology is different from the collision-induced star formation in RCW120 where theregion was formed within the cavity created by the collision.We suggest the difference is due to the density-bound distribution of the large cloud in S116-117-118, which made theregions expand toward the less dense outer part of the cloud rather than the denser inside.The present case is important in two ways. One is that it demonstrates unambiguously formation of a molecular cavity by cloud-cloud collision without O stars.Another is that it allows us to watch details of the collision compressed layer with no influence of ultraviolet photons of O stars; in particular, the highly filamentary distribution is seen as is consistent with the numerical simulations of cloud-cloud collision that predicted filamentary distributions in the shocked gas <cit.>. To summarize, we conclude that the threeregions are ionized by the O stars formed by triggeringin the cloud-cloud collision. This is a case whereregions expanded outside the collision-created cavity. The collision time scale is estimated to be ∼1 Myr. A mass accretion rate of 3 × 10^-4based on MHD simulations of cloud-cloud collision <cit.> explains O star formation in 10^5 yr, significantly smaller than the collision duration, which is consistent with the O star formation in the late phase of the collision. NANTEN2 is an international collaboration of 11 universities: Nagoya University, Osaka Prefecture University, University of Bonn, University of Cologne, Seoul National University, University of Chile, University of New South Wales, Macquarie University, University of Sydney, University of Adelaide, and University of ETH Zurich. The Mopra radio telescope is part of the Australia Telescope National Facility. The University of New South Wales, the University of Adelaide, and the National Astronomical Observatory of Japan (NAOJ) Chile Observatory supported operations. The ASTE telescope is operated by NAOJ. This work was financially supported by Grants-in-Aid for Scientific Research (KAKENHI) of the Japanese society for the Promotion of Science (JSPS, grant No. 15H05694).Finally, we are grateful to the referee for his/her thoughtful comments.[Anderson et al.(2009)]and09 Anderson, L. D., Bania, T. M., Jackson, J. M., et al. 2009, , 181, 255[Beaumont & Williams(2010)]bea10 Beaumont, C. 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J., et al. 2003, , 342, 1117[Mizuno & Fukui(2004)]miz04 Mizuno, A., & Fukui, Y. 2004, Milky Way Surveys: The Structure and Evolution of our Galaxy, 317, 59[Okamoto et al.(2017)]oka17 Okamoto, R., Yamamoto, H., Tachihara, K., et al. 2017, , 838, 132[Panagia(1973)]pan73 Panagia, N. 1973, , 78, 929[Strong et al.(1988)]str88 Strong, A. W., Bloemen, J. B. G. M., Dame, T. M., et al. 1988, , 207, 1[Takahira et al.(2014)]tak14 Takahira, K., Tasker, E. J., & Habe, A. 2014, , 792, 63[Torii et al.(2015)]tor15 Torii, K., Hasegawa, K., Hattori, Y.,et al. 2015, , 806, 7[Tóth et al.(2014)]tot14 Tóth, L. V., Marton, G., Zahorecz, S., et al. 2014, , 66, 17[Walborn et al.(2002)]wal02 Walborn, N. R., Howarth, I. D., Lennon, D. J., et al. 2002, , 123, 2754[Wolfire & Cassinelli(1987)]wol87 Wolfire, M. G., & Cassinelli, J. P. 1987, , 319, 850[Zavagno et al.(2007)]zav07 Zavagno, A., Pomarès, M., Deharveng, L., et al. 2007, , 472, 835[Zinnecker & Yorke(2007)]zin07 Zinnecker, H., & Yorke, H. W. 2007, , 45, 481 | http://arxiv.org/abs/1706.08720v2 | {
"authors": [
"Yasuo Fukui",
"Akio Ohama",
"Mikito Kohno",
"Kazufumi Torii",
"Shiji Fujita",
"Yusuke Hattori",
"Atsushi Nishimura",
"Hiroaki Yamamoto",
"Kengo Tachihara"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170627082941",
"title": "Molecular clouds toward three Spitzer bubbles S116, S117 and S118: Evidence for the cloud-cloud collision which formed the three \\HII \\ regions and a 10-pc scale molecular cavity"
} |
theoremTheorem[section] *theorem*Theorem lemma[theorem]Lemma proposition[theorem]Proposition corollary[theorem]Corollary claim[theorem]Claim define[theorem]Definition∅∅∅ | http://arxiv.org/abs/1706.08575v1 | {
"authors": [
"John McKay",
"Anne Gelb",
"Vishal Monga",
"Raghu Raj"
],
"categories": [
"math.NA",
"cs.CV"
],
"primary_category": "math.NA",
"published": "20170626194259",
"title": "Using Frame Theoretic Convolutional Gridding for Robust Synthetic Aperture Sonar Imaging"
} |
Contamination of polymethylmethacrylate by organic quantum emitters Andre Neumann^1,*, Jessica Lindlau^1,*, and Alexander Högele December 30, 2023 =================================================================== This paper is devoted to assess the presence of some regularities in the magnitudes of the earthquakes in Italy between January 24^th, 2016 and January 24^th, 2017, and to propose an earthquakes cost indicator. The considered data includes the catastrophic events in Amatrice and in Marche region. To our purpose, we implement two typologies of rank-size analysis: the classical Zipf-Mandelbrot law and the so-called universal law proposed by Cerqueti and Ausloos (2016). The proposed generic measure of the economic impact of earthquakes moves from the assumption of the existence of a cause-effect relation between earthquakes magnitudes and economic costs. At this aim, we hypothesize that such a relation can be formalized in a functional way to show how infrastructure resistance affects the cost. Results allow us to clarify the impact of an earthquake on the social context and might serve for strengthen the struggle against the dramatic outcomes of such natural phenomena.Keywords: Earthquake, magnitude, economic cost, Zipf-Mandelbrot law, rank-size analysis, Italy.§ INTRODUCTION Seismologists have carefully clustered the world in different non-overlapping zones on the basis of the probability that the zone experiences an earthquake. Such natural phenomena might cause very dramatic damages to the human activities and kill several people. Thus, policymakers should adopt anti-seismic building strategies, mainly in zones with a high seismic risk. Unfortunately, some countries come from a political history of myopic decisions in this respect, and Italy is an illustrative example of them. This paper aims at exploring the Italian earthquakes occurred in 2016 and early 2017, with a specific reference to the big ones in Amatrice (August, 24^th) and Visso (October, 26^th – two times – and 30^th) along with the large amount of minor earthquakes before and after them. The considered period is 365 days, from January 24, 2016 to January 24, 2017, along which we observe 978 seismic events within a Richter magnitude range: [3.1 - 6.5].We decide to exclude observations with magnitudes smaller than 3.1 for many reasons. First of all, this paper deals with formulations of damages' cost indicators of the earthquakes and according to the United States Geological Survey, a seismic event with magnitude less than 3.1 has very low probability to cause observable damages. Secondly, the restriction to magnitudes not smaller than 3.1 allows to face the incomplete catalog problem. Indeed, we are analyzing a peculiar time period from a seismic point of view. Such a period has given a lot of work to the Italian National Institute of Geophysics and Vulcanology (INGV) because of the high number of earthquakes concentrated in very short time and of the intensity of them. In fact, after the mainshock of Amatrice, SISMIKO, the coordinating body of the emergency seismic network at INGV, was activated to install a temporary seismic network integrated with the existing permanent network in the epicentral area, but the risk that many aftershocks were not registered or not revised remains high (see Moretti et al., 2016). On this point, some scholars are actively working on the estimation of the catalog completeness. For example, Marchetti et al. (2016) have estimated M_c = 2.7 for the revised catalog of the seismic events occurred immediately after the Amatrice's earthquake. In accord to Marchetti et al's work, M_c could rise to a maximum level of 3.1 (on this topic see also Chiaraluce et al., 2017). Moreover, our dataset has no particular peaks apart from those showed in Figure <ref> after August 24^th. Then, from the 24/01/2016 to 23/08/2016, we can consider M_c = 2.5, in accord to Romashkova and Peresan (2013) and Schorlemmer et al. (2010). Thus, the considered restriction to magnitudes greater than 3.1 let prudentially the catalog incompleteness problem be quite negligible in the reference period without affecting the cost analysis of the earthquakes. We propose here a rank-size approach for analyzing the earthquakes' magnitudes sequence just described in order to assess the presence of data regularities. The rank-size relationship has been explored for several sets of data and it is still at the center of the scientific debate. At its inception, power law and Pareto distribution with unitary coefficient, introduced in Zipf (1935, 1949) and denoted from there as Zipf law, has been suitably employed to provide a best fit of the rank-size connections in the field of linguistics. After the first applications, several contributions supporting the validity of the Zipf law have appeared in the literature. In this respect, we just mention some recent important papers: Ioannides and Overman (2003), Gabaix and Ioannides (2004), Dimitrova and Ausloos (2015), Cerqueti and Ausloos (2015) in the context of economic geography; Montemurro (2001) and Piantadosi (2014) in linguistic; Axtell (2001), Fujiwara (2004), Bottazzi at al (2015) in the business size field; Li and Yan (2002) in biology; Levene, Borges and Loizou (2001) and Maillart et al (2008) in informatics; Manaris et al (2005) and Zanette (2006), in the context of music; Huang et al (2008) in the context of fraud detection; Blasius and Tönjes (2009) in the gaming field. For a wide review of rank-size analysis see Pinto et al (2012). However, some cases of rank-size relationships fail to be well-fitted by Zipf law (see e.g. Rosen and Resnick, 1980; Peng, 2010; Ioannides and Skouras, 2013; Matlaba et al., 2013). By one side, such examples support the acknowledged lack of a theoretical ground for this statistical regularity (see Fujita et al., 1999; Fujita and Thisse, 2000); by the other side, they represent a further hint for proceeding with the methodological research, and construct more general laws. Indeed, under a pure methodological point of view, several extensions of the Zipf law have been introduced. The most prominent examples are the Zipf-Mandelbrot law (ZML, hereafter; see Mandelbrot, 1953, 1961; Fairthorne, 2005) and the Lavalette law (LL hereafter; see Lavalette, 1966), which have been proven to provide a spectacular fit of rank-size relations, even when Zipf law fails to do it (see e.g. Cerqueti and Ausloos, 2015). In this paper, we implement two general rank-size procedures: the above-mentioned ZML and a universal law (UL from now on), which is an extension of the LL to a five parameters rule that has been recently introduced by Cerqueti and Ausloos (2016). All fits have been carried out through a Levenberg-Marquardt algorithm (Levenberg 1944, Marquardt 1963, Lourakis2005) with a restriction on the parameters that have to be positive. Furthermore, we have also discussed the economic costs of the earthquakes. At this aim, we propose a new generic cost indicator based on a suitable transformation of magnitudes into costs. As we will see, such an indicator moves from the best fit procedures implemented in the rank-size analysis phase, and it might be effectively used for finalizing policies for the management of seismic risks. We show how the cost indicator can be computed in the special case of the analyzed earthquakes. Rank-size relations have been introduced for the explanation of seismological data and for the earthquakes magnitudes (see e.g. Jaume, 2000; Wu, 2000;Mega et al, 2003; Newman, 2005; Saichev and Sornette, 2006; Pinto et al, 2012; Aguilar-San Juan and Guzman-Vargas, 2013). However, this is the first paper which treats very recent Italian seismic events under this perspective. Moreover, to the best of our knowledge, there are no contributions in the literature on the construction of a cost indicator for earthquakes based on the rank-size laws. In order to validate the obtained results, extra investigations on two different datasets have been performed. The first deals with a more global analysis on the basis of a suitable enlargement of the dtaset. At this aim, we notice that an important change of Italian seismic network is occurred in 16^th April, 2005, when the new network for seismic events collection has been activated. From that date the data elaboration system has sensibly increased and, in order to deal with the incompleteness catalog problem, the accepted average M_c has been set to 2.5 (see Romashkova and Peresan 2013, and Schorlemmer et al. 2010). Therefore, we have performed the rank-size analysis on the data from the INGV catalog in the period ranging from 16/04/2005 to 31/03/2017, with the restriction to magnitudes not smaller than 2.5. The second extra investigation is developed to face the effects of space variables. In this case, the considered dataset has been created by selecting the earthquakes with epicenters in the eight adjacent provinces involved in the seismic sequence started with the Amatrice's earthquake: Macerata, Perugia, Rieti, Ascoli Piceno, L'Aquila, Teramo, Terni and Fermo (and respective coasts), from 24/01/2016 to 24/01/2017. In so doing, we are in line with geophysicists who claim that taking a small region and a short time period let the space effects be not relevant (see e.g. De Natale et al., 1988). It is interesting to note that, as we will see, the local analysis is not too different from the original one in terms of the cardinality of the dataset, in that the most part of the earthquakes in the reference period in Italy has occurred in such eight provinces. The rest of the paper is organized as follows: Section <ref> is devoted to the description of the data and of the methodologies used for performing the analysis. This section illustrates also the procedure adopted for the identification of the earthquakes costs and for the development of the cost indicator. Section <ref> investigates the robustness of the reached results by presenting the study of the global and local datasets. Section <ref> proposes the results of the analysis, along with a critical discussion of them. Last Section concludes and offers directions for future research.§ DATA AND METHODOLOGYThis section is devoted to the description of the data on the magnitudes of the earthquakes occurred in Italy in 2016 and early 2017. Furthermore, it contains the illustration of the methodological tools used for analysis. §.§ Data Our dataset is composed by the magnitudes of the earthquakes registered in Italy during the period: January 24^th, 2016 - January 24^th, 2017. The definition of the magnitude of an earthquake and the employed dataset are taken from the website of the INGV (the Italian National Institute of Geophysics and Vulcanology see <<http://cnt.rm.ingv.it/>>). Such a definition is based on the different measurement methods used from seismograms, each of them being also tailored on a specific magnitude range and epicentral distance. For the details on the concept of magnitude, please refer to the website of the INGV (see <<http://cnt.rm.ingv.it/en/help/>>). Specifically, the considered period starts at the first hour of January 24^th,2016 and ends on the midnight of January 24^th, 2017, hence including relevant earthquakes like those registered in Amatrice, on August 24^th (magnitude equals to 6) Umbria and Marche regions on October 26^th (two times) and 30^th of 2016 (magnitudes 5.4, 5.9 and 6.5 respectively), and the most recent on January 18^th 2017, in L'Aquila (three times, magnitude 5.5, 5.4 and 5.1). To have an idea of the seismic activity of the analyzed period, see Figure <ref>.The number of the available data is of high relevance. Indeed, the number of registered seismic events over the considered period is 59190, which gives to the reader the dimension of how often earthquakes are registered in this period in Italy, in particular in the Center of Italy, since the majority of the earthquakes are located there. Data on depth of the epicenters and on their localization are also available, but they are not treated in this study. They are left for future researches.We need to point out that there is a catalog incompleteness problem, in that the main events might hide several minor subsequent aftershocks. In order to deal with such catalog incompleteness problem, we restrict the analysis to the seismic events of magnitude not smaller than 3.1 (see Section <ref> for a detailed discussion of this point). Therefore, the number of observations reduces to 978. Table <ref> collects the main statistical indicators of the data and Figure <ref> represent the probability density function of the considered time series. Notice that Figure <ref> contains also the best fit of a power law function with the empirical distribution of the sizes of the earthquakes. This supports an empirical evidence, already pointed out by previous studies (see e.g. Kagan, 2010). Some comments on the statistical characteristics can be found in Section <ref>. §.§ Methodologies The magnitude of an earthquake represents the size of the rank-size analysis. Since the target of the analysis is to construct an aggregated costs indicator, magnitudes are not taken as they are. Indeed, the same earthquake can produce different levels of damages if it follows a long list of foreshocks or not: in the former case, the earthquake insists over an already solicited territory, while in the latter one it is the first shake and human activities have not previous solicitations. Therefore, each earthquake has been temporally contextualized – suppose, it has occurred at time t – and we have transformed its magnitude z into z̃=η(n,z_1,…, z_n,Δ t)× z, where η(n,z_1,…, z_n,Δ t) is a parameter dependent on the number n of the foreshocks whose magnitudes are assumed to be z_1,…, z_n and occurred in the time interval [t-Δ t,t]. The parameter η(n,z_1,…, z_n,Δ t) is marginally increasing with respect to z_1, …, z_n and n and marginally decreasing with respect to Δ t, and it is not smaller than 1. In fact, if the territory has experienced several foreshocks of large magnitude in a small time range before t, then the damages created by the earthquake are comparable with those of an isolated earthquake with magnitude z̃>z. With a reasonable abuse of notation, we refer hereafter simply to magnitudes, having in mind z̃ instead of z. The single earthquakes have been ranked in decreasing order, so that rank r=1 corresponds to the highest registered magnitude while r=978 is associated to the lowest value of the considered phenomenon, which is 3.1. Then, in general, low ranks are the ones associated to the strongest seismic events in terms of magnitudes, while high ranks point to the earthquakes with small magnitudes. Here we implement two times the best fit procedure to assess whenever the size-magnitude z might be view as a function of the rank r. The considered fit functions are the ZML and the UL. The former can be written asz̃∼ f_ZML(r)=α(r+β)^-γ,while the latter isz̃∼ f_UL(r)=k(N+1-r+ψ)^ξ[N(r+ϕ)]^λ,where α, β, γ must be calibrated on the size data when (<ref>) is used, while k, ψ, ξ, ϕ, λ are those to be calibrated if the fit procedure is as in (<ref>). The parameter N corresponds to the number of observations, and it is N=978 for this specific case. To implement the rank-size analysis and derive the proposed aggregated cost indicator we need to provide an explicit shape of the parameter η(n,z_1,…, z_n,Δ t). In order to meet space constraints[The proposal of other scenarios and their analysis is available upon request.], we present here the analysis of the unbiased scenario of η(n,z_1,…, z_n,Δ t)=1, for each n,z_1,…, z_n,Δ t. In this case we are in absence of amplification effects. Since we aim at constructing an aggregated cost indicator, this situation has an intuitive reasoning: indeed, it is the case with the lowest level of damages – all the earthquakes are treated as isolated ones – and let clearly understand how the outcomes of a missing anti-seismic policy can be negative, even in the lucky case of absence of propagation effects. Under the considered scenario, we have z̃=z. The economic indicator is obtained by transforming the magnitude of an earthquake into the cost associated to such an earthquake. In this respect, as already said above, the decision of taking magnitudes not smaller than 3.1 lies also in the evidence that a very low-magnitude earthquake does not produce damages. We assume that costs are positive and increasing for magnitudes greater than a certain threshold z̅≥ 3.1, and they are null below it. The value of the critical threshold z̅ is strongly affected by the way in which infrastructures and buildings are constructed on the seismic territory. Neglecting the adoption of anti-seismic building procedures leads to destructive earthquakes even at low magnitudes, i.e. when z̅ has a small value. Under a general perspective, we use the rank-size laws written in (<ref>) and (<ref>) in order to transform magnitudes into costs. This will lead to the definition of two different cost indicators, as we will see. We define C_⋆:[0,+∞) →[0,+∞) such that C_⋆(z)=H(f_⋆(r)), where ⋆=ZML, UL. Quantity C_⋆(z) is the cost associated to an earthquake with magnitude z when the best fit is performed through function f_⋆ and H:[0,+∞) →[0,+∞) increases in [z̅, +∞) and is null in [0,z̅). Under the rank-size law perspective, the identification of a critical magnitude z̅ is associated to the identification of a critical rank r̅ such that z ≤z̅ if and only if r ≥r̅. Such a critical rank varies if one takes (<ref>) and (<ref>). To distinguish them, we will refer to the intuitive notation of r̅_ZML and r̅_UL. The cost indicator Γ associated to the collection of the considered earthquakes is defined as the aggregation of their individual costs. We include in such an aggregation also the presence of a maximum for the level of magnitude of an earthquake, and we denote it by Z_MAX. In fact, we point out that the greatest magnitude ever registered is 9.5 of the Great Chilean earthquake in 1960. To be prudential, we will set a theoretical Z_MAX=10 even if the empirical maximum is 6.5, as reported in the applications (see Table <ref>). Thus, we setΓ_ZML = ∫_z̅^Z_MAXC_ZML(z)dz = ∫_0^r̅_ZMLH(α̂(r+β̂)^-γ̂) dr,andΓ_UL = ∫_z̅^Z_MAXC_UL(z)dz = ∫_0^r̅_ZMLH(k̂(N+1-r+ψ̂)^ξ̂[N(r+ϕ̂)]^λ̂) dr,which represent the cost indicators for the fits in (<ref>) and (<ref>), respectively, and where ⋆̂ is the calibrated parameter ⋆, according to the best fit procedure. The Γ's depend on the value of z̅, once all the rest is fixed. Of course, the cost indicators decrease as z̅ increases, and they are null when z̅=Z_MAX. We propose three scenarios for the selection of function H:(i)H(z)= exp(z), ∀ z ∈ [z̅, Z_MAX];0,∀ z ∈ [0,z̅);(ii)H(z)=z, ∀ z ∈ [z̅, Z_MAX];0,∀ z ∈ [0,z̅); (iii)H(z)= ln(z), ∀ z ∈ [z̅, Z_MAX];0,∀ z ∈ [0,z̅); The considered scenarios are representative of three very different realities for the economic costs. Indeed, the exponential case (item (i)) is the one providing a severe penalization of the high magnitudes in terms of costs; differently, the logarithm (item (iii)) is the function assigning a lower value to the costs for high magnitudes and the linear case (item (ii)) is the middle case between these extremes. To identify the considered cases, we will insert an intuitive superscript to the cost indicator so that, for example, Γ^(ii)_ZML is the Γ_ZML obtained when H is as in item (ii). § ROBUSTNESS CHECKIn order to validate the obtained findings, we here investigate the problem by using two different datasets: a global and a local one. In the global case, we present the analysis on a bigger dataset by assuming that enlarging the considered time window let the average magnitude completeness be closer to 2.5, in accord to Romashkova and Peresan (2013), and Schorlemmer et al. (2010). In so doing, we provide a validation of the results.So, we have downloaded from the same source (INGV), 13239 observations detected from April 16^th, 2005 to March 31^st, 2017 with magnitude not smaller than 2.5. The initial data is consistently selected, in that it coincides with the change of the Italian earthquake survey by INGV. Table <ref> contains a summary statistics of the dataset and in Figure <ref> there is the probability density function of the data. As for the original sample, Figure <ref> shows that a power law is a good approximation of the empirical distribution of the earthquakes (see e.g. Kagan, 2010). Table <ref> illustrates the parameters of the best fit estimation obtained by applying the processes described in Section <ref> on this global dataset. For a visual inspection of the estimated model, refer to Figures <ref> and <ref>, which contain the original data and the fitted model of the calibration performed with Eq. (<ref>) and (<ref>) respectively. In the local case, we explore the spatial effects by running the same procedure described in Section <ref> on the restricted area of the provinces of Macerata, Perugia, Rieti, Ascoli Piceno, L'Aquila, Teramo, Terni and Fermo (for the estimation precision of the epicenters see Amato and Mele, 2008) that are relevant for the 2016 Amatrice earthquake sequence (see Gruppo di Lavoro INGV sul Terremoto in Centro Italia, 2016). The reference period is the same of the original study: from January 24^th, 2016 to January 24^th 2017, with 849 observations. This local analysis is in line, from a methodological point of view, with seismological researches which state that taking small zones and short time periods leads to negligible space effects (see e.g. De Natale et al., 1988). Notice that the local analysis serves as validating the robustness of the study of the considered sample. This said, it is also important to stress that the identification of an earthquake as a product of spatio-temporal correlations among shakes is not relevant for implementing the rank-size analysis and, subsequently, for deriving the aggregated cost indicator. Indeed, we are not interested on the reasoning behind the occurrence of an earthquake but only on the fact that it has occurred and on the knowledge of its magnitude. To be sure that we avoid the catalog incompleteness and in order to make the analysis comparable with the one object of this paper, we take in consideration magnitudes not smaller than 3.1 (Marchetti et al., 2016). It is very important to note that the local dataset contains about the 87% of the earthquakes of the original sample. Thus, results of the local analysis in line with those obtained for the original sample are expected. The statistical summary of the reduced dataset is reported in Table <ref> while the density function of the registered magnitudes is presented in Figure <ref>. Also in this case, Figure <ref> evidences that the empirical distribution of the earthquakes follows a power law (see e.g. Kagan, 2010). Table <ref> contains the parameters of the best fit estimation obtained by applying the processes described in Section <ref> on the local data. For a visual inspection of the estimated model, Figures <ref> and <ref> contain the original data and the fitted model of the calibration performed with Eq. (<ref>) and (<ref>) respectively. § RESULTS AND DISCUSSIONTable <ref> offers a preliminary view of the phenomenon under investigation. Since the empirical distribution of the sizes of the earthquakes can be well-fitted through a power law, as expected, the mean and the median of the magnitude distribution are different. This suggests the presence of asymmetry. The positional indicators show that the most part of the observations takes values close to 3.3. Furthermore, the variability indexes confirm that the values are rather concentrated near the distribution's center. The positive skewness suggests a right-tailed shape, and the value of the kurtosis indicates a leptokurtic distribution. The leptokurtic property of the data is due to the presence of outliers (see Figure <ref>). As mentioned above, the best fit procedures on (<ref>) and (<ref>) are performed over the dataset considering magnitudes not smaller than 3.1 for the reasons discussed in Section <ref> and Section <ref>. Results are presented in Table <ref> where the calibrated parameters and the R^2's are reported. For a visual inspection of the goodness of fit, refer to Figures <ref> and <ref>.The analysis evidences a first important fact that is the presence of outliers at low ranks. They do not affect the performance of the fitting procedures with (<ref>) or (<ref>), and consequently we cannot note substantial discrepancies in using ZML or UL for the dataset containing the earthquakes from 24/01/2016 to 24/01/2017 in Italy.Looking at Section <ref>, we can compare our results with those obtained for the global and the local datasets and check the coherence of our findings.The local analysis excludes 149 observations with magnitudes mainly allocated in the high rank and only one of magnitudes around 5. The exclusions do not change too much the estimations, and the parameters and the R^2's remain rather similar to those presented for the case of the original sample. Such a similarity appears to be more evident for the ZML fit, hence supporting that the UL approximates the data in a more convincing way and is more sensitive to data variation (see Tables <ref> and <ref>). In particular, the upper side of Tables <ref> and <ref> shows a ZML best fit calibration with β̂ close to zero and a small value of γ̂ because the fitted model captures at the best the effect of the low ranks. Consequently, α̂ is close to the highest registered magnitude. Visual inspection is also appealing (see Figures <ref> and <ref> for the ZML case and Figures <ref> and <ref> for the UL case). This suggests the negligible presence of space effects in performing the rank-size analysis and computing the cost indicators. The situation is notably different for the case of the dataset with enlarged time window (see Table <ref>).In this case, we observe an increment of the relative number of magnitudes at high ranks, hence leading to a calibration which is more distorted from the small magnitude events and loses representation capacity at lowest ranks, even in presence of some outliers at low ranks. The opportunity to catch the effects of the lowest ranked outliers is due to ψ in (<ref>) (see Cerqueti and Ausloos, 2016) which increases in the case of sizes at low ranked magnitudes close to the medium ranked sizes. By comparing the levels of the parameter ψ̂ from Tables<ref>, <ref> and <ref>, one can observe the increment in the global case. Notice that a small value of ψ̂ stands for a fit which can capture the high ranked data effect without flattering the part of the curve at a low rank. Moreover, the parameter ϕ in (<ref>) acts in the same way of ψ, but to capture the effects of the lowest outliers. Thus, in presence of high ranked outliers the value of ϕ increases. Consistently with this idea, ϕ̂ is equal to 9.52 for the case of the enlarged time window and it is null in the other cases. A slight improvement of the goodness of fit is shown by the R^2 of the enlarged case, even if it moves from 0.98 to 0.99. So, the goodness of fit is generally so high that a discrepancy between observed data and fit curves are not appreciable (see Figures <ref>, <ref> and <ref> for the ZML case and Figures <ref>, <ref> and <ref> for the UL case). We also notice that the highest (lowest) level of the magnitudes estimated through (<ref>) and (<ref>), namely Ẑ^ZML_Max and Ẑ^UL_Max (Ẑ^ZML_Min and Ẑ^UL_Min), respectively, adds further arguments for supporting the goodness of fit. In fact, we have found Ẑ^ZML_Max = 6.21, Ẑ^UL_Max = 6.18, Ẑ^ZML_Min = 3.07 and Ẑ^UL_Min = 3.07. For the maximum points curves are slightly below the maximum empirical observation of 6.5, while for minimum we have the same value very close to 3.1, hence suggesting an analogous behavior at the highest rank. To sum up, we argue that the ZML and UL show similar behaviors in fitting the original catalog and the one associated to the local dataset, hence giving a substantial lack of space effects. The analysis of catalog with M_c=2.5 and wider time windows highlights that the UL fit is more appropriate to represent the data, even if the goodness of fit remains unchanged. Thus, data show an analogous regularity property in both of cases of short and long period, and this suggests that results provided for the original sample are robust to enlargement of the period. The incompleteness catalog problem has been faced in both of cases by truncating to a low level of magnitude, in accord to seismological literature. For what concerns the economic costs indicators, some integrals can be easily computed in closed form, while other ones will be estimated. We have Γ^(ii)_ZML = ∫_0^r̅_ZMLα̂(r+β̂)^-γ̂ dr = α̂/1-γ̂[(r̅_ZML+β̂)^1-γ̂-β̂^1-γ̂]Γ^(iii)_ZML =∫_0^r̅_ZMLln(α̂(r+β̂)^-γ̂)dr= ln(α̂)·r̅_ZML--γ̂·[(r̅_ZML+β̂){ln (r̅_ZML+β̂)-1}-β̂{ln (β̂)-1}];Γ^(iii)_UL = ∫_0^r̅_ULln(k̂·(N+1-r+ψ̂)^ξ̂[N(r+ϕ̂)]^λ̂) dr = lnk̂·r̅_UL++ξ̂[-(N+1-r̅_UL+ψ̂){ln(N+1-r̅_UL+ψ̂)-1} +(N+1+ψ̂){ln(N+1+ψ̂)-1}] - -λ̂·[ln (N)·r̅_UL +(r̅_UL+ϕ̂){ln(r̅_UL+ϕ̂)-1}-ϕ̂{ln(ϕ̂)-1}]. The other cases of cost indicators Γs are properly estimated through standard numerical techniques. Specifically, the generic interval [0,r̅] is discretized in S sub–intervals with a discretization step Δ r, so thatr_0=0,r_s=r_s-1+Δ r,r_S=r̅.From such a discretization, the generic integrals defining the Γ's are approximated as follows:Γ=∫_0^r̅H(r)dr ∼Δ r ·∑_s=1^S H(r_s).Now, recall that a specific value of r̅ is associated to a value of z̅. Thus, we can compare the cost indicators in terms of the threshold magnitudes z̅. Figure <ref> allows the comparison among the cases of Γ_ZML's and Γ_UL's as z̅ varies, respectively. The discretization step used for integral approximation in (<ref>), (<ref>) and (<ref>) is taken as Δ r= 0.01.Cost indicators are decreasing functions of z̅, as expected. The value of z̅ that represents a measure of the Italian infrastructures' ability of resisting to earthquakes.The costs decays have no differences in the behaviours considering the two fit functions (see Figure <ref>).As expected, for both of cases of Eq. (<ref>) and (<ref>), the most expensive case emerges by transforming magnitudes into cost with the exponential function Γ^(i), while the logarithmic transformation of the magnitudes leads to the lowest level of cost indicator and the sensitiveness to increments of z̅ are less evident. The Γ^(ii)'s and Γ^(iii)'s decay quite simultaneously, even if starting by different point, and converge to zero, while Γ^(i)_ZML and Γ^(i)_UL tend to rapidly reduce the cost until z̅ is around 3.7 (by a visual inspection). After this threshold the curves' inclination decrease very slowly denoting resistance to damages reduction. Furthermore, the exponential transformations of estimated magnitude flatten after about z̅ = 3.5. Moreover, one can observe a change in the concavity of the curves Γ^(i)'s around magnitude 5.7. After such a value, the curves decrease rapidly to zero. This finding suggests that the aggregated economic costs of the earthquakes collapse rapidly above a large enough threshold, and this should be viewed as a hint to the policymakers of implementing strategies for letting the no-damage zone above such a magnitude threshold. In order to visualize the robustness of the results obtained with this cost analysis, in Figures <ref> and <ref> we also present the different curves obtained from the different dataset presented in Section <ref>. Panel (a) is the case of the original sample, (b) is the local analysis and (c) is the global one. For the cases of the cost indicators calibrated on the Eq. (<ref>), see Figure <ref>. We can note that (a) and (b) have the same shapes, but (b) is a little bit scaled due to the fact that the zones individuated entails the exclusion of some seismic events. The decays are the same but the curves of the (b) case reach zero first. A motivation can be found in the exclusion of an important earthquake of magnitude around 5.5 in the local dataset, hence leading to slightly cheaper damages. Case (c) is referred to a wider time window (about 12 years) and to a dataset with M_c=2.5 on average. Consequently, as expected, the increased amount of minor earthquakes rises the cost mainly in the left side of the curve. In this case, null costs are achieved at magnitudes around 5.5. This misrepresentation is due to the functional form of ZML, being the percentage of high-magnitudes phenomena over the considered series very low. The costs analysis performed with the employment of Eq. (<ref>) are reported in Figure <ref>. For cases (a) and (b), the same arguments carried out above can be applied. The null costs are achieved for a magnitude in case (b) smaller than that of case (a), due to the removal of one important seismic event in the local dataset. The (c) case is different. There one can appreciate the relevant capacity of the UL in representing the data. In fact the zeroing of the costs occurs near magnitude 6.5, which is the real value of the highest registered earthquake. To conclude, the definition of economic costs performed over the original sample (see Figures <ref> and <ref>, panel (a)) can be reasonably considered valid because they coherently represent the logic of the phenomena that we are studying. Furthermore, the implemented selection of the local dataset does not change the substance of the findings, hence supporting the negligibility of space effects in the considered sample (see De Natale et al. (1988)). Furthermore, results are robust also in terms of the catalog incompleteness problem, in that taking magnitudes not smaller than 3.1 and 2.5 has a very weak effect on the total cost aggregation. § CONCLUSIONS This paper deals with a rank-size analysis of earthquakes' magnitudes occurred in Italy from24^th January, 2016 to 24^th January, 2017. Two different fit functions are proposed: the ZML (see Eq <ref>) and the UL (see Eq. <ref>). It is shown that the earthquakes data exhibit a strong rank-size regularity and that the both functions exhibit a remarkable goodness of fit.The five parameters UL (<ref>) improves the fit – even if in a not so significant way – only when an enlargement in time and magnitude of the dataset is implemented. In this case, UL is more capable than ZML to capture the effect of higher earthquakes. To e consistent under a seismological perspective, both problems of incomplete catalog and of space effects have been treated. Moreover, a new formulation of economic cost indicators has been introduced. Such a conceptualization moves from the presence of a critical threshold for the magnitude which distinguishes earthquakes in terms of damages. The definition of economic costs performed over the original sample (see Figures <ref> and <ref>, panel (a)) can be reasonably considered valid because they coherently represent the logic of the phenomena that we are studying. Furthermore, the implemented selection of the local dataset does not change the substance of the findings, hence supporting the negligibility of space effects in the considered sample (see De Natale et al. (1988)). Results are robust also in terms of the catalog incompleteness problem, in that taking magnitudes not smaller than 3.1 and 2.5 has a very weak effect on the total cost aggregation. The analysis of the cost indicators explains clearly that the reduction of the earthquakes' impact on infrastructures should be pursue by letting the no-damages magnitude growing (see Figures <ref>, <ref> and <ref>). More than this, the discussion of three different scenarios for the individual cost of an earthquake with a given magnitude illustrates also the way in which such a reduction takes place. 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Human Behavior and the Principle of Least Effort. | http://arxiv.org/abs/1708.06735v1 | {
"authors": [
"Valerio Ficcadenti",
"Roy Cerqueti"
],
"categories": [
"physics.soc-ph",
"cond-mat.stat-mech"
],
"primary_category": "physics.soc-ph",
"published": "20170626122851",
"title": "Earthquakes economic costs through rank-size laws"
} |
http://arxiv.org/abs/1706.08726v1 | {
"authors": [
"Gergely Markó",
"Urko Reinosa",
"Zsolt Szép"
],
"categories": [
"hep-ph",
"hep-th"
],
"primary_category": "hep-ph",
"published": "20170627084613",
"title": "Padé approximants and analytic continuation of Euclidean Phi-derivable approximations"
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|
[Electronic address: ][email protected] Institute of Science and Technology Austria, 3400 Klosterneuburg, AustriaNational Aerospace Centre, Luqa LQA 9023, Malta[Electronic address: ][email protected] Department of Physics, University of Malta, Msida MSD 2080, Malta There has been significant interest recently in using complex quantum systems to create effective non-reciprocal dynamics. Proposals have been put forward for the realization of artificial magnetic fields for photons and phonons; experimental progress is fast making these proposals a reality. Much work has concentrated on the use of such systems for controlling the flow of signals, e.g., to create isolators or directional amplifiers for optical signals. In this paper, we build on this work but move in a different direction. We develop the theory of and discuss a potential realization for the controllable flow of thermal noise in quantum systems. We demonstrate theoretically that the unidirectional flow of thermal noise is possible within quantum cascaded systems. Viewing an optomechanical platform as a cascaded system we here that one can ultimately control the direction of the flow of thermal noise. By appropriately engineering the mechanical resonator, which acts as an artificial reservoir, the flow of thermal noise can be constrained to a desired direction, yielding a thermal rectifier. The proposed quantum thermal noise rectifier could potentially be used to develop devices such as a thermal modulator, a thermal router, and a thermal amplifier for nanoelectronic devices and superconducting circuits.Manipulating the flow of thermal noise in quantum devices André Xuereb December 30, 2023 =========================================================§ INTRODUCTIONThe control of thermal noise in complex systems has straightforward applications to the miniaturization of technology; as devices become smaller and smaller it is essential to steer thermal noise away from hot spots towards sinks where it may be disposed of (see, e.g., Ref. <cit.>). Recently a significant effort has emerged that is devoted to design a new generation of thermal-based nanoscale devices such as thermal rectifiers <cit.>, thermal logic gates <cit.>, thermal diodes <cit.>, and thermal transistors <cit.>. When quantum systems are coupled together, the thermal noise associated with the reduced state of each component is affected by the coupling, leading to a flow a thermal noise (see Appendix); controlling this thermal noise is essential in the context of quantum technologies, such as quantum computers <cit.> and simulators <cit.>, especially because of the fragility of quantum states and quantum correlations <cit.> that is well-known from the literature. Coupled quantum systems can also be used to transfer signals; a signal input to one quantum system can appear at the output of another (see Appendix). A basic building block for controlling how such signals flow around a complex system takes the form of devices that are non-reciprocal, in which transmission of a signal from one point to another is qualitatively different <cit.>, in amplitude or phase, from transmission in the reverse direction. An interesting line of research has emerged recently that aims to use complex mechanical, electromagnetic, or quantum-optical systems to create effective optical isolators <cit.> or other kinds of non-reciprocity <cit.>. Several theoretical analyses <cit.> of such systems have been published and experimental demonstrations <cit.> reported, illustrating a rich variety of mechanisms for achieving the desired non-reciprocity. In their simplest form, several such mechanisms are based on coupled quantum systems that also share a common bath <cit.>. These can be conceptually connected to techniques discussed several years ago under the guise of cascaded quantum systems <cit.>.In this paper we will combine cascaded quantum systems, non-reciprocal devices, and controlling the flow of thermal noise to achieve a thermal rectifier. We analyze a quantum system consisting of two fields between which we set up non-reciprocal transport. Our analysis differs from what is known in the literature because we are interested not in the transport of signals, but in the transport of thermal noise between the two fields. We also use recently-developed techniques <cit.> for analysing the flow of excitations between quantum systems and their heat baths to better understand how our system manipulates the flow of thermal noise. Our work thus considers thermal noise not as a nuisance complicating our analysis, but as the object of that analysis. Our results show that the temperature of a third bath can be used to increase or decrease the thermal noise of one system without affecting the other, paving the way to quantum thermal transistors.We proceed as follows. First, we describe an effective quantum optics model based on the cascaded quantum systems formalism [Model(a)]. This yields general expressions that have a transparent physical meaning. We then develop an optomechanical model where a mechanical oscillator is coupled to two electromagnetic cavities [Model(b,c)]. We can show that these two systems behave identically with respect to quantum states that are broadband compared to coherent signals but still contained within the bandwidth of the mechanical oscillator. This allows us to use our general expressions to derive conclusions about this specific system. We then discuss how the effect we explore manifests itself in experiment, with reference to achievable parameters. Finally, we conclude with a short discussion on the implications of our model.§ EFFECTIVE QUANTUM OPTICS MODELTo start off, we briefly summarize what is known about cascaded quantum systems; our aim is to build a model for the system shown in Model(a) to discuss its operation as a non-reciprocal thermal device. We start by considering two harmonic oscillators, associated with annihilation operators ĉ_1 and ĉ_2, respectively. Let Ĥ_sys govern the free dynamics of these systems. We impose cascaded dynamics (see Sec. 12.1 of Ref. <cit.>) onto these systems, i.e., we assume that the output of oscillator 1 forms the input of oscillator 2 via some channel, whereas the output of oscillator 2 does not feed back into oscillator 1. Define γ_1>0 and γ_2>0 as the coupling rates of the two systems, respectively, to this channel. Together with the two standard input–output relations ĉ_out,i=ĉ_in,i+√(γ_i)ĉ_i, where i=1,2, we must therefore add the restriction ĉ_in,2=ĉ_out,1. Next we can follow Ref. <cit.> in obtaining the Langevin equations governing the dynamics of this system, converting them into Itō stochastic differential equations, and from there deriving a master equation. In the following we denote by N̅_3 the average occupation number of an effective common bath, and we take there to be no classical driving field associated with this bath. This master equation can be rewritten in Lindblad form to yield ρ̇=-ħ[Ĥ_sys+Ĥ_hop,ρ]+(N̅_3+1)κ_3𝒟_ĉ_3[ρ]+N̅_3κ_3𝒟_ĉ_3^†[ρ], where 𝒟_ĉ[ρ]=ĉρĉ^†-12{ρ,ĉ^†ĉ} is the standard dissipative Lindblad term, Ĥ_hop=ħ2√(γ_1γ_2)(ĉ_1^†ĉ_2-ĉ_1ĉ_2^†) is a hopping Hamiltonian, κ_3=γ_1+γ_2 is a collective damping rate, and ĉ_3=(√(γ_1)ĉ_1+√(γ_2)ĉ_2)/√(κ_3) is a collective bosonic annihilation operator that obeys [ĉ_3,ĉ_3^†]=1. The physical content of this master equation is rather straightforward: To produce the non-reciprocal effect required of a cascaded system, the two oscillators must be coupled by a direct coherent hopping term as well as to a common bath; see Model(b). To account for an arbitrary phase ϕ in the hopping between the two oscillators, we replace ĉ_2→ e^ϕĉ_2 throughout, yielding Ĥ_hop=ħ2√(γ_1γ_2)(e^ϕĉ_1^†ĉ_2-e^-ϕĉ_1ĉ_2^†) and ĉ_3=(√(γ_1)ĉ_1+√(γ_2)e^ϕĉ_2)/√(κ_3). This master equation results in equations of motion that are maximally non-reciprocal with respect to ĉ_1 and ĉ_2, which is due to a coherent cancellation (addition) of the hopping between the direct term and through the bath in the direction 2→1 (1→2). The phase-matching condition required to ensure this cancellation or addition is encoded in a - sign in the coherent hopping Hamiltonian, compared to a + sign in the dissipation-related operator ĉ_3. For further generality, we must add terms to this master equation. First, we modify the hopping Hamiltonian to Ĥ_hop=ħ2√(γ_1γ_2)(e^ϕĉ_1^†ĉ_2-e^-ϕĉ_1ĉ_2^†)+ħ(Fĉ_1^†ĉ_2+F^∗ĉ_1ĉ_2^†), where F is an arbitrary complex constant; full non-reciprocity requires F=0. Second, we add a bath for each oscillator:ρ̇=-ħ[Ĥ_sys+Ĥ_hop,ρ]+∑_i=1^3{(N̅_i+1)κ_i𝒟_ĉ_i[ρ]+N̅_iκ_i𝒟_ĉ_i^†[ρ]}.In the following we will use this master equation to describe any system composed of two oscillators that are coupled directly to one another, to a common thermal bath, and to two individual thermal baths [Model(c)]. We will show that an effective model where the coupling between two electromagnetic cavities and their common bath are induced by a third, mechanical, mode is equivalent to the one described here.To proceed, we convert the master equation to its equivalent quantum Langevin equations (see Appendix): We derive the mean-field equations of motion from FullME, obtain the operator equations by adding noise terms using the fluctuation–dissipation theorem, and then Fourier-transform to the frequency domain:-ω[ ĉ_1; ĉ_2 ] = [-ω_1-γ_1+κ_12- F; - F^∗-√(γ_1γ_2)e^ϕ-ω_2-γ_2+κ_22 ][ ĉ_1; ĉ_2 ]+[ √(κ_1)ĉ_in,1; √(κ_2)ĉ_in,2 ]+[√(γ_1); √(γ_2)e^ϕ; ]ĉ_in,3.Under the white-noise assumption, these zero-mean noise operators are such that ⟨ĉ_in,i(t)ĉ_in,j^†(t^')⟩=(N̅_i+1)δ_i,jδ(t-t^'), ⟨ĉ_in,i^†(t)ĉ_in,j(t^')⟩=N̅_iδ_i,jδ(t-t^'), and ⟨ĉ_in,i(t)ĉ_in,j(t^')⟩=0 (i,j=1,2,3). Since Langevin is a linear system of equations, a full description of the state at any point in time requires only the first and second moments of the quadrature operators x̂_i=(ĉ_i+ĉ_i^†)/√(2) and p̂_i=-(ĉ_i-ĉ_i^†)/√(2) (i=1,2). It can be shown that the covariance matrix V of this system obeys the Lyapunov equation V̇=A· V+V· A^T+N, where the drift matrix A is related to the matrix in the first term of Langevin and the noise matrix N is related to the second and third terms of this same equation. When the eigenvalues of A all have negative real parts, a unique solution to V exists. In our case, we define n̅_i=⟨ĉ^†_iĉ_i⟩ and Δ:=ω_2-ω_1, and simplify our expressions by taking κ_1=κ_2=γ_1=γ_2=:κ. We want to compare our system to one in which the two oscillators lack any direct coupling or common bath. Simply removing the common bath and the link between the oscillators fundamentally alters the nature of the system, as it changes the number of baths each oscillator is connected to. For a physically meaningful comparison we must modify the bath parameters appropriately. In this disconnected scenario, which is physically equivalent to taking |Δ|→∞ in the above expressions whilst keeping F, κ, and N̅_i (i=1,2,3) fixed, the steady-state occupation numbers are m̅_i=12(N̅_i+N̅_3) (i=1,2,3); note that the m̅_i are independent of F and that m̅_3=N̅_3. Define Δ n_i:=n̅_i-m̅_i (i=1,2) to quantify the difference between the two scenarios, whose explicit expressions we reproduce elsewhere (see Appendix). For simplicity let us look at the maximally non-reciprocal case (F=0), whereby Δ n_1=0 and Δ n_2=2κ^24κ^2+Δ^2(m̅_1-m̅_3).This very clearly shows that, whatever the value of m̅_1-m̅_2, we find an increase (decrease) in n̅_2 over the disconnected case for m̅_1>m̅_3 (m̅_1<m̅_3), whereas n̅_1 is unaffected by the presence of the other oscillator. It is interesting to note that this conclusion remains unchanged if we have m̅_2=m̅_1. In other words, even if the two oscillators equilibrate to the same temperature in the disconnected case, the channel will cause an excess or depleted the flow of thermal noise towards oscillator 2 that depends only on the temperature difference between oscillators 1 and 3. Dn2 shows that the temperature of the common bath can be used as a control knob to modulate the flow of thermal noise into or out of the second oscillator. Note that, for F=0 the temperature of the first oscillator is unaltered. The temperature of the second oscillator can be lower (blue), the same (green), or higher (red) in comparison to the disconnected scenario depending on the temperature of the common bath and Δ which, e.g., can be chosen to reduce the flow of thermal noise into oscillator 2 even when all coherent signals flow from oscillator 1 to 2. The case for F≠0 is shown in Fig. S.1 of the Appendix. Regardless of the temperature difference between the two oscillators and the direction of signal flow, the thermal noise flowing into the second oscillator can be increased or decreased.We next turn our attention to an experimentally-feasible optomechanical platform that can realize this model. We shall use terminology related to platforms operating in optical domain, but all our results hold identically for microwave-based systems. Our results are important for interfacing with such systems, since the thermal occupation of the electromagnetic field at microwave frequencies is often non-negligible.§ OPTOMECHANICAL REALIZATIONOur aim in this section is to employ a mechanical degree of freedom interacting with two optical fields, acting as a controllable reservoir. The result is an optomechanical system that works as a thermal rectifier, with the temperature of the mechanical oscillator bath controlling the steady-state temperature of the second optical field. A schematic realization of this optomechanical system is sketched in Model(c). Here we consider an optomechanical platform consisting of two optical cavities with resonance frequencies ω_i (i=1,2), which interact simultaneously with a mechanical resonator with frequency ω_m, and where the single-photon optomechanical coupling strength between the oscillator and the ith cavity is g_i (i=1,2). The direct photon hopping rate between the cavities is denoted by J, which is assumed real for simplicity. The Hamiltonian governing the unitary evolution of this system is given by <cit.>Ĥ=ħω_mb̂^†b̂+∑_i=1,2ħ[ω_i â^†_iâ_i+g_i(b̂+b̂^†)â^†_iâ_i]+ħ J(â_1â_2^†+â_1^†â_2)+∑_i=1,2ħℰ_i(â_ie^-ω_dt+h.c.),where â_i (with [â_i,â_j^†]=δ_ij) are the annihilation operators of the cavity fields and b̂ is the mechanical annihilation operator. The first and second terms of OPMHamiltonian describe the free Hamiltonians of the mechanical and cavity fields, respectively; the third term indicates the optomechanical coupling between the cavities and the mechanical resonator; and the fourth term shows the cavity–cavity photon hopping. The last term represents the driving of each cavity i by a coherent electromagnetic field with frequency ω_d, which we assume to be the same for both cavities, and amplitude ℰ_i. We note that our analysis also applies to systems where two mechanical modes are used to generate non-reciprocal coupling between two electromagnetic cavities. Recent realizations of such systems <cit.> illustrate the feasibility of implementing non-reciprocal transport of thermal noise and signals.In a rotating frame with respect to ω_d, and after adding losses by means of dissipative Lindblad terms as in the preceding section, the dynamics of the system can be fully characterized by the quantum Langevin equations of motion (i=1,2) â̇_i =-(Δ_i+κ_i2)â_i- Jâ_i̅- g_iâ_i(b̂+b̂^†)+ℰ_i+√(κ_i)â_in,i, ḃ̂̇ =-(ω_m+γ_m2)b̂-∑_i=1,2g_iâ_i^†â_i+√(γ_m)b_in,m,where Δ_i:=ω_i-ω_d, 1̅=2, and 2̅=1. Here, κ_i:=κ_int,i+κ_ext,i are the linewidths of the cavities in which κ_int,i and κ_ext,i are the intrinsic and extrinsic linewidths, respectively. Intrinsic losses and input quantum noise are associated with the zero-mean noise operators â_int,i and â_ext,i, respectively; we can conveniently define â_in,i:=(√(κ_ext,i)â_ext,i+√(κ_int,i)â_int,i)/√(κ_i). The damping of the mechanical resonator is given by γ_m. The zero-mean quantum fluctuations â_in,i and b̂_in,m satisfy the usual white noise correlations (see Appendix). QLE can be solved linearization around the classical steady state of the system. We define the zero-mean cavity field fluctuation operators δâ_i:=â_i-α_i where α_i=2ℰ_ie^ϕ_i/√(4Δ_i^2+κ_i^2) are the steady-state solutions, ignoring a small change in Δ_i due to a static shift in the position of the mechanical oscillator, and assuming |α_i|≫1. If the driving frequencies are chosen such that Δ_i≈ω_m and the system is in the sideband-resolved regime, i.e., ω_m≫κ_i, it is possible to use the rotating-wave approximation to drop the rapidly-rotating terms oscillating at ±ω_m. This allows to eliminate the mechanical degree of freedom, whereby the equations can be approximated in the frequency domain by -ω[ δâ_1; δâ_2 ]=[ -Δ_1-κ_1/2-G_1^2χ_m(ω) - J-χ_m(ω)G_1G_2e^-ϕ;- J-χ_m(ω)G_1G_2e^ϕ -Δ_2-κ_2/2-G_2^2χ_m(ω) ][ δâ_1; δâ_2 ]+[ √(κ_1)â_in,1; √(κ_2)â_in,2 ]+[G_1√(γ_m)χ̃_m(ω); G_2√(γ_m)χ̃_m(ω)e^ϕ ]b̂_in,m where G_i=g_iα_i is the effective optomechanical coupling rate and the mechanical susceptibility is defined as χ_m(ω)=1/[γ_m/2-(ω-ω_m)]. To simplify matters, we chose the phase reference such that G_1 is real and set G_2→ G_2e^ϕ (where the G_2 on the right-hand side is real). We also defined χ̃_m(ω):=χ_m(ω)|χ_m(Ω)|/χ_m(Ω), where Ω is some frequency of interest. This procedure detailed elsewhere (see Appendix).OMfreqdom reveals that in general the photon hopping between cavities is not symmetric—note that the off-diagonal terms of the drift matrix on the right-hand side of the equation are not complex conjugates of one another. This means that by properly choosing the system parameters one can break the reversibility of the thermal photon hopping between the cavities and set up a preferred direction for the flow of thermal noise. For example, a situation of full non-reciprocity at frequency Ω, where the photon hopping is entirely suppressed in the direction 2→1, may be obtained by choosing the parameters such that J=χ_m(Ω)G_1G_2e^-ϕ.Consider, now, a quantum state centered around frequency Ω in the rotating frame and whose bandwidth Γ is much smaller than γ_m, such that χ_m(ω)=χ̃_m(ω)≈χ_m(Ω), constant over the bandwidth of the signal. Under these `large bandwidth' conditions, when γ_m≫Γ, all the parameters entering OMfreqdom can be held constant, and this equation therefore becomes identical to Langevin, with the following replacements: ω_i→Δ_i+G_i^2{χ_m(Ω)}, γ_i→2G_i^2{χ_m(Ω)}, and F→ J-χ_m(Ω)G_1G_2e^-ϕ. For example, perfect non-reciprocity requires J=G_1G_2[(γ_m2)^2+(Ω-ω_m)^2]^-1/2, with ϕ chosen such that F=0. A detailed discussion of the equivalence between the two systems is presented elsewhere (see Appendix). We can therefore apply the formalism developed previously to conclude that any thermal noise in the signal will be suppressed in one direction only. By manipulating the properties of the mechanical oscillator, e.g., using an auxiliary optical field, one may control the flow of thermal energy in the electromagnetic signal transmitted between the two cavities. An in-depth analysis (see Appendix) may be performed to derive the flow of excitations between the system and the three baths it is connected to. Figure S.2 in the Appendix shows that changing the temperature of either resonator does not affect the flow of excitations between the other resonator and its own bath. Any excess flow between the resonators is therefore borne exclusively by their common bath and the link between them. The net flow, given by the sum of the flows to all baths, is shown to be equal to zero, as required for physical consistency.This proposed thermal rectifier can be implemented using an on-chip microwave electromechanical system based on a lumped-element superconducting circuit with a drumhead capacitor <cit.> or a dielectric nanostring mechanical resonator <cit.>. We assume the following experimentally feasible parameters: Optomechanical coupling rates of G_1=G_2=2π×7 kHz, cavities resonant at 2π×5 GHz and having damping rates of κ_1=κ_2=2π×2 MHz, mechanical resonance frequency ω_m=2π× 6 MHz and damping rate γ_m=2π×100 Hz. Inductive or capacitive coupling between microwave resonators can yield J=2π×1 MHz. An auxiliary cavity can be used to change the isolation bandwidth γ_m. The ambient temperature of the microwave and mechanical resonators can be kept below 10 mK by using cryogen-free dilution refrigerators. Optomechanical cooling can be used to cool the mechanical resonator down to ca. 0.5 phonons (260 K). For these parameters the temperature of resonator 2 is lower with respect to the disconnected case, and depends linearly on that of resonator 1. Furthermore, the temperature of resonator 1 is independent of that of resonator 2.§ CONCLUSIONSWe have investigated a generic framework to describe non-reciprocal transport in compound quantum systems. In contrast to several previous studies, we chose to concentrate on the transport of thermal states rather than coherent signals. Our framework can easily be mapped to a prototypical optomechanical realization, which we discussed explicitly in the text. We have also shown how, with parameters typical of present-day microwave optomechanical experiments, the effects we describe should be visible in a proof-of-concept experiment. In the context of quantum measurements and emerging quantum technologies, these techniques and ideas will find use in the manipulation of flow of thermal noise inside quantum devices for phonon-based signal processing and computation, as well as in the construction of quantum-limited amplification systems that perform measurements on sensitive quantum devices without adding thermal noise. Our system can be realized with state-of-the-art technology both in optical <cit.> and microwave <cit.> domains, and is potentially suited to control the flow of thermal noise in nanoscale devices and to design a new generation of thermal rectifiers, thermal diodes, and transistors. Our work could facilitate noise control and remote cooling of nanoelectronic devices and superconducting circuits using in situ-engineerable thermal sinks with possible applications in emerging quantum technologies such as quantum computers and simulators.§ ACKNOWLEDGMENTSWe acknowledge funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 732894 (FETPRO HOT). SB acknowledges support under the Marie Skłodowska-Curie Actions programme, grant agreement No. 707438 (MSCA-IF-EF-ST SUPEREOM).§ DEFINING TEMPERATURE, FLOW OF THERMAL NOISE, AND SIGNAL FLOWOne can define a temperature of a harmonic oscillator as the temperature of the equivalent thermal state. More precisely, what we call “temperature” is the temperature of the heat bath with which a harmonic oscillator reaches equilibrium, such that its equilibrium state is a thermal state with a given mean occupation number. Symbolically, we define the thermal state at temperature T for a harmonic oscillator at frequency ω through the relationρ(T):=e^-ħωâ^†â/(k_BT)/{e^-ħωâ^†â/(k_BT)},where â is the annihilation operator for the oscillator and k_B is Boltzmann's constant. Such a state can uniquely be parametrised either through the temperature T or through its mean occupation numbern̅:={â^†â ρ(T)}=1/e^ħω/(k_BT)-1.Since this relation is one-to-one there is no ambiguity in using either quantity. When the state ρ of a harmonic oscillator is such that ρ=ρ(T) for some T≥0 we refer to this value of T as the temperature of the oscillator. We restrict our discussion to T≥0, which implies that n̅≥0. The mean occupation number of a thermal state is associated with the variance of its quadratures x̂=(â+â^†)/√(2) and p̂=(â-â^†)/(√(2)):√(⟨x̂^2⟩-⟨x̂⟩^2)=√(⟨p̂^2⟩-⟨p̂⟩^2)=n̅+12.This relation between the variances and the mean occupation number allows us to use n̅ as a proxy for the thermal noise in the state of the oscillator. When the oscillator is coupled to an output channel, this thermal noise can be observed as noise in the output signal, with amplitude that increases monotonically with n̅.Given two coupled oscillators, each connected to their own heat bath, the flow of thermal noise can be defined qualitatively through its effect on the average occupation number of the reduced thermal states of the two oscillators. In this simple model, the average occupation number of the thermal state describing the oscillator with the cooler bath increases when the two oscillators are coupled; this can be described as a flow of thermal noise to this oscillator. This definition, which has a straightforward physical interpretation, forms the basis of our work.One can drive either of the oscillators using a monochromatic force, which can be called a coherent “signal,” and monitor how that drive affects the state of the other oscillator. In the language of quantum optics, this is equivalent to displacing the state of one oscillator and seeing how that translates to a displacement of the other oscillator. To give this model a more physically-relevant foundation, let us introduce one input–output channels associated with each oscillator; our heat baths can also serve this purpose. One can derive the relevant input–output relations that quantify how a coherent signal in an input channel connected to one oscillator is transferred to an output channel connected to the other oscillator. We refer to this process as signal flow.§ THEORETICAL BACKGROUND OF OPTOMECHANICAL SYSTEMSIn this section, we explore the theoretical model describing the optomechanical system discussed in the main text. A schematic of the optomechanical thermal noise rectifier is sketched in Fig. 1(c) of the main text, where we consider an optomechanical cavity platform in the form of two cavities, with resonance frequencies ω_i (i=1,2), which interact simultaneously with a mechanical resonator with frequency ω_m in which the single-photon optomechanical coupling rate for the interaction between cavity i and the mechanical element is given by g_i. The direct photon hopping rate between the cavities is defined by J. The Hamiltonian of the system is given by Ĥ=ħω_mb̂^†b̂+∑_i=1,2ħ[ω_i â^†_iâ_i+g_i(b̂+b̂^†)â^†_iâ_i] +ħ J(â_1â_2^†+â_1^†â_2)+∑_i=1,2ħℰ_i(â_ie^-ω_d,it+h.c.),where â_i (with [â_i,â_j^†]=δ_i,j) is the annihilation operator of cavity i=1,2, and b̂ is the mechanical annihilation operator. The first and second terms of the Hamiltonian (<ref>) show the free energy of the mechanical and cavity fields, respectively, while the third term indicates the the optomechanical coupling between the cavities and mechanical resonator. The fourth term stands for the direct cavity–cavity interaction Hamiltonian, where photon hopping occurs with rate J. The last term shows that each cavity is driven by a coherent external source with amplitude ℰ_i and frequency ω_d,i. In the rotating frame with respect to the drive frequencies, the above Hamiltonian reduces toĤ=ħω_mb̂^†b̂+∑_i=1,2ħ[Δ_i â^†_iâ_i+g_i(b̂+b̂^†)â^†_iâ_i] +ħ J(â_1â_2^†+â_1^†â_2)+∑_i=1,2ħℰ_i(â_i+h.c.),where Δ_i:=ω_i-ω_d,i, and for simplicity we have assumed ω_d,1=ω_d,2.After adding losses by means of dissipative Lindblad terms, as in the preceding section, the dynamics of the system can be fully characterized by the quantum Langevin equations of motion <cit.> â̇_1 =-(Δ_1+κ_12)â_1- Jâ_2- g_1â_1(b̂+b̂^†)+ℰ_1+√(κ_1)â_in,1, â̇_2 =-(Δ_2+κ_22)â_2- Jâ_1- g_2â_2(b̂+b̂^†)+ℰ_2+√(κ_2)â_in,2, and ḃ̂̇ =-(ω_m+γ_m2)b̂-∑_i=1,2g_iâ_i^†â_i+√(γ_m)b_in,m,where κ_i:=κ_int,i+κ_ext,i are the linewidths of the cavities in which κ_int,i and κ_ext,i are the intrinsic and extrinsic linewidths, respectively. Intrinsic losses and input quantum noise are associated with the uncorrelated zero-mean noise operators â_int,i and â_ext,i, respectively, whereby we can conveniently define â_in,i:=(√(κ_ext,i)â_ext,i+√(κ_int,i)â_int,i)/√(κ_i). The damping of the mechanical resonator is given by γ_m. The zero-mean quantum fluctuations â_in,i and b̂_in,m satisfy the correlations ⟨Ô_in,i(t)Ô_in,i^†(t^')⟩=(N̅_i+1)δ(t-t^'), ⟨Ô_in,i^†(t)Ô_in,i(t^')⟩=N̅_iδ(t-t^'), and ⟨Ô_in,i(t)Ô_in,i(t^')⟩=0 where i=1,2 for Ô=â, and i=m for Ô=b̂; N̅_i=1/{exp[ħω_i/(k_BT_i)]-1} are the thermal photon (phonon) occupancies of the cavities (mechanical resonator) for i=1,2 (i=m) at temperature T_i. For i=1 and 2, N̅_i can be obtained from a weighted sum of the mean occupation numbers associated with â_int,i and â_ext,i.In the strong-drive regime one can linearize these equations around the classical steady state of the cavities. We define δâ_i:=â_i-α_i where α_i=2ℰ_ie^ϕ_i/√(4Δ_i^2+κ_i^2) are the steady-state solutions, ignoring a small change in Δ_i due to a static shift in the position of the mechanical oscillator. These δâ_i are the cavity field fluctuation operators, and under the linearization approximation have zero mean. When |α_i|≫1, QLEStart can be approximated by δ̇â̇_1 =-(Δ_1+κ_12)δâ_1- Jδâ_2- G_1(b̂+b̂^†)+√(κ_1)â_in,1, δ̇â̇_2 =-(Δ_2+κ_22)δâ_2- Jδâ_1- G_2(b̂+b̂^†)+√(κ_2)â_in,2, and δ̇ḃ̂̇ =-(ω_m+γ_m2)b̂-(G_1^∗δâ_1+G_2^∗δâ_2+h.c.)+√(γ_m)b_in,m,where G_i=g_iα_i are the multi-photon optomechanical coupling rates.We choose driving frequencies such that Δ_i=ω_m. In addition, we assume operation in the sideband-resolved regime, i.e., ω_m≫κ_i. Under these conditions around cavity resonance frequency it is possible to use the so-called rotating-wave approximation to simplify QLELin. This allows to eliminate δb̂ from the equations, whereby they can be approximated in the frequency domain by-ω[ δâ_1; δâ_2 ]=[ -Δ_1-κ_1/2-| G_1|^2χ_m(ω)- J-χ_m(ω)G_1G_2^∗; - J-χ_m(ω)G_1^∗ G_2 -Δ_2-κ_2/2-| G_2|^2χ_m(ω) ][ δâ_1; δâ_2 ]+[ √(κ_1)â_in,1; √(κ_2)â_in,2 ]+[ - G_1√(γ_m)χ_m(ω); - G_2√(γ_m)χ_m(ω) ]b̂_in,mwhere the mechanical susceptibility is defined as χ_m(ω)=1/[γ_m/2-(ω-ω_m)].§ CONVERSION BETWEEN CASCADED AND OPTOMECHANICAL FORMALISMSIn this section we give more detail on the equivalence between the two different sets of Langevin equations. Let us start by listing the two sets of expressions in frequency space. First, the optical cavities of the optomechanical system can, under the rotating-wave approximation, be described by:-ω[ δâ_1; δâ_2 ] = [ -Δ_1-κ_12-G_1^2χ_m(ω)- J-G_1G_2χ_m(ω)e^-ϕ; - J-G_1G_2χ_m(ω)e^ϕ -Δ_2-κ_22-G_2^2χ_m(ω) ][ δâ_1; δâ_2 ]+[ √(κ_1)â_in,1; √(κ_1)â_in,2 ]+[- G_1√(γ_m)χ_m(ω); - G_2√(γ_m)χ_m(ω)e^ϕ ]b̂_in,m,where G_1 is assumed to be real and non-negative and the replacement G_2→ G_2e^ϕ, with real and non-negative G_2 on the right-hand side, has already been effected. We can also perform the simple gauge transformation b̂_in,m→ e^-νb̂_in,m, where ν={χ_m(Ω)} and Ω is some fixed frequency of interest, to write-ω[ δâ_1; δâ_2 ] = [ -Δ_1-κ_12-G_1^2χ_m(ω)- J-G_1G_2χ_m(ω)e^-ϕ; - J-G_1G_2χ_m(ω)e^ϕ -Δ_2-κ_22-G_2^2χ_m(ω) ][ δâ_1; δâ_2 ]+[ √(κ_1)â_in,1; √(κ_1)â_in,2 ]+[ G_1√(γ_m)χ̃_m(ω); G_2√(γ_m)χ̃_m(ω) e^ϕ ]b̂_in,m,defining χ̃_m(ω):=χ_m(ω)|χ_m(Ω)|/χ_m(Ω)=e^-νχ_m(ω).Next, we write the cascaded system equations in frequency space. This procedure assumes that the various coefficients that appear are not time- or frequency-dependent. For generality, we shall use θ as the phase angle between ĉ_1 and ĉ_2. Thus,-ω[ ĉ_1; ĉ_2 ] = [-ω_1-γ_1+κ_12- F; - F^∗-√(γ_1γ_2)e^θ-ω_2-γ_2+κ_22 ][ ĉ_1; ĉ_2 ]+[ √(κ_1)ĉ_in,1; √(κ_2)ĉ_in,2 ]+[√(γ_1); √(γ_2)e^θ; ]ĉ_in,3.Let us compare the two sets of expressions to obtain an equivalence; we shall, for the time being, ignore the frequency-dependence of χ_m(ω). First, it is clear thatω_i = Δ_i+G_i^2{χ_m(ω)}(i=1,2).From the first term on the right-hand side of each expression, we can also deduce thatγ_i = 2G_i^2{χ_m(ω)}(i=1,2).Next,F=J-χ_m(ω)G_1G_2e^-ϕ.Continuing further, we can deduce two equations for F^∗ that must hold simultaneously: F^∗=J- G_1G_2χ_m(ω)e^ϕ+√(γ_1γ_2)e^θ and F^∗=J+ G_1G_2χ_m^∗(ω)e^ϕ. From these two equations we deduce that√(γ_1γ_2)e^θ=2G_1G_2{χ_m(ω)}e^ϕ.Finally, comparing the second noise term on the right-hand side of the Langevin equations, we find√(γ_1)= G_1√(γ_m)|χ_m(ω)|, and √(γ_2)e^θ= G_2√(γ_m)|χ_m(ω)| e^ϕ.The equations for γ_i (i=1,2) are only equivalent ifθ=ϕ,and2{χ_m(ω)}=γ_m|χ_m(ω)|^2.However, we note that this always holds, sinceχ_m(ω)=1/γ_m2-(ω-ω_m),whereby{χ_m(ω)} =γ_m2/(γ_m2)^2+(ω-ω_m)^2, and |χ_m(ω)|^2 =1/(γ_m2)^2+(ω-ω_m)^2.Incidentally, note that {χ_m(ω)}≥0, which guarantees that the γ_i (i=1,2) are also non-negative. It can now be seen that the two expressions are perfectly equivalent if we ignore the ω-dependence of χ_m(ω). In order to discuss the flow of thermal noise through the system we argue as follows. We are interested in thermal state whose bandwidth Γ is small relative to γ_m, and which is centred at frequency Ω. With this in mind we can now state (i=1,2 throughout):Cascaded system ↔Optomechanical platform ĉ_i↔δâ_i ĉ_in,i ↔â_in,i ĉ_in,3 ↔b̂_in,m ω_i↔Δ_i+G_i^2{χ_m(Ω)}=Δ_i+G_i^2(Ω-ω_m)/(γ_m2)^2+(Ω-ω_m)^2 γ_i↔ 2G_i^2{χ_m(Ω)}=G_i^2γ_m/(γ_m2)^2+(Ω-ω_m)^2 θ ↔ϕF↔ J-χ_m(Ω)G_1G_2e^-ϕ=J- G_1G_2e^-ϕ/γ_m2-(Ω-ω_m)At this stage, note that none of these coefficients depends on ω, as formally required for the expressions derived using the cascaded systems formalism to be valid. It is easy to read off that perfect non-reciprocity requires F=0, i.e.,J=G_1G_2/√((γ_m2)^2+(Ω-ω_m)^2),with ϕ chosen appropriately, constrained by the demand that J is real.In the large-bandwidth limit (γ_m→∞) we obtain a perfect equivalence, since χ_m(ω)=2/γ_m is then no longer a function of frequency. Under these conditions, we can writeCascaded system ↔Optomechanical platform ĉ_i↔δâ_i ĉ_in,i ↔â_in,i ĉ_in,3 ↔b̂_in,m ω_i↔Δ_iγ_i↔4G_i^2/γ_m θ ↔ϕF↔ J-2 G_1G_2e^-ϕ/γ_m § OCCUPATION NUMBERSBased on the assumption that none of the coefficients entering the cascaded system calculation is time- or frequency-dependent, it is relatively straightforward to obtain the steady-state occupation numbers for the two oscillators. First, define Δ:=ω_2-ω_1 for simplicity, and let N̅_1, N̅_2, and N̅_3 be the occupation numbers for the baths defined by b̂_in,1, b̂_in,2, and b̂_in,m, respectively. Furthermore, let us simplify matters by taking κ_1=κ_2=γ_1=γ_2=:κ Then,n̅_1=2| F|^2(N̅_1+N̅_2+N̅_3)+{Fe^θ}Δ(N̅_1-N̅_3)+2{Fe^θ}^2N̅_3+2{Fe^θ}κ(N̅_1+3N̅_3)+(4κ^2+Δ^2)(N̅_1+N̅_3)/2[3| F|^2+4κ(κ+{Fe^θ})+{Fe^θ}^2+Δ^2],andn̅_2=2| F|^2(N̅_1+N̅_2+N̅_3)-{Fe^θ}Δ(N̅_2-N̅_3)+2{Fe^θ}^2N̅_3+2{Fe^θ}κ(N̅_2+3N̅_3)+(4κ^2+Δ^2)(N̅_2+N̅_3)/2[3| F|^2+4κ(κ+{Fe^θ})+{Fe^θ}^2+Δ^2]+κ(2{Fe^θ}+κ)(N̅_1-N̅_3)/3| F|^2+4κ(κ+{Fe^θ})+{Fe^θ}^2+Δ^2.For perfect non-reciprocity we set F=0 and obtainn̅_1=12(N̅_1+N̅_3),andn̅_2=12(N̅_2+N̅_3)+κ^2(N̅_1-N̅_3)/4κ^2+Δ^2.Simplifying the latter further for the resonant case, Δ=0, we obtainn̅_2^(Δ=0)=14(N̅_1+2N̅_2+N̅_3)=12(N̅_2+n̅_1).The seemingly anomalous factor 12 in the expressions for n̅_1 and n̅_2^(Δ=0) comes from the fact that the two oscillators are both connected to a third bath. Indeed, for a fair comparison, we can consider the two oscillators connected to two baths each, but devoid of any direct coupling or common baths. In this (“disconnected”) scenario, which is physically equivalent to taking |Δ|→∞ in the above expressions whilst keeping F, κ, and N̅_i (i=1,2,3) fixed, the steady-state occupation numbers are, instead,m̅_1 =12(N̅_1+N̅_3), and m̅_2 =12(N̅_2+N̅_3).Thus, if we again allow Δ to be general,n̅_1 =m̅_1, and n̅_2 =m̅_2+κ^2(N̅_1-N̅_3)/4κ^2+Δ^2.This very clearly shows that, whatever the value of N̅_1-N̅_2, we find an increase (decrease) in n̅_2 over the disconnected case for N̅_1>N̅_3 (N̅_1<N̅_3), whereas n̅_1 is unaffected by the presence of the other oscillator. It is interesting to note that this conclusion remains unchanged if we have N̅_2=N̅_1.For general F and Δ we findn̅_1-m̅_1 =| F|^2(-N̅_1+2N̅_2-N̅_3)-[{Fe^θ}(2{Fe^θ}+κ)-{Fe^θ}Δ](N̅_1-N̅_3)/2[3| F|^2+4κ(κ+{Fe^θ})+{Fe^θ}^2+Δ^2], and n̅_2-m̅_2 =| F|^2(2N̅_1-N̅_2-N̅_3)-[{Fe^θ}(2{Fe^θ}+κ)+{Fe^θ}Δ](N̅_2-N̅_3)+2κ(2{Fe^θ}+κ)(N̅_1-N̅_3)/2[3| F|^2+4κ(κ+{Fe^θ})+{Fe^θ}^2+Δ^2].We can generalize Fig. 2 in the main text for the case of imperfect non-reciprocity, obtaining Dn. These two figures show the versatility of the system at hand, where the control parameters Δ, F, and m̅_3 can be used to set the temperature difference of either oscillator with respect to the disconnected system. As expected, the first oscillator can never be cooled, but it is indeed possible to cool the second oscillator. This shows that reduced net thermal noise flow can be set up to oscillator 2, despite the fact that, when F=0, all coherent signals flow from oscillator 1 to oscillator 2.§ RATE OF FLOW OF EXCITATIONS INTO AND OUT OF THE BATHSIn this section we will briefly summarize a technique that can be used to obtain knowledge of the full counting statistics of the exchange of excitations between a quantum system and a heat bath. The development of this technique can be traced in recent literature (see Refs. <cit.> and references therein); the focus here is on its application to Gaussian states evolving under the action of dynamics that preserves their Gaussian nature (see Appendix B in Ref. <cit.>). The basis of the technique rests on the definition of a biased covariance matrix V_s, which under steady-state conditions satisfies the relation0=[A-F_-(s)]· V_s+V_s·[A-F_-(s)]^T+V_s· F_+(s)· V_s+N,where A is the drift matrix and N the noise matrix that is obtained from the noise terms entering the corresponding Langevin equations; both are defined in the main text. We define the auxiliary matrices F_±(s) through the relation F_±(s)=⊕_j=1^Nδ_i,j[ f_j±(s) 0; 0 f_j±(s) ],where i=1,2,3 is the noise channel of interest and f_j±(s)=γ_j[(N̅_j+1)(e^-s-1)±N̅_j(e^s-1)], where γ_j is the rate through which the system is coupled to bath j and N̅_j the mean number of excitations of this bath. The ordinary case, where V_s reduces to the usual covariance matrix V, results from taking s=0, whereupon f_j±(0)=0, F_±(0)=0, and the algebraic Riccati equation, RiccatiVs, reduces to the usual steady-state Lyapunov equation, as used the main text. The full counting statistics of the counting process associated with the excitations being exchanged between the system and bath i can be obtained through the large-deviation functionθ(s)=12{F_+(s)· V_s-F_-(s)},with the nth derivative of θ(s) evaluated at s=0 being related to the nth moment of the counting process, η^(n), through the relationη^(n)=(-1)^n[∂_s^nθ(s)|_s=0].For convenience we drop the superscript when referring to the first moment and define η_i (i=1,2,3) to be the first moment of the counting process—i.e., the average rate of flow of excitations—between the system and bath i. A concise expression can be obtained for these first moments that does not make reference to V_s directly:η_i=-12{F_+^'· V-F_-^'},where F_±^' is the first derivative of F_±, evaluated at s=0:F_±^'=⊕_j=1^Nδ_i,j[ f_j±^'0;0 f_j±^' ],with f_j±^'=-γ_j[N̅_j(1∓1)+1]. Applying this procedure to the generic system described in the main text, we obtain rather unwieldy expressions. However, in the simplified situation where the γ_i and κ_i are all equal to κ, and F=0, we find η_1 =κ(n̅_3-n̅_1), η_2 =κ[2κ^2/4κ^2+Δ^2(n̅_1-n̅_3)+(n̅_3-n̅_2)], and η_3 =κ[2κ^2/4κ^2+Δ^2(n̅_3-n̅_1)+(n̅_1-n̅_3)+(n̅_2-n̅_3)], such that ∑_jη_j=0, as required when accounting for all the heat baths connected to a system. Under these simplified conditions we can see that η_1 (η_2) does not depend on N̅_2 (N̅_1). This is shown explicitly in ExcitationFlows for the parameters used in the main text. 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"authors": [
"Shabir Barzanjeh",
"Matteo Aquilina",
"André Xuereb"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170627211607",
"title": "Manipulating the flow of thermal noise in quantum devices"
} |
Rejuvenating Functional Responses with Renewal Theory [ ===================================================== Applying a newly developed tangent-circle method (TCM), we derive a volume density map of HI gas in the inner Galaxy as a function of galacto-centric distance R and height Z. The HI hole around the Galactic Center (GC) is shown to have a crater-shaped wall, which coincides with the brightest ridge of the North Polar Spur and emanates from the 3-kpc expanding ring. The crater structure is explained by sweeping of the halo gas by a shock-wave from the GC. The unperturbed HI halo outside 3 kpc is shown to be in hydrostatic equilibrium, obeying the sech^2 Z/h density law with a scale height h ∼ 450 pc. § INTRODUCTION Among the number of expanding features in the Galactic disk, the most massive object is known as the 3-kpc ring in the HI- and CO-line observations (Cohen and Davies 1976; Bania 1977; Oort 1977 for review; Dame and Thaddeus 2008; Garcia et al. 2014). The near-side arm of the ring is approaching, and therefore, expanding at velocity of ∼ 53 (Cohen and Davies 1976; Oort 1977), and the far-side is receding at ∼ 56 (Dame and Thaddeus 2008). For such a motion the ring has been interpreted as an expanding shock front driven by an explosion at the Galactic Center (GC) (Sanders and Prendergast 1974; Oort 1977; Sofue 1976, 1977; 1984). Coinciding with the ring, a giant HI hole has been found in the halo around theGC, which was interpreted as a void swept by the nuclear wind associated with the Fermi Bubbles (Lockman 1984; Lockman and 2016). As the Milky Way is evidenced to be a barred galaxy, the ring's motion has been more widely believed to be due to an oval flow in the bar potential (Binney et al. 1991; Rodriguez-Fernandez and Combes 2008). On the other hand, the explosion hypothesis has recently been highlighted according to accumulating evidences and arguments for energetic phenomena in the GC such as the high-temperature plasma (Uchiyama et al. 2013), bipolar hyper shells (BHS) in radio and X-rays (Sofue 1994, 2000, 2017;Bland-Hawthorn and Cohen 2003; Sofue et al. 2016; and the literature therein for observations), and the Fermi Bubbles in γ-rays (Su et al. 2010; Bordoloi et al. 2017). Also, if the Galaxy is barred, the activity in the GC is inevitable as the consequence of bar-shocked gas accretion.In this paper, we investigate the relationship of the HI hole discovered by Lockman and (2016) with the 3-kpc HI ring and the North Polar Spur, and discuss their origin based on the GC explosion model.§ RADIO AND HI SPURS Lockman and McClure-Griffiths (2016) showed that the HI hole has a clear wall at R∼ 3 kpc, recognized as a remarkable change of vertical extent in the channel maps at ∼± 130 at tangent longitude of l∼ 20.The integrated intensity (column density) map along the tangent circle showed a bipolar void surrounding the Fermi Bubbles, and they explained the structure as due to galactic wind-driven HI hole. Figure <ref> shows mosaic channel maps at ± 130 produced from the HI Galactic All-Sky Survey (GASS; McClure-Griffiths et al. 2009; Kalberla et al. 2010). Symmetric HI spurs emanate from the tangential directions of the 3-kpc ring at l∼± 20. Using the GASS HI survey data, we measured the brightness of the wall to be T∼ 1-2 K, less bright than the main disk, but the latitudinal extension is as large as b ∼± 10, reaching altitudes Z∼± 1.3 kpc.In the figure we indicate the radio continuum spurs by the dashed lines, which show the North Polar Spur (NPS), NPS-West, South Polar Spur (SPS), and SPS-West as identified by Sofue (2000). The spurs are commonly traced in galactic surveys of the synchrotron radio emission (Haslam et al. 1982; Reich et al. 2001; Planck Collaboration et al. 2016).Figure <ref> enlarges the same at =130 , where we overlay the background-filtered 1420 MHz radio continuum map taken from Sofue and Reich (1979). Also shown is the X-ray ridge in the R7 (1.4 keV) band as obtained from a ROSAT image (Snowden et al. 1997), which coincides with the radio NPS, located slightly inside. The figure demonstrates a remarkable coincidence of the NPS with the 3-kpc HI ring and the wall of the HI hole.§ VERTICAL DENSITY PROFILEFigure <ref> shows vertical (b-directional) cross sections of the HI brightness T at different radial velocities, which show two components with narrow and wide scale heights. We fit the distribution by two gas layers in gravitational equilibrium in the Z direction expressed asρ =ρ_0 sech^2 (Z/h) where h is the hydrostatic scale height of the layer (Spitzer 1942). Accordingly, the brightness temperature of the HI emission at a givenis expressed by T =Σ_i T_isech^2 ( Z/ _i), where T_i and _i are the temperature at the midplane and the scale height of the i-th components, respectively. Here, i=1 and 2 represent the disk and halo components, respectively.By the χ^2 method we determined T_i and _i at each longitude. In the inner region with terminal velocity higher than >150 , the b profiles are fitted by one-disk with small scale height. On the other hand at lower velocities than ∼ 130 , or at larger longitude than l ∼ 20, the b profiles are fitted by two components with narrow and wide scale heights. The midplane temperature and scale heights toward the 3-kpc ring are determined to be T_1=22 K and _1=0.56 (74 pc) for the disk, and T_2=5.4 K and _2=3.2 (450 pc) for the halo component. The halo component is thus well fitted by the sech^2 function, indicating that the gas is in gravitational equilibrium. The tight disk component, having _1 ∼ 74 pc, represents the main HI disk in the current galactic studies (e.g. Nakanishi and Sofue 2003, 2005, 2015), corresponding to a velocity dispersion of σ_1∼ 5-10 .The halo component, showing large scale height of _2 ∼ 450 pc, has an order of magnitude less intensity compared to the disk, and is well represented by the hydrostatic equilibrium profile. In order for the gas to be vertically extended to this height, the velocity dispersion must be as large as σ_2∼ 12-25 . The HI halo can be traced up to b∼± 10 above the detection limit of T∼ 0.01 K, indicating that the outskirt is extending to height of Z∼± 1.3 kpc§ VOLUME DENSITY MAP BY TANGENT-CIRCLE METHOD (TCM)The column density of HI and H_2 gases is related to the velocity integrated intensity of the brightness temperature T_i byN_i=X_i ∫ T_i dv,and to the local volume density n_i byn_i=X_i T_i dvdr,where X_i is the conversion factor for HI (i=1) and CO (i=2). We here consider the gas distribution along the tangent circle, which traces the gas having radial velocities equal to the terminal velocities. Using the relation r=R_0cos l, R=rtan l=R_0 sinl, and dl/dr= 1/R_0sin l at the tangent points, we obtainn_i=X_i T_i (V_0R_0 - dVdR)cot l. We here assume that the HI gas is optically thin, as we are interested in the halo gas. We can thus transform the observed brightness temperature T_i on the LV diagram along the terminal velocity ridge into the local volume density n_i as a function of the radius R. We call this method the tangent-circle method (TCM). The method can avoid the degenerate depth problem at the tangent points when calculating the volume density from the intensity (column density).We applied the TCM to the HI line data from the GASS (McClure-Griffiths et al. 2009; Kalberla et al. 2010), Leiden-Argentine-Bonn (LAB) HI survey (Kalberla et al. 2005), and CO line data from the Columbia galactic plane CO-line survey (Dame et al. 2001). In order to calculate dV/dR, we adopt the most recent Galactic constants (R_0=8 kpc and V_0=238 : Honma et al. 2015) and rotation curve (Sofue 2016). The conversion factors are taken to be X_ HI=1.82× 10^18 H cm^-2 and X_ CO=2.0× 10^20 H_2 cm^-2 =4.0× 10^20 H cm^-2. Figure <ref> shows the distribution map of the H density, n_ H, along the tangent circle in the (l,b) and (R,Z) plane. Thereby, the TCM was applied to the terminal velocity ridge approximated by a straight line in the LV diagram parallel to the tangent ridge around l∼ 20. By this approximation, the map is not accurate at |l|>∼ 40 and |l|<∼ 5. The map is essentially the same as the column density map obtained by Lockman and (2016). However, it gives the volume density distribution, which exhibits a sharper and clearer-cut wall thanks to the pin-pointing nature of the solar-circle gas by the TCM.In figure <ref> we compare the HI density profiles at fixed latitudes with those of the CO-line and radio continuum emissions. Panels (a)-(c) show:(a) Radio continuum excess at 408 (Haslam et al. 1982) and 1420 MHz (Reich et al. 2001) at b=5. The radio excess is defined by =/-1,, whereis the brightness temperature andis the smoothed intensity in an area δ l ×δ b = 5× 1 around each data point. (b) HI density profiles at b=+3, where the full and dashed lines show the results for the GASS and LAB data, respectively. The difference between the two results are due to the difference in the beam widths as well as the noise levels. The increasing scatter toward the GC is due to the cot l effect in equation <ref> on the errors.(c) HI and molecular hydrogen densities along the galactic plane at b=0 from GASS and Columbia CO survey data.The HI crater is now evident in the profiles at b=+3 (∼ 400 pc). It is remarkable that the crater edges exactly coincide with the radio continuum peaks of the NPS and NPS-W. About the same HI profiles were obtained at b∼ 3 - 10 as well as at negative latitudes except for the decreasing density with |b|. The HI density inside the hole is as low as n_ HI≤∼ 5× 10^-3. At the edge of the hole, the intensity suddenly increases to a sharp peak at R=2.9 kpc, making a clear-cut wall. The peak density in the wall is n_ HI∼ 0.07. The full radial width at half-density of the wall is measured to be Δ R ≃ 0.15 kpc in the first quadrant (l≥ 0). The HI profile in the galactic plane (b=0) is similar, but much milder. The intensity peaks representing the 3-kpc ring appear at R∼ 2.8 kpc, coinciding with the HI walls. The peak HI density is n_ HI∼ 2.5, and the full width of the ring is Δ R ≃ 0.3 kpc. It is found that the peak of the 3-kpc ring is more evident at negative longitudes.The molecular disk is clumpy (figure <ref>(a) red lines), while it still exhibits peaks at R∼ 2.7 kpc with density of n_ H_2∼ 13. The molecular fraction (molecular density/total density) in the peaks (wall) is f_ mol≃ 0.8, lower than that in the surrounding regions with f_ mol≃ 0.9, in agreement with the variation of molecular fraction in the inner Galaxy obtained by Sofue and Nakanishi (2016).Using the measured parameters we calculated the total mass of the 3-kpc ring, assuming a perfect circle around the GC with a constant peak density. We also calculated the kinetic energy, assuming that the expansion velocity is V_ expa=53 . The derived quantities are listed in table <ref>.§ A MODEL FOR THE HI CRATERWe now examine if the crater structure in the HI halo can be explained by sweeping of the halo gas by an explosive event in the GC based on the bipolar-hyper-shell (BHS) model of the NPS (Sofue 2000; Sofue et al. 2016). The propagation of a shock wave from the Center is calculated using the Sakashita's (1971) method to trace radial ray paths of an adiabatic shock wave.The unperturbed gas disk and halo are assumed to be composed of stratified layers with density distributions represented by the hydrostatic equilibrium in the Z direction (Spitzer 1942). It is assumed that the disks are further embedded in an intergalactic gas with uniform, low-density gas. We express the density distribution asρ=Σρ_i sech^2 ( Z/_i).Here, i=1, 2, and 3 represent the disk, HI halo, and a constant background. We here take ρ_1=1, ρ_2=0.1, and ρ_3=10^-5 H cm^-3, and_1=50, _2=500 pc, and _3=∞. Figure <ref> shows the calculated result for an initial injection energy E_0=1.8× 10^55 erg. The shock front is drawn in the (R,Z) plane every 1 My up to 10 My. As the shock wave expands, the front shape becomes elongated in the vertical direction due to the steep pressure gradient toward the halo. As the shock wave is blown off into the halo, the front shape gets dumbbell shaped, making a BHS. The dumbbell's equator is sharply pinched by the dense disk at the galactic plane. At elapsed time of t∼ 10 My, the BHS front approximately mimics the NPS, SPS, NPS-W and SPS-W. The expanding velocity of the front at intermediate latitudes b ∼ 10-20 corresponding to the main NPS ridge is ∼ 300 . This velocity is coincident with the required velocity to heat the shocked gas to a temperature ∼ 10^7 K responsible for the observed X-ray emission in the NPS (Snowden et at al. 1997; Sofue et al. 2016). Note, however, the shock-heated gas inside the shock front is no more neutral (HI), but is ionized to X-ray temperatures, and is not observed in the HI line emission. The HI wall outside the shock front is in a pre-shock compression stage, and is observed as the expanding HI ring. The ring's expansion is still slow, as observed to be expanding at ∼ 50 , and approximately obeys the normal galactic rotation.In figure <ref> we schematically summarize the view about the radio and X-ray NPS (BHS), 3-kpc ring, HI hole and wall, HI halo, and the dense main (HI+H_2) disk. The enlarged illustration is drawn by referring to the hydrodynamical BHS model, showing that the dense disk is kept unperturbed inside the global front. The view is consistent with the hydrodynamic simulation shown by the inserted reproduction from Sofue et al. (2016) at t= 10 My. In the present model, the Fermi Bubble is considered to be a younger object of a few My related to the innermost expanding ring in the GC (Sofue 2017).§ DISCUSSION§.§ SummaryWe derived the volume density map of HI gas along the tangent circle, and showed that the HI halo has a large crater-shaped structure around the GC. Thereby, we developed the TCM for pin-pointing the solar-circle gas without suffering from the kinematically degenerate depth problem in the tangent-circle direction.The wall of the HI crater positionally coincides with the brightest emission ridge of the NPS and the tangential direction of the 3-kpc expanding ring.While the HI hole inside the crater is almost empty, the outside halo is not disturbed, being kept in hydrostatic equilibrium (sech^2 Z/h_2) with a scale height h_2∼ 0.45 kpc. The origin of the crater structure is explained by a shock wave model based on the giant explosion hypothesis at the GC. Figure <ref> summarizes the observed structures in HI, CO, radio, X-ray, and γ rays. §.§ Some difficulties in the modelsIn our model the shock front expands into the halo, whereas the dense galactic disk is not strongly disturbed. In order to accelerate the 3 kpc ring to the observed velocity, Sanders and Prendergast (1974) had to assume an energy as large as ∼ 3× 10^58 erg, but such a huge explosion totally destroyed the disk. Therefore, the hydrodynamical models cannot reproduce the expanding motion of the 3-kpc ring. Such a local acceleration could be possible by refraction of waves transmitting the halo and focusing onto the disk as suggested by the MHD wave propagation model (Sofue 1977, 1984).This kinematics problem is not encountered by the bar hypothesis (e.g. Binney et al. 1991), where no energetic explosion is required to produce the non-circular motion. On the other hand, the largely extended vertical structure of the HI wall and the empty hole in the HI halo might not be easy to explain by the bar.§.§ Low-velocity HI shell It has been suggested that low-velocity HI gas at || <∼ 50 apparently surrounds the NPS (Heiles et al. 1980). However, it is not clear if the HI is indeed related, because the low velocity HI map is full of bright, often much brighter, local HI shells and filaments, making it difficult to confirm their true relation.If the HI outer shell is indeed associated with the NPS, the following scenario would be possible. The swept-up gas by the BHS is compressed to form high-temperature X-ray gas, while the densest front cools down to neutral gas and drops toward the disk. Since the snow-plowed gas has not enough angular momentum, the galactic rotation is somehow canceled by the dropping gas. 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"authors": [
"Yoshiaki Sofue"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170627105930",
"title": "Giant HI Hole inside the 3-kpc Ring and the North Polar Spur - The Galactic Crater -"
} |
Faculty of Engineering, Environment and Computing, Coventry University, UK [email protected] Institut für Informatik II, Universität Bonn, Germany {errami,weber}@cs.uni-bonn.de CNRS, Mathématiques, Université de Lille, Villeneuve d'Ascq, France [email protected] DIMNP UMR CNRS/UM 5235, University of Montpellier, France [email protected] University of Lorraine, CNRS, Inria, and LORIA, Nancy, France [email protected] MPI Informatics and Saarland University, Saarbrücken, Germany [email protected] Symbolic Versus Numerical Computation and Visualization of Parameter Regions for Multistationarity of Biological NetworksMatthew England1Hassan Errami2Dima Grigoriev3Ovidiu Radulescu4 Thomas Sturm5,6 Andreas Weber2today =========================================================================================================================We investigate models of the mitogenactivated protein kinases (MAPK) network, with the aim of determining where in parameter space there exist multiple positive steady states. We build on recent progress which combines various symbolic computation methods for mixed systems of equalities and inequalities. We demonstrate that those techniques benefit tremendously from a newly implemented graph theoretical symbolic preprocessing method. We compare computation times and quality of results of numerical continuation methods with our symbolic approach before and after the application of our preprocessing. § INTRODUCTIONThe mathematical modelling of intra-cellular biological processes has been using nonlinear ordinary differential equations since the early ages of mathematical biophysics in the 1940s and 50s <cit.>. A standard modelling choice for cellular circuitry is to use chemical reactions with mass action law kinetics, leading to polynomial differential equations. Rational functions kinetics (for instance the Michaelis-Menten kinetics) can generally be decomposed into several mass action steps. An important property of biological systems is their multistationarity which means having multiple stable steady states. Multistationarity is instrumental to cellular memory and cell differentiation during development or regeneration of multicellular organisms and is also used by micro-organisms in survival strategies. It is thus important to determine the parameter values for which a biochemical model is multistationary. With mass action reactions, testing for multiple steady states boils down to counting real positive solutions of algebraic systems. The models benchmarked in this paper concern intracellular signaling pathways. These pathways transmit information about the cell environment by inducing cascades of protein modifications (phosphorylation) all the way from the plasma membrane via the cytosol to genes in the cell nucleus. Multistationarity of signaling usually occurs as a result of activation of upstream signaling proteins by downstream components <cit.>. A different mechanism for producing multistationarity in signaling pathways was proposed by Kholodenko <cit.>. In this mechanism the cause of multistationarity are multiple phosphorylation/ dephosphorylation cycles that share enzymes. A simple, two steps phosphorylation/dephosphorylation cycle is capable of ultrasensitivity, a form of all or nothing response with no multiple steady states (Goldbeter–Koshland mechanism). In multiple phosphorylation/dephosphorylation cycles, enzyme sharing provides competitive interactions and positive feedback that ultimately leads to multistationarity <cit.>.Our study is complementary to works applying numerical methods to ordinary differential equations models used for biology applications. Gross et al. <cit.> used polynomial homotopy continuation methods for global parameter estimation of mass action models. Bifurcations and multistationarity of signaling cascades was studied with numerical methods based on the Jacobian matrix <cit.>. Other symbolic approaches to multistationarity either propose necessary conditions or work for particular networks <cit.>. Our work here follows <cit.>, where it was demonstrated that determination of multistationarity of an 11-dimensional model of a mitogen-activated protein kinases (MAPK) cascade can be achieved by currently available symbolic methods when numeric values are known for all but potentially one parameter.We show that the symbolic methods used in <cit.>, viz. real triangularization and cylindrical algebraic decomposition, and also polynomial homotopy continuation methods, benefit tremendously from a graph theoretical symbolic preprocessing method. This method has been sketched by Grigoriev et al. <cit.> and has been used for a “hand computation,” but had not been implemented before.For our experiments we use the model already investigated in <cit.> and a higher dimensional model of the MAPK cascade.§ THE SYSTEMS FOR THE CASE STUDIESFor our investigations we use models of the MAPK cascade that can be found in the Biomodels database[<http://www.ebi.ac.uk/biomodels-main/>] as numbers 26 and 28 <cit.>. We refer to those models as and , respectively. §.§ , which we have studied also in <cit.>, is given by the following set of differential equations. We have renamed the species names as x_1, …, x_11 and the rate constants as k_1, …, k_16 to facilitate reading:x_1 = k_2 x_6 + k_15 x_11 - k_1 x_1 x_4 - k_16 x_1 x_5 x_2 = k_3 x_6 + k_5 x_7 + k_10 x_9 + k_13 x_10 - x_2 x_5 (k_11 + k_12) - k_4 x_2 x_4 x_3 = k_6 x_7 + k_8 x_8 - k_7 x_3 x_5 x_4 = x_6 (k_2 + k_3) + x_7 (k_5 + k_6) - k_1 x_1 x_4 - k_4 x_2 x_4 x_5 = k_8 x_8 + k_10 x_9 + k_13 x_10 + k_15 x_11 - x_2 x_5 (k_11 + k_12) - k_7 x_3 x_5 - k_16 x_1 x_5 x_6 = k_1 x_1 x_4 - x_6 (k_2 + k_3) x_7 = k_4 x_2 x_4 - x_7 (k_5 + k_6) x_8 = k_7 x_3 x_5 - x_8 (k_8 + k_9) x_9 = k_9 x_8 - k_10 x_9 + k_11 x_2 x_5 x_10= k_12 x_2 x_5 - x_10 (k_13 + k_14) x_11= k_14 x_10 - k_15 x_11 + k_16 x_1 x_5The Biomodels database also gives us meaningful values for the rate constants, which we generally substitute into the corresponding systems for our purposes here: k_1 = 0.02,k_2 = 1,k_3 = 0.01,k_4 = 0.032,k_5 = 1,k_6 = 15,k_7 = 0.045,k_8 = 1,k_9 = 0.092,k_10 = 1,k_11 = 0.01,k_12 = 0.01,k_13 = 1,k_14 = 0.5,k_15 = 0.086,k_16 = 0.0011.Using the left-null space of the stoichiometric matrix under positive conditions as a conservation constraint <cit.> we obtain three linear conservation laws:x_5+ x_8 + x_9 + x_10 + x_11 = k_17, x_4+ x_6 + x_7 = k_18,x_1+ x_2 + x_3 + x_6 + x_7 + x_8 + x_9 + x_10 + x_11 = k_19,where k_17, k_18, k_19 are new constants computed from the initial data. Those constants are the parameters that we are interested in here.The steady state problem for the MAPK cascade can now be formulated as a real algebraic problem as follows. We replace the left hand sides of all equations in (<ref>) with 0 and substitute the values from (<ref>). This together with (<ref>) yields a system of parametric polynomial equations with polynomials in [k_17,k_18,k_19][x_1,…,x_11]. Since all entities in our model are strictly positive, we add to our system positivity conditions k_17>0, k_18>0, k_19>0 and x_1>0, …, x_11>0. In terms of first-order logic the conjunction over our equations and inequalities yields a quantifier-free Tarski formula. §.§ The system with number 28 in the Biomodels database is given by the following set of differential equations. Again, we have renamed the species names into x_1, …, x_16 and the rate constants into k_1, …, k_27 to facilitate reading: x_1= k_2 x_9 + k_8 x_10 + k_21 x_15 + k_26 x_16 - k_1 x_1 x_5 - k_7 x_1 x_5 - k_22 x_1 x_6 - k_27 x_1 x_6 x_2 = k_3 x_9 + k_5 x_7 + k_24 x_12 - k_4 x_2 x_5 - k_23 x_2 x_6x_3= k_9 x_10 + k_11 x_8 + k_16 x_13 + k_19 x_14 - k_10 x_3 x_5 - k_17 x_3 x_6 - k_18 x_3 x_6x_4 = k_6 x_7 + k_12 x_8 + k_14 x_11 - k_13 x_4 x_6x_5 = k_2 x_9 + k_3 x_9 + k_5 x_7 + k_6 x_7 + k_8 x_10 + k_9 x_10 + k_11 x_8 + k_12 x_8 -k_1 x_1 x_5 - k_4 x_2 x_5 - k_7 x_1 x_5 - k_10 x_3 x_5 x_6= k_14 x_11 + k_16 x_13 + k_19 x_14 + k_21 x_15 + k_24 x_12 + k_26 x_16 -k_13 x_4 x_6 - k_17 x_3 x_6 - k_18 x_3 x_6 - k_22 x_1 x_6 - k_23 x_2 x_6 - k_27 x_1 x_6 x_7= k_4 x_2 x_5 - k_6 x_7 - k_5 x_7 x_8 =k_10 x_3 x_5 - k_12 x_8 - k_11 x_8 x_9 = k_1 x_1 x_5 - k_3 x_9 - k_2 x_9 x_10 =k_7 x_1 x_5 - k_9 x_10 - k_8 x_10 x_11 =k_13 x_4 x_6 - k_15 x_11 - k_14 x_11 x_12=k_23 x_2 x_6 - k_25 x_12 - k_24 x_12 x_13 =k_15 x_11 - k_16 x_13 + k_17 x_3 x_6 x_14= k_18 x_3 x_6 - k_20 x_14 - k_19 x_14 x_15=k_20 x_14 - k_21 x_15 + k_22 x_1 x_6 x_16=k_25 x_12 - k_26 x_16 + k_27 x_1 x_6The estimates of the rate constants given in theBiomodels database are: k_1 = 0.005,k_2 = 1,k_3 = 1.08,k_4 = 0.025,k_5 = 1,k_6 = 0.007,k_7 = 0.05,k_8 = 1,k_9 = 0.008,k_10 = 0.005,k_11 = 1,k_12 = 0.45,k_13 = 0.045,k_14 = 1,k_15 = 0.092,k_16 = 1, k_17 = 0.01,k_18 = 0.01,k_19 = 1,k_20 = 0.5, k_21 = 0.086,k_22 = 0.0011,k_23 = 0.01,k_24 = 1, k_25 = 0.47,k_26 = 0.14,k_27 = 0.0018.Again, using the left-null space of the stoichiometric matrix under positive conditions as a conservation constraint <cit.> we obtain the following:x_6 + x_11 + x_12 + x_13+ x_14 + x_15 + x_16 = k_28,x_5 + x_7 + x_8 + x_9 + x_10 =k_29 , x_1 + x_2 + x_3 + x_4 + x_7 + x_8 + x_9 + x_10 + x_11 +x_12 + x_13 + x_14 + x_15 + x_16 =k_30,where k_28, k_29, k_30 are new constants computed from the initial data.We formulate the real algebraic problem as described at the end of Sect. <ref>. In particular, note that we need positivity conditions for all variables and parameters.§ GRAPH-THEORETICAL SYMBOLIC PREPROCESSING The complexity, primarily in terms of dimension, of polynomial systems obtained with steady-state approximations of biological models plus conservation laws is comparatively high for the application of symbolic methods. It is therefore highly relevant for the success of such methods to identify and exploit particular structural properties of the input. Our models have remarkably low total degrees with many linear monomials after some substitutions for rate constants. This suggests to preprocess with essentially Gaussian elimination in the sense of solving single suitable equations with respect to some variable and substituting the corresponding solution into the system.Generalizing this idea to situations where linear variables have parametric coefficients in the other variables requires, in general, a parametric variant of Gaussian elimination, which replaces the input system with a finite case distinction with respect to the vanishing of certain coefficients and one reduced system for each case. With and considered here it turns out that the positivity assumptions on the variables are strong enough to effectively guarantee the non-vanishing of all relevant coefficients so that case distinctions are never necessary.On the other hand, those positivity conditions establish an apparent obstacle, because we are formally not dealing with a parametric system of linear equations but with a parametric linear programming problem. However, here the theory of real quantifier elimination by virtual substitution tells us that it is sufficient that the inequality constraints play a passive role. Those constraints must be considered when substituting Gauss solutions from the equations, but otherwise can be ignored <cit.>.Parametric Gaussian elimination can increase the degrees of variables in the parametric coefficient, in particular destroying their linearity and suitability to be used for further reductions. As an example consider the steady-state approximation, i.e., all left hand sides replaced with 0, of the system in (<ref>), solving the last equation for x_5, and substituting into the first equation. The natural question for an optimal strategy to Gauss-eliminate a maximal number of variables has been answered positively only recently <cit.>: draw a graph, where vertices are variables and edges indicate multiplication between variables within some monomial. Then one can Gauss-eliminate a maximum independent set, which is the complement of a minimum vertex cover. Fig. <ref> shows that graph for , where {x_4,x_5} is a minimal vertex cover, and all other variables can be linearly eliminated. Similarly, for we find {x_5,x_6} as a minimum vertex cover. Recall that minimum vertex cover is one of Karp's 21 classical NP complete problems <cit.>. However, our instances considered here and instances to be expected from other biological models are so small that the use of existing approximation algorithms <cit.> appears unnecessary. We have used real quantifier elimination, which did not consume measurable CPU time; alternatively one could use integer linear programming or SAT-solving.It is a most remarkable fact that a significant number of biological models in the databases have that property of loosely connected variables. This phenomenon resembles the well-known community structure of propositional satisfiability problems, which has been identified as one of the key structural reasons for the impressive success of state-of-the-art CDCL-based SAT solvers <cit.>.We conclude this section with the reduced systems as computed with our implementation in Redlog <cit.>. For we obtain x_5 >0, x_4 > 0, k_19 > 0, k_18> 0, k_17 > 0 and1062444 k_18 x_4^2 x_5 + 23478000 k_18 x_4^2 + 1153450 k_18 x_4 x_5^2 + 2967000 k_18 x_4 x_5+ 638825 k_18 x_5^3 + 49944500 k_18 x_5^2 - 5934 k_19 x_4^2 x_5 - 989000 k_19 x_4 x_5^2- 1062444 x_4^3 x_5 - 23478000 x_4^3 - 1153450 x_4^2 x_5^2- 2967000 x_4^2 x_5- 638825 x_4 x_5^3 - 49944500 x_4 x_5^2 =0,1062444 k_17 x_4^2 x_5 + 23478000 k_17 x_4^2 + 1153450 k_17x_4 x_5^2 + 2967000 k_17 x_4 x_5 + 638825 k_17 x_5^3 + 49944500 k_17 x_5^2 - 1056510 k_19 x_4^2 x_5 - 164450 k_19 x_4 x_5^2 - 638825 k_19 x_5^3 - 1062444 x_4^2 x_5^2 - 23478000 x_4^2 x_5 - 1153450 x_4 x_5^3- 2967000 x_4 x_5^2 - 638825 x_5^4 - 49944500 x_5^3 =0. For we obtain x_6 >0, x_5 > 0, k_30 > 0, k_29> 0, k_28 > 0 and3796549898085 k_29 x_5^3 x_6 + 71063292573000 k_29 x_5^3 + 106615407090630 k_29 x_5^2 x_6^2 + 479383905861000 k_29 x_5^2 x_6 + 299076127852260 k_29 x_5 x_6^3 + 3505609439955600 k_29 x_5 x_6^2+ 91244417457024 k_29 x_6^4 + 3557586742819200 k_29x_6^3 - 598701732300 k_30 x_5^3 x_6- 83232870778950 k_30 x_5^2 x_6^2 - 185019487578700 k_30 x_5x_6^3 - 3796549898085 x_5^4 x_6- 71063292573000 x_5^4 - 106615407090630 x_5^3 x_6^2- 479383905861000 x_5^3 x_6 - 299076127852260 x_5^2 x_6^3 - 3505609439955600 x_5^2 x_6^2 - 91244417457024 x_5 x_6^4 - 3557586742819200 x_5 x_6^3 =0,3796549898085 k_28 x_5^3 x_6 + 71063292573000 k_28 x_5^3 + 106615407090630 k_28 x_5^2 x_6^2 + 479383905861000 k_28 x_5^2 x_6 + 299076127852260 k_28 x_5 x_6^3 + 3505609439955600 k_28 x_5 x_6^2 + 91244417457024 k_28 x_6^4 + 3557586742819200 k_28 x_6^3 - 3197848165785 k_30 x_5^3 x_6- 23382536311680 k_30 x_5^2 x_6^2 - 114056640273560 k_30 x_5 x_6^3 - 91244417457024 k_30 x_6^4 - 3796549898085 x_5^3 x_6^2 - 71063292573000 x_5^3 x_6 - 106615407090630 x_5^2 x_6^3 - 479383905861000 x_5^2 x_6^2 - 299076127852260 x_5 x_6^4- 3505609439955600 x_5 x_6^3 - 91244417457024 x_6^5 - 3557586742819200 x_6^4 =0.Notice that no complex positivity constraints come into existence with these examples. All corresponding substitution results are entailed by the other constraints, which is implicitly discovered by using the standard simplifier from <cit.> during preprocessing.§ DETERMINATION OF MULTIPLE STEADY STATES We aim to identify via grid sampling regions of parameter space where multistationarity occurs. Our focus is on the identification of regions with multiple positive real solutions for the parameters introduced with the conservation laws. We will encounter one or three such solutions and allow ourselves for biological reasons to assume monostability or bistability, respectively. Furthermore, a change in the number of solutions between one and three is indicative of a saddle-node bifurcation between a monostable and a bistable case. A mathematically rigorous treatment of stability would, possibly symbolically, analyze the eigenvalues of the Jacobian of the respective polynomial vector field. We consider two different approaches: first a polynomial homotopy continuation method implemented in Bertini, and second a combination of symbolic computation methods implemented in Maple. We compare the approaches with respect to performance and quality of results for both the reduced and the unreduced systems. §.§ Numerical ApproachWe use the homotopy solver Bertini <cit.> in its standard configuration to compute complex roots. We parse the output of Bertini using Python, and determined numerically, which of the complex roots are real and positive using a threshold of 10^-6 for positivity. Computations are done in Python with Bertini embedded.For System we produced the two plots in Fig. <ref> using the original system and the two in Fig. <ref> using the reduced system. The sampling range for k_19 was from 200 to 1000 by 50. In the left plots the sampling range for k_17 is from 80 to 200 by 10 with k_18 fixed at 50. In the right plots the sampling range for k_18 is 5 to 75 by 5 with k_17 fixed to 100. We see two regions forming according to the number of fixed points: yellow discs indicate one fixed point and blue boxes three.The diamonds indicate numerical errors where zero (red) or two (green) fixed states were identified.We analyse these further in Sect. <ref>.For we produced the two plots in Fig. <ref> using the original system. The sampling range for k_30 was from 100 to 1600 by 100. In the left plots the sampling range for k_28 is from 40 to 160 by 10 with k_29 fixed at 180. In the right plots the sampling range for k_29 is from 120 to 240 by 10 with k_28 fixed to 100. The colours and shapes indicate the number of fixed points as before. For the reduced system Bertini (wrongly) could not find any roots (not even complex ones) for any of the parameter settings. The situation did not change when going from adaptive precision to a very high fixed precision. However, we have not attempted more sophisticated techniques like providing user homotopies. We analyse these results further in Sect. <ref>. §.§ Symbolic ApproachOur next approach will still use grid sampling, but each sample point will undergo a symbolic computation. The result will still be an approximate identification of the region (since the sampling will be finite) but the results at those sample points will be guaranteed free of numerical errors. The computations follow the strategy introduced in <cit.>. This combined tools from the Regular Chains Library[<http://www.regularchains.org/>] available for use in Maple. Regular chains are the triangular decompositions of systems of polynomial equations (triangular in terms of the variables in each polynomial). Highly efficient methods for working in complex space have been developed based on these (see <cit.>for a survey).We make use of recent work by Chen et al. <cit.> which adapts these tools to the real analogue: semi-algebraic systems. They describe algorithms to decompose any real polynomial system into finitely many regular semi-algebraic systems: both directly and by computation of components by dimension. The latter (the so called lazy variant) was key to solving the 1-parameter MAPK problem in <cit.>. However, for the zero dimensional computations of this paper there is only one solution component and so no savings from lazy computations.For a given system and sample point we apply the real triangularization (RT) on the quantifier-free formula (as described at the end of Sect. <ref>: a quantifier free conjunction of equities and inequalities) evaluated with the parameter estimates and sample point values. This produces a simplified system in several senses. First, as guaranteed by the algorithm, the output is triangular according to a variable ordering. So there is a univariate component, then a bivariate component introducing one more variable and so on.Secondly, for all the MAPK models we have studied so far, all but the final (univariate) of these equations has been linear in its main variable. This thus allows for easy back substitution.Thirdly, most of the positivity conditions are implied by the output rather than being an explicit part of it,in which case a simpler sub-system can be solved and back substitution performed instantly.§.§.§For the original version of the output of RT was a component consisting of 11 equations and a single inequality. The equations were in ascending main variable according to the provided ordering (same as the labelling). All but the final equation is linear in its main variable, with the final equation being univariate and degree 6 in x_1. The output of the triangularization requires that this variable be positive, x_1>0, with the positivity of the other variables implied by solutions to the system.So to proceed we must find the positive real roots of the degree 8 univariate polynomial in x_1: counting these will imply the number of real positive solutions of the parent system. We do this using the root isolation tools in the Regular Chains Library.This whole process was performed iteratively for the same sampling regime as Bertini used to produce Fig. <ref>.We repeated the process on the reduced version of the system. The triangularization again reduced the problem to univariate real root isolation, this time with only one back substitution step needed. As to be expected from a fully symbolic computation, the output is identical and so again represented by Fig. <ref>. However, the computation was significantlyquicker with this reduced system. More details are given in the comparison in Sect. <ref>.§.§.§The same process was conducted on . As with the system was triangular with all but the final equation linear in its main variable; this time the final equation is degree 8. However, unlike two positivity conditions were returned in the output meaning we must solve a bivariate problem before we can back substitute to the full system. Rather than just perform univariate real root isolation we must build a Cylindrical Algebraic Decomposition (CAD) (see, e.g., <cit.> and the references within) sign invariant for the final two equations and interrogate its cells to find those where the equations are satisfied and variable positive. Counting these we find always 1 or 3 cells, with the latter indicating bistability. This is similar to the approach used in <cit.>, although in that case the 2D CAD was for one variable and one parameter. We used the implementation of CAD in the Regular Chains Library <cit.> with the results producing the plots in Fig. <ref>.For the reduced system we proceeded similarly. A 2D CAD still needed to be produced after triangularization and so in this case there was no reduction in the number of equations to study with CAD via back substitution.However, it was still beneficial to pre-process CAD with real triangularization: the average time per sample point with pre-processing (and including time taken to pre-process) was 0.485 seconds while without it was 3.577 seconds.§.§ ComparisonFigure <ref>, Fig. <ref>, and Fig. <ref> all refer to .The latter, produced using the symbolic techniques in Maple, is guaranteed free of numerical error.We see that computing with the reduced system rather than the original system allowed Bertini to avoid such errors: the rouge red and green diamonds in Fig. <ref>. However, in the case of the reduction led to catastrophic effects for Bertini: built-in heuristics quickly (and wrongly) concluded that there are no zero dimensionalsolutions for the system, and when switching to a positive dimensional run also no solutions could be found.Bertini computations (v1.5.1) were carried out on a Linux64 bit Desktop PC with Intel i7. Maple computations (v2016 with April 2017 Regular Chains) were carried out on a Windows 7 64 bit Desktop PC with Intel i5.For the pairs of plots together contain 476 sample points. Table <ref> shows timing data.We see that both Bertini and Maple benefited from the reduced system: Bertini took a third of the original time while the speedup for Maple was even greater: a tenth of the original.Also, perhaps surprisingly, the symbolic methods were quicker than the numerical ones here.For the speed-up enjoyed by the symbolic methods was even greater (almost 100 fold).However, for this system Bertini was significantly faster.The symbolic methods used are well known for their doubly exponential computational complexity (in the number of variables) so it is not surprising that as the system size increases there so should the results of the comparison. We see some other statistical data for the timings in Maple: the standard deviation for the timings is fairly modest but in each row we see there are outliers many multiples of the mean value and so the median is always a little less than the mean average. §.§ Going FurtherOf course, we could increase the sampling density to get an improved idea of the bistability region, as in Fig. <ref> and Fig. <ref>. However, a greater understanding comes with 3D sampling.We have performed this using the symbolic approach described above, at a linear cost proportional to the increasednumber of sample points.This was completed for : the region in question is bounded to both sides in the k_17 and k_18 directions but extends infinitelyabove in k_19.With the k_19 range bound at 1000 the region is bounded by extending k_17 to 800 and k_18 to 600. For obtaining exact bounds (in one parameter) see <cit.>.Sampling in 20 seconds for k_17 and k_18 and 50 seconds for k_19 produced a Maple point plot of 20400 in 18 minutes. Figure <ref> shows 2D captures of the 3D bistable points and Fig. <ref> the convex hull of these, produced using the convex package[<http://www.math.uwo.ca/ mfranz/convex/>].We note the lens shape seen in the orientation in the left plots is comparable with the image in the original paper of Markevich et al. <cit.>.§ CONCLUSION AND FUTURE WORK We described a new graph theoretical symbolic preprocessing method to reduce problems from the MAPK network. We experimented with two systems and found the reduction offered computation savings to both numerical and symbolic approaches for the determination of multistationarity regions of parameter space. In addition, the reduction avoided instability from rounding errors in the numerical approach to one system, but uncovered major problems in that approach for the other. An interesting side result is that, at least for the smaller system, the symbolic approach can compete with and even outperform the numerical one, demonstrating how far such methods have progressed in recent years.In future work we intend to combine the results of the present paper and our recent publication <cit.> to generate symbolic descriptions of the bistability region beyond the 1-parameter case. Other possible routes to achieve this is to consider the effect of the various degrees of freedom with the algorithms used. For example, we have a free choice of variable ordering: has 11 variables corresponding to 39 916 800 possible orderings while has 16 variables corresponding tomore than 10^13 orderings. Heuristics exist to help with this choice <cit.> and machine learning may be applicable <cit.>. Also, since MAPK problems contain many equational constraints an approach as described in <cit.> may be applicable when higher dimensional CADs are needed. § ACKNOWLEDGEMENTSD. Grigoriev is grateful to the grant RSF 16-11-10075. H. Errami, O. Radulescu, and A. Weber thank the French-German Procope-DAAD program for partial support of this research. M. England and T. Sturm are grateful to EU H2020-FETOPEN-2015-CSA 712689 SC2. Research Data Statement: Data supporting the research in this paper is available from http://doi.org/10.5281/zenodo.807678doi:10.5281/zenodo.807678. splncs_srt | http://arxiv.org/abs/1706.08794v1 | {
"authors": [
"Matthew England",
"Hassan Errami",
"Dima Grigoriev",
"Ovidiu Radulescu",
"Thomas Sturm",
"Andreas Weber"
],
"categories": [
"cs.SC",
"I.1.4"
],
"primary_category": "cs.SC",
"published": "20170627114535",
"title": "Symbolic Versus Numerical Computation and Visualization of Parameter Regions for Multistationarity of Biological Networks"
} |
Université d'Avignon, Campus Jean-Henri Fabre- 301, rue Baruch de Spinoza,BP 21239 F-84 916 AVIGNON Cedex 9. [email protected] University of Zanjan, Faculty of Science, Department of Mathematics, University blvd, Zanjan, [email protected] work is funded by the French Ministry of Foreign Affairs, through Campus France.Grant number 878286E We describe the orbits of the irreducible action of PSL(2, ℝ) on the 3-dimensional Einstein universe ^1,2. This work completes the study in <cit.>, and is one element of the classification of cohomogeneity one actions on ^1,2 (<cit.>). On the irreducible action of PSL(2, ℝ) on the 3-dimensional Einstein universe.Masoud Hassani December 30, 2023 =============================================================================== § INTRODUCTION §.§ Einstein universeLet ℝ^2,n+1 denote a (n+3)-dimensional real vector space equipped with a non-degenerate symmetric bilinear form 𝔮 with signature (2,n+1). The nullcone of ℝ^2,n+1 isΝ^2,n+1={v∈ℝ^2,n+1∖{0}: 𝔮(v)=0} .The (n+1)-dimensional Einstein universe ^1,n is the image of the nullcone Ν^2,n+1 under the projectivization:ℙ:ℝ^2,n+1∖{0}⟶ℝℙ^n+2.The degenerate metric on Ν^2,n+1 induces a O(2,n+1)-invariant conformal Lorentzian structure on Einstein universe. The group of conformal transformations on ^1,n is O(2,n+1) <cit.>.A lightlike geodesic in Einstein universe is a photon. A photon is the projectivisation of an isotropic 2-plane in ^2,n+1.The set of photons through a point p∈^1,n denoted by L(p) is the lightcone at p. The complement of a lightcone L(p) in Einstein universe is the Minkowski patch at p and we denote it by Mink(p). A Minkowski patch is conformally equivalent to the (n+1)-dimensional Minkoski space ^1,n <cit.>.The complement of the Einstein universe in ℝℙ^n+2 has two connected components: the (n+2)-dimensional Anti de-Sitter space ^1,n+1 and the generalized hyperbolic space ^2,n: the first (respectively the second) is the projection of the domain ℝ^2,n+1 defined by {𝔮<0 } (respectively {𝔮>0 }).An immersed submanifold S of^1,n+1 or ℍ^2,n is of signature (p, q, r) (respectively^1,n) if the restriction of the ambient pseudo-Riemmanian metric (respectively the conformal Lorentzian metric) is of signature (p,q,r), meaning that the radical has dimension r, and that maximal definite negative and positive subspaces have dimensions p and q, respectively. If S is nondegenerate, we forgot r and simply denote its signature by (p,q).§.§ The irreducible representation of PSL(2, ℝ)A subgroup of O(2,n+1) is irreducible if it preserves no proper subspace of ^2,n+1. By <cit.>, up to conjugacy, SO_∘(1,2)≃ PSL(2,) is the only irreducible connected Lie subgroup of O(2,3).On the other hand, for every integer n, it is well known that, up to isomorphism, there is only one n-dimensional irreducible representation of PSL(2,). For n=5, this representation is the natural action of PSL(2, ℝ) on the vector space =_4[X,Y] of homogeneous polynomials of degree 4 in two variables X and Y. This action preserves the following quadratic form𝔮(a_4X^4+a_3X^3Y+a_2X^2Y^2+a_1XY^3+a_0Y^4)=2a_4a_0-1/2a_1a_3+1/6a_2^2.The quadratic form 𝔮 is nondegenerate and has signature (2,3). This induces an irreducible representation PSL(2,)→ O(2,3) <cit.>. The irreducible action of PSL(2,) on the 3-dimensional Einstein universe ^1,2 admits three orbits: * An 1-dimensional lightlike orbit, i.e. of signature (0,0, 1)* A 2-dimensional orbit of signature (0, 1, 1),* An open orbit (hence of signature (1,2)) on which the action is free.The 1-dimensional orbit is lightlike, homeomorphic to ^1, but not a photon. The union of the 1-dimensional orbit and the 2-dimensional orbit is an algebraic surface, whose singular locus is precisely the 1-dimensional orbit. It is the union of all projective lines tangent to the 1-dimensional orbit. Figure <ref> describes a part of the 1 and 2-dimensional orbits in the Minkowski patch Mink(Y^4). We will also describe the actions on Anti de-Sitter space and the generalized hyperbolic space ^2,2: The orbits of PSL(2,) in the Anti de-sitter component ^1,3 are Lorentzian, i.e. of signature (1,2). They are the leaves of a codimension 1 foliation. In addition, PSL(2,) induces three types of orbits in ^2,2: a 2-dimensional spacelike orbit (of signature (2,0)) homeomorphic to the hyperbolic plane ^2, a 2-dimensional Lorentzian orbit (i.e., of signature (1,1)) homeomorphic to the de-Sitter space ^1,1, and four kinds of 3-dimensional orbits where the action is free: * one-parameter family of orbits of signature (2,1) consisting of elements with four distinct non-real roots,* one-parameter family of Lorentzian (i.e. of signature (1,2)) orbits consisting of elements with four distinct real roots,* two orbits of signature (1,1,1),* one-parameter family of Lorentzian (i.e. of signature (1,2)) orbits consisting of elements with two distinct real roots, and a complex root z in ℍ^2 making an angle θ smaller than 5π/6 with the two real roots. F. Fillastre indicated to us an alternative description for the last case stated in Theorem <ref>: these orbits correspond to polynomials whose roots in ℂℙ^1 are ideal vertexes of regular ideal tetraedra in ^3. § PROOFS OF THE THEOREMS Let f be an element in . We consider it as a polynomial function from ^2 into . Actually, by specifying Y=1, we consider f as a polynomial of degree at most 4. Such a polynomial is determined, up to a scalar, by its roots z_1, z_2, z_3, z_4 in ^1 (some of these roots can be ∞ if f can be divided by Y). It provides a natural identification between () and the set _4^1 made of 4-tuples (up to permutation) (z_1, z_2, z_3, z_4) of ^1 such that if some z_i is not in ℝ^1, then its conjugate z̅_i is one of the z_j's. This identification is PSL(2,)-equivariant, where the action of PSL(2,) on _4^1 is simply the one induced by the diagonal action on (^1)^4. Actually, the complement of ^1 in ^1 is the union of the upper half-plane model ℍ^2 of the hyperbolic plane, and the lower half-plane. We can represent every element of_4^1 by a 4-tuple (up to permutation) (z_1, z_2, z_3, z_4) such that:– either every z_i lies in ^1,– or z_1, z_2 lies in ^1, z_3 lies in ℍ^2 and z_4 = z̅_3,– or z_1, z_2 lies in ℍ^2 and z_3 = z̅_1, z_4 = z̅_2.Theorems <ref> and <ref> will follow from the following proposition: Let [f] be an element of (). Then: * it lies in ^1,2 if and only if it has a root of multiplicity at least 3, or two distinct real roots z_1, z_2, and two complex roots z_3, z_4 = z̅_3, with z_3 in ℍ^2 and such that the angle at z_3 between the hyperbolic geodesic rays [z_3, z_1) and [z_3, z_2) is 5π/6,* it lies in ^1,3 if and only it has two distinct real roots z_1, z_2, and two complex roots z_3, z_4 = z̅_3, with z_3 in ℍ^2 and such that the angle at z_3 between the hyperbolic geodesic rays [z_3, z_1) and [z_3, z_2) is > 5π/6,* it lies in ℍ^2,2 if and only if it has no real roots, or four distinct real roots, or a root of multiplicity exactly 2, or it has two distinct real roots z_1, z_2, and two complex roots z_3, z_4 = z̅_3, with z_3 in ℍ^2 and such that the angle at z_3 between the hyperbolic geodesic rays [z_3, z_1) and [z_3, z_2) is < 5π/6. Proof of Proposition <ref>. Assume that f has no real root. Hence we are in the situation where z_1, z_2 lie in ℍ^2 and z_3 = z̅_1, z_4 = z̅_2. By applying a suitable element of PSL(2, ℝ), we can assume z_1=i, and z_2= r i for some r>0. In other words, f is in the PSL(2, ℝ)-orbit of (X^2+Y^2)(X^2 + r^2Y^2). The value of 𝔮 on this polynomial is 2×1× r^2 + 1/6(1+r^2)^2 >0, hence [f] lies in ℍ^2,2.Hence we can assume that f admits at least one root in ^1, and by applying a suitable element of PSL(2, ℝ), one can assume that this root is ∞, i.e. that f is a multiple of Y.We first consider the case where this real root has multiplicity at least 2:f = Y^2(aX^2 + bXY + cY^2)Then, 𝔮(f) = 1/6a^2: it follows that if f has a root of multiplicity at least 3, it lies in ^1,2, and if it has a real root of mulitplicity 2, it lies in ℍ^2,2.We assume from now that the real roots of f have multiplicity 1. Assume that all roots are real. Up to PSL(2, ℝ), one can assume that these roots are 0, 1, r and ∞ with 0 < r < 1.f(X,Y) = XY(X-Y)(X-rY) = X^3Y - (r+1)X^2Y^2 + rXY^3Then, 𝔮(f) = -1/2r + 1/6(r+1)^2 = 1/6(r^2-r+1) > 0. Therefore f lies in ℍ^2,2 once more. The only remaining case is the case where f has two distinct real roots, and two complex conjugate roots z, z̅ with z ∈ℍ^2. Up to PSL(2,ℝ), one can assume that the real roots are 0, ∞, hence:f(X,Y) = XY(X-zY)(X-z̅Y)=XY(X^2 - 2|z|cosθ XY + |z|^2Y^2)where z = |z|e^iθ. Then:𝔮(f) = 2|z|^2/3(cos^2θ - 3/4)Hence f lies in ^1,2 if and only if θ = π/6 or 5π/6. The proposition follows easily. □ In order to determine the signature of the orbits induced by PSL(2,) in (), we consider the tangent vectors induced by the action of 1-parameter subgroups of PSL(2,). We denote by E, P and H, the 1-parameter elliptic, parabolic and hyperbolic subgroups stabilizing i, ∞ and {0,∞}, respectively. Proof of Theorem <ref>. It follows from Proposition <ref> that there are precisely three PSL(2,ℝ)-orbits in ^1,2:– one orbit 𝒩 comprising polynomials with a root of multiplicity 4, i.e. of the form [(sY - tX)^4] with s,t ∈ℝ. It is clearly 1-dimensional, and equivariantly homeomorphic to ^1 with the usual projective action of PSL(2,ℝ). Since d/dt|_t=0(Y-tX)^4 = -4XY^3 is a 𝔮-null vector, this orbit is lightlike (but cannot be a photon since the action is irreducible),– one orbit ℒ comprising polynomials with a real root of multiplicity 3, and another real root. These are the polynomials of the form [(sY - tX)^3(s'Y - t'X)] with s, t, s', t' ∈ℝ. It is 2-dimensional, and it is easy to see that it is the union of the projective lines tangent to 𝒩. The vectors tangent to ℒ induced by the 1-parameter subgroups P and E at [XY^3]∈ℒ are v_P=-Y^4 and v_E=3X^2Y^2+Y^4. Obviously, v_P is orthogonal to v_E and v_E+v_P is spacelike. Hence ℒ is of signature (0,1,1).– one open orbit comprising polynomials admitting two distinct real roots and a root z in ℍ^2 making an angle 5π/6 with the two real roots in ∂ℍ^2. The stabilizers of points in this orbit are trivial since an isometry of ℍ^2 preserving a point in ℍ^2 and one point in ∂ℍ^2 is necessarily the identity. □ Proof of Theorem <ref>. According to Proposition <ref>, the polynomials in ^1,3 have two distinct real roots, and a complex root z in ℍ^2 making an angle θ greater than 5π/6 with the two real roots. It follows that the action in ^1,3 is free, and that the orbits are the level sets of the function θ. Suppose that M is a PSL(2,)-orbit in ^1,3. There exists r∈ such that [f]=[Y(X^2+Y^2)(X-rY)]∈ M. The orbit induced by the 1-parameter elliptic subgroup E at [f] is γ(t)=[(X^2+Y^2)((sin tcos t-r sin^2t)X^2-(sin tcos t+r cos^2t)Y^2+(cos^2t-sin^2t+2r sin tcos t)XY)] .Then 𝔮(dγ/dt|_t=0)=-2-2r^2<0. This implies, as for any submanifold of a Lorentzian manifold admitting a timelike vector, that M is Lorentzian, i.e., of signature (1,2).The case of ℍ^2,2 is the richest one. According to Proposition <ref> there are four cases to consider:* No real roots. Then f has two complex roots z_1, z_2 in ℍ^2 (and their conjugates). It corresponds to two orbits: one orbit corresponding to the case z_1 = z_2: it is spacelike and has dimension 2. It is the only maximal PSL(2,ℝ)-invariant surface in ℍ^2,2 described in <cit.>. The case z_1 ≠ z_2 provides a one-parameter family of 3-dimensional orbits on which the action is free (the parameter being the hyperbolic distance between z_1 and z_2). One may assume that z_1=i and z_2=r i for some r>0. Denote by M the orbit induced by PSL(2,) at [f]=[(X^2+Y^2)(X^2+r^2Y^2)]. The vectors tangent to M at [f] induced by the 1-parameter subgroups H, P and E are: v_H=-4X^4+ 4r^2Y^4, v_P=-4X^3Y-2(r^2+1)XY^3,v_E=2(r^2-1)X^3Y+2(r^2-1)XY^3, respectively. The timelike vector v_H is orthogonal to both v_P and v_E. It is easy to see that the 2-plane generated by {v_P,v_E} is of signature (1,1). Therefore, the tangent space T_[f]M is of signature (2,1).* Four distinct real roots.This case provides a one-parameter family of 3-dimensional orbits on which the action is free - the parameter being the cross-ratio between the roots in ^1. Denote by M the PSL(2,)-orbit at [f]=[XY(X-Y)(X-rY)] (here as explained in the proof of Proposition <ref>, we can restrict ourselves to the case 0 < r < 1). The vectors tangent to M at [f] induced by the 1-parameter subgroups H, P, and E are:v_H=-rY^4+2(r+1)XY^3-3X^2Y^2, v_P=-2X^3Y+2rXY^3,v_E=X^4-rY^4+3(r-1)X^2Y^2+2(r+1)XY^3-2(r+1)X^3Y,respectively. A vector x=av_H+bv_P+cv_E is orthogonal to v_P if and only if 2ra+b(r+1)+c(r+1)^2=0. Set a=(b(r+1)+c(r+1)^2)/-2r in𝔮(x)=2ra^2+3/2b^2+(7/2(r^2+1)-r)c^2+2(r+1)ab+2(r+1)^2+ac(2r^2-r+5). Consider 𝔮(x)=0 as a quadratic polynomial F in b. Since 0<r<1, the discriminant of F is non-negative and it is positive when c≠ 0. Thus, the intersection of the orthogonal complement of the spacelike vector v_P with the tangent space T_[f]M is a 2-plane of signature (1,1). This implies that M is Lorentzian, i.e., of signature (1,2).* A root of multiplicity 2. Observe that if there is a non-real root of multiplicity 2, when we are in the first "no real root" case. Hence we consider here only the case where the root of multiplicity 2 lies in ^1. Then, we have three subcases to consider: – two distinct real roots of multiplicity 2: The orbit induced at X^2Y^2 is the image of the PSL(2,)-equivariant map ^1,1⊂(_2[X,Y])⟶^2,2,[L]↦ [L^2], where _2[X,Y] is the vector space of homogeneous polynomials of degree 2 in two variables X and Y, endowed with discriminant as a PSL(2,)-invariant bilinear form of signature (1,2) <cit.>. (Here, L is the projective class of a Lorentzian bilinear form on ^2). The vectors tangent to the orbit at X^2Y^2 induced by the 1-parameter subgroups P and E are v_P=-2XY^3 and v_E=2X^3Y-2XY^3, respectively. It is easy to see that the 2-plane generated by {v_p,v_E} is of signature (1,1). Hence, the orbit induced at X^2Y^2 is Lorentzian.– three distinct real roots, one of them being of multiplicity 2: Denote by M the orbit induced by PSL(2,) at [f]=[XY^2(X-Y)]. The vectors tangent to M at [f] induced by the 1-parameter subgroups H, P and E are: v_H=-2XY^3, v_P=Y^4-2XY^3, v_E=Y^4-X^4-2X^2Y^2+X^3Y-XY^3,respectively. Obviously, the lightlike vector v_H+v_P is orthogonal to T_[f]M. Therefore, the restriction of the metric on T_[f]M is degenerate. It is easy to see that the quotient of T_[f]Mby the action of the isotropic line (v_H+v_P) is of signature (1,1). Thus, M is of signature (1,1,1).– one real root of multiplicity 2, and one root in ℍ^2: Denote by M the orbit induced by PSL(2,) at [f]=[Y^2(X^2+Y^2)]. The vectors tangent to M at [f] induced by the 1-parameter subgroups H, P and E are v_H=4Y^4,v_P=-2XY^3,v_E=2X^3Y+2XY^3, respectively. Obviously, the lightlike vector v_H is orthogonal T_[f]M. Therefore, the restriction of the metric on T_[f]M is degenerate. It is easy to see that the quotient of T_[f]Mby the action of the isotropic line (v_H) is of signature (1,1). Thus M is of signature (1,1,1).* Two distinct real roots, and a complex root z in ℍ^2 making an angle θ smaller than 5π/6 with the two real roots. Denote by M the orbit induced by PSL(2,) at [f]=[Y(X^2+Y^2)(X-rY)]. The vectors tangent to M at [f] induced by the 1-parameter subgroups H, P and E are: v_H=-4rY^4-2X^3 Y+2XY^3, v_P=-3X^2Y^2+2r XY^3-Y^4,v_E=X^4-Y^4-2rX^3Y-2rXY^3, respectively. The following set of vectors is an orthogonal basis for T_[f]M where the first vector is timelike and the two others are spacelike.{(7r+3r^3)v_H+(6-2r^2)v_P+(5+r^2)v_E,4v_P+ v_E,v_H}.Therefore, M is Lorentzian, i.e., of signature (1,2). □00Barbot T. Barbot, V. Charette, T. Drumm, W.M. Goldman, K. Melnick, A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys, (2008) 179-229.Col B. Collier, N, Tholozan, J. Toulisse, The geometry of maximal representations of surface groups into SO(2,n).https://arxiv.org/abs/1702.08799.Scala A.J. Di Scala, T. Leistner, Connected subgroups of SO(2,n) acting irreducibly on ^2,n. Inst. Hautes Études Sci. Publ. Math. No. 78 (1993) 187-206.Fra C. Frances, Géometrie et dynamique lorentziennes conformes. Thèse, E.N.S. Lyon (2002).Masoudthesis M. Hassani, Cohomogeneity one actions on the three-dimensional Einstein universe. Work in Progress. | http://arxiv.org/abs/1706.08813v1 | {
"authors": [
"Masoud Hassani"
],
"categories": [
"math.DG",
"math.MG",
"math.RT"
],
"primary_category": "math.DG",
"published": "20170627122148",
"title": "On the irreducible action of psl(2, r) on the 3-dimensional einstein universe"
} |
Methanol in HD 101584 Dept. of Space, Earth and Environment, Chalmers Univ. of Technology, Onsala Space Observatory, SE-43992 Onsala, [email protected], Karl-Schwarzschild-Str. 2, D-85748 Garching bei München, GermanyJoint ALMA Observatory, Alonso de Cordova 3107, Vitacura, Santiago de Chile, ChileESO, Alonso de Cordova 3107, Vitacura, Santiago, Chile Dept. of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, SwedenThe circumstellar environments of objects on the asymptotic giant branch and beyond are rich in molecular species. Nevertheless, methanol has never been detected in such an object, and is therefore often taken as a clear signpost for a young stellar object. However, we report the first detection of CH_3OH in a post-AGB object, HD 101584, using ALMA. Its emission, together with emissions from CO, SiO, SO, CS, and H_2CO, comes from two extreme velocity spots on either side of the object where a high-velocity outflow appears to interact with the surrounding medium. We have derived molecular abundances, and propose that the detected molecular species are the effect of a post-shock chemistry where circumstellar grains play a role. We further provide evidence that HD 101584 was a low-mass, M-type AGB star. First detection of methanol towards a post-AGB object, HD 101584 H. Olofsson1 W.H.T. Vlemmings 1 P. Bergman 1 E.M.L. Humphreys 2 M. Lindqvist 1 M. Maercker 1 L. Nyman 3,4 S. Ramstedt 5 D. Tafoya1Received 14 May 2017; accepted 19 June 2017 =======================================================================================================================================================================================================================================§ INTRODUCTION The circumstellar envelopes (CSEs) of asymptotic giant branch (AGB) and post-AGB objects have turned out to be rich in different molecular species; more than 100 are now detected. This is the effect of a number of different processes, such as stellar atmosphere equilibrium chemistry, extended atmosphere non-equilibrium chemistry, and photo-induced circumstellar chemistry <cit.>. The post-AGB objects have a special niche in terms of chemistry because of the increased internal UV light and the presence of shocks where fast winds interact with slower-moving material. The result being that they often show molecular species that are not detected (or tend to be much weaker) in AGB CSEs; for example, a number of ions <cit.>. One species, methanol (CH_3OH), has escaped every attempt at detection in an AGB-related object <cit.>, despite detectable predicted abundances <cit.>. It is therefore taken as a clear signpost for a young stellar object (YSO) <cit.>. However, in the course of a chemical study of an interesting post-AGB object HD 101584 using ALMA, we have detected methanol for the first time in an AGB-related object, and in this Letter we discuss its origin together with detections of CO, SiO, SO, CS, and H_2CO.HD 101584 is a bright star (V ≈ 7^ m) of spectral type A6Ia <cit.>. It was shown to have a large far infrared excess and an evolutionary status at, or shortly after, the end of the AGB was proposed by <cit.> and further corroborated by <cit.>. It has also been shown to be a binary system <cit.>. The distance is estimated to be 0.7 kpc, but recent Gaia data suggests a somewhat larger distance, 0.9 – 1.8 kpc. The HST images show only a diffuse circumstellar medium in dust-scattered light <cit.>, but its circumstellar gas characteristics are remarkable.<cit.> used ALMA ^12CO(J = ) data to identify a narrow, ≈ 10 long, high-velocity molecular outflow directed at PA ≈ 90^∘. Its velocity range covers almost 300 km s^-1 and has a Hubble-like gradient. The outflow is seen almost along its axis. The kinematical age is estimated to be500 yr. There is an hour-glass structure surrounding the outflow, and a complex structure within 1 (in radius) of HD 101584, most likely a torus-like component centered on a circumbinary disc. Here we provide further evidence, based on circumstellar isotopolog ratios, for the post-AGB nature of HD 101584.§ OBSERVATIONS The ALMA data were obtained during Cycles 1 and 3 with 35 to 43 antennas of the 12 m main array in two frequency settings in Band 6, one for the ^12CO(J = ) line and one for the ^13CO(J = ) line (only Cycle 1). In both cases, the data set contains four 1.875 GHz spectral windows with 3840 channels each. The baselines range from 13 to 12934 m.<cit.> concluded that very little flux is lost even in the ^12CO() ALMA data, in particular at the extreme velocities and close to the systemic velocity.Bandpass calibration was performed on J1107-4449, and gain calibration on J1131-5818 (Cycle 1) and J1132-5606 (Cycle 3). Flux calibration was done using Ceres and Titan. Based on the calibrator fluxes, we estimate the absolute flux calibration to be accurate to within 5%.The data were reduced using CASA 4.5.2. After corrections for the time and frequency dependence of the system temperatures, and rapid atmospheric variations at each antenna using water vapour radiometer data, bandpass and gain calibration were done. For the ^12CO(J = ) setting, data obtained in three different configurations were combined. Subsequently, for each individual tuning, self-calibration was performed on the strong continuum. Imaging was done using the CASA clean algorithm after a continuum subtraction was performed on the emission line data. The final line images were created using natural weighting. Complementary ^12CO J =and , and C^17O and C^18O J =line data were obtained using APEX. The Swedish heterodyne facility instruments APEX-1,2 <cit.> and APEX-3were used together with the facility FFT spectrometer covering about 4 GHz. The observations were made from August to October 2015 in dual-beamswitch mode with a beam throw of 2. Regular pointing checks were made on strong CO line emitters and continuum sources. Typically, the pointing was found to be consistent with the pointing model within 3. The antenna temperature, T_A^⋆, is corrected for atmospheric attenuation. A Jy/K conversion of 40 and 48 was adopted for APEX-1,2 and APEX-3, respectively. The uncertainty in the absolute intensity scale is estimated to be about ± 20%.Finally, in our analysis we make use of the ^12CO and ^13CO J =data published by <cit.>. § DISCUSSION §.§ The evolutionary nature of HD 101584The most likely interpretation of the A6Ia spectral classification and the circumstellar characteristics is that HD 101584 is a post-AGB object,but an evolved massive supergiant or possibly a star of young age remain alternatives. However, HD 101584, being a high-latitude source, is not associated with any star-forming region or molecular cloud. Nevertheless, since here we claim the first detection of methanol in an AGB-related object, we provide some further evidence that supports the post-AGB nature of HD 101584.The circumstellar chemistry (detections of e.g., SO, SO_2, and OCS, Olofsson et al. in prep.), the strong 1667 MHz OH maser <cit.>, and the presence of a silicate feature <cit.> all strongly favor the idea that the circumstellar medium of HD 101584 is O-rich (C/O < 1). A stellar ^12C/^13C ratio that is not consistent with that in the local interstellar medium, 45–70 <cit.>, nor with that of the Sun, 87 <cit.>, would be a strong argument for an AGB-related object. We provide such an estimate where opacity and chemistry are expected to have little effect by using the APEX and ALMA detections of the ^12C^17O and ^13C^17O J =lines, respectively (Fig. <ref>).We have determined the intensities of the emission coming from the very central region, within 05 of HD 101584 in the ALMA data and using the narrow feature in a Gaussian line decomposition of the single-dish data (Table <ref>). The integrated line intensity ratio is 13±6, suggesting a low ^12C/^13C ratio in line with matter that has been processed in the CNO-cycle and brought to the surface in an AGB star;<cit.>, for example, find circumstellar ^12CO/^13CO ratios in the range 6–30 for M-type AGB stars, an effect believed to be due to evolution on the Red Giant Branch.The ^17O/^18O ratio is a measure of the initial mass of an AGB star provided that it is not affected by hot-bottom-burning (HBB), that is, M_ i4 M_⊙ <cit.>. During HBB the ^17O/^18O ratio rapidly becomes very high <cit.>.The intensity ratio of the APEX C^17O and C^18O J =lines (Fig. <ref>), 0.20±0.08, is expected to be a good measure of the ^17O/^18O ratio. This shows that HD 101584 has not gone through HBB, and the low ratio suggests an initial mass of ≈ 1 M_⊙ <cit.>. This low mass is also consistent with the fact that HD 101584 has not evolved into a carbon star, and with the present-day mass estimate of <cit.> provided that only the core mass remains. Thus, the inferred isotope ratios provide strong evidence that HD 101584 was a low-mass, M-type AGB star. §.§ The extreme velocity spotsThe high-velocity outflow is marked with a number of emission spots along its extent, particularly prominent in the SiO() data (Fig. <ref>). The end-points at about 4 W and 4 E of the central star are visible as distinct features at the extreme velocities υ_ LSR ≈ –100 and 185 km s^-1, respectively, in many of the detected molecular lines (we refer to <cit.>). Notably, the H_2CO (3_03–2_02, 3_22–2_21, and 3_21–2_20) line emissions emanate most strongly from these extreme-velocity spots (EVSs) (Fig. <ref>). This suggests special conditions, either in chemical or excitation terms, and the presence of shocked gas is likely. With an inclination angle of 10^∘, the physical distances between the star and the EVSs are ≈ 3×10^17 cm, that is, ≈ 0.1 pc or 20000 au. §.§ The detection of methanol We detect two CH_3OH lines, the J_K = 4_2–3_1 line at 218.440 GHz and the 8_-1–7_0 line at 229.759 GHz (Fig. <ref>). In the frequency range covered by our ALMA data, these are also the strongest lines in, for example, star-forming regions <cit.>, where the 8_-1–7_0 line has a tendency to show maser emission <cit.>, but there is no indication of this in our data. At the sensitivity obtained, the CH_3OH line emission comes exclusively from the EVSs (Fig. <ref>), further supporting the idea that the conditions are special in these regions. To our knowledge, this is the first detection ever of methanol in an AGB-related object. We note that IRAS 19312+1950 has been detected in methanol, but it has been convincingly shown to be a young object rather than an AGB star <cit.>. Likewise, methanol was detected towards IRC+10420, but this is an evolved massive star <cit.>. §.§ Molecular abundances We need a physical model for an EVS to derive the molecular abundances through a radiative transfer analysis. The observational data provide limited information here except that the emission is largely confined to a region of size1, and the emission from the rarer species seems to come from a region at least two times smaller than this. The geometry is also uncertain. In order to get at least order-of-magnitude estimates we assume a spherical clump with an inner higher-density, higher-temperature region (015 in diameter) surrounded by an outer lower-density, lower-temperature envelope (1 in diameter; required to mainly reproduce the CO line intensities); we refer to Table <ref>. Radiative excitation due to central star light (too distant) and dust emission inside the clump (too low optical depth) can be safely ignored.Based on this physical model we solve the radiative transfer using an Accelerated-Lambda-Iteration code, taking into account excitation through collisions with H_2. Collisional coefficients for CH_3OH were taken from <cit.>. We have not included the torsionally excited states of methanol, which may lead to an underestimate of its abundance by up to a factor of two. We use as input the line intensities for the 4 E EVS where the emission is slightly stronger (Table <ref>). The size of the higher-density region is constrained by requiring that the ^28SiO/^29SiO abundance ratio equals 20 (the solar value) since an AGB star is not expected to alter this ratio. Its size fits well the observed sizes of the most intense molecular line emissions of, for example, H_2CO and CH_3OH. The densities and kinetic temperatures are reasonably constrained by the observed CO, H_2CO, and CH_3OH line intensities (Table <ref>).The resulting fractional abundances with regards to H_2 are listed in Table <ref>. The ^12CO abundance (5×10^-4) is at the level expected in an O-rich circumstellar gas (full association of CO and solar values for O and C results in a fractional CO abundance of 5×10^-4). Furthermore, the ^12CO/^13CO ratio is ≈ 15, that is, in very good agreement with the value derived in Sect. <ref>. Importantly, this means that the EVS material is dominated by circumstellar gas (possibly swept-up from the previous AGB wind), not by swept-up interstellar material. This is strengthened by the H_2CO/H_2^13CO line intensity ratio, 12^+12_-6. The C^16O/C^18O ratio of 210 (the solar value is 480, <cit.>) is somewhat low since AGB stars are expected to destroy rather than produce ^18O, but considering our simple model we draw no conclusions based on this result.The estimated gas mass of the 4 E EVS is 4×10^-3 M_⊙. A crude estimate shows that such a clump would produce a 1.3 mm continuum flux density that is lower than the noise limit in our ALMA data.§.§ Chemistry We focus here on the results for the inner region of the 4 E EVS. The SO (1×10^-6), SiO (5×10^-6), and CS (1×10^-6; after correcting the ^13CS abundance by the estimated ^12C/^13C ≈ 10) abundances all lie in the range reported for AGB CSEs <cit.>, while the H_2CO abundance (2×10^-6; assuming an ortho-to-para ratio of 3) is about an order of magnitude higher than in IK Tau <cit.>. CH_3OH has an estimated abundance of 3×10^-6 (assuming an E-to-A ratio of 1), almost two orders of magnitude higher than towards the supergiant IRC+10420 <cit.>. However, the large distance between HD 101584 and the EVS (3×10^17 cm) combined with a reasonable expansion velocity of the AGB wind (15 km s^-1) indicate a time scale of ≈ 6500 yr. This strongly suggests that due to photodissociation all detected species, except CO which is self-shielding, have their origin in the EVS.A comparison with the results for OH231.8+4.2, a post-AGB object with a rich molecular setup (incl. SO, SiO, CS, and H_2CO) and a number of characteristics similar to those of HD 101584 <cit.>, shows that also here some of the species are particularly abundant in regions where shocks are likely present, although these are not associated with the outer extremes of its high-velocity outflow. Another interesting comparison can be made with the results for the high-velocity outflows of young stellar objects (YSOs), where in particular the features at the extreme velocities resemble the EVS emission features of HD 101584 <cit.>. Interestingly, the detected abundant molecules, including CH_3OH, are largely to be the same <cit.>; as are the masses and temperatures of the clumps.It is therefore tempting to compare with the work on the chemistry of such outflows, for example, that of L1157-B1 <cit.>. In this case a chemical model where gas-grain interaction, including freeze-out and chemical processing over ≈ 10^5 yr, and subsequent release of the formed species by a C-shock, works reasonably well to explain the observed abundances <cit.>. However, there are notable differences with HD 101584.In the latter, the shock works on pre-existing circumstellar grains, not interstellar grains coated in a proto-stellar environment. Further, in HD 101584 the observed CH_3OH/H_2CO abundance ratio is ≈ 1 as opposed to a value of ≈ 20 for L1157-B1 <cit.>. Finally, H_2S is detected in L1157-B1 <cit.>, while this is not the case for the EVSs of HD 101584 despite the fact that the H_2S(2_20–2_11) line, emanating from the central region of HD 101584, is almost as strong as the ^12CO(2–1) line (about 30% of its strength) (Olofsson et al., in prep). An alternative explanation could be evaporation of pre-existing planetary system objects, but also here the observed CH_3OH/H_2CO abundance ratio is very different, for example, ≈ 10 in comets <cit.>.Lacking an obvious explanation, but building on the similarity with the YSO high-velocity outflows, we propose that the circumstellar grains around HD 101584 have had time to develop some surface chemistry, the result of which is liberated when the high-velocity gas hits the circumstellar medium.§ CONCLUSIONS We have, for the first time, detected CH_3OH towards an AGB-related object, HD 101584. Among other things, this is interesting since the detection of methanol is normally taken as being characteristic of star-forming activity. The detections of CS, SO, SiO, H_2CO, and CH_3OH in the EVSs of HD 101584 follow a very similar pattern of molecular detections in the extreme velocity flows of YSOs. However, there are significant differences both in the environmental conditions and in the observed (relative) abundances. Nevertheless, based on the similarity, we propose that the detected molecular species in the EVSs have their origin in a post-shock chemistry where circumstellar grains play a role; the details of this situation, however, remain to be elucidated. It would be interesting to perform searches for CH_3OH in other post-AGB objects where shocks are likely to be present, for example, the water fountain sources <cit.>. Finally, we have added evidence that HD 101584 is a post-AGB object, the remnant of a solar-mass M-type AGB star.HO and WV acknowledge support from the Swedish Research Council. WV acknowledges support from the ERC through consolidator grant 614264. This Letter makes use of the following ALMA data: ADS/JAO.ALMA#2012.1.00248.S and #2015.1.00078.S. 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A., et al. 2011, , 740, L3[Willacy & Millar(1997)]willmill97 Willacy, K. & Millar, T. J. 1997, , 324, 237 § OBSERVATIONAL RESULTS | http://arxiv.org/abs/1706.08254v1 | {
"authors": [
"H. Olofsson",
"W. H. T. Vlemmings",
"P. Bergman",
"E. M. L. Humphreys",
"M. Lindqvist. M. Maercker",
"L. Nyman",
"S. Ramstedt",
"D. Tafoya"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170626071724",
"title": "First detection of methanol towards a post-AGB object, HD101584"
} |
[email protected]@physics.gatech.eduSchool of Physics, Georgia Institute of Technology,Atlanta, Georgia 30332-0430 A constructive theoretical platform for the description of quantum space-time crystals uncovers for Ninteracting and ring-confined rotating particles the existence of low-lying states with proper space-time crystalbehavior. The construction of the corresponding many-body trial wave functions proceeds first via symmetry breaking at the mean-field level followed by symmetry restoration using projection techniques.The ensuing correlated many-body wave functionsare stationary states and preserve the rotational symmetries, and at the same time they reflect the point-groupsymmetries of the mean-field crystals. This behavior results in the emergence of sequences of select magic angularmomenta L_m. For angular-momenta away from the magic values, the trial functions vanish. Symmetry breaking beyondmean field can be induced by superpositions of such good-L_m many-body stationary states. We show thatsuperposing a pair of adjacent magic angular momenta states leads to formation of special broken-symmetry statesexhibiting quantum space-time-crystal behavior. In particular, the corresponding particle densities rotate around thering, showing undamped and nondispersed periodic crystalline evolution in both space and time. The experimentalsynthesis of such quantum space-time-crystal wave packets is predicted to be favored in the vicinityof ground-stateenergy crossings of the Aharonov-Bohm-type spectra accessed via an externally applied magnetic field. These results are illustrated here for Coulomb-repelling fermionic ions and for a lump of contact-interaction attracting bosons. Trial wave functions for ring-trapped ions and neutral atoms:Microscopic description of the quantum space-time crystal Uzi Landman 21 September 2017 ========================================================================================================================§ INTRODUCTION Groundbreaking experimental progress<cit.>in the field of trapped ultracold ions and neutral atoms, inparticular the unprecedented control of interpaticle interactions and the attainment of ultracold temperatures,offer these systems as prime resources for experimental realization of the emergent exciting concept of aquantumspace-time crystal (QSTC). Inspired by the relativistic 3+1-dimensions analogy <cit.>,the QSTC idea extends translational symmetry breaking (SB) to encompass both the spatial and time dimensions. Indeed, the original QSTC proposal<cit.> motivated an abundance of scientific discussion, commentary, and exploration <cit.>. The original QSTC was proposed in the form of crystalline spatial-particle-density arrangements <cit.>, orother solitonic-type (charge-density-wave) formations <cit.> revolving around a ring-shaped ultracold trapwithout dispersion or damping.Although significant experimental progress has been reported toward this goal <cit.>, formation of aQSTC in this experimental configuration is yet to be demonstrated. At the same time, experimental progress for a“weaker class” <cit.> of discrete-time-crystals <cit.> limitedexclusively to the time domain has been reported <cit.>, employing time-periodically-driven spinsystems. Contributing to this state of affairs are limitationsof earlier theoretical treatments of the QSTCthat were discussed extensively in previous commentary <cit.>, e.g., limiting oneselfto mean-field (MF) dynamics <cit.>, or considering solely the energetics of states with good total angularmomenta which (as a matter of principle) have uniform spatial densities <cit.>. To throw further lighton the nature and properties of QSTCs, it is imperative that a formulation and implementation of appropriatemany-body trial wave functions for the QSTC on a ring be advanced. The sought-after trial wave functions shouldexplore for a finite system of N particles the interplay <cit.> between the mean-field symmetry-brokenstates, which are not eigenstates of the total angular momentum L̂, and the exact symmetry-preserving (goodtotal-angular-momentum) states.Here, we introduce such trial wave functions and analyze their spectra and combined spatially dispersionless andtemporally undamped evolution,which are the defining characteristics of a QSTC. Contrasting with these findings,previous beyond-mean-field theoretical studies <cit.> that investigated spatialsolitonic formations in finite boson systems in one dimension or on a ring have revealed drastically different behaviors, such as increasing dispersion with time accompanied by a revival at the initial position of the propagated inhomogeneous wave packet <cit.>.We employ a beyond-mean-field methodology of symmetry restoration via projection techniques, introduced by uspreviously <cit.>for two-dimensional semiconductor quantum dots (with and without an applied magnetic field B). Themultilevel symmetry-breaking and symmetry-restoration approach which we persue provides acomplete theoretical framework for treating symmetry breaking aspects in finite systems, without reference to the N →∞ limit. Indeed this approach originated, and is widely employed, in nuclear physics and chemistry<cit.>.The paper is organized as follows. In Section II we introduce and illustrate the hierarchical, multilevel methodology that we use for the construction of the trial wave functions for the microscopic many-body Hamiltonian of few ultracold ring-confined interacting particles. Following a short synopsis of the method, we discuss first in sectionII A the mean-field, broken-symmetry state, and subsequently in section II B a beyond-mean-field level is outlined, entailing symmetry-restoration via the use of an angular momentum projection technique.This results in many-bodystationary-state good-angular-momentum solutions of the microscopic Hamiltonian. These (projected) symmetry-restoredstates show uniform particle density around the ring. However, simultaneously they posses hidden crystalline symmetrieswhich can be revealed through the analysis of the corresponding conditional probability densities. In section III Cwe complete our exposition of the construction of the QSTC trial wave functions by analyzing the properties ofsuperpositions of pairs of the above-noted symmetry-restored (projected) stationary states [see Eq. (<ref>)] thatare favored to mix in the vicinity of crossings of Aharonov-Bohm-type spectra of ground-state energies versusapplied magnetic-field (Fig. <ref>). The particle density corresponding to such superposed wavefunctions reveals crystalline structure on the ring.Numerical solutions using the trial wave functions are illustrated and analyzedfor the case of few (even and odd in number) Coulomb-repelling fermionic ions, and for a lump of contact-interactingattractive bosons. When evolved with the microscopic many-body Hamiltonian, these trial functions exhibit, for boththe fermionic repelling ions and attracting bosons, undamped and non-dispersive space and time crystalline periodicevolution – that is, they exhibit breaking of both the space and time symmetries. Section III is devoted to further elaborationon three main topics. In section III A we discuss the symmetryproperties of the symmetry-restored (projected) wave functions and the selection rules for their “magic” angularmomenta. Section III B analyzes the properties of the initial wave packets and their associated time evolution, andsection III C comments on the relation between the constructed trial functions (in particularthe aforementionedsymmetry-restored stationary states) and the wavefunctions obtained through exact-diagonalization[configuration-interaction (CI)] solutions of the microscopic many-body Hamiltonian. We conclude in section IV with a summary of our work, including a brief listing of recent progress achieved indeveloping experimental techniques for preparation and measurement of ring-confined ultracold particles. TheAppendices give tables of numerical results (rotational energies for different magic angular momenta, and moments ofinertia) for the systems investigated in the paper, as well as explicit expressions for the conditional probabilitydistribution and single particle density. § THE INTERPLAY BETWEEN SYMMETRY-BROKEN AND SYMMETRY-PRESERVING STATES: GROUP-THEORETICAL FORMULATIONIn connection with the QSTC, we consider three levels of many-body trial wave functions: (1) A Slater determinant forlocalized fermions (or permanent for localized bosons) on the ring. We denote this wave function by Ψ^ SB; itcorresponds to the unrestricted Hartree-Fock (UHF), or Gross-Pitaevskii (GP), mean-field step <cit.> that exhibits symmetry breaking of the space degrees of freedom. Ψ^ SB does not preserve the total angular momentum. Out of the three levels in the hierarchical scheme (see below), it is the trial wave functionclosest to the familiar concept of a classical Wigner crystal <cit.>.(2) A stationary multideterminantal (multipermanental)wave function Φ^ PROJ_L characterized by a good total angular momentum ħ L, which is generatedby applying a projection operator P_L (see below) onΨ^ SB. This step goes beyond the MF approximation and restores (as required) the quantum many-body Hamiltonian symmetries in the stationary-state solutions. Unlike Ψ^ SB, Φ^ PROJ_L exhibits anazimuthally uniform single-particle density [SPD, ρ( r,t)], which is also time-independent (stationary). Previously, we referred to such projected wave functions Φ^ PROJ_L as quantum rotating Wignermolecules <cit.>.(3) Coupling between the stationary states (brought about by a perturbation which we term in the following as“the pinning agent”) results in a superposition of two projected wave functions with different angular momentaL_1 and L_2, leading to formation of a pinned Wigner molecule (PWM), i.e., Φ^ PWM(L_1,L_2; t=0) =αΦ^ PROJ_L_1 + β e^iϕ(t=0)Φ^ PROJ_L_2, where ϕ(t=0) can be set to zero without loss of generality, and α^2+β^2=1. In the following we illustrate the case of α=β =1/√(2) (the physics of the PWM maintains for otherchoices of the mixing coefficients).For selected magic (see below) L_1 and L_2, Φ^ PWM(L_1,L_2; t=0) represent a special family of quantalwave packets with broken azimuthal symmetry.Consequently their corresponding ρ( r,t=0) are not uniform, forminginstead a crystal-like particle density pattern, with kN, k=1,2,3,… possible peaks for N fermionic ionsand 1,2,3,… possible peaks for N attractive bosons. When the pinning agent is lifted, theΦ^ PWM(L_1,L_2;t) evolve in time undamped according to the exact many-body quantum Hamiltonian dynamics,i.e., the phase ϕ will vary as ϕ(t)=(E_2-E_1) t/ħ, and the associated ρ( r,t) willoscillate at any given space point with a time period T=τ/n; τ=2πħ/|E_1-E_2|, E_1(2)being the energies of the stationary states Φ^ PROJ_L_i with i=1,2, respectively.For either statistics (fermions or bosons), n = N for repelling ions and n = 1 for attractive particles;as aformentioned, here we discuss explicitly Coulomb-repelling fermionic ions and contact-interacting attractivebosons. Such undamped and dispersionless periodic time variation is not possible for the MF Hartree-Fock (orGross-Pitaevskii) wave packet Ψ^ SB, because it contains all the possible angular momenta when expandedin the complete basis set of the stationary wave functions Φ^ PROJ_L. Additionally, the MF wave functionslose <cit.> their single-determinant (single-permanent) character under the exact time evolution. The many-body Hamiltonian of N identical particles in a ring-type trap threaded by a constantmagnetic field B isH= ∑_i=1^N( ( p_i-η A_i)^2/2M + (r_i-R)^2/2l_0^2/(ħω_0)) + ∑_i < j V( r_ij),where A( r)= B× r/2 is the vector potential in the symmetric gauge, r=√(x^2+y^2), ω_0 is the frequency of the trap, R=√(X^2+Y^2) is the ring radius,the oscillator length l_0=√(ħ/(Mω_0)), and r_ij=| r_i- r_j|. B can be thefamiliar magnetic field in the case of charged ions (when η=e/c), or a synthetic one in the case of ultracold neutral atoms <cit.>. §.§ First level: The mean-field-ansatz, symmetry-broken crystalline stateIn view of the strong inter-particle interactions (large values of the parameters R_W and R_δ, see below)associated with the proposed experimental realizations of the QSTC's <cit.>, resulting, for ultracoldquasi-1D-ring-trapped particles, in formationof a Wigner crystal (in the case of Coulomb repelling ultracold ions) or a lump (in the case of contactattracting bosons), we construct here the symmetry-broken initial state via the use of an ansatz. This ansatz exploresthe localized nature of the N space orbitals (corresponding to the N ring-trapped particles) from which thesingle broken-symmetry determinant (permanent) is formed in the corresponding fermion (boson) systems.Such ansatz proved most adequate to approximate broken-symmetry mean-field solutions (reflecting individual particle localization) in previous studies of fermions and bosons in harmonically confined quantum dots and other 2Dsystems <cit.>. We describe each particle localized at position R_j as a displaced Gaussian functionu( r,R_j) =1/√(π)λexp( -( r- R_j)^2/2λ^2 -i φ( r, R_j; B) ),with λ=√(ħ/(MΩ)); Ω=√(ω_0^2+ω_c^2/4) whereω_c=η B/M is the cyclotron frequency. The phase in Eq. (<ref>) is due to the gaugeinvariance of magnetic translations <cit.>) and is given by φ( r, R_j; B)= (xY_j-yX_j)/(2l_B^2), with l_B=√(ħ /(η B)) being the magnetic length. For simplicity, in the following we provide examples for only three cases: (i) that of N fully polarized fermionic ions with oddN, (ii) that of N fully polarized fermionic ions with even N, and (iii) that of N spinless bosons interactingvia an attractive contact potential. As will be shown explicitly, case (ii) presents different characteristics comparedto case (i).In the case of ultracold ions repelling each other via the Coulomb interaction, we take theR_j=R_ eq e^2π (j-1) i/N, j=1,2,…,N to coincidewith the equilibrium positions (forming a regular polygon) of N classical charges inside the annularconfinement specified in Eq. (<ref>). Then R_eq (> R) is given by the real solution of the cubic equation aw^3+bw^2+d=0, where a=1, b=-R, d=-l_0^3 R_W S_N/4, with the Wigner parameter (the ratio between the characteristic interparticle repulsion and the kinetic zero-point energy of thering-confined particle), R_W=e^2/(l_0 ħω_0) <cit.> and S_N=∑_j=2^N 1/sin[(j-1)π/N]. Then the corresponding MF wave function, Ψ^ SB, is the determinant formed by the N orbitalsu( r_i,R_j).In the case of N ultracold neutral bosons attracting each other with a contact interaction-g δ( r_i- r_j),the atoms are localized at the same position, and thus R_ eq=R and R_j=R e^iθ_0, j=1,2,…,N. Then the MF wave function, Ψ^ SB, is the product (permanent) of the orbitals u( r_i, R e^iθ_0). The parameter corresponding to R_W is given here byR_δ =g M/ħ^2.§.§ Second level (beyond mean field): The projected, symmetry-restored stationary stateA stationary many-body state that preserves the total angular momentum, as well as the rotational symmetry of theannular trap, can be projected out of the symmetry-broken Ψ^ SB by applying the projector operatorP_L,P_L = 1/2π∫_0^2π e^i γ (L-L̂) dγ,where L̂=∑_i=1^N l̂_i, i=1,2,…,N, and ħL̂ is the total angular-momentumoperator. Then the projected many-body state is given byΦ^ PROJ_L =1/2π∫_0^2π dγΨ^ SB(γ) e^iγ L. P_L is analogous to the projector operators used in chemistry for molecular orbitals governed by pointgroup symmetries <cit.>. Such projection operators are constructedthrough a summation over the characters of the point group <cit.>;the phases e^iγ L are the characters of the rotational group in two dimensions <cit.>and the operator e^-i γL̂ is the corresponding generator of 2D rotations. Alternatively, Eq.(<ref>) may be viewed as a linear superposition of all the (energy-degenerate) symmetry-broken statesΨ^ SB(γ), azimuthally rotated by γ. Due to the rotational symmetry, the coefficients of this superposition, i.e., the phases e^iγ L, can be determined a priori, without the need to diagonalize a Hamiltonian matrix.The projected energies, associated with the stationary wave functions Φ^ PROJ_L, are given byE^ PROJ(L)=∫_0^2π h(γ) e^iγ Ldγ/∫_0^2π n(γ)e^iγ L dγ,where h(γ)=⟨Ψ^ SB(0)| H| Ψ^ SB(γ) ⟩,and the norm overlapn(γ)=⟨Ψ^ SB(0)| Ψ^ SB(γ) ⟩enforces proper normalization of Φ^ PROJ_L. Note that the original double integration reduces to asingle integration over γ because P_L^2= P_L, [ P_L, H]=0. We note that the unresticted HF energies for the ansatz determinant (or permanent), Ψ^ SB, before projection are simply given byE_ UHF=h(0)/n(0). We have carried out numerical calculations to determine the rotational spectrum of the Φ^ PROJ_L's. For the calculation of h(γ) and n(γ), we use the rules for determinants composed of nonorthogonalorbitals; see, e.g., Ref. <cit.>. Similar rules apply for permanents. The numerical calculations are facilitated by the fact that the one-body and two-body matrix elements between the orbitalsu( r,R_j) have closed analytic expressions <cit.>. In all three cases [(i) odd number of repelling fermions, (ii) even number of repelling fermions, and (iii)attractive bosons], and for all values of N ≤ 10, large localization parameters R_W ≥ 200 andR_δ≥ 50, and large ratios R/l_0 ≥ 40 that we studied, we found that indeed the numerically calculatedenergies of the Φ^ PROJ_L's according to Eq. (<ref>) (see, e.g., Tables <ref>, <ref>, and<ref> in Appendix <ref>) can be well-fitted by that of an Aharonov-Bohm-typespectrum associated with a quantum many-body rigid rotor (see also <cit.>), i.e., E^ PROJ(L) ≈ V_ int + C_R (L - N Φ/Φ_0)^2.V_ int approximates the ground-state energy of the few-particle system and takes different values fordifferent many-body wave functions. The numerically determined coefficient C_R is essentially a constant (see below). Φ=π R_ eq^2 B is the magnetic flux through the ring and Φ_0=h/η is the magnetic flux quantum. Thevalues of the angular momenta L are not arbitrary. Because of the crystalline symmetries, as well as the symmetricor antisymmetric behavior under particle exchange, they are given by proper sequences of magic angular momenta L_m(see section <ref> below for further discussion). In particular, all values of angular momenta are allowed forthe case of attractive bosons, i.e., L_m=0,± 1,± 2,…. For the case of fully polarized repelling fermionswith N odd or spinless repelling bosons with any N, the allowed angular momenta are restricted to the sequenceL_m=kN, with k=0,± 1,± 2,….For the case of fully polarized repelling fermions with N even, the allowed angular momenta are given by a differentsequence L_m=(k+1/2)N, with k=0,± 1,± 2,….Due to the very large values of R_W and R_δ, the value of C_R is very close to that of aclassical rigid rotor, corresponding to N point particles in their equilibrium configuration inside the annularconfinement, i.e., C_R ≈ C_R^ cl=ħ^2/[2I(R_ eq)], with inertia moment I(R_ eq)=NM R_ eq^2. As typical examples, in Table <ref>, Table <ref>, and Table <ref>of Appendix <ref>, we list calculated energies according to Eq. (<ref>) for an odd number N=7 and an even number N=8 of fermionic ions, as well as for N=10 attractive bosons, respectively; R_W=1000for the repelling ions and R_δ=50 for the attractive bosons. The ratio f̃≡ C_R/C_R^ cl≈ 1 for all L < 165 for ions, and for all L ≤ 30 for attractive bosons. As aforementioned, the rigid-rotor-type spectrum in Eq. (<ref>) was explored earlier in theQSTC literature <cit.>; however, by itself it does not lead to the derivation of appropriate QSTC wavefunctions. The demonstrated agreement between the microscopically calculated rotational part of the spectrum[Eq. (<ref>)] and the analytic second term in Eq. (<ref>) expected for a QSTC <cit.> validates the expressions Φ^ PROJ_L introduced in Eq. (<ref>) as proper trial wave functions for the QSTC.We showed previously that the limit of a quantum rigid rotor for a system of strongly interacting particles canalso be reached in external confinements with geometries other than the ring geometry. In particular, the rigid-rotor limit for R_W=200 and in a fully two-dimensional parabolic confinement was demonstrated for two electrons in Ref.<cit.> using exact many-body wave functions and for a few electrons in Ref. <cit.> using the same ansatz as in Eq. (<ref>) here. In this universal rigid-rotor limit, the rotational part of the spectra is naturally similar. However, the presence of strong many-body correlations (which result from thebeyond-mean-field, multi-determinant, or multi-permanent, nature of the wave function) is reflected in the actualnumerical values of the first term, V_ int, in Eq. (<ref>).In our scheme, which allows for an expanded variational freedom by employing unrestricted orbitals at the mean-fieldsingle-determinant, or single-permanent, level (i.e., a different orbital for each particle), the ground-state energy is lowered at every step (see in particular Fig. 1 in Ref. <cit.>): restricted HF (symmetry conserving) → unrestricted HF (symmetry breaking) → symmetry restoration via projection techniques (anexample of the full scheme can be seen in Fig. 5 of Ref. <cit.>). This scheme for repelling bosons translates as: symmetry-conserving Gross-Pitaevskii → symmetry-breaking ansatz of unrestrictedpermanent → symmetry restoration via projection techniques. We note that allowing the single orbital of the Gross-Pitaevskii equation to break the azimuthal symmetry of the ring leads <cit.> tohigher-energy solitonic Bose-Einstein condensate branches in the rotational part of the spectrum, which are sharplydifferent from the QSTC wave functions introduced in this paper. As a specific example of the lowering of the ground-state energy in our scheme, we report that the energy for theunrestricted ansatz determinant [see Eq. (<ref>)] in the case of N=8 ultracold ions on a ring of radius R=200l_0 with R_W=1000 (case described in Table <ref> of Appendix A) is E_ UHF=118.1771ħω_0, while the corresponding restored-symmetry ground state has indeed a lower energy V_ int= 117.9271 ħω_0. This lowering of the total energy is immense compared to the quantum of the rotational motion C_R^ cl=1.5614 × 10^-6ħω_0 (see caption of TABLE <ref> in Appendix A). An illustrative case of the rigid-rotor rotational spectra encoded in the second termin Eq. (<ref>) is displayed in Fig. <ref>. A main feature of these spectra are the crossing points (several of them encircled) between pairs of curves with different L_m's. The crossings define special magnetic-field values, Φ̃/Φ_0=(L_1+L_2)/(2N),in the neighborhood of which the system is particularly susceptible to symmetry breaking via theintermixing of two angular momenta and the ensuing generation of the PWM wave packets [see Eq. (<ref>)].Because the symmetry-restored (projected) wave function Φ^ PROJ [Eq. (<ref>)] preserves thegroup-theoretical requirements of the continuous 2D rotational group, its single-particle density is azimuthallyuniform. However, the crystalline order of the original MF (symmetry-broken) wave function Ψ^ SB is notdestroyed in the symmetry-restoration step; instead, it mutates into a hidden order, which however can be revealed via the conditional probability distribution (CPD) (density-density correlation function). The CPD isgiven byD( r, r_0) = ⟨Φ^ PROJ_L|∑_i≠ jδ( r_i, r) δ( r_j, r_0)|Φ^ PROJ_L ⟩.The CPD provides the probabilty of finding a particle in position r assuming that another one is located at the fixed point r_0. Substitution of the expression [Eq. (<ref>)] that defines Φ^ PROJ_L, yields for D( r, r_0) a double integral over the azimuthal anglesγ_1 and γ_2; this integral expression is given in the Appendix <ref>. Fig. <ref>(a) displays an illustrative example of the hidden order in the symmetry-restored wave function Φ^ PROJ_L. The CPD in Fig. <ref>(a) exhibits well localized features; it contrasts with the uniformhorizontal black dashed lines in Figs. <ref>(a) and <ref>(b) which describe ρ( r,t)'s ofΦ^ PROJ_L along the perimeter of the ring trap (at a radius R_ eq). We stress that the fixed point r_0 in the CPD is arbitrary, i.e., the four peaks in the CPD in Fig.<ref>(a) readjust to a different choice of θ_0 so that the relative distance between them and the arrowremain unchanged. Fig. <ref>(b) displays the SPD of the original state Ψ^ SB(γ=0.1π)(a determinant) for N=5 fermionic ions, exhibiting explicitly the symmetry breaking at the mean-field level.§.§ Third level (beyond mean field): Periodic time evolution of the spatially inhomogeneousρ( r,t) associated with the wavepacket Φ^ PWM(L_1,L_2; t=0)As aforementioned, the two-state wave packet in Eq. (<ref>) is not an eigenstate of the total angularmomentum, and thus it is not a stationary state when the pinning agent is lifted; such a pinning agent could beimplemented, for example, as a distortion of the circular geometry of the trap confinement, or as a modulation ofthe trap potential in the azimuthal direction along the ring <cit.>. (A sudden variation of the magneticfield can also transform an eigenstate Φ^ PROJ_L(B_1) at a given B_1 value to a superposition ofΦ^ PROJ_L(B_2) states at another B_2 value <cit.>.) The resulting timeevolution is associated with a time-dependent phase ϕ(t) as discussed previously. Here we will showexplicitly that ϕ(t) represents an undamped rotation of spatially inhomogeneous ρ( r,t)'s around the ring, so that the many-body Φ^ PWM(L_1,L_2; t) exhibit the desired behavior of a QSTC. Thesuccessful theoretical identification and experimentally implemented superposition of two appropriate many-bodyspin eigenstates of the Ising Hamiltonian (resulting in a “spin Schrödinger-Cat” state) were keys to the emulation of the “weaker class” of discrete time crystals <cit.>.The ρ( r,t) of Φ^ PWM(L_1,L_2; t) is defined asρ( r;t) = ⟨Φ^ PWM(L_1,L_2; t) | ∑_i=1^N δ( r_i- r) |Φ^ PWM(L_1,L_2; t) ⟩. As in the case of the CPD, ρ( r;t) entails a double integral over the azimuthal angles γ_1 and γ_2;the lengthy expression is given in Appendix <ref>.Fig. <ref> displays the periodic time evolution of ρ( r,t)'s for two illustrativeΦ^ PWM(L_1,L_2; t) cases, one for N=5 Coulomb repelling fermionic ions [Fig. <ref>(a)] withL_2-L_1=N and the other for N=7 neutral bosons with L_2-L_1=1 [Fig. <ref>(b)] interacting via anattractive contact interaction. The ρ( r,t)'s were calculated at times t_j=jτ/4, whereτ = 2 πħ /(|E^ PROJ(L_2)-E^ PROJ(L_1)|); the actual usedj's label the ρ( r,t) curves. The number of humps exhibited by the PWM ρ( r,t)'s in Fig. <ref>(a)and Fig. <ref>(b) is equal to that in the original MF densities, i.e., N for the repelling-fermions PWMand one for the attractive-bosons lump. The period of the PWM ρ( r,t)'s is T=τ/N for repelling ions and T=τ for attractive bosons. Finally, Fig. <ref>(c) demonstrates a different state of matter, i.e., multi-harmonic excitations of the QSTC exhibiting a multiple number of density humps, i.e., k N and k (with k=2,3,…),corresponding to Φ^ PWM(L_1,L_2; t)'s with L_2-L_1=k N for repelling fermions and with L_2-L_1 =k for attractive bosons, respectively.We note that the PWM broken-symmetry state introduced here to describe a QSTC has an energy intermediate betweenE_1 and E_2 because α^2+β^2=1 (i.e., E^ PWM=α^2 E_1 + β^2 E_2). In particular, at thecrossing point of the two parabolas (where E_1=E_2=E_ cross), one has always E^ PWM=E_ cross. This contrasts with the behavior of the energy of the non-crystalline states studied in Refs. <cit.>, which lies always well above the crossing point.§ DISCUSSION §.§ Symmetries of the trial wave functions, magic angular momenta, and rigidityDespite the fact that the trial wave functions in Eq. (<ref>) are agood approximation to the rotational-symmetry-preservingmany-body eigenstates, they do embody and reflect in an optimum way the crystalline point-group symmetries (familiarfrom bulk crystals).Specifically, the C_N point-group symmetry of the “classical” crystal, which is accounted for through the kernel of symmetry-broken MF determinants (or permanents) Ψ^ SB, is reflected in the fact that the trial wave functions Φ^ PROJ_L are identically zero except for a subset of magic angularmomenta L_m. In the case of N repelling particles, the magic total angular momenta can be determined by considering thepoint-group symmetry operator R̂(2π/N) ≡exp (-i 2πL̂ /N) that rotates on the ringsimultaneously the localized particles by an angle 2π/N.In connection to the state Φ^ PROJ_L, the operator R̂(2π/N) can be invoked in two different ways,namely either by applying it on the “intrinsic” part Ψ^ SB or the “external” phase factorexp(i γ L) (see Ch. 4-2c Ref. <cit.>). One gets in the case of fermionsR̂(2π/N) Φ^ PROJ_L = (-1)^N-1Φ^ PROJ_L,from the first alternative andR̂(2π/N) Φ^ PROJ_L = exp(-2π L i/N) Φ^ PROJ_L,from the second alternative. The (-1)^N-1 factor in Eq. (<ref>) results from the fact that the 2π/Nrotation is equivalent to exchanging N-1 rows in theΨ^ SB determinant. Now ifΦ^ PROJ_L ≠ 0, the only way that Eqs. (<ref>) and (<ref>) can be simultaneously true is if the condition exp (2π L i/N)=(-1)^N-1 is fulfilled. This leads to the following sequence of magic angularmomenta,L_m = k N; k=0,± 1, ± 2, ± 3, …,for N odd, andL_m = (k + 1/2) N; k=0, ± 1,± 2,± 3, …,for N even.Because a permanent is symmetric under the interchange of two rows, the corresponding magic L_m's forspinless bosons are given by the sequence in Eq. (<ref>) for both odd and even numbers of localized bosons. Regarding the numerical aspects, the fact that Φ^ PROJ_L is zero for non-magic L values results in the vanishing(within machine precision) of the normalization factor ∫_0^2π n(γ)e^iγ L dγin Eq. (<ref>). As a result only the physically meaningful energies associated withmagic angular momenta are given in Table <ref>, Table <ref>, and Table <ref> of Appendix <ref>.We stress that the properties and physics associated with magic-angular-momentum yrast states are well known in theliterature of 2D quantum dots <cit.>.Of immediate relevance to this paper is the enhanced energy stabilization that they acquire in their neighborhood (thus characterized often as “cusp” states) in the regime of strong interactions (i.e., for large R_W or R_δ). This energy stabilization can be explicitly seen in Fig. 15 of Ref. <cit.>, where the triplet state correponds to the fully polarized case for two electrons with magic angular momenta L_m=(2k+1), k=0,± 1, ± 2, …. The fact that large energy gaps do develop between themagic-angular-momentum rotational yrast states and the other (excited) states is also well established in the QSTC literature; for the case of ultracold ions on a ring, see Refs. <cit.>, and for the case of the bosonic lump, see Ref. <cit.>.In this paper, we consider fully polarized fermions only, that is cases when S=S_z=N/2, where S is the total spin and S_z is its projection. Consideration with our methodology of the other spin values S_z < N/2is straightforward; it requires, however, restoration of both the total spin S^2 and the total angular momentum. An explicit example for N=3 fermions is discussed in Ref. <cit.>.In addition to the magic angular momenta, the properties of the original crystalline structurebuilt-in in Ψ^ SB are reflected in the high-degree of rigidity exhibited by the symmetry-preserving Φ^ PROJ_L. As demonstrated previously, the SPD of Φ^ PROJ_L is uniform, but the CPD ofΦ^ PROJ_L reveals the now hidden crystalline structure of N strongly repellingparticles on the ring.The rigidity of Φ^ PROJ_L is manifested in that the CPDs have the same N-hump shape and areindependent of the actual value of the magic angular momenta, as well as of the fermion or boson statistics and of whether the number N of fermions is odd or even.This rigidity is a consequence of the strong two-body interaction and cannot be found in many-body wave functions associated with weak interparticle interactions.§.§ Initial wave packets and associated time evolution The focus of this paper is the construction of a symmetry-preserving wave function Φ^ PROJ_L associated with the finite-crystal symmetry-broken determinant (or permanent) Ψ^ SB. However, it is instructive to inquire about the reverse process, that is how to represent the symmetry-broken crystal as a superposition in the complete basis formed by the symmetry-preserving Φ^ PROJ_L's. Indeed one can write the expansionΨ^ SB = ∑_LC_L Φ^ PROJ_L, where [using Eq. (<ref>)] the expansion coefficients are given byC_L = 1/2π∫_0^2π dγ e^-iγ L n(γ), and the norm overlap n(γ) was defined in Eq. (<ref>). Of course the index L runs over the appropriate sequence of magic angular momenta as discussed in Section <ref>.Eqs. (<ref>) and (<ref>) illustrate the fact that with respect to the exact many-body (linear) Schrödinger equation the symmetry-broken-crystal wavefunction Ψ^ SB is a wave packet and not a stationary eigenstate. This is also in general true for allSB mean-field solutions, whether they are solutions of the unresticted Hartree-Fock equations in the case of confined electrons (e.g., in quantum dots <cit.>), or they are the familiar solitonic solutions of theGross-Pitaevskii equations in onedimensional bosonic systems. Due to the rapid experimental control,the latter cases are currently attracting alot of attention. Indeed motivated by experiments that suggest the need to go beyond-mean field dynamics, the number of related theoretical investigations has burgeoned <cit.>.These theoretical studies investigate how an initial state approximating a solution of the nonlinearGross-Pitaevskii equation evolves in time under the exact many-body Hamiltonian. For both the cases of dark <cit.> (a hole in matter density)and bright <cit.> solitons (an excess in matter density, like the case of the lump consideredin this paper), these studies are finding a “universal” behavior of dispersion in space, decay in time, andtime revival at the initial positionof the soliton. This behavior can be understood by taking into consideration the expansion in Eq. (<ref>). In fact, each energy mode in Eq. (<ref>)will evolve in time according to its own phase exp(-iE_L t/ħ),and the interaction between all of them results in a decay-type behavior. Moreover, the initial occupation amplitudes C_L^2 of the different modes are unequal, and as a resultprobability flows from the higher occupied modes to those with lower initial occupations, which leads to a dispersive behavior. However, because the system is finite, there exists a Poincaré period, and the system will eventually experience a revival <cit.>.For achieving a QSTC, we propose here a different initial wave packet, i.e., a two-mode one with equal weights, asspecified in Eq. (<ref>). As explicitly demonstrated through numerical claculations, such a two-mode initial wavepacket preserves at all times, and without damping, the spatial and temporal periodicities expected from the classical finite crystal. We note that the consideration of two-mode Schrödinger-catstates is a key element in the theory of the discrete time crystal, where the focus is to enable a sloshing behavior between these specialized paired states by minimizing interactions to the rest of the system. In fact, our symmetry-preserving trial functions Φ^ PROJ_L [see Eq. (<ref>)] can be viewed as a more complex class of Schrödinger-cat states. This analogy is straightforward for the case of the mirror superposition used in Ref. <cit.>, which can be reproduced from expression (<ref>) as a limiting case by using only two angles γ_1=0 and γ_2=π. §.§ Relation to configuration-interaction (CI) wave functions As mentioned previously, the symmetry-preserving trial functions Φ^ PROJ_L are identically zero for values of L different from the magic angular momenta L_m; see section <ref>. Naturally the exact many-body spectrum has a plethora of additional states with good L, which however cannot be reached with the approach in this paper. Indeed this approach is tuned to extracting from the complete spectrum only the ground states that correspond to non-vibrating classical finite crystal arrangements. The remaining many-body states can be reached by using the CI approach, which is in principle an exact methodology when converged; the CI is oftenreferred to as exact diagonalization (EXD). The CI approach is computationally expensive, but comparisons betweenthe symmetry-restored trial functions and the CI wave functions have been used by us to demonstrate the numericalaccuracy of the symmetry-restored wave functions, as well as to clarify their special place in the whole spectrum,namely that for particular magnetic-field ranges they can become the global ground state, as is the case with the Aharonov-Bohm spectrum in Fig. <ref> of the present paper. Higher-in-energy CIsolutions with different L (and also with L=L_m) do incorporate vibrational and other type of internalexcitations, and as a result a superposition of two random CI states with good L_1 and L_2 will not necessarilyexhibit the crystalline single-particle-density structure of exactly N humps.Systematic comparisons between symmetry-restored states and CI wave functions have been carried out by us previously for the case of a few electrons confined in parabolic quantum dots. Although the external confining potential and particle species in parabolic QDs are different from the case of the ring traps considered here, thesymmetry properties of the many-body wave functions are universal. Thus the analysis presented in our previous QDstudies can be used to gain further insights to the results for the QSTC presented in section <ref>.In particular, Fig. 6 and Fig. 7 of Ref. <cit.> offer an explicit illlustration of the factthat N-humped crystalline SPD structures arise only when both L_1 and L_2 coincide with magic angular momenta and the associated CI wave functions correspond to global ground states in some range of magneticfields (see Fig. 5 and section III in Ref. <cit.>).In our previous studies of QDs, excellent agreement was found between the total energies of symmetry-restored trialfunctions and the corresponding CI energies for both the cases with or without an applied magnetic field, astestified by the many reported direct numerical comparisons. We mention here a few specific examples, i.e.,Table III and IV in Ref. <cit.>, Table IV in Ref. <cit.>, and Fig. 4 in Ref. <cit.>.Such systematic numerical comparisons between symmetry-restored and CI wave functions for the case ofring-trapped ultracold ions and neutral bosons are outside the scope of the present paper. However, they will be reported in subsequent publications <cit.>, including the case away from the quasi-1D regime(when the dependence on the ring width becomes important). § CONCLUSIONSThe discussion <cit.>, motivated by the criticism <cit.> of theoriginal <cit.> QSTC proposals (which were based on ground states), spurred speculations aboutnon-equilibrium low-lying states as possible instruments for describing QSTCs.For N rotating particles on a ring, and using the theory ofsymmetry breaking and symmetry restoration via projection techniques <cit.>, this paper succeeded in explicitly uncovering the existence of low-lying states with QSTC behavior, by introducingbeyond-MF appropriate trial many-body wave functions (see Fig. <ref>). Along with its conceptual andmethodological significance, we propose to focus experimental attention on selected applied magnetic field valueswhere the Aharonov-Bohm-type spectra corresponding to different magic angular momenta are most susceptible to mixing(Fig. <ref>), resulting in rotating pinned-Wigner-molecule many-body states found here to exhibit QSTCbehavior. This constructive platform fills an apparent gap in the quest for ultracold ring-confined ions orneutral-atom QSTCs. We recall that although the original proposals for the quantum space-time crystal <cit.> suggestedrealization of the concept through the use of ultracold few ring-trapped particles, this is yet to be achievedexperimentally. Nevertheless, for the case of ultracold ions, several publications have reported significant progressin controlling aspects of a quantum rotor on a ring. In particular, the abilityto generate and control symmetry-breaking through pinning of the rotating ion crystal has been demonstratedby using up to 15 ^40Ca^+ ions in a ring with a microfabricatedsilicon surface Paul trap <cit.>, or 3 ^40Ca^+ ions in a 2D ring-type configuration in a linear Paul trap <cit.>. To fully inplement and control the QSTC trial functions presented in this paper, the ionsneed to be cooled down to nearthe ground states. In this respect, Ref. <cit.> has achieved temperatures∼ 3 mK (for a trap with a radius of ∼ 60 μm), while Ref. <cit.> reported temperatures in the nanometer range (for an effective ring radius in the 6 to 8 micrometer range).It is expected that cooling techniques and procedures will be further optimized and will be successful in the near future in producing near-ground-statetemperatures, as is exemplified by a very recent publication <cit.>. An essential requirement, met by our theory, is that it is imperative that the proposed beyond-mean-field many-body trial wave functions (i.e., beyond the UHF or GP treatments) for predicting proper quantum space-time-crystalbehavior of particles moving on a ring will be based on solutions to theinteracting particles Schrödinger equation that posses good angular momenta, as well as exhibit (hidden)ordering that reflects an underlying finite crystalline symmetry. This is achieved in our theorythrough the first two construction stages, namely, the unrestricted Hartree-Fock solution followed by anangular momentum projection, yielding the function Φ^ PROJ_L [Eq. (<ref>)]. It is then proposed by usthat these projected and stationary many-body wave functions are susceptible to mixing, see Eq. (<ref>),favored to occur in the vicinity of crossings of Aharonov-Bohm-type spectra of ground-state energies vs.applied magnetic field (see circles in Fig. <ref>). This mixing results in non-stationary low-lying states that, when evolved with the many-body Hamiltonian, yield undamped and non-dispersing periodic oscillations in both space andtime. Work supported by the Air Force Office of Scientic Research under Award No. FA9550-15-1-0519.Calculations were carried out at the GATECH Center for Computational Materials Science. § NUMERICAL CALCULATIONS OF THE MANY-BODY ROTATIONAL ENERGIES E^ PROJ(L_M) [EQ. (<REF>)]Tables <ref>, <ref>, and <ref> below present three illustrative examples of the rotational energyspectra E^ PROJ(L_m) according to numerical calculations of the many-body expression inEq. (<ref>) of the main text. The captions explainhow the numerical C_R in Eq. (<ref>) is extracted from the computed values of E^ PROJ(L_m). C_R is found to be very close to the classical rigid-rotor valueC_R^ cl=ħ^2/[2I(R_ eq)]. § CONDITIONAL PROBABILITY DISTRIBUTIONThe explicit expression for the CPDs of the symmetry-restored wave functions Φ^ PROJ_L[see Eq. (<ref>)] is given byD( r, r_0) = ∫_0^2π dγ_1 ∫_0^2π dγ_2 e^i(γ_1-γ_2)L∑_k≠ m,l≠ n(G^ln_km(γ_1,γ_2) ∓ G^nl_km(γ_1,γ_2) )S^km_ln(γ_1,γ_2)/2 π∫_0^2π n(γ) e^iγ L dγ, A.1where G^ln_km (γ_1,γ_2)= 1/π^2 λ^4exp( -( r- R_k(γ_1))^2 + ( r- R_l(γ_2))^2 + ( r_0- R_m(γ_1))^2 + ( r_0- R_n(γ_2))^2/2 λ^2) ×exp( ix(Y_k(γ_1)-Y_l(γ_2))+ y(X_l(γ_2)-X_k(γ_1))+x_0 (Y_m(γ_1)-Y_n(γ_2))+y_0(X_n(γ_2)-X_m(γ_1))/2 l_B^2), A.2 and the S^km_ln(γ_1,γ_2)'s are two-row (km)-two-column (ln) cofactors of the determinant(minors of the permanent) constructed out of the overlaps of the localized space orbitals u( r,R_j) [Eq. (<ref>)]. The ∓ sign in Eq. (<ref>) corresponds to fermions or bosons. § SINGLE-PARTICLE DENSITYThe explicit expression for the SPDs of the broken-symmetry wave packets Φ^ PIN(L_1,L_2;t)[see Eq. (<ref>)] is given byρ( r;t) = ∫_0^2π dγ_1 ∫_0^2π dγ_2( α^2 e^i(γ_1-γ_2)L_1 + αβ e^i(γ_1 L_1-γ_2 L_2-ϕ(t)) + αβ e^i(γ_1 L_2+ϕ(t)-γ_2 L_1) + β^2 e^i(γ_1-γ_2) L_2) ∑_kl F_kl(γ_1,γ_2) S^k_l(γ_1,γ_2)/2 π∫_0^2π n(γ)( α^2 e^iγ L_1 + β^2 e^iγ L_2)dγ, A.3where F_kl (γ_1,γ_2)= 1/πλ^2exp( -( r- R_k(γ_1))^2 + ( r- R_l(γ_2))^2/2 λ^2) ×exp( -iy(X_l(γ_2)-X_k(γ_1))+ x(Y_k(γ_1)-Y_l(γ_2))/2 l_B^2), A.4 and the S^k_l(γ_1,γ_2)'s are one-row (k)-one-column (l) cofactors of the determinant(minors of the permanent) constructed out of the overlaps of the localized space orbitals u( r,R_j)[Eq. (<ref>)]. 99 bloc08 I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008). blat12 R. Blatt and C.F. Roos, Quantum simulations with trapped ions, Nature Phys. 8, 277 (2012), spie14 N. Goldman, G. 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A 96, 013417 (2017). | http://arxiv.org/abs/1706.09006v3 | {
"authors": [
"Constantine Yannouleas",
"Uzi Landman"
],
"categories": [
"cond-mat.quant-gas",
"nucl-th",
"quant-ph"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170627184345",
"title": "Trial wave functions for ring-trapped ions and neutral atoms: Microscopic description of the quantum space-time crystal"
} |
=1 | http://arxiv.org/abs/1706.08978v1 | {
"authors": [
"Keith K. Ng",
"Robert B. Mann",
"Eduardo Martin-Martinez"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th"
],
"primary_category": "quant-ph",
"published": "20170627180006",
"title": "Over the horizon: distinguishing the Schwarzschild spacetime and the $\\mathbb{RP}^3$ spacetime using an Unruh-DeWitt detector"
} |
Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Ave., Cambridge CB3 0HE, UKKavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UKDipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I-50019, Sesto Fiorentino (Firenze), ItalyINAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, ItalyINAF - Osservatorio Astronomico di Trieste, via G. Tiepolo 11, I34124 Trieste, Italy Dipartimento di Fisica e Astronomia, Università di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, ItalyInstituto de Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 70-264, Mexico D.F., 04510 MexicoINAF-Osservatorio Astronomico di Brera, via Brera 28, I-20121, Milano, Italy SNS - Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, ItalyINAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monteporzio Catone, ItalyDipartimento di Matematica e Fisica, Università Roma Tre, via della Vasca Navale 84, I-00146 Roma, ItalyEuropean Southern Observatory, Karl-Schwarzschild-str. 2, 85748 Garching bei München, GermanySchool of Physics and Astronomy, The Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, IsraelDepartment of Physics, University of North Texas, Denton, TX 76203, USA European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, ChileExcellence Cluster 'Universe', Boltzmannstr. 2, D-85748 Garching bei München, Germany We present new ALMA observations aimed at mapping molecular gas reservoirs through the CO(3-2) transition in three quasars at z≃2.4, LBQS 0109+0213, 2QZ J002830.4-281706, and [HB89] 0329-385. Previousobservations of these quasars showed evidence for ionised outflows quenching star formation in their hostgalaxies.Systemic CO(3-2) emission has been detected only inone quasar, LBQS 0109+0213, wherethe CO(3-2) emission is spatially anti-correlated with the ionised outflow,suggesting thatmost of the molecular gas may have been dispersed or heated in the region swept by the outflow. In all three sources, including the one detected in CO, our constraints on the molecular gas mass indicate a significantly reduced reservoir compared to main-sequence galaxies at the same redshift, supporting anegative feedback scenario. In the quasar 2QZ J002830.4-281706, we tentatively detect an emission line blob blue-shifted by v∼-2000 km/s with respect to the galaxy systemic velocity and spatially offset by 0.2 (1.7 kpc) with respect to the ALMA continuum peak.Interestingly, such emission feature is coincident in both velocity and space with the ionised outflow as seen in .This tentative detection must be confirmed with deeper observations but, if real, it could represent the molecular counterpart of the ionised gas outflow driven by the Active Galactic Nucleus (AGN). Finally, in all ALMA maps we detect the presence of serendipitous line emitters within a projected distance ∼ 160 kpc from the quasars. By identifying these features with the CO(3-2) transition, we find that the serendipitous line emitters would be located within |Δ v|<500 km/s from the quasars, hence suggesting an overdensity of galaxies in two out of three quasars. Carniani et al. AGN feedback on molecular gas reservoirs in quasars at z∼2.3 AGN feedback on molecular gas reservoirs in quasars at z∼2.4 S. Carniani1,2, A. Marconi3,4, R. Maiolino1,2, C. Feruglio5, M. Brusa6,7, G. Cresci4, M. Cano-Díaz8, C. Cicone9, B. Balmaverde10,F. Fiore11, A. Ferrara10, S. Gallerani10, F. La Franca12, V. Mainieri13, F. Mannucci4,H. Netzer14, E. Piconcelli11,E. Sani16,R. Schneider10, O. Shemmer15, L. Testi13,4,17December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTION Both the growth of super-massive black holes (SMBHs) and star formation history of galaxies are regulated by the supply of cold gasavailable in the host. The molecular gas reservoir can be replenishedthrough either accretion of cold gas from the halo orwet mergers.Intense bursts of star formation, such as those observed in sub-millimetre galaxies (SMGs), andin the host galaxies of powerful active galactic nuclei (AGN) can be induced by mergers, interactions and disk instabilities <cit.>. Several studies have suggested that SMGs and quasars (QSOs) represent two distinct stages of galaxy evolution <cit.>. SMGswould correspond to the starburst phase when galaxies are dust obscured and therefore emit mainlyat far-infrared wavelengths. QSOs are unobscured systemswhere the gas has been expelled by energetic outflows, which eventually quench both SMBH growth and star formation (SF) <cit.>.The discovery ofubiquitous massive, powerful galaxy-wide outflowsin QSO host galaxies supports the QSO feedback scenariodepicted above.Studies based on millimetre observations of local QSO hosts have estimated molecular outflow mass-loss rates exceeding the star formation rates by almost two orders of magnitude in the most powerful sources <cit.>.Massive outflows can deplete the host galaxies of theircold gas content in approximately a few Myrs, that is, on timescales even shorter than the depletion time scales due to gas consumption by star formation <cit.>. However, althoughthese observations are in overall agreement with AGN feedback models <cit.>, we are still missing the smoking gun evidence that theAGN-driven outflows are effectively quenching star formation: what we are seeking is a clear and unambiguous indication that star formation is indeed inhibited in the galaxy regions swept by the outflows.Several SINFONI/VLT observations of z∼1.5-2.5 QSOs indicate the presence of fast, galaxy-wide ionised outflows with a conical morphology that are spatially anti-correlated withthe brightest actively star forming region in the host galaxy <cit.>. These results suggest that the fast winds aresimultaneously expelling gas from the host galaxies and quenching star formation in the regionswept by the outflow. However,we note that, at optical wavelengths, observations may be affected by differential extinction effects,and so we cannot fully rule out the presence of obscured emission powered by star formation in the region affected by the ionised outflow.In conclusion, it is still debated whether theobserved absence ofstar formation signatures in the outflow region isreal and related to gas depletionby feedback or ifit is caused by dust obscuration. In this context, observations at (sub-)millimetre wavelengths are crucial to definitely establish whether star formation is inhibited by fast outflows.Through the carbon monoxide (CO) line emissionwe can directly trace the cold molecular gas that fuels the star formation activity, and thus confirm or reject negative-feedback scenarios.<cit.> reported the detection of CO(3-2) emission with the IRAM Plateau de Bure Interferometer (PdBI) in XID2028, one of the z∼2 QSOs exhibiting spatial anti-correlation between narrowemission, tracingstar formation, and ionised AGN-driven outflows <cit.>. Themodest molecular massinferred from the CO(3-2) line detection indicates that the gas in the host galaxy has been already depletedor dispersed by QSO feedback. However the angular resolution (∼4) of these PdBI observations is not sufficient to spatially resolve the CO(3-2) emission inXID2028,hence not allowing an accurate determination of the location of the molecular gas reservoir with respect to the ionised outflow <cit.>. We haverecently undertaken an Atacama Large Millimetre/submillimetre Array (ALMA) programme targeting two QSOs of the sample by <cit.>, LBQS0109+0213 (hereafter ),2QZJ002830.4-281706 (hereafter ),and [HB89] 0329-385 (hereafter HB8903)in which thespatial distribution of their narrowandemissionswith respect to the location of theoutflow supports a negative-feedback scenario <cit.>.The aim of the ALMA observations presented in this paper is to map the molecular gas in the host galaxiesthrough the CO(3-2) emission (rest frequency ν_ rest=345.8 GHz) and compare the spatial distribution of the molecular gas with that of the fast-outflowing ionised gas. The paper is organised as follows: Sect. <ref> describes thetarget properties, and Sect. <ref>summarises the ALMA observations. In Sect. <ref>, <ref>, and <ref> we presentour results on , , and HB8903, respectively. A discussion of the molecular gas content inall host galaxies is included in Sect. <ref> and, our conclusions are summarised in Sect. <ref> and <ref>. In this work we adopt a ΛCDM cosmological model with H_0 = 67.3 kms^-1Mpc^-1 ,Ω_M = 0.315, Ω_Λ = 0.685 <cit.>. According tothis model, 1 at z = 2.4 corresponds to aphysical scale of 8.35 kpc.§ SAMPLE SELECTION The selected targets, , , and HB8903, are part of a large high-luminosity (L>10^47 erg/s) QSO sample at z>2<cit.>,characterised byequivalent widths of EW>10 Å andrelatively bright in the H band, that is, H<16.5 mag. The properties of the three targets are listed in Table <ref>. To investigatethe properties of the AGN-driven outflows,we have observed the three QSOs in the H (1.45-1.85 μm) and K (1.95-2.45 μm) bands with the Spectrograph for INtegral Field Observations in the Near Infrared (SINFONI)in seeing limited mode(angular resolution ∼0.6). The kinematical analysis of theemission line revealed fast (>1000 km/s) ionised outflows extended a few kpc from the galaxy centre <cit.>. In addition,the nuclear spectrum of , which has also been recently observed with SINFONI assisted by adaptive optics (angular resolution ∼0.15), ischaracterisedby a broad, blueshiftedabsorption trancing nuclear outflowing gas with density higher than 10^9 cm^-3 and velocity up to 10,000 km/s <cit.>.Intriguingly, the presence ofextended outflows appears to be spatially anti-correlated with the narrowemission componenttracing star formation in the host galaxy <cit.>. These results have been interpreted as evidencefor negative feedback in action,where star formation is quenched in the region where AGN-driven outflows interact with the host galaxy.If excluding dust-extinction effects, the reduction of star formation activity in the outflow region can be caused by a lack of a substantial molecular gas reservoir, that may have been expelled by the outflow itself, or to heating and turbulence effects related to the feedback process that may lower the star formation efficiency of the gas.§ OBSERVATIONSLBQS0109, 2QZJ0028, and HB8903 were observedat the Band 3 frequencies (∼100 GHz, corresponding to λ∼3 mm) with the ALMA array between July 2015 and July 2016. Theon-source timewas about 40 minutes with40-44 12-m antennas for all sources.The antennas were distributed in asemi-compact configuration with a maximum baseline length of ∼ 1.5 km. The average precipitable water vapour (PWV) values during the observations were 3 mm, 2.2 mm and 0.8 mmm for the three targets, respectively. The millimetre observations, carried out in frequency division mode, have a total bandwidth of 7.5 GHz divided into four spectral windows of ∼1.875 GHz with a channel width of 1.9 MHz (∼ 5.7 km/s).One of the four spectral windows was tuned to the expected central frequency of the CO(3-2) line, that is, 103.2 GHz for LBQS0109,101.6 GHz for 2QZJ0028, and100.5 GHz for HB8903. The redshifts of the three QSOs were estimated from thenarrowandemission lines <cit.>.The data were calibratedusing the CASA software version v4.5.2 <cit.>. The phase calibrators were J0038-2459 and J0108+0135 for LBQS0109 and , respectively. Ceres and J0238+166 were used as flux calibrators, whilebandpass calibrations were carried out throughthe observations of J2258-2758 and J0238+1636. The flux, bandpass and phase calibrator for HB8903 was J0334-4008.All final images were reconstructed by using the CASA task clean. Continuum images at 3 mm were obtained using the line-free channels of the four spectral windows.By using a natural weighting we achievedfor all sources a sensitivity of 12 /beam for the first two QSOs and 18 /beamfor HB8903. The final images have an angular resolution of about 0.6, which corresponds to 5 kpc at z∼2.4, and a spatial scale per pixel of0.1. We subtracted thecontinuum emission by fitting an UV - plane model to the line-free channels of each spectral window using the uvcontsub task. Wegenerated the final cubes from the continuum subtracted data using the clean task with the parameter weighting = briggs and robust = 0.5, which offers a compromise between high-resolution and highest sensitivity per beam. In all sources, we achieved a 1σ sensitivity of 240-280 /beam per spectral bin of 30 km/s with a beam size of about 0.6×0.5.The angular resolution of ALMA imagesmatches well that ofthe SINFONI observations (∼0.6). The source size and the flux density of the continuum emission of the three sources are inferred by fitting a 2D elliptical Gaussian profile to the visibility data in CASA byusing the uvmodelfit task. The line properties were estimated in the image plane instead. In this work we are interested in comparing ALMA and SINFONI observations,hence we verified the astrometry accuracy of bothdatasets. The absolute positions of theQSOs in the ALMA field are consistent with the Sloan-Digital-Sky-Survey (SDSS)and Two-Micron-All-Sky-Survey (2MASS) positions within the astrometric uncertainty of about 0.1.As no astrometric calibrations of SINFONI were observed for the three targets, we had to align the peak of the H- and K-band SINFONI continuum emission with the centroid position obtained from 2MASS images in the same bands (H and K). § LBQS0109TheALMA 3 mm continuum emission map ofis shown inpanel (a) of Fig. <ref> overlaid onto the SINFONI H-bandcontinuum image.has a flux density of 165and is detectedat the ∼14σ level (σ=12 ) in the ALMA continuum map.The emission at 3 mm is spatially resolved with a beam-deconvolved size of (0.5±0.1)×(0.3±0.2) with P.A. = (85±11) (Table <ref>). We note that the emission is elongated in the same direction (east-west) of the ionised outflow traced by the broad component <cit.>, which is indicated as a blue arrow inFig. <ref>a.The radio emission of log_10(L_8.4/ WHz^-1) = 24.83±0.22 at 8.4 GHz<cit.> measured with the Very Large Array indicates thatis just below the limit to be classifiedas radio-loud QSO (log_10(L_8.4/ WHz^-1)>25). Therefore the elongated continuum emission at 3 mm may be associated with a radio jet co-spatial with the ionised outflow.However, two photometric measurements at 3 mm, which corresponds toλ_ rest∼0.9 mm in the rest frame,and at 8.4 GHz (λ_ rest∼ 10 mm), are not sufficient to perform a spectral energy distribution (SED) fitting decomposition.Indeed a typicalgalaxy SED from radio-to-infrared wavelengths can be modelled as a linear sum of dust continuum, thermal bremsstrahlungand synchrotron emission <cit.> but we would need more photometric data to disentangle the various components. Therefore it is not possible to establish whether dust or synchrotron emission dominates the sub-mm continuum in .Assuming that the continuum emission at 3 mm is mainly associated with dust thermal emission, we estimated an upper limit on the dust content of . Since dust israrely optically thick at millimetre wavelengthsa part from a few extreme starburst galaxies <cit.>, we have adopted the optically thin approximation for our unobscured QSO. The total dust massis thus given by:M_ dust = S_ν D_ L^2/B_ν(T_d)κ_ν, where S_ν is the flux density at the rest frame frequency ν, D_ L is the luminosity distance of the target, B_ν(T_d) is the black-body function at the dust temperature T_d and κ_ν is theabsorption coefficient. Following <cit.>, the equation can be simply rewritten as M_ dust = 3.25×10^9 e^0.048ν/T_d-1/ν^3κ_ν(S_ν/ mJy)(D_L/ Mpc)^2, where the rest frame frequency ν is in GHz, κ_ν is in cm^2 g^-1 and T_d is in K. We adopt a κ_ν = 0.45×(ν/250 GHz)^β cm^2 g^-1 with a fixed emissivity index β=2.0.We note that κ_ν depends on the properties of dust grain and can suffer from large uncertainties.Using a dust temperature T_d=40-60 K <cit.>, thedust mass from the 3 mm continuum emission is around 5-8×10^8 . We stress that the inferred dust mass is more likely an upper limit because a fraction of the continuum emission in this wavelength band can be also associated to non-thermal synchrotron radiation. In addition, the ALMA continuum map of the field surrounding reveals the presence of three serendipitous sources within 15 (∼120 kpc) from the QSO and with a signal-to-noise ratio (S/N)>4. The three sources are also visible with a level of confidence > 5σ in the linechannels.By assuming that theline emission detectedat the positions of these sources corresponds to the CO(3-2) transition, we propose that they are physically associated with , with a Δ z=0.002. The properties inferred from the analysis of these serendipitous detections are reported in Appendix <ref>.The three detections do not have optical counterparts in the SDSS images. A detailed discussion of this over-density system is presented in Sect. <ref>.The CO(3-2) spectrum ofis shown in Fig. <ref>b, and the vertical greendotted line shows the expected central wavelength corresponding to the redshift measured from the SINFONI data by using the narrowemission. Thefull-width at half maximum (FWHM) and centroidof the CO(3-2) line in , derived from a single Gaussian fit, are 400±60 km/s and λ_ obs=2.9094±0.0004 mm (z_ CO(3-2) = 2.3558±0.0005). We note that both the FWHM and the redshift are consistentwith the narrowandcomponents (FWHM_ [OIII] = 490±90 km/s,FWHM_ Hα = 250±200 km/s, z_ [OIII]=2.3558±0.0008, and z_ Hα=2.357±0.002)tracing SF in the host galaxy <cit.>.The agreement strongly indicates that most of the CO emission is tracing molecular gas in the host galaxy. Panel (c) of Fig. <ref> shows the map integrated over the line emission in which the peak is detected at ∼7σ (σ=0.03 Jy/beam km/s).The integrated flux, extracted from the region of the map with a level of confidence higher than 2σ, is S_CO(3-2)Δ v = 0.34±0.03 Jy km/s, corresponding to a line luminosity of L_CO(3-2) = (1.4±0.2)×10^7at z=2.35.The CO line emission is spatially resolvedwith an estimated size of (0.93±0.08)×(0.82±0.07), that is (7.7±0.7)kpc ×(6.8±0.6) kpc. Although the CO(3-2) line emission is resolved by our ALMA observations, we cannot perform a detailed pixel-by-pixel kinematic analysisbecause of thelow S/Nof the data. However, we note that the CO line extracted in the southern region has aFWHM= 280 km/s that is smaller than that measured in the nuclear region (see Table 1).This discrepancy will be discussed in more detail in Sect. <ref>. §.§ Molecular and stellar mass estimates Consistently with recenthigh-redshift (z>2) observations of AGN host galaxies <cit.>, we assumean r_31≡ L^'_CO(3-2)/L^'_CO(1-0) ratio of,=1.0±0.5, yielding an estimated CO(1-0) line luminosity of, L^'_ CO(1-0) = (1.0±0.6)×10^10 K km/s pc^2 for .The conversion factorbetween CO(1-0) line luminosity and H_2 mass depends on the interstellar medium conditions.In general an =4 /K km/s pc^2 is assumed for main-sequence (MS) galaxies and an =0.8/K km/s pc^2 value is adopted for compact luminous systems, such as starburst galaxies, SMGs and QSOs <cit.>.Since the molecular ISM conditions of our targets are still unknown, weestimate two limiting values for the molecular gas mass (),corresponding to the two choices mentioned above. Theresultingvalues are listed in Table <ref> andtheir associated statistical errorsinclude both ALMA flux calibration and uncertainties. We now explore the consequences of thepossibility that the host galaxy oflies on themain sequence (MS) of star forming galaxies. Typical MS galaxies at z∼2 have molecular gas fractions of f_ mol-gas=/(+)≃ 0.44 <cit.>.Forand assuming =4 / km/s pc^2, this gas fraction would result in a stellar mass estimate of, =1.3×10^11 . This value, combined with the BH mass of =10^10inferred by <cit.>, yields for /≃0.1. This ratio is much larger than those observed in massive galaxies in the local Universe <cit.>. A similar / has been recentlyinferred by <cit.> for an unobscured AGN at z = 3.328, CID947,where it is believed that the SMBH has grown more efficiently than the host galaxy.Inthe star-formation activity may have been shut-off due to the negative-feedback exerted by the QSO, as we further argue below. It is also possible of course that the host galaxy ofis not on the z∼2 MS, in which case the above estimate of / would not be valid. §.§ Morphology of the CO(3-2) emissionTheCO(3-2) map ofshown in Fig. <ref>c exhibits a complex morphology: the molecular emission is notdistributed symmetrically around the QSO. Figure <ref> shows the CO(3-2) spectra extracted from four regions placed at different positions with respect to the location of the QSO.The CO(3-2) emission in the south-west regionis almost absent, while the line spectra extracted fromthe other three regions have similar fluxes and profiles.We note that the synthesised beam is oriented from north-east to south-west. Thus, the signal visible towards the west and south relative to centre suggests that the molecular emission the molecular emission is either spatially unresolved or faintalong the direction of the ALMA beam. In the left panel of Fig. <ref> we compare thedistribution of CO(3-2) with the velocity map of the broadblue-shiftedcomponent tracing the ionised outflow in the QSO host galaxy <cit.>. TheCO(3-2)emissionis partially dislocated with respect to the regions where the outflow traced byis fastest.In addition, thechannel map, obtained integrating the continuum subtracted SINFONI datacube onthe blue wing, indicates that the ionised outflow is elongated from north-east to south-west (cyan contours in the middle panel) where the CO(3-2) emission is faint (or spatially unresolved). Finally, the right panel of Fig. <ref> shows the surface brightnessof the narrowcomponent tracing SF in the host galaxy and the CO(3-2) flux map in white contours.We refer to <cit.> for further arguments supporting the identification of the narrowemission with emission powered by star formation in the quasar host galaxy. It is interesting to note a similarity between the CO and the narrowsurface brightness distributions. Both emission lines are faint or absentalong the direction of the ionised outflow, while they are clearly visible in the other regions.These results support a scenario in which fast outflows are cleaning up the galaxy of its moleculargas, hence quenching SF inthe region where the outflow breaks in the host galaxy ISM. In Sect. <ref> we noted that the CO(3-2) profile extracted from an apertureplaced south of the QSO (see Fig. <ref>) is narrower than that observed in the nuclear region.Such a discrepancy suggests that the motion of the gas in the external regions is less turbulentthan in the QSO centre which is influenced by the nuclear fast winds.This residual gas fuels the SF in the region of the host galaxy is not affected by AGN-driven outflows. The current ALMA CO(3-2) observations tracemolecular gas only in region within 2 kpc from the centre, while the narrowcomponent is extended up to ∼8 kpc. Unfortunately, higher sensitivity ALMA observations would be needed to compare the distribution of molecular gas andemission in the external regionsat a distance >2 kpc from the QSO, and to verify whether the extended structures areconsistent with the Schmidt-Kennicutt relation between SFR and gas density <cit.>.§ 2QZJ0028Panel (a) of Fig. <ref> shows in white contours the spatially unresolved continuum emission map of at 3 mm, while the coloured background is the SINFONI continuum emission in the H band. The peak at 3 mmhas aS/N=14 and the integrated flux density of the source is 170±12 . We cannot perform a radio-to-FIR SED fitting decomposition because we have only one photometric point.However, by assuming that the emission at 3 mm is mainly associated to dust thermal emission we can infer the dust mass as we did forin Sect. <ref>. We thus estimate M_ dust = 6-9×10^8 We have performed a blind search in the ALMA continuum map around the QSO and we have detected a millimetre continuum source with aconfidence level of 5σ and a flux density of 61at 3 mm. Such source is located at a distance of 118 kpc from the quasar (Appendix <ref>). At the redshift ofwe do not detect any CO(3-2) emission line at a significance level higher than 3σ (panel (b) Fig. <ref>). By assuming a CO line width similar to that measured in(FWHM=400 km/s), we can estimate a 3σ upper limit on the CO(3-2) integrated line flux of 0.09 Jy km/s, which corresponds to an upper limit on the CO(3-2) line luminosity of 0.6×10^7 . Following the same method as in Sect. <ref>,we derive two different 3σ upper limit estimates for the total molecular gas mass, based on differentprescriptions:M_ CO(=0.8)<0.2× 10^10andM_ CO(=4) < 1.2× 10^10 . §.§ Possible association of offset CO(3-2) emission with a molecular outflowThe total spectrum extracted at the location offrom a beam-sized aperture shows an emission feature at a velocity of ∼-2000 km/s relative to theexpected CO(3-2)frequency based on the QSO redshift (Fig. <ref>b).Figure <ref> (panel c) shows the map extracted from the spectral range centred at 2.9316 mm (∼-2000 km/s) and with a spectral width of 250 km/s. This map clearly shows an unresolved source whose peak is detected with S/N=5.2 and is spatially offset by ∼0.2 (1.3 kpc) towards south-east relative to the QSO centre.Although we cannot completely rule out that this detection is spurious, we note that (as discussed in Appendix A) the number of positive peaksat >5σ is 1.5 times larger than the number of negative peaks, suggesting that one third of the positive peaks might represent real sources. Similarly, the emission feature at ∼1000 km/s should not be considered as real because in the integrated map the emission peak has a S/N of only 3.8; at that S/N the number of positive peaks is similar to that of negative ones, strengthening the idea that it is simply a noise fluctuation. In any case, deeper observations are needed to confirm the reliability of our CO detection at the velocity of ∼-2000 km/s. Under the assumption that such detection is real and associated to the CO(3-2) transistion, Table <ref> summarises the properties of the line. Boththe central velocity and the positional offset of theblue-shifted CO feature are consistent with the velocity (∼2300 km/s) and location of the ionised outflow tracedby the broademission.This is clearly shown in left panel of Fig. <ref> where we compare the flux map of this tentative CO(3-2) component with the ionised gas velocity map obtained from the SINFONI observations. Such a remarkable agreement strongly suggests that the blue-shifted CO(3-2) component detected by ALMA is real and that it traces a molecular outflow.The middle panel of Fig. <ref> shows the flux map of the CO(3-2) emission outlined over the ALMA continuum emission (background image).In this panel, we plot the surface brightness of the broademission lineobtained by collapsing the SINFONI data-cube in a velocity range -2500<v<-2300 km/s, where our spectro-astrometry technique has revealed the presence of an extended outflow <cit.>. The CO blueshifted emission overlaps with the blueshiftedemission suggesting thatwe may be tracing an outflowing molecular component associated with the ionised outflow.The radius of the ionised outflow, estimated by using the spectroastrometry method, is ∼0.7 kpc, which is consistent, within astrometric error (∼0.9 kpc),with the spatial offset (∼1.6 kpc) measured between the line and continuum emission at 3 mm.The non detection of CO emission at systemic velocity and the host galaxy position can be explained by the different excitation of the molecular gas in the star formation regions and in the outflow. For instance, from recent ALMA observations of a jet-driven molecular outflow, <cit.> found that the CO(4-3) emission ismore excitedalong the jet propagation axis than in the rest of the galaxy disk. If the same excitation ratio describes the case ofthen the emission at the systemic velocity would not be detected with our observations. Blueshifted CO emission like our own without any counterpart at the systemic velocity has been detected in quasars by <cit.>.On the other hand, the non detection of CO emission at the systemic velocity may also indicate that a large fraction of the molecular gas in the host galaxy is accelerated by the AGN-driven outflow and the sensitivity of the current ALMA observations is not sufficient to detect the residual quiescentgas,even assuming the same excitation ratio for the outflow and star-formation regions. Future deeper ALMA observations of higher and lower rotational CO transitions are fundamental to confirm or rule out the hypothesis that the blue-shifted emission is real and traces molecular outflows in .Outflowing clumps have already been observed by <cit.> in a QSO atz∼6.4 (SDSS J1148+5251). They find clumps of [CII] emission extended up to ∼30 kpc from the nucleus and with velocities >1000 km/s. In addition, the spectralfitting to the [CII] extended emission exhibits the presence of narrow(σ_v∼100-200 km/s) and fast (v>1000 km/s) clumps similar to thatobserved in . In the local Universe, the presence of outflowing clumpsof molecular gas has been revealed by CO(2-1) and CO(3-2) observations of Markarian 231<cit.>, a QSO host and ultra-luminous IR galaxy (ULIRG) in the local Universe, as well as in a few other nearby QSOs <cit.>.In several cases, the CO line profiles show the presence of a blue and red wing composed by several `bumps' with different intensity and velocity. Such profiles may be generated by molecular outflows with a multi-clump morphology. The sensitivity of our current ALMA observations islikely not sufficient toappreciate both the blue and red wingsof the CO line as observed in local molecular outflows <cit.>, butit allows us to marginally detect only the brightest knot of theclumpy molecular outflow.Since the velocity and the positional offset are consistent with those of the outflow, wehypothesise that most of the blueshiftedemission is co-spatial withthe molecular outflow clump. We also note that the CO emission is located in the region where theemission is missing (right panel of Fig. <ref>). Such an anti-correlation supports the notion that the AGN-driven outflowdrives gas out of the galaxy and exhausts the fuel necessary to SF.An alternative interpretation to the outflow scenario could be that the detected blue-shifted CO emission is associated with a merging companion galaxy. However, our SINFONI observations <cit.>do not show any merging signature.Additional data arerequired to further dismiss or validate this possibility. § HB8903Figure <ref>a shows the 3 mm continuum emission of HB8903, which is detected witha high S/N of approximately 300.The total 3 mm flux density of the QSO is 5.738±0.018 mJy (including calibration uncertainties) that is, about 30 times higher than that measured in thetwo previous QSOs (see Table <ref>). From a 2D-Gaussian fitting in the UV-plane we estimate a beam-deconvolved size of about 0.1.HB8903 has been identified asa radio-loud QSO <cit.>with a luminosityof log_10(L_8.4/ W Hz^-1) ≃ 27.7 at 8.4 GHz <cit.>. Therefore, the 3 mm flux is probably dominated by synchrotron emission which does not allow us an estimate of the far-infrared emission associated with the dust.Similarly to the analysis performed in the previous sections, we estimate an upper limit on the dust mass of M_ dust = 2-3×10^10 , depending on dust temperature. The CO(3-2) line is not detected at the location of the QSO as shown in thepanels (b) and (c) of Fig. <ref>.To estimate a 3σ upper limit on the line flux, we assume a line width as large as that observed in(FWHM = 400 km/s) yielding S_ CO(3-2)Δ v < 0.08 Jy km/s (Table <ref>).We then infer an upper limit on the molecular gas mass of M_ gas(=0.8) < 0.2×10^10 and M_ gas(=4) < 1.0×10^10 for the two different CO-to-H_2 conversion factor, respectively.In the ALMA field of view of HB8903, wealso detect two additional sources with likely molecular line emission: one located 4.9(∼40 kpc at z = 2.44) to the north-east of the QSO and the other 16.6(∼140 kpc at z = 2.44) to the south-west of the QSO. The properties of these galaxies are discussed in Appendix <ref>.As already observed in LBQS0109, even the QSO HB8903 could be located in a overdensity at z≃2.4 (see Sect. <ref>) § LACK OF MOLECULAR GAS Figure <ref> shows the best-fit relations by <cit.> between molecular gas mass and SFRfor massive (> 10^10 ) main-sequence galaxies and starburst at low and high redshift (z<4).The SFRs ofour three QSOsare estimated from the narrow emission <cit.> assuming a Chabrier initial mass function <cit.>. Since theemissions are not corrected for reddening, the inferred SFRs are lower limits for both targets. In the -SFR plane,all three sources areplaced below the relation extrapolated for star-forming galaxies. In fact, a main-sequence star-forming galaxy with SFR=50has a molecular gas mass of =4×10^10that is similar to (=4) of , but it is at least five times higher if we assume a conversion factor =0.8 (see Table <ref>). The inferred molecular gas masses are comparable with the expectation based on <cit.>.We can estimate the depletion timescale, which is defined as the rate at which the gas is converted into stars: = /SFR. Because the SFR fromare lower limits, we infer an upper limit of <160-800 Myrfor LBQS0109, <30-120 Myr for 2QZJ0028 and <20-110 Myr for HB8903, depending on .The depletion timescales are similar to those observed in starburstand SMG galaxies <cit.> and reddened QSOs<cit.> at z∼2.5, suggesting that the three host galaxies may still be in a starburst phase andthe star formation activity is not affected by AGN-driven outflows. However the low molecular gas mass may also indicate that a fraction of the gas reservoiris expelled away from the galaxy by AGN-driven feedback, and, at the same time, the fast winds inducehigh pressure in the rest of the gas, triggering star formation in the region unaffected by AGN activity <cit.>. This scenario is similar to that observed in the QSO XID2028 at redshift z∼1.5, where the presence of ionised outflow has been observed throughemission <cit.> and thesmall gas reservoir, respect to a MS star-forming galaxy with similar , is explained by negative-feedback <cit.>. Also, <cit.> have recently reported a lower gas fraction for a sample of AGN at z∼1.5 compared with a sample of galaxies without an AGN matched in redshift, stellar mass, and star-formation rate. In addition, <cit.> have found thatthe molecular gas depletion timescale and the molecular gas fraction of a sample of 15 galaxies hosting powerful AGN driven winds are between three and ten times smaller than those of main-sequence galaxies with similar star-formation rate, stellar mass and redshift. According to such negative-feedback scenario, the molecular gas should be removed in the host region with the high velocity outflow. Instead, the anti-correlation between CO emission and ionised outflow direction indicatesthat a fraction of the gas has been already expelled from the galaxy. Deeper ALMA observations will confirm this scenario also in . Although the three QSOs have similar properties, such as SFR, , , the CO(3-2) at the systemic velocity ofthe host galaxies is visible only in one of the three targets. This discrepancy may be due to a different CO(3-2) excitation in the other two host galaxies. Inand HB8903 the CO(3 - 2) may be less excited than that inand the current sensitivity is not sufficient to detect the emission line in the host galaxy. In addition thedense molecular clouds invested by AGN-driven wind develop Kelvin-Helmholtz instabilities <cit.>. These instabilitiesdevelop shocks responsible for higher gas excitation likely resulting into the strong CO(3-2) blueshifted emission that we observe in . A similar result is recently reported by <cit.> and <cit.> who observed CO(4-3) and CO(2-1) emission in a local Seyfert galaxy, IC 5063. Most of the CO(4-3) emission has been detected in the outflow regions, while the CO(2-1) is mainly emitted in the host galaxy location. In the outflow regions the CO(4-3)/CO(2-1) flux ratio approaches 16 (×3 higher than that observed in the rest of the host galaxy ).In this regard, we also note that CO observations in the distant Universe (z>1) show that the CO spectral line energy distribution of normal star-forming galaxies are less excited than those ofSMGs and QSOs. The ratio between mid-J and low-J CO transition measured in SMGs and QSOs is higher than that observed in normal galaxies by a factor >1.5 <cit.>. § OVERDENSITYALMA observations have revealed thepresence of six companion sources within a projected distance ∼ 160 kpc from the quasars (Appendix A).In five out of the six sources we also detect a line emissionthat may be identified with CO(3-2) transition at similar redshifts of the QSOs.The CO(3-2) lines in thesesources have luminosities of 0.6-23 K km/s pc^2, resulting in a molecular masses of 0.5-90×10^10 that are even higher than those measured in the QSOs themselves. The molecular mass and the high SFR (∼1000), inferred from the continuum emission is comparable to those observed in starbursts and submillimetre galaxies (SMGs). In one case, the line emission is even spatially resolved by the ALMA beam and the gradient of velocity indicates a dynamical mass of M_ dyn=2×10^11sin^2(i), which is similar to those observed in high-z SMGs <cit.>. Recent galaxy evolution models predict that the rate of galaxy mergers and interactions increases in the redshift range 1<z<3, driving extreme starburst events and rapid accretion onto the massive black holes in the galaxy centre <cit.>. These predictions have been supported by new extragalactic surveys at millimetre and submillimetre ranges having uncovered a population of dusty star-forming galaxies at high redshift. <cit.> found an overdensity of submillimetre galaxies in 17 out of 49 QSOs at redshift z∼2.A similar scenario has been recently reported by <cit.> who detected two millimetre-bright galaxieswithin 200 kpc from a QSO at z=2.5.An overdensity system is also observed in BR 1202-0725, which is mainly composed by a QSO and a submillimetre galaxyat z∼4.7<cit.>. Overall, these results support the hypothesis that submillimetre galaxies and QSOs represent different stages of galaxy evolution after a merger <cit.>. The detections of massive companions sources in the ALMA field of view have been also observed in QSOs at higher redshifts <cit.> indicating that major mergers are important drivers for rapid early SMBH growth.The detections of these seredentipitous sources in our ALMA observations suggest that the three QSOs are located in a overdensity. Assuming that a SMBH of 10^10is associated to a dark halo of mass of10^13<cit.>, we estimate a virial radius of about 500 kpc for the three QSOs. This is larger than the projected distance between the QSOs and the serendipitous companions. We therefore conclude that the QSOsand the serendipitous sources mayrepresent a complex merging system at redshift z∼2.3-2.5. Futuremillimetre observations at different wavelength bands will confirm the redshift of the serendipitous galaxiesand their nature.§ SUMMARYWe have presented new ALMA 3mm observations aimed at mapping CO(3-2) in three z∼2.4 quasars, , , and HB8903, showing evidence for ionised outflows quenching star formation <cit.>. Below, we summarise the main results of this work:* The ALMA observations reveal the presence of serendipitous galaxies, three of those are detected both incontinuum (at 3 mm) and inline emission, within a projected distance of 160 kpc from the QSOs.Assuming the emission line detected in these galaxies can identified with the CO(3-2) transition, we conclude thatand HB8903reside inoverdense systems, as often found for QSOs at similar and higher redshifts.* The CO(3-2) emission at the systemic velocity of the QSO is detected in only oneof the three targets, that is, .The CO profile has a velocity and line width consistent with the narrowandcomponents tracing SF in the host galaxy.In addition the CO emission is spatially resolved by the ALMA beam and is not symmetrically distributed around the location of the QSOs, butabsent or faint in the outflow region. This is suggestive of a scenario in which the AGN-driven outflow is removing the ionised and molecular gas from the host galaxy. * Inwe tentatively detect a faint CO(3-2) emission blueshifted by 2000 km/s relative to the redshift of the host galaxy and spatially coincident with the ionised outflow emission. If confirmed by follow-up observations, this CO emission may be tracing amolecular cloud at high velocity that has been ejected away from the galaxy by AGN-driven outflows.Also, our analysis would suggest that the molecular gas in the outflow region is more highly excited than the rest of molecular gas in the host galaxy. An alternative interpretation to the outflow scenario could be that the detected CO emission is associated to a faint companion galaxy. Future deeper ALMA observations of CO(3-2) and higher (or lower) rotation transition will be fundamental in confirming the redshift of this detection and analyse the excitation state of the molecular gas. If the outflow scenario will be confirmed, the new ALMA observations will allow us to estimate the molecular outflow mass rate and compare this value with that estimated from ionised gas in the same QSO.* Assuming a =0.8 / km/s pc^2, the inferred molecular gas mass in both host galaxies is clearly below what observed inMS galaxies with similar SFR and consistent with those observed in other high-z QSO and SMGs.We conclude that AGN-driven outflows in our sample are removing ionised and molecular gas from the host galaxy and quenching the star formation.The interaction of the fast winds with thehost galaxy is clearlyvisiblein the outflow region, where the CO emission is faint. This result supports our previous studiesof these QSOs showingemission quenched in the outflow region.§ ACKNOWLEDGEMENTS This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.0.00965.S; which can be retrieved from the ALMA data archive: https://almascience.eso.org/ alma-data/archive. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. SC and RM acknowledge financial support from the Science and Technology Facilities Council (STFC). RM acknowledges ERC Advanced Grant 695671 “QUENCH”. MB acknowledges support from the FP7 Career Integration Grant “eEASy”: Supermassive black holes through cosmic time: from current surveys to eROSITA-Euclid Synergies" (CIG 321913).CC acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 664931. RS acknowledges support from the European Research Council under the European Union (FP/2007-2013)/ERC GrantAgreement n. 306476. CF acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curiegrant agreement No 664931.aa § OVERDENSE SYSTEMSThe ALMA observations serendipitously reveal line and continuum sources in the field ofand HB8903 quasars, (Fig. <ref> and <ref>),located within a projected radius of ∼ 160 kpc from the centre of the two QSO. From the continuum emission we infer a SFR>900for all sources and a M_ dust = 1-6 ×10^8 .In Table <ref> we list the properties of these detections and, hereafter, we use the terms LBQS0109-A, -B ,-C, and HB8903-A, -Btorefer to the serendipitous sources around LBQS0109 andHB8903, respectively.If theemission line are identified with CO(3-2) transition, the serendipitous sources are in a redshift range of Δ z = 0.005 (|Δ v|<500 km/s) relative to the respective QSOs.We estimate a CO(3-2) luminosity of 0.6-23×10^10 K km/s pc^2 for these sources and a molecular gas mass of 0.5-90×10^10 , depending of(Table <ref>), which is higher than molecular gas masses inferred for the three QSOs. The SFR and M_ gas of these serendipitous sources are consistent with those observed inSMGs (Fig. <ref>).Assuming a dark halo mass of 10^13 , which is reasonable for a galaxy hosting a massive BH of mass 10^10 , we note that the serendipitousgalaxies arewithin the virial radius (∼500 kpc) of the central QSO. Bothand HB8903 could represent an overdense system at z≃2.4. From a 2D Gaussian fitting we estimate the size of the CO(3-2) emission in all serendipitous galaxies andtwoout of five sources turn out to be spatially resolved (LBQS0109-A and LBQS0109-B). However, because the S/N-per-pixel in the LBQS0109-B is too low (<10), we performa pixel-by-pixel kinematic analysisonly on LBQS0109-A.The results of the kinematic analysis are reported in Fig. <ref> that shows a clear north-southvelocity gradient of ∼500 km/s over ∼1 (8.3 kpc). If it were due to simple rotation, this would imply a dynamical mass without inclination angle i correction of 2×10^11sin^2(i) . We do not detect any significant (S/N>5) line emission in the ALMA field-of-view ofexcept the line emission candidate spatially offset by 0.2 discussed in Sect. <ref>.We observe a continuum emission with a S/N=5 located at ∼14 from the QSO.The continuum emission (Fig. <ref>) is not spatially resolved and its flux density at 3 mm is 61±12 . The comparison of our ALMA detections withnumber counts studies at similar wavelengths indicates that all QSOs reside in a significant over-density.In fact, theoretical results obtained by simulations and semi-analytical models predict a number counts at 3 mm of N(S>100)≃100^-2 <cit.>, yielding to a N(S>100)≃0.02 FOV^-1_ ALMA where FOV_ ALMA is the field-of-view area of our ALMA maps (∼2500 arcsec^2). Similar number counts values are also observed in recent ALMA studies at 1.3 mm <cit.>.The number of serendipitous sources detect in our ALMA maps is larger than that expected by simulation and millimetre surveys, hence supporting that the three QSOs are located in a overdense region. Properties of serendipitous sources. LBQS0109-A LBQS0109-B LBQS0109-C HB8903-A HB8903-B 2QZJ0028-A RA01:12:17.08 01:12:17.4901:12:17:23 3:31:06.77 3:31:06.09 00:28:31.812 DEC 2:29:32.992:29:52.262:29:63.64 -38:24:02.79 -38:24:20.95 - 28.16.51.616S_ 3mm [] 115±12 52±12 48±12 36±18 54±18 61±12M_ dust [10^8 ]^a 4-6 1-3 1-3 1-2 2-3 2-3 SFR []^a 2800 1200 1100 900 1400 1500 λ_ CO(3-2) [mm] 2.90642±0.00008 2.9112±0.0003 2.9045±0.00032.9779±0.00062.9843±0.0005 -FWHM_ CO(3-2) [km/s]190±20 590±70300±20 400±100 400±100 - S_ CO(3-2)Δ v [Jy km/s] 1.72±0.06 7.16±0.050.19±0.05 0.27±0.030.32±0.04 - L^'_ CO(3-2) [10^10 K km/s pc^2] 5.2±0.2 22.9±0.2 0.58±0.150.88±0.09 1.04±0.13 - L_ CO(3-2) [10^7 ] 6.9±0.2 29.1±0.2 0.8±0.21.26±0.13 1.38±0.17 - M_ CO(=0.8) [10^10 ]^b4±2 18±90.5±0.3 0.7±0.4 0.8±0.5 -M_ CO(=4) [10^10 ]^b21±11 90±502.3±1.7 3±2 4±2 -^a Under the assumption that the continuum emission at 3 mm is completely associatedto thermal dust continuum emission. We assume a T_d=40-60 K and a β=2.0. ^bUnder the assumption that the line detection is associated to the CO(3-2) transition and assuming a =1.0±0.5. The statistical errors associated to the molecular gas includeuncertainties.§ RELIABILITY OF THECO(3-2) LINE DETECTION IN .In Sect. <ref> we show the detection of a faint line emission spatially offset by 0.2 from the centroid ofcontinuum emission at 3 mmand blusefhited of ∼-2000 km/s respect to the redshift of . In this section we discuss more in detail the significance of this candidate lineemission.We checked whether negative sources are detected with the same significance or not. We performed a blind search for positive and negative line emitters within the ALMA primary beam areaand within a velocity range |v|< 2000 km/s relative to the redshift of QSO. We searched for line emitters with line widthranging from 200 km/s to 500 km/s.We extracted 16 positive and 10 negative emission line with a level of confidence higher 5σ. The number of positive peaks at S/N> 5 is 50% higher than the number of negative peaks, hence it indicates that few of these detections could be not noise fluctuations. In Fig. <ref> is shown the spectrum of the blueshifted line extracted at the location of the emission peak. Focusing on the spectrum, we note that there are no negative peaks with a level of confidence above 3σ suggesting that the positive blueshifted line, which is detected with a S/N>4, is not due noise fluctuations. These tests and the coincidence thatboth velocity and location of the line emitted are consistent with the ionised outflow support the reliability of this detection. | http://arxiv.org/abs/1706.08987v2 | {
"authors": [
"S. Carniani",
"A. Marconi",
"R. Maiolino",
"C. Feruglio",
"M. Brusa",
"G. Cresci",
"M. Cano-Díaz",
"C. Cicone",
"B. Balmaverde",
"F. Fiore",
"A. Ferrara",
"S. Gallerani",
"F. La Franca",
"V. Mainieri",
"F. Mannucci",
"H. Netzer",
"E. Piconcelli",
"E. Sani",
"R. Schneider",
"O. Shemmer",
"L. Testi"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170627180331",
"title": "AGN feedback on molecular gas reservoirs in quasars at $z\\sim$2.4"
} |
Adaptive prefix-assignment for symmetry reduction]An adaptive prefix-assignment techniquefor symmetry reduction^† Aalto University, Department of Computer Science [email protected] [email protected] [email protected] ^ Current address: University of Helsinki, Department of Computer Science [email protected] paper presents a technique for symmetry reduction that adaptively assigns a prefix of variables in a system of constraints so that the generated prefix-assignments are pairwise nonisomorphic under the action of the symmetry group of the system.The technique is based on McKay's canonical extension framework[J. Algorithms 26 (1998), no. 2, 306–324]. Among key features of the technique are(i) adaptability—the prefix sequence can be user-prescribed and truncated for compatibility with the group of symmetries;(ii) parallelizability—prefix-assignments can be processed in parallel independently of each other;(iii) versatility—the method is applicable whenever the group of symmetries can be concisely represented as the automorphism group of a vertex-colored graph; and (iv) implementability—the method can be implemented relying on a canonical labeling map for vertex-colored graphs as the only nontrivial subroutine. To demonstrate the practical applicability of our technique, we have prepared an experimental open-source implementation of the techniqueand carry out a set of experiments that demonstrate ability to reduce symmetryon hard instances. Furthermore, we demonstrate that the implementationeffectively parallelizes to compute clusters with multiple nodes viaa message-passing interface. [ Jukka Kohonen^† December 30, 2023 =====================§ INTRODUCTION§.§ Symmetry reductionSystems of constraints often have substantial symmetry. For example,consider the following system of Boolean clauses:(x_1∨ x_2) ∧ (x_1∨x̅_3∨x̅_5) ∧ (x_2∨x̅_4∨x̅_6) .The associative and commutative symmetries of disjunction and conjunction induce symmetries between the variables of (<ref>), a fact thatcan be captured by stating that the group Γ generated by the two permutations (x_1 x_2)(x_3 x_4)(x_5 x_6) and (x_4 x_6) consists of all permutations of the variablesthat map (<ref>) to itself. That is, Γ is theautomorphism group of the system (<ref>), cf. Section <ref>.Known symmetry in a constraint system is a great asset from theperspective of solving the system, in particular since symmetryenables one to disregard partial solutions that are isomorphic toeach other under the action of Γ on the space of partial solutions.Techniques for such isomorph rejection [ A term introduced by J. D. Swift <cit.>;cf. Hall and Knuth <cit.> fora survey on early work on exhaustive computer search and combinatorial analysis.] <cit.> (alternatively, symmetry reduction or symmetry breaking)are essentially mandatory if one desires an exhaustive traversal ofthe (pairwise nonisomorphic) solutions of a highly symmetric system ofconstraints, or if the system is otherwise difficult to solve, for example,with many “dead-end” partial solutions compared with the actual number of solutions.A prerequisite to symmetry reduction is that the symmetries are known.In many cases it is possible to automatically discover and compute thesesymmetries to enable practical and automatic symmetry reduction. In this context the dominant computational approach for combinatorial systems of constraints is to represent Γ via the automorphism group ofa vertex-colored graph that captures the symmetries in the system.Carefully engineered tools for working with symmetries of vertex-coloredgraphs <cit.> andpermutation group algorithms <cit.>then enable one to perform symmetry reduction. For example, for purposesof symmetry computations we may represent (<ref>) as thefollowing vertex-colored graph: [ < g r a p h i c s > ] In particular, the graph representation (<ref>) enablesus to discover and reduce symmetry to avoid redundant work when solving the underlying system (<ref>). §.§ Our contributionThe objective of this paper is to present a novel technique for symmetryreduction on systems of constraints. The technique is basedon adaptively assigning values to a prefix of the variables so that theobtained prefix-assignments are pairwise nonisomorphic under the actionof Γ. The technique can be seen as an instantiation ofMcKay's <cit.> influential canonical extension framework forisomorph-free exhaustive generation.To give a brief outline of the technique,suppose we are working with a system of constraints over a finite set Uof variables that take values in a finite set R. Suppose furthermore thatΓ≤(U) is the automorphism group of the system.Select k distinct variables u_1,u_2,…,u_k in U.These k variables form the prefix sequence considered bythe method. The technique works by assigning values in R to the variables ofthe prefix, in prefix-sequence order, with u_1 assigned first, then u_2,then u_3, and so forth, so that at each step the partial assignmentsso obtained are pairwise nonisomorphic under the action of Γ. For example, in (<ref>) the partial assignments x_1↦ 0, x_2↦ 1 and x_1↦ 1, x_2↦ 0 are isomorphic since (x_1 x_2)(x_3 x_4)(x_5 x_6)∈Γmaps one assignment onto the other; in total there are three nonisomorphic assignments to the prefix x_1,x_2 in (<ref>), namely(i) x_1↦ 0, x_2↦ 0,(ii) x_1↦ 0, x_2↦ 1,and (iii) x_1↦ 1, x_2↦ 1. Each partial assignment that represents an isomorphism class can then be used to reduce redundant work whensolving the underlying system by standard techniques— in the nonincremental case, the system is augmented with a symmetry-breaking predicate requiring that one of the nonisomorphic partial assignments holds, while in the incremental setting <cit.> the partial assignments can be solved independently or even in parallel.Our contribution in this paper lies in how the isomorph rejection isimplemented at the level of isomorphism classes of partial assignments by careful reduction to McKay's <cit.> isomorph-free exhaustive generation framework. The keytechnical contribution is that we observe how to generate the partialassignments in a normalized form that enables both adaptability (that is, the prefix u_1,u_2,…,u_k can be arbitrarily selectedto match the structure of Γ) and precomputation ofthe extending variable-value orbits along a prefix.Among further key features of the technique are: * Implementability. The technique can be implemented by relying on a canonical labeling mapfor vertex-colored graphs(cf. <cit.>) as the only nontrivial subroutine that is invoked once for each partial assignment considered.* Versatility. The method is applicable whenever the group of symmetries can beconcisely represented as a vertex-colored graph;cf. (<ref>) and (<ref>). This is useful inparticular when the underlying system has symmetries that are not easilydiscoverable from the final constraint encoding, for example, due to the factthat the constraints have been compiled or optimized [For a beautiful illustration, we refer to Knuth's <cit.> example of optimum Boolean chains for 5-variable symmetric Boolean functions—from each optimum chain it is far from obvious that the chain represents a symmetric Boolean function. (See also Example <ref>.)] from a higher-level representation in a symmetry-obfuscating manner. A graphical representation can represent such symmetry directly andindependently of the compiled/optimized form of the system.* Parallelizability. As a corollary of implementing McKay's <cit.> framework, the technique does not need to store representatives of isomorphism classesin memory to perform isomorph rejection, which enables easy parallelizationsince the partial assignments can be processed independently of each other. The required mathematical preliminaries on symmetry and McKay's framework are reviewed in Sections <ref> and <ref>, respectively.The main technical contribution of this paper is developed in Section <ref> where we present the prefix-assignment technique that we will subsequently extend to account for value symmetry in Section <ref>. Our development in Sections <ref> and <ref> relies on an abstract group Γ, with the understanding that a concrete implementation can be designed e.g. in terms of a vertex-colored graph representation, as will be explored in Section <ref>.To demonstrate the practical applicability of our technique, we have prepared an open-source parallel implementation <cit.>. The implementation is structured as a preprocessor that works with an explicitly given graph representation and utilizes the nauty <cit.> canonical labeling software for vertex-colored graphs as a subroutine to prepare an exhaustive collection of nonisomorphic prefix assignments relative to a user-supplied prefix of variables, and the Message PassingInterface (MPI) for parallelization. Further details of the implementation are presented in Section <ref>.In Section <ref>, we report on a set of experiments that(i) demonstrate the ability to reduce symmetry on hard instances,(ii) study the serendipity of an auxiliary graph for encoding the symmetries in an instance,(iii) give examples of instances with hard combinatorial symmetry whereour technique performs favorably in comparison with earlier techniques, and(iv) study the parallel speedup obtainable when we distributethe symmetry reduction task to a compute cluster with multiple compute nodes.§.§ Earlier workA classical way to exploit symmetry in a system of constraints isto augment the system with so-called symmetry-breaking predicates (SBP) that eliminate either some or all symmetric solutions <cit.>. Such constraints are typically lexicographic leader (lex-leader) constraintsthat are derived from a generating set for the group of symmetries Γ.Among recent work in this area,Devriendt et al. <cit.> extend the approachby presenting a more compact way for expressing SBPs and a method for detecting “row interchangeabilities”. Itzhakov and Codish <cit.> present a method forfinding a set of symmetries whose corresponding lex-leader constraintsare enough to completely break symmetries in search problems on small (10-vertex) graphs; this approach is extended by Codish et al. <cit.> by adding pruning predicates that simulate thefirst iterations of the equitable partition refinement algorithmof nauty <cit.>. Heule <cit.> shows that small completesymmetry-breaking predicates can be computed by considering arbitraryBoolean formulas instead of lex-leader formulas.Our present technique can be seen as a method for producingsymmetry-breaking predicates by augmenting the system of constraints withthe disjunction of the nonisomorphic partial assignments. The main difference to the related work above is that our technique does not produce the symmetry-breaking predicate from a set of generators forΓ but rather the predicate is produced recursively, and with the possibility for parallelization, by classifying orbit representatives upto isomorphism using McKay's <cit.> framework. As such ourtechnique breaks all symmetry with respect to the prescribed prefix, but comes at the cost of additional invocations of graph-automorphism andcanonical-labeling tools. This overhead and increased symmetry reduction in particular means that our technique is best suited for constraintsystems with hard combinatorial symmetry that is not easily capturable from a set of generators, such as symmetry in combinatorial classification problems <cit.>. In addition to McKay's <cit.>canonical extension framework, other standard frameworks for isomorph-freeexhaustive generation in this context include orderly algorithms due to Faradžev <cit.> and Read <cit.>, as well as the homomorphism principle for group actions due to Kerber andLaue <cit.>.It is also possible to break symmetry within a constraint solver duringthe search by dynamically adding constraints that rule out symmetric parts ofthe search space (cf. <cit.>). If we use the nonisomorphic partial assignments produced by our techniqueas assumption sequences (cubes) in the incremental cube-and-conquer approach <cit.>, our technique can be seen as a restricted way of breaking the symmetries in the beginning of the search, with thebenefit—as with cube-and-conquer—thatthe portions of the search space induced by the partialassignments can be solved in parallel, either with complete independenceor with appropriate sharing of information (such as conflict clauses)between the parallel nodes executing the search. For further work in dynamicsymmetry breaking, cf. <cit.>.For work on isomorphism and canonical labeling techniques,cf. <cit.>.§ PRELIMINARIES ON GROUP ACTIONS AND SYMMETRYThis section reviews relevant mathematical preliminaries and notational conventions for groups, group actions, symmetry, and isomorphism for oursubsequent development.(Cf. <cit.> for further reference.) §.§ Groups and group actionsLet Γ be a finite group and let Ω be a finite set(the domain) on which Γ acts. For two groups Λ and Γ,let us write Λ≤Γ to indicate that Λ is a subgroupof Γ. We use exponential notation for group actions, and accordingly our groups act from the right. That is, for an object X∈Ωand γ∈Γ, let us write X^γ for the object in Ωobtained by acting on X with γ. Accordingly, we have X^(βγ)=(X^β)^γ forall β,γ∈Γ and X∈Ω. For a finite set V, let us write (V) for the group of all permutations of V with composition of mappings as the group operation. §.§ Orbit and stabilizer, the automorphism groupFor an object X∈Ω let us writeX^Γ={X^γ:γ∈Γ}for the orbit of X under the action of Γ andΓ_X={γ∈Γ:X^γ=X}≤Γfor the stabilizer subgroup of X in Γ.Equivalently we say that Γ_X is the automorphism group of Xand write (X)=Γ_X whenever Γ is clear from the context; if we want to stress the acting group we write _Γ(X). We write Ω/Γ={X^Γ:X∈Ω} for the set of all orbits of Γ on Ω.For Λ≤Γ and γ∈Γ, let us write Λ^γ=γ^-1Λγ ={γ^-1λγ:λ∈Λ}≤Γ for the γ-conjugate of Λ.For all X∈Ω and γ∈Γ we have (X^γ)=(X)^γ.That is, the automorphism groups of objects in an orbit are conjugates of each other. §.§ IsomorphismWe say that two objects are isomorphic if they are onthe same orbit of Γ in Ω.In particular, X,Y∈Ω are isomorphicif and only if there exists an isomorphism γ∈Γfrom X to Y that satisfies Y=X^γ. An isomorphism from an object to itself is an automorphism.Let us write (X,Y) for the set of all isomorphisms from X to Y.We have that (X,Y)=(X)γ=γ(Y) where γ∈(X,Y) is arbitrary.Let us write X≅ Y to indicate that X and Y are isomorphic.If we want to stress the group Γ under whose action isomorphismholds, we write X≅_Γ Y. §.§ Elementwise action on tuples and sets Suppose that Γ acts on two sets, Ω and Σ. We extend the action to the Cartesian product Ω×Σelementwise by defining (X,S)^γ=(X^γ,S^γ)for all (X,S)∈Ω×Σ and γ∈Γ.Isomorphism extends accordingly; for example, we say that (X,S) and (Y,T)are isomorphic and write (X,S)≅ (Y,T) if there exists aγ∈Γ with Y=X^γ and T=S^γ.Suppose that Γ acts on a set U.We extend the action of Γ on U to an elementwise action of Γon subsets W⊆ U by setting W^γ={w^γ:w∈ W} for allγ∈Γ and W⊆ U. In what follows we will tacitly work with these elementwise actions on tuples and sets unless explicitly otherwiseindicated. §.§ Canonical labeling and canonical formA function κ:Ω→Γ is acanonical labeling map for the action of Γ on Ω if (K)for all X,Y∈Ω it holds that X≅ Y impliesX^κ(X)=Y^κ(Y) (canonical labeling).For X∈Ω we say that X^κ(X) isthe canonical form of X in Ω.From (K) it follows that isomorphic objects have identical canonical forms,and the canonical labeling map gives an isomorphism that takes an object toits canonical form. We assume that the act of computing κ(X) for agiven X produces as a side-effect a set of generators for the automorphism group (X). § MCKAY'S CANONICAL EXTENSION METHOD This section reviews McKay's <cit.> canonical extension methodfor isomorph-free exhaustive generation. Mathematically it will be convenientto present the method so that the isomorphism classes are captured as orbitsof a group action of a group Γ, and extension takes place in one stepfrom “seeds” to “objects” being generated, with the understanding that the method can be applied inductively in multiple steps so that the “objects” of the current step become the “seeds” for the next step.For completeness and ease of exposition, we also give a correctness proof forthe method. We stress that all material in this section is well known. (Cf. <cit.>.) §.§ Objects and seedsLet Ω be a finite set of objects and let Σ be a finite set of seeds. Let Γ be a finite group that acts on Ωand Σ. Let κ be a canonical labeling map for the action of Γ on Ω.§.§ Extending seeds to objects.Let us connect the objects and the seeds by means of a relation e⊆Ω×Σ that indicates which objects can be built from which seeds by extension. For X∈Ω and S∈Σ we say that X extends S and write XS if (X,S)∈ e. We assume the relation e satisfies(E1)e is a union of orbits of Γ, that is, e^Γ=e (invariance), and(E2) for every object X∈Ω there exists a seed S∈Σ suchthat XS (completeness).For a seed S∈Σ, let us write e(S)={X∈Ω:XS} for the set of all objects that extend S. §.§ Canonical extensionWe associate with each object a particular isomorphism-invariant extension by which we want to extend the object from a seed. A function M:Ω→Σ is a canonical extension map if (M1)for all X∈Ω it holds that (X,M(X))∈ e (extension), and(M2) for all X,Y∈Ω we have that X≅ Y implies(X,M(X))≅ (Y,M(Y)) (canonicity).That is, (M1) requires that X is in fact an extension of M(X) and (M2) requires that isomorphic objects have isomorphic canonical extensions. In particular, X↦ (X,M(X)) isa well-defined map from Ω/Γ to e/Γ. §.§ Generating objects from seedsLet us study the following procedure, which is invoked for exactly one representative S∈Σ from each orbit inΣ/Γ:(P)Let S∈Σ be given as input. Iterate over all X∈ e(S).Perform zero or more isomorph rejection tests on X and S.If the tests indicate we should accept X, visit X. Let us first consider the case when there are no isomorph rejection tests. The procedure (P) visits every isomorphismclass of objects in Ω at least once. To see that every isomorphism class is visited, let Y∈Ω be arbitrary.By (E2), there exists a T∈Σ with YT. By our assumption on how procedure (P) is invoked, T is isomorphic to a unique S such thatprocedure (P) is invoked with input S. Let γ∈Γ be anassociated isomorphism with S^γ=T. By (E1) and YT,we have XS for X=Y^γ^-1. By the structure of procedure (P)we observe that X is visited and X≅ Y. Since Y was arbitrary,all isomorphism classes are visited at least once. Let us next modify procedure (P) so that any two visits to the same isomorphism class of objects originate from the same procedure invocation.Let M:Ω→Σ be a canonical extension map. Whenever we construct X by extending S in procedure (P),let us visit X if and only if (T1) (X,S)≅ (X,M(X)).The procedure (P) equipped with the test (T1) visits every isomorphism class of objects in Ω at least once. Furthermore,any two visits to the same isomorphism class must(i) originate by extension from the same procedure invocation on input S, and(ii) belong to the same (S)-orbit of this seed S.Suppose that X is visited by extending S and Y is visited by extending T, with X≅ Y. By (T1) we must thus have(X,S)≅(X,M(X)) and (Y,T)≅(Y,M(Y)). Furthemore, from (M2)we have (X,M(X))≅ (Y,M(Y)). Thus, (X,S)≅ (Y,T) andhence S≅ T. Since S≅ T, we must in fact have S=T by our assumption on how procedure (P) is invoked. Since X and Y were arbitrary,any two visits to the same isomorphism class must originate by extension from the same seed. Furthermore, we have (X,S)≅ (Y,S) and thus X≅_(S) Y. Let us next observe that every isomorphism class of objects is visited at leastonce. Indeed, let Y∈Ω be arbitrary. By (M1), we have YM(Y).In particular, there is a unique S∈Σ with S≅ M(Y) such that procedure (P) is invoked with input S. Let γ∈Γ be an associated isomorphism with S^γ=M(Y). By (E1), we have XS for X=Y^γ^-1. Furthermore, X≅ Y implies by (M2) that(X,M(X))≅ (Y,M(Y))=(X^γ,S^γ)≅ (X,S), so (T1) holdsand X is visited. Since X≅ Y and Y was arbitrary, everyisomorphism class is visited at least once. Let us next observe that the outcome of test (T1) is invariant on each (S)-orbit of extensions of S. For all α∈(S)we have that(T1) holds for (X,S) if and only if(T1) holds for (X^α,S).From X≅ X^α and (M2) we have(X,M(X))≅ (X^α,M(X^α)).Thus, (X,S)≅ (X,M(X)) if and only if (X^α,S) =(X^α,S^α) ≅ (X,S) ≅ (X,M(X)) ≅ (X^α,M(X^α)). Lemma <ref> in particular implies that we obtain complete isomorph rejection by combining the test (T1) with a further test that ensures complete isomorph rejection on (S)-orbits.Towards this end, let us associate an arbitrary order relation on every(S)-orbit on e(S). Let us perform the following further test:(T2) X=min X^(S).The following lemma is immediate from Lemma <ref>and Lemma <ref>.The procedure (P) equipped with the tests (T1) and (T2)visits every isomorphism class of objects in Ω exactly once.§.§ A template for canonical extension mapsWe conclude this section by describinga template of how to use an arbitrary canonical labeling mapκ:Ω→Γ to construct a canonical extension map M:Ω→Σ. For X∈Ω construct the canonical form Z=X^κ(X). Using the canonical form Z only, identify a seedT with ZT. In particular, such a seed must exist by (E2).(Typically this identification can be carried out by studying Z and finding an appropriate substructure in Z that qualifies as T. For example, T may be the minimum seed in Σ that satisfies ZT.Cf. Lemma <ref>.)Once T has been identified, set M(X)=T^κ(X)^-1. The map X↦ M(X) above is a canonical extension map.By (E1) we have XM(X) because Z^κ(X)^-1=X,T^κ(X)^-1=M(X), and ZT.Thus, (M1) holds for M. To verify (M2), let X,Y∈Ω withX≅ Y be arbitrary. Since X≅ Y, by (K) we haveX^κ(X)=Z=Y^κ(Y).It follows that M(X)=T^κ(X)^-1 and M(Y)=T^κ(Y)^-1, implying that γ=κ(X)κ(Y)^-1 is an isomorphism witnessing (X,M(X))≅ (Y,M(Y)). Thus, (M2) holds for M.§ GENERATION OF PARTIAL ASSIGNMENTS VIA A PREFIX SEQUENCE This section describes an instantiation of McKay's method that generates partial assignments of values to a set of variables U one variable at a time following a prefix sequence at the level of isomorphism classes given by the action of a group Γ on U. We postpone the extension to include variables to Section <ref>. Let R be a finite set where the variables in U take values. §.§ Partial assignments, isomorphism, restrictionFor a subset W⊆ U of variables, let us say thata partial assignment of values to W is a mapping X:W→ R.Isomorphism for partial assignments is induced by the following group action.Let γ∈Γ act on X:W→ R by settingX^γ:W^γ→ R whereX^γ is defined for all u∈ W^γ by X^γ(u)=X(u^γ^-1) .The action (<ref>) is well-defined.We observe that for the identity ϵ∈Γ of Γ we have X^ϵ=X. Furthermore, for all γ,β∈Γ andu∈ W^γβ=(W^γ)^β we have X^γβ(u) =X(u^(γβ)^-1) =X((u^β^-1)^γ^-1) =X^γ(u^β^-1) =(X^γ)^β(u) . For an assignment X:W→ R, let us write X=W for the underlying set of variables assigned by X. Observe that the underline map is a homomorphism of group actions in the sense that X^γ=X^γholds for all γ∈Γ and X:W→ R. For Q ⊆X, let us write X|_Q for the restriction of X to Q.§.§ The prefix sequence and generation of normalized assignmentsWe are now ready to describe the generation procedure. Let us begin by prescribing the prefix sequence.Let u_1,u_2,…,u_k be k distinct elements of U and let U_j={u_1,u_2,…,u_j} for j=0,1,…,k. In particular we observe that U_0⊆ U_1⊆⋯⊆ U_kwith U_j∖ U_j-1={u_j} for all j=1,2,…,k.For j=0,1,…,k let Ω_j consist of all partial assignmentsX:W→ R with W≅ U_j. Or what is the same, using theunderline notation, Ω_j consists of all partial assignments X withX≅ U_j.We rely on canonical extension to construct exactly one object from eachorbit of Γ on Ω_j, using as seeds exactly one object from each orbit of Γ on Ω_j-1, for each j=1,2,…,k. We assume the availability of canonical labeling mapsκ:Ω_j→Γ for each j=1,2,…,k.Our construction procedure will work with objects that are in a normal form to enable precomputation for efficient execution of the subsequenttests for isomorph rejection. Towards this end,let us say that X∈Ω_j is normalized if X=U_j.It is immediate from our definition of Ω_jand (<ref>) that each orbit in Ω_j/Γcontains at least one normalized object. Let us begin with a high-level description of the construction procedure, to be followed by the details of the isomorph rejection tests and a proof of correctness. Fix j=1,2,…,k andstudy the following procedure, which we assume is invoked for exactly one normalized representative S∈Ω_j-1 from each orbitin Ω_j-1/Γ:(P')Let a normalized S∈Ω_j-1 be given as input.For each p∈ u_j^(U_j-1) and each r∈ R,construct the assignment X:U_j-1∪{p}→ Rdefined by X(p)=r and X(u)=S(u) for all u∈ U_j-1.Perform the isomorph rejection tests (T1') and (T2') on X and S. If both tests accept, visit X^ν(p) where ν(p)∈(U_j-1)normalizes X. From an implementation point of view,it is convenient to precompute the orbit u_j^(U_j-1)together with group elements ν(p)∈(U_j-1) foreach p∈ u_j^(U_j-1) that satisfyp^ν(p)=u_j. Indeed, a constructed X withX=U_j-1∪{p} can now be normalized by acting withν(p) on X to obtain a normalized X^ν(p) isomorphic to X.§.§ The isomorph rejection tests Let us now complete the description of procedure (P') by describing the two isomorph rejection tests (T1') and (T2'). This subsection only describes the tests with an implementation in mind, the correctness analysis is postponed to the following subsection.Let us assume that the elements of U have been arbitrarily ordered and that κ:Ω_j→Γ is a canonical labeling map. Suppose that X has been constructed by extending a normalized S with X=S∪{p}=U_j-1∪{p}.The first test is: (T1')Subject to the ordering of U, select the minimum q∈ Usuch that q^κ(X)^-1ν(p)∈ u_j^(U_j). Accept if and only if p≅_(X) q^κ(X)^-1. From an implementation perspective we observe that we can precompute the orbit u_j^(U_j). Furthermore, the only computationally nontrivial part of the test is the computation of κ(X) since we assume that weobtain generators for (X) as a side-effect of this computation. Indeed, with generators for (X) available, it is easy to compute the orbits U/(X) and hence to test whether p≅_(X) q^κ(X)^-1.Let us now describe the second test:(T2')Accept if and only if p=min p^(S) subject to the ordering of U. From an implementation perspective we observe thatsince S is normalized we have(S)≤(S)=(U_j-1) and thus the orbitu_j^(U_j-1) considered by procedure (P') partitions intoone or more (S)-orbits. Furthermore, generators for (S) arereadily available (due to S itself getting accepted in the test (T1')at an earlier level of recursion), and thus the orbitsu_j^(U_j-1)/(S) and their minimum elements are cheap to compute.Thus, a fast implementation of procedure (P') will in most cases executethe test (T2') before the more expensive test (T1').We display below a possible search tree forthe system of clauses (<ref>) and the prefix sequence x_3,x_4,x_5,x_6. Each node in the search tree displays the prefix assignment X (top), its canonical version X^κ(X) (middle) and its normalized version X^ν(p) (bottom). The variables have the Boolean domain {,} and the assignments are given in the literal form; for example, we write x̅_3 x_4 for the assignment {x_3 ↦, x_4 ↦}.< g r a p h i c s > The nodes with a red cross are nodes eliminated by the test (T1') and the ones with a blue cross are eliminated by the test (T2'). (For convenience of display, these eliminated nodes are only drawn inthe first three levels above.)For instance, the node with X = x̅_3 x_4 is eliminated by the test (T1') because the minimum q such that q^κ(X)^-1ν(x_4)∈ x_4^(U_2)={x_3,x_4} when κ(X) = {x_3 ↦ x_3,x_4 ↦ x_4} and ν(x_4) = {x_3 ↦ x_3,x_4 ↦ x_4} is x_3 and x_4 ≇_(X) q^κ(X)^-1=x_3 as (X) = {ϵ}.On the other hand, the node with X = x̅_3 x̅_6 is eliminated by the test (T2') as x_6 ≠min x_6^(x̅_3) and x_6^(x̅_3) = {x_4,x_6}.We observe that the search tree has dead-end nodes that do not extend to any full prefix assignment. §.§ CorrectnessWe now establish the correctness of procedure (P') together withthe tests (T1') and (T2') by reduction to McKay's frameworkand Lemma <ref>.Fix j=1,2,…,k.Let us start by defining the extension relatione⊆Ω_j×Ω_j-1for all X∈Ω_j and S∈Ω_j-1 bysetting XS if and only if there exists a γ∈Γ such thatX^γ=U_j,S^γ=U_j-1, and X^γ|_U_j-1=S^γ. This relation is well-defined in the context of McKay's framework:The relation (<ref>) satisfies (E1) and (E2).To establish (E1), let X∈Ω_j and S∈Ω_j-1 be arbitrary. It suffices to show that for all β∈Γ we have XS if and only if X^βS^β. Let β∈Γ be arbitrary. By (<ref>), for all γ∈Γ we have X^γ=U_jif and only ifX^β)^β^-1γ=X^ββ^-1γ =X^γ=U_j. Similarly,for any γ∈Γ we have S^γ=U_j-1if and only ifS^β)^β^-1γ=S^ββ^-1γ =S^γ=U_j-1.Finally, for any γ∈Γ that satisfiesX^γ=U_j and S^γ=U_j-1 (equivalently, β^-1γ satisfies X^β)^β^-1γ=U_j and S^β)^β^-1γ=U_j-1), we haveX^γ|_U_j-1=S^γ if and only if(X^β)^β^-1γ|_U_j-1= X^γ|_U_j-1= S^γ= (S^β)^β^-1γ. To establish (E2), observe thatfor an arbitrary X∈Ω_j there exists a γ∈Γ withX^γ=U_j, andthus XS holds for S=T^γ^-1,where T is obtained from Y=X^γ by deleting the assignment tothe variable u_j.The following lemma establishes that the iteration inprocedure (P') constructs exactly the objects X∈ e(S); cf. procedure (P).Let S∈Ω_j-1 be normalized. For all X∈Ω_j we have XS if and only if there exists ap∈ u_j^(U_j-1) withX=U_j-1∪{p} and X|_U_j-1=S.From (<ref>) we have that XS if and only ifthere exists a γ∈Γ with X^γ=U_j,S^γ=U_j-1, and X^γ|_U_j-1=S^γ.Since S is normalized, we have S=U_j-1 and hence U_j-1^γ=S^γ=U_j-1. Thus,γ∈(U_j-1) and X|_U_j-1 =X^γγ^-1|_U_j-1 =(X^γ|_U_j-1)^γ^-1 =(S^γ)^γ^-1 =S .Thus, to establish the “only if” direction of the lemma, takep=u_j^γ^-1, and for the “if” direction,take γ∈(U_j-1) with p^γ=u_j. Next we show the correctness of the test (T1') by establishing that it is equivalent with the test (T1) for a specific canonical extension function M. Towards this end, let us use the assumed canonical labeling mapκ:Ω_j→Γ to build a canonical extensionfunction M using the template ofLemma <ref>. In particular, given an X∈Ω_j as input with X=U_j-1∪{p},first construct the canonical form Z=X^κ(X). In accordancewith (T1'), select the minimum q∈ Usuch that q^κ(X)^-1ν(p)∈ u_j^(U_j).Now construct M(X) from X bydeleting the value of q^κ(X)^-1.The mapping X↦ M(X) is well-defined and satisfies both (M1) and (M2). From (<ref>) we have both(Z)≤(Z) andZ^κ(X)^-1ν(p)=U_j. Thus,(Z)^κ(X)^-1ν(p)≤(Z)^κ(X)^-1ν(p) =(U_j) .It follows that the choice of q depends on Z and u_j but not on thechoices of κ(X) or ν(p). Furthermore, we observe thatq∈Z and q^κ(X)^-1∈X. Thus, the construction of M(X) is well-defined and (M2) holds byLemma <ref>. To verify (M1), observe that since q^κ(X)^-1ν(p)∈ u_j^(U_j), there exists an α∈(U_j) with q^κ(X)^-1ν(p)α=u_j. Thus, for γ=ν(p)α we have X^γ =(U_j-1∪{p})^ν(p)α =U_j^α=U_j, M(X)^γ =(U_j∖{q^κ(X)^-1ν(p)})^α =U_j-1, andX^γ|_U_j-1=M(X)^γ.Thus, from (<ref>) we have XM(X) and thus (M1) holds.To complete the equivalence between (T1') and (T1),observe that since X and p determine Sby X|_X∖{p}=S,and similarly X and q^κ(X)^-1 determine M(X) by X|_X∖{q^κ(X)^-1}=M(X), the test (T1) is equivalent to testing whether(X,p)≅(X,q^κ(X)^-1) holds, that is, whether p≅_(X) q^κ(X)^-1 holds. Observe that this is exactly the test (T1'). It remains to establish the equivalence of (T2') and (T2). We start with a lemma that captures the (S)-orbits considered by (T2). For a normalized S∈Ω_j-1the orbits in e(S)/(S) are in a one-to-one correspondence with the elements of(u_j^(U_j-1)/(S))× R.From (<ref>) we have(S)≤(S)=(U_j-1) since S is normalized. Furthermore, Lemma <ref> implies that every extension X∈ e(S) is uniquely determined bythe variable p∈ u_j^(U_j-1)∩X and thevalue X(p)∈ R. Since the action (<ref>)fixes the values in R elementwise, for any X,X'∈ e(S)we have X≅_(S)X' if and only if both p≅_(S) p'and X(p)=X'(p'). The lemma follows. Now order the elements X∈ e(S) based on the lexicographic ordering of the pairs (p,X(p))∈ u_j^(U_j-1)× R.Since the action (<ref>)fixes the values in R elementwise, we have that (T2') holdsif and only if (T2) holds for this ordering of e(S). The correctness of procedure (P') equipped with the tests (T1') and (T2') now follows from Lemma <ref>. §.§ Selecting a prefixThis section gives a brief discussion on how to select the prefix. Let U_k={u_1,u_2,…,u_k} be the set of variables in theprefix sequence. It is immediate that there exist |R|^k distinctpartial assignments from U_k to R. Let us write R^U_k for the set of these assignments. The group Γ now partitions R^U_kinto orbits via the action (<ref>), and it suffices to consider at most one representative fromeach orbit to obtain an exhaustive traversal of the search space, up toisomorphism. Our goal is thus to select the prefix U_k so that the setwisestabilizer Γ_U_k has comparatively few orbits on R^U_kcompared with the total number of such assignments.In particular, the ratio of the number of orbits |R^U_k/Γ_U_k| to the total number of mappings |R|^k can be viewed as a proxy forthe achieved symmetry reduction and as a rough [ Here it should be noted that executing the symmetry reduction carries initself a nontrivial computational cost. That is, there is a tradeoff between the potential savings in solving the system gained by symmetry reductionversus the cost of performing symmetry reduction. For example, if theinstance has no symmetry and Γ is a trivial group, then executingsymmetry reduction merely makes it more costly to solve the system.] proxy for the speedup factor obtained compared with no symmetry reductionat all.§.§ SubroutinesBy our assumption, the canonical labeling map κ produces as a side-effect a set of generators for the automorphism group (X) for a given input X. We also assume that generators for the groups (U_j) for j=0,1,…,k can be precomputed by similar means. This makes the canonical labeling map essentially the only nontrivial subroutine needed to implement procedure (P'). Indeed, the orbit computations required by tests (T1') and (T2') are implementable by elementary permutation group algorithms <cit.>. Section <ref> describes how to implement κ by reduction to vertex-colored graphs. [Reduction to vertex-colored graphs is by no means the only possibility to obtain the canonical labeling map to enable (P'), (T1'), and (T2'). Another possibility would be to represent Γ directly as a permutation group and use dedicated permutation-group algorithms <cit.>. Our present choice of vertex-colored graphs is motivated by easy availability of carefully engineered implementations for working with vertex-colored graphs.]§ VALUE SYMMETRIESThe previous section considered prefix-assignment generation subject to an action of a group Γ on the set of variables U. In this section, we extend the framework so that it captures symmetries in values assigned to variables, or value symmetries. Towards this end, weextend the domain that records the symmetries from U to U× R,where R is the set of values that can be assigned to the variables in U. Accordingly, in what follows we assume that the group Γ actson U× R as well as on U, the latter by restriction. The action of the group Γ on U× R may not be completelyarbitrary, however, because we want partial assignments X:W→ R with W⊆ U to remain well-defined functions under the actionof Γ. This property is naturally captured bythe wreath product group (R)≀(U) and its natural actionon U× R.§.§ The wreath product and its actions We will follow the convention that (R)≀(U) acts on U× R by first acting on U and then on R.For accessibility and convenience, we review our conventions in detail.The group (R)≀(U) consists of all pairs (π,σ),where π∈(U) is a permutation of U andσ:U→(R) associates a permutation σ(u)∈(R)with each element u∈ U. In particular, (R)≀(U) has order |U|!·(|R|!)^|U|.The product of two elements (π_1,σ_1),(π_2,σ_2)∈(R)≀(U) is defined by (π,σ)=(π_1,σ_1)(π_2,σ_2), whereπ=π_1π_2and for all u∈ U we setσ(u)=σ_1(u^π_2^-1)σ_2(u) .The inverse of an element (π,σ)∈(R)≀(U) is thus given by (π,σ)^-1=(ρ,τ), where ρ=π^-1and for all u∈ U we haveτ(u)=σ(u^π)^-1 . An element (π,σ)∈(R)≀(U) acts on an element u∈ U byu^(π,σ)=u^πand on a pair (u,r)∈ U× R by(u,r)^(π,σ)=(u^π,r^σ(u^π)) .Here in particular the intuition is that we first act on (u,r) with π to obtain (u^π,r), and then act with σ(u^π)to obtain (u^π,r^σ(u^π)). Extend theaction (<ref>) elementwise to subsets W⊆ U. §.§ Partial assignments and isomorphism Let Γ be a subgroup of (R)≀(U) and let Γ act on Uand U× R by (<ref>) and (<ref>), respectively. Furthermore, we let an element γ=(π,σ)∈Γact on a partial assignment X:W→ R with W⊆ U toproduce the partial assignment X^γ:W^π→ Rdefined for all u∈ W^π byX^γ(u) = X(u^π^-1)^σ(u) . In analogy with Lemma <ref>, let us verify that the value-permuting action (<ref>) is well-defined.The action (<ref>) is well defined. We observe that for the identityϵ∈Γ of Γ≤(R)≀(U), we have X^ϵ=X.Furthermore, for all γ_1=(π_1,σ_1)∈Γ, γ_2=(π_2,σ_2)∈Γ, andu∈ W^γ_1γ_2=(W^γ_1)^γ_2,by (<ref>),(<ref>), and (<ref>), we haveX^γ_1γ_2(u)= X(u^(π_1π_2)^-1)^σ_1(u^π_2^-1)σ_2(u) = X(u^π_2^-1π_1^-1)^σ_1(u^π_2^-1)σ_2(u) = X^γ_1(u^π_2^-1)^σ_2(u) = (X^γ_1)^γ_2(u).Let us recall that for X:W→ R we write X=W for the underlying set of variables assigned by X. In analogy with Section <ref>, the underline map is a homomorphism of group actions that satisfies (<ref>) for the action (<ref>) and the action (<ref>) extended elementwise to subsets of U. Isomorphism for partial assignments is now induced by theaction (<ref>). §.§ Generating normalized assignments Working with the group action (<ref>),let u_1,u_2,…,u_k be k distinct elements of U, and let U_j={u_1,u_2,…,u_j} for j=0,1,…,k. Let Ω_j consist of all partial assignments X:W→ R with W≅ U_j. We construct exactly one object form each orbit of Γ on Ω_j, using as seeds exactly one object from each orbit of Γ on Ω_j-1, for each j=1,2,…,k, assuming the availability of canonical labeling maps κ:Ω_j→Γ. We say the assignment X∈Ω_j is normalized if X=U_j.We now present a version of the procedure (P') modified for the group action (<ref>). Let us fix j=1,2,…,k. We assume that the procedure is invoked for exactly one normalized representative S∈Ω_j-1 from each orbit in Ω_j-1/Γ. (P”) Let a normalized S∈Ω_j-1 be given as input.For eachp∈ u_j^(U_j-1) and each r∈ R, construct the assignment X:U_j-1∪{p}→ R defined by X(p)=r and X(u)=S(u) for all u∈ U_j-1. Perform the isomorph rejection tests (T1') and (T2”) on X and S. If both tests accept, visit X^ν(p) where ν(p)∈(U_j-1) normalizes X.In particular, procedure (P”) has two differences compared with procedure (P'). First, the underlying group actionis (<ref>). Second, the test (T2') has been replaced with a new test (T2”) to account for more extensive orbitsof pairs (p,r) under the action of (S). §.§ The isomorph rejection tests Assume that the elements of U, R, and U× R have been arbitrarilyordered and that κ:Ω_j→Γ is a canonical labeling map.Suppose that X has been constructed by extending a normalized S with X=S∪{p}=U_j-1∪{p} and X(p)=r.Let us first recall the test (T1') for convenience: (T1') Subject to the ordering of U, select the minimum q∈ Usuch that q^κ(X)^-1ν(p)∈ u_j^(U_j). Accept if and only if p≅_(X) q^κ^-1(X).The new isomorph rejection test is as follows: (T2”) Accept if and only if (p,r)=min(p,r)^(S) subject to the ordering of U× R.§.§ Correctness We now establish the correctness of the modified procedure (P”). Fix j=1,2,…,k. Define the extension relation e⊆Ω_j×Ω_j-1 as in (<ref>).This relation is well-defined in the context of McKay's framework under the modified group action. The relation (<ref>) satisfies (E1) and (E2) when the group action is as defined in (<ref>). Identical to Lemma <ref> since (<ref>) holds for the action (<ref>) and the action (<ref>) extended elementwise to subsets of U. The correctness analysis of the test (T1') proceeds identically as inSection <ref>, relying on (<ref>)in the proof of Lemma <ref>. To establish the correctness of the new test (T2”), we first observe that the counterpart of Lemma <ref> holds for the modified group action.Let S∈Ω_j-1 be normalized. For all X∈Ω_j, we have XS if and only if there exists ap∈ u_j^(U_j-1) withX=U_j-1∪{p} and X|_U_j-1=S.First observe that (<ref>) holdsfor the action (<ref>). Then proceed as in the proof of Lemma <ref>. Let us now proceed to the counterpart of Lemma <ref>.For a normalized S∈Ω_j-1, the orbits in e(S)/(S) are in a one-to-one correspondence with the orbits in(u_j^(U_j-1)× R)/(S). Lemma <ref> implies that every extension X∈ e(S) is uniquely determined by the variablep∈ u_j^(U_j-1)∩X and the value X(p)∈ R. That is, the elements in e(S) are in one-to-one correspondence withelements in u_j^(U_j-1)× R. Let (S) act on e(S) via (<ref>); thisaction is well-defined by Lemma <ref> and (E1) sincefor all α∈(S) we have XS if and onlyif X^αS. Let (S) act on u_j^(U_j-1)× R via (<ref>); this action is well-defined because S is normalized and hence(S)≤(S)=(U_j-1) holds by (<ref>).Let X,Y∈ e(S) be arbitrarywith X = U_j-1∪{p} andY = U_j-1∪{q}. We now claim that X≅_(S) Y holds under the action (<ref>) if and only if (p,X(p)) ≅_(S) (q,Y(q)) holds under the action(<ref>). To see this, first observe that for all α∈(S) we haveU_j-1^α=S^α=S=U_j-1 by (<ref>) since S is normalized.Furthermore, X|_U_j-1=Y|_U_j-1=S.Thus, by (<ref>) it holds thatfor all α=(π,σ)∈(S) with π∈(U) and σ:U→(R) we have Y=X^α if and onlyif q=p^π andY(q)=X^α(q)=X(q^π^-1)^σ(q)=X(p)^σ(p^π).Or what is the same by (<ref>), if and only if(q,Y(q))=(p^π,X(p)^σ(p^π))=(p,X(p))^α.Order the elements X∈ e(S) based on the lexicographic ordering of the pairs (p,X(p))∈ u_j^(U_j-1)× R. We now have that (T2”) holdsif and only if (T2) holds for this ordering of e(S). The correctness of procedure (P”) equipped with the tests (T1') and (T2”) now follows from Lemma <ref>.§ REPRESENTATION USING VERTEX-COLORED GRAPHSThis section describes one possible approach to represent the group ofsymmetries Γ≤(U) of a system of constraints over a finiteset of variables U taking values in a finite set R. Our representation of choice will be vertex-colored graphs over a fixedfinite set of vertices V. In particular, isomorphisms between such graphsare permutations γ∈(V) that map edges onto edges and respect the colors of the vertices; that is, every vertex in V maps to a vertex of the same color under γ.It will be convenient to develop the relevant graph representations in steps, starting with the representation of the constraint system and then proceeding to the representation of setwise stabilizers and partial assignments. Theserepresentations are folklore (see e.g. <cit.>) and arepresented here for completeness of exposition only.§.§ Representing the constraint systemTo capture Γ≅(G) via a vertex-colored graph G withvertex set V, it is convenient to represent the variables U directlyas a subset of vertices U⊆ V such that no vertex in V∖ Uhas a color that agrees with a color of a vertex in U. We then seek agraph G such that (G)≤(U)×(V∖ U) projected to U is exactly Γ. In most cases such a graph G is conciselyobtainable by encoding the system of constraints with additional verticesand edges joined to the vertices representing the variables in U.We discuss two examples. Consider the system of clauses (<ref>) and its graphrepresentation (<ref>). The latter can be obtained asfollows. First, introduce a blue vertex for each of the six variablesof (<ref>). These blue vertices constitute the subset U.Then, to accommodate negative literals, introduce a red vertex joined byan edge to the corresponding blue vertex representing the positive literal.These edges between red and blue vertices ensure that positive and negativeliterals remain consistent under isomorphism. Finally, introduce a green vertex for each clause of (<ref>) with edges joining the clausewith each of its literals. It is immediate that we can reconstruct (<ref>) from (<ref>) up to labeling ofthe variables even after arbitrary color-preserving permutation of thevertices of (<ref>). Thus, (<ref>)represents the symmetries of (<ref>). Let us next discuss an example where it is convenient to represent thesymmetry at the level of original constraints rather than at the levelof clauses. Consider the following system of eight cubic equations over 24 variablestaking values modulo 2: [ x_11y_11z_11 + x_12y_12z_12 + x_13y_13z_13 = 0 x_21y_11z_11 + x_22y_12z_12 + x_23y_13z_13 = 0; x_11y_11z_21 + x_12y_12z_22 + x_13y_13z_23 = 0 x_21y_11z_21 + x_22y_12z_22 + x_23y_13z_23 = 1; x_11y_21z_11 + x_12y_22z_12 + x_13y_23z_13 = 1 x_21y_21z_11 + x_22y_22z_12 + x_23y_23z_13 = 1; x_11y_21z_21 + x_12y_22z_22 + x_13y_23z_23 = 1 x_21y_21z_21 + x_22y_22z_22 + x_23y_23z_23 = 1 ]This system seeks to decompose a 2 × 2 × 2tensor (whose elements appear on the right hand sides of the equations) into a sum of three rank-one tensors. The symmetries of addition and multiplication modulo 2 imply that the symmetries of the system can berepresented by the following vertex-colored graph: < g r a p h i c s > Indeed, we encode each monomial in the system with a product-vertex, andgroup these product-vertices together by adjacency to a sum-vertex torepresent each equation, taking care to introduce two uniquely coloredconstant-vertices to represent the right-hand side of each equation. Remark. The representation built directly from the system of polynomial equationsin Example <ref> concisely captures the symmetries in the systemindependently of the final encoding of the system (e.g. as CNF) for solvingpurposes.In particular, building the graph representation from such a finalCNF encoding (cf. Example <ref>) results in a less compact graph representation and obfuscates the symmetries of the original system,implying less efficient symmetry reduction. §.§ Representing the valuesIn what follows it will be convenient to assume that the graph Gcontains a uniquely colored vertex for each value in R.(Cf. the graph in Example <ref>.) That is, we assume that R⊆ V∖ U and that (G)projected to R is the trivial group. §.§ Representing setwise stabilizers in the prefix chainTo enable procedure (P') and the tests (T1') and (T2'), we require generators for (U_j)≤Γ for each j=0,1,…,k.More generally, given a subset W⊆ U, we seek to compute a set ofgenerators for the setwise stabilizer_Γ(W)=Γ_W={γ∈Γ:W^γ=W},with W^γ={w^γ:w∈ W}. Assuming we have available a vertex-colored graph G that represents Γby projection of _(V)(G) to U, let us define the graphG↑W by selecting one vertex r∈ R and joining each vertexw∈ W with an edge to the vertex r. It is immediate that_(V)(G↑W) projected to U is precisely_Γ(W).§.§ Representing partial assignmentsLet X:W→ R be an assignment of values in R to variables in W⊆ U. Again to enable procedure (P') together withthe tests (T1') and (T2'), we require a canonical labeling κ(X) and generators for the automorphism group (X).Again assuming we have a vertex-colored graph Gthat represents Γ, let us define the graph G↑X by joining each vertex w∈ W with an edge to the vertex X(w)∈ R.It is immediate that _(V)(G↑X) projected to U isprecisely _Γ(X). Furthermore, a canonical labeling κ(X) can be recovered from κ(G↑X) and thecanonical form (G↑X)^κ(G↑X). §.§ Using tools for vertex-colored graphsGiven a vertex-colored graph G as input, practical tools exist for computing a canonical labeling κ(G)∈(V) and a set of generators for (G)≤(V). Such tools includebliss <cit.>, nauty <cit.>,and traces <cit.>. Once the canonical labeling and generators are available in (V) it is easy to map backto Γ by projection to U so that corresponding elementsof Γ are obtained. § PARALLEL IMPLEMENTATIONThis section outlines the parallel implementation of our technique into a tool called . The implementation is written in C++ andstructured as a preprocessor that works with an explicitly given graphrepresentation. In the absence of such an input graph, the graph isconstructed automatically from CNF as described inSection <ref>. The nauty <cit.> canonical labeling software for vertex-colored graphs is utilized as a subroutine. §.§ Backtracking search for partial assignmentsThe backtracking search for partial assignments is implemented usinga stack that stores nodes of the search tree.(Recall Example <ref> for an illustration of a search tree.)Each node X in the stack represents thecomplete subtree of the search tree rooted at X. Initially, the stackconsists of the empty assignment, which represents the entire search tree. Throughout the search, we maintain the invariant that the nodes stored in the stack represent pairwise node-disjoint subtrees of the search tree, whichenables us to work through the contents of the stack in arbitrary order andto distribute the contents of the stack to multiple compute nodes as necessary;we postpone a detailed discussion of the distribution of the stack andparallelization of the search to Section <ref>.Viewed as a sequential process,the search proceeds by iterating the following work procedure until the stack is empty: (W)Pop an assignment X_ℓ from the stack. Unless X_ℓis the empty assignment (that is, unless ℓ=0), it will have the formX_ℓ:U_ℓ-1∪{p_ℓ}→ R for some p_ℓ∈ u_ℓ^(U_ℓ-1). Furthermore, X_ℓextends the normalized assignment S_ℓ-1=X|_U_ℓ-1.Execute the test (T1') on X_ℓ and S_ℓ-1.If the test (T1') fails, reject the subtree of X_ℓ from furtherconsideration. If either ℓ=0 or the test (T1') passes, then normalizeX_ℓ to obtain S_ℓ=X_ℓ^ν(p_ℓ) with S_ℓ:U_ℓ→ R. At this point S_ℓ has been accepted as the unique representative of its isomorphism class. If ℓ=k, then output the full prefix assignment S_ℓ. If ℓ≤ k-1, proceed to consider extensions of S_ℓ at level ℓ+1 as follows.Iterate over each variable-value pair (p_ℓ+1,r) withp_ℓ+1∈ u_ℓ+1^(U_ℓ) and r∈ R.Construct the assignment X_ℓ+1: U_ℓ∪{p_ℓ+1}→ Rby setting X_ℓ+1(p_ℓ+1)=r and X_ℓ+1(u)=S_ℓ(u)for all u∈ U_ℓ. For each constructed X_ℓ+1, performthe test (T2'). If the test (T2') passes, push X_ℓ+1 to the stack.We observe that the procedure (W) above implements procedure (P') using the stack to maintain the state of the search. In particular, when a single worker process executes the search, we obtain a standard depth-firsttraversal of the search tree. However, we also observe that procedure (W)pushes all the child nodes of S_ℓ to the stack before consulting the stack for further work. This enables multiple worker processes,all executing procedure (W), to work in parallel, if we take care to ensurethat (i) push and pop operations to the stack are atomic, and (ii) the termination condition is changed from the stack being empty to the stackbeing empty and all worker procedures being idle. Furthermore, aspresented in more detail in what follows, we can distribute the stack acrossmultiple compute nodes by appropriately communicating push and pop requestsbetween nodes. §.§ Parallelization and distributing the stack We parallelize the search using the OpenMPI implementation <cit.> of the Message Passing Interface (MPI) <cit.>. We provide two different communication modes, both of which rely on a master–slave paradigm with N processes. The master process with rank 0 distributes the work to N-1 worker processes that, in turn, communicate their resultsback to the master process. We now proceed with a more detailed description of the two communication modes. Master stack mode. In the simpler of the two modes, the master process stores the entire stack. The worker processes interact with the master directly, making push and pop requests to the master process via MPI messages.While inefficient in terms of communication and in terms of potentiallyoverwhelming the master node, this mode provides load balancing that is empirically adequate for a small number of compute nodes and instances whose search tree is not too wide.Hierarchical stack mode. The hierarchical stack mode divides the N-1 worker nodes into M classes,each of which is associated with a subset of levels of the search tree.Each worker process maintains a local stack for nodes at their respective levels. Whenever a worker process pushes an assignment, the assignment is stored in the local stack if the level of theassignment belongs to the levels associated with the node; otherwise,the assignment is communicated to the master process which then pushesthe assignment to the global stack maintained in the master process. Whenever a worker process pops an assignment, the worker process firstconsults its local stack and pops the assignment from the local stack if an assignment is available; otherwise, the worker process makes a poprequest to the master process, which supplies an assignment from the global stack as soon as an assignment of one of the levels associated with the worker becomes available. This strategy helps in cases where the search tree becomes very wide; in our experiments, we found that a simple thresholding into one low-level process that processes levels 1,2,…,t, and N-2 high level processes that process levels t+1,t+2,…,k was sufficient. For both modes of communication, the master process keeps track of theworker processes that are idle, that is, workers that have sent poprequests that have not been serviced. If all workers are idle and theglobal stack is empty, the master process instructs all worker processesto exit and then exits itself. These communication modes serve as a proof-of-concept of the practicalparallelizability of our present technique for symmetry reduction.For parallelization to very large compute clusters, we expect that moreadvanced communication strategies will be required(see, for example, <cit.> or <cit.>);however, the implementation of such strategies is beyond the scope ofthe present work.§ EXPERIMENTS This section documents an experimental evaluation of our parallelimplementation of the adaptive prefix-assignment technique. Our main objective is to demonstrate the effective parallelizability of the approach,but we will also report on experiments comparing the performance of our tool(without parallelization) with existing tools that do not parallelize.§.§ InstancesLet us start by defining the families of input instances used in ourexperiments. First, we study the usefulness of an auxiliarysymmetry graph with systems of polynomial equations aimed at discovering thetensor rank of a small m× m× m tensor T=(t_ijk) modulo 2, with t_ijk∈{0,1} and i,j,k=1,2,… m.Computing the rank of a given tensor is NP-hard <cit.>. [Yet considerable interest exists to determine tensor ranks of small tensors, in particular tensors that encode and enable fast matrix multiplication algorithms; cf. <cit.>. For numerical work on discovering small low-rank tensor decompositions, cf. <cit.>.] In precise terms, we seek to find the minimum r such that there exist three m× r matrices A,B,C∈{0,1}^m× r such that for all i,j,k=1,2,…,m we have∑_ℓ=1^r a_iℓb_jℓc_kℓ=t_ijk 2 .Such instances are easily compilable into CNF with A,B,C constituting three matrices of Boolean variables so that the task becomes to find the minimum r such that the compiled CNF instance is satisfiable. Independently of the target tensor T, such instances have a symmetry group of order at least r! due to the fact that the columns of the matrices A,B,C can be arbitrarily permuted so that (<ref>) maps to itself.In our experiments, we select the entries of T uniformly at random so that the number of 1s in T is exactly n. We use the first three rows of the matrix A as the prefix sequence.As a further family of instances with considerable symmetry, we study the Clique Coloring Problem (CCP) that yields empirically difficult-to-solve instances for contemporary SAT solvers <cit.>. For positive integer parameters n, s, and t, the CCP asks whether there exists an undirected t-colorable graph on n nodes such that the graph contains a complete graph K_s as a subgraph. Such instances are unsatisfiable if s>t. The particular encoding that we use (see <cit.>) is as follows. Introduce variables x_i,j for 1≤ i,j ≤ n with i≠ j to indicate the presence of an edge joining vertex i and j, variables y_p,j for 1≤ p ≤ s with 1≤ j≤ n to indicate that vertex j occupies slot p in a clique, and variables z_i,k for 1≤ i ≤ n and 1 ≤ k ≤ t to indicate that vertex i has color k. The clauses are * ⋀_1≤ p ≤ s⋁_1 ≤ j ≤ n y_p,j ,* ⋀_1≤ p≤ s⋀_1≤ q ≤ s : p≠ q⋀_1 ≤ j ≤ ny_p,j∨y_q,j ,* ⋀_1≤ p≤ s⋀_1≤ q ≤ s : p≠ q⋀_1 ≤ i ≤ n⋀_1 ≤ j ≤ n : i ≠ jy_p,i∨y_q,j∨ x_i,j ,* ⋀_1≤ k≤ t⋀_1 ≤ i ≤ n⋀_1 ≤ j ≤ n : i ≠ jz_i,k∨z_j,k∨x_i,j, and* ⋀_1≤ i ≤ n⋁_1 ≤ k ≤ t z_i,k .We consider unsatisfiable instances with parameters s ∈{5,6}, t = s-1, and let n vary from 15 to 20 in the case of s=5 and 12 to 24 when s=6. We use the variables y_1,1,y_1,2,…,y_1,nas the prefix sequence. The auxiliary graph for encoding the symmetries is constructed as follows. Introduce a vertex for each variable x_i,j, for each variable y_p,j, and for each variable z_i,k.These vertices are colored with three distinct colors,one color for each type of variable. Next, introduce three types of auxiliary vertices, with each type colored with its own distinct color. Introduce vertices 1,2,…,n for the n nodes, vertices1',2',…,s' for the s clique slots, and vertices 1”,2”,…,t” for the t node colors. Thus, in total the graph consists of n(n-1)+sn+tn+n+s+t vertices colored with six distinct colors.To complete the construction of the auxiliary graph,introduce edges to the graph so that each variable x_i,j is joined tothe nodes i and j, each variable y_p,j is joined to clique slot p'and to the node j, and each variable z_i,k is joined to the node iand to the node color k”.We study the parallelizability of our algorithm using two input instances with hard combinatorial symmetry.The first instance, which we call R(4,4;18) in what follows, isan unsatisfiable CNF instance that asks whether there exists an 18-node graphwith the property that neither the graph nor its complement contains thecomplete graph K_4 as a subgraph. That is, we ask whether the Ramsey numberR(4,4) satisfies R(4,4) > 18 (in fact, R(4,4)=18 <cit.>). No auxiliary graph is provided to accompany this instance.The second instance consists of an empty CNF over 36 variables together with an auxiliary graph that encodes the isomorphism classes of 9-node graphs by inserting a variable vertex in the middle of each of the 92=36edges of the complete graph K_9. Applyingwitha length-36 prefix sequence (listing the 36 variable vertices in any order)yields a complete listing of all the 274668 isomorphism classes of 9-nodegraphs. The number of isomorphism classes of graphs of order n is thesequence A000088 in the Online Encyclopedia of Integer Sequences. §.§ Hardware and software configurationThe experiments were performed on a cluster of Dell PowerEdge C4130 compute nodes, each equipped with two Intel Xeon E5-2680v3 CPUs(12 cores per CPU, 24 cores per node) and 128 GiB (8×16 GiB) of DDR4-2133 main memory,running the CentOS 7 distribution of GNU/Linux. Comparative experiments were executed by allocating a single core on a single CPU of a compute node.All experiments were conducted as batch jobs using thebatch scheduler, and running between one to fourphysical nodes, with one to 24 cores allocated in each node, using oneMPI process per core.OpenMPI version 2.1.1 was used as the MPI implementation. §.§ Symmetry reduction tools and SAT solversWe report on three methods for symmetry reduction: (1) no reduction (“”),(2)version 2.1-152-gb937230-dirty[We thank Bart Bogaerts for implementing custom graph input in .] <cit.>, (3) our technique (“”) with a user-selected prefix. Three different SAT solvers were used in the experiments:andversion <cit.>,andversion 4.1 <cit.>.We use the incremental solvertogether with the incremental CNF output of .§.§ Experiments on parallel speedupThis section documents experiments that study the wall-clock running timeof symmetry reduction using our toolas we increase the number of CPU cores and compute nodes participating in parallelsymmetry reduction. The range of the experiments was between one to fourcompute nodes, with one to 24 cores allocated in each node.One MPI process was launched per core. Each node was exclusively reservedfor the experiment. In addition to the wall-clock running time, wemeasure the total reserved time that is obtained by recording,for each core, the length of the time interval the core is reservedfor an experiment, and taking the sum of these time intervals.The total reserved time conservatively tracks the total resources consumedby an experiment in a batch job environment regardless of whether eachallocated core is running or idle. The results of our parallel speedup experiments are displayed in Figure <ref>. The top-left plot in the figure displays the parallel speedup (ratio of parallel wall-clock running time to sequential running time) of running our toolon the instance R(4,4;18) with a length-33 prefix sequence as a function of the number of cores used for one, two, and four allocated compute nodes. We also display the line y=x for reference to compare against perfect linear speedup.As the number of cores grows, in the top-left plot we observe linear scaling of the speedup as a function of the number of cores. The slope of the speedup yet remains somewhat short of the perfect y=x scaling. This is most likely due to the use of the master stack mode and associated communication overhead. The top-right plot displays the total reserved time to demonstrate the total resource usage in addition to the parallel speedup.Table <ref> displays the number of canonical partial assignments at different levels of the search tree exploredby . The two plots in the middle row of Figure <ref> display the parallel speedup and the total reserved time of executing our toolon the instance A000088 (with n=9 and a length-36 prefix sequence) in the master stack mode. This instance requires extensive stack access with many easy instances of canonical labeling (cf. Table <ref> and compare withTable <ref>); accordingly we observe poor speedup fromparallelization in the master stack mode. The two plots in the bottom row of Figure <ref> show an otherwise identical experiment but now executed in hierarchical stack mode with the threshold parameter set to t=21, in which case both the parallel speedup obtained and the totalresource usage become substantially better.When the number of processes is small, Figure <ref> revealsinefficiency in terms of the total reserved time compared with a larger number of processes. This inefficiency is explained by two factors.First, when the number of processes is small, a significant fraction of the total reserved time is used by the master process which does not contribute work to the exploration of the search tree but does consume reserved time from the start to the end of the computation.As soon as more worker processes start exploring the search tree, thetotal reserved time decreases because the time consumed by the master processdecreases. Second, in hierarchical stack mode, a small number of processesmeans that some of the worker nodes processing lower levels of the searchtree can run out of work—but will still consume total reserved time—asassignments in these levels are exhausted, while the small number of processesassigned to work on the higher levels of the tree still remain at work.This bottleneck can be alleviated by increasing the number of workersassociated with the higher levels. §.§ Experiments comparing with other tools We compared our present toolagainst thetool <cit.>.Sincedoes not parallelize, no parallelization was usedin these experiments and all experiments were executed using a singlecompute core. All running times displayed in the tables that followare in seconds, with “t/o” indicating a time-out of 25 hours of wall-clocktime. Other compute load was in general present on the compute nodes wherethese experiments were run. Table <ref> shows the results of a tensor rankcomputation modulo 2 for two random tensors T with m=5, n=9 and m=5,n=20 with (top table) and without (bottom table) an auxiliary graph.When m=5 and n=9, the tensor has rank 8 and decompositions for rank 7and 8 are sought. When m=5 and n=20, the tensor has rank 9 anddecompositions of rank 8 and 9 are sought. For both tensors we observedecreased running time due to symmetry reduction. Comparing the top and bottom tables, we observe the relevance of the graph representation of the symmetries in (<ref>), which are not easily discoverable from the compiled CNF.As the auxiliary graph, we used the graph representation of thesystem (<ref>), constructed as in Example <ref>.Table <ref> shows the results of applyingand our toolas preprocessors for solving instances ofthe Clique Coloring Problem. We observe that for sufficiently large instances, our tool is faster thanin the combined runtime ofpreprocessor and solver.Table <ref> compares running times ofon instances of the Clique Coloring Problem (i) usingthe graph automatically constructed from CNF, and (ii) using a tailored auxiliary graph constructed as described in Section <ref>.For these instances, the available symmetry can be easily discovereddirectly from the CNF encoding, but we observe that the use of the tailoredauxiliary graph does result in faster preprocessing times for .§.§ AcknowledgementsThe research leading to these results has receivedfunding from the European Research Council under the European Union'sSeventh Framework Programme (FP/2007-2013) / ERC Grant Agreement 338077“Theory and Practice of Advanced Search and Enumeration” (M.K., P.K., and J.K.). We gratefully acknowledge the use of computational resources providedby the Aalto Science-IT project at Aalto University.We thank Tomi Janhunen and Bart Bogaerts for useful discussions. A preliminary conference abstract of this paper appeared in Junttila T., Karppa M., Kaski P., Kohonen J. (2017) An Adaptive Prefix-Assignment Technique for Symmetry Reduction. In: Gaspers S., Walsh T. (eds) Theory and Applications of Satisfiability Testing – SAT 2017. SAT 2017. Lecture Notes in Computer Science, vol 10491. Springer, Cham. amsplain | http://arxiv.org/abs/1706.08325v2 | {
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§.0.5em§.§.0.5em§.§.§.0.5em[2] -5pt -5pt 17.cm 21cmmyheadingsF. Ebobisse, P. Neff A fourth order gradient plasticity model based on Kröner's incompatibility tensor equationsection | http://arxiv.org/abs/1706.08770v3 | {
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"title": "A fourth order gauge-invariant gradient plasticity model for polycrystals based on Kröner's incompatibility tensor"
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3m* † pa-ra-me-tri-sa-tionComputer Communicationsuc3m]Iñaki Ucarcorrauth [corrauth]Corresponding author [email protected]]Carlos Donato [email protected]]Pablo Serrano [email protected]]Andres Garcia-Saavedra [email protected],imdea]Arturo Azcorra [email protected],imdea]Albert Banchs [email protected][uc3m]Universidad Carlos III de Madrid, 28911 Leganés, Spain [imdea]IMDEA Networks Institute, 28918 Leganés, Spain [nec]NEC Labs Europe, 69115 Heidelberg, Germany[copy]©2017. This manuscript version is made available under the http://creativecommons.org/licenses/by-nc-nd/4.0/CC-BY-NC-ND 4.0 license. DOI: http://doi.org/10.1016/j.comcom.2017.07.00210.1016/j.comcom.2017.07.002Rate adaptation and transmission power control in 802.11 WLANs have received a lot of attention from the research community, with most of the proposals aiming at maximising throughput based on network conditions. Considering energy consumption, an implicit assumption is that optimality in throughput implies optimality in energy efficiency, but this assumption has been recently put into question. In this paper, we address via analysis, simulation and experimentation the relation between throughput performance and energy efficiency in multi-rate 802.11 scenarios. We demonstrate the trade-off between these performance figures, confirming that they may not be simultaneously optimised, and analyse their sensitivity towards the energy consumption parameters of the device. We analyse this trade-off in existing rate adaptation with transmission power control algorithms, and discuss how to design novel schemes taking energy consumption into account. WLAN 802.11 rate adaptation transmission power control energy efficiency § INTRODUCTIONIn recent years, along with the growth in mobile data applications and the corresponding traffic volume demand, we have witnessed an increased attention towards “green operation” ofnetworks, which is required to support a sustainable growth of the communication infrastructures. For the case of wireless communications, there is the added motivation of a limited energy supply (i.e., batteries), which has triggered a relatively large amount of work on energy efficiency <cit.>. It turns out, though, that energy efficiency and performance do not necessarily come hand in hand, as some previous research has pointed out <cit.>, and that a criterion may be required to set a proper balance between them.This paper is devoted to the problem of rate adaptation (RA) and transmission power control (TPC) in 802.11 WLANs from the energy consumption's perspective. RA algorithms are responsible for selecting the most appropriate modulation and coding scheme (MCS) to use, given an estimation of the link conditions, and have received a vast amount of attention from the research community (see e.g. <cit.> and references therein). In general, the challenge lies in distinguishing between those loses due to collisions and those due to poor radio conditions, because they should trigger different reactions. In addition, the performance figure to optimise is commonly the throughput or a related one such as, e.g., the time required to deliver a frame.On the other hand, network densification is becoming a common tool to provide better coverage and capacity. However, densification brings new problems, especially for 802.11, given the limited amount of orthogonal channels available, which leads to performance and reliability issues due to RF interference. In consequence, some RA schemes also incorporate TPC, which tries to minimise the transmission power (TXP) with the purpose of reducing interference between nearby networks. As in the case of “vanilla” RA, the main performance figure to optimise is also throughput. It is generally assumed that optimality in terms of throughput also implies optimality in terms of energy efficiency. However, some previous work <cit.> has shown that throughput maximisation does not result in energy efficiency maximisation, at least for 802.11n. However, we still lack a proper understanding of the causes behind this “non-duality”, as it may be caused by the specific design of the algorithms studied, the extra consumption caused by the complexity of MIMO techniques, or any other reason. In fact, it could be an inherent trade-off given by the power consumption characteristics of 802.11 interfaces, and, if so, RA-TPC techniques should not be agnostic to this case.This work tackles the latter question from a formal standpoint. A question which, to the best of the authors' knowledge, has never been addressed in the literature. For this purpose, and with the aim of isolating the variables of interest, we present a joint goodput (i.e., the throughtput delivered on top of 802.11) and energy consumption model for single 802.11 spatial streams in the absence of interfering traffic. Packet losses occur due to poor channel conditions and RA-TPC can tune only two variables: MCS and TXP.Building on this model, we provide the following contributions: (i) we demonstrate through an extensive numerical evaluation that energy consumption and throughput performance are different optimisation objectives in 802.11, and not only an effect of MIMO or certain algorithms' suboptimalities; (ii) we analyse the relative impact of each energy consumption component on the resulting performance of RA-TPC, which serves to identify the critical factors to consider for the design of RA-TPC algorithms; (iii) we experimentally validate our numerical results; and (iv) we assess the performance of several representative RA-TPC algorithms from the energy consumption's perspective.The rest of this paper is organised as follows. In Section <ref>, we develop the theoretical framework: a joint goodput-energy model built around separate previous models. In Section <ref>, we provide a detailed analysis of the trade-off between energy efficiency and maximum goodput, including a discussion of the role of the different energy parameters involved. We support our numerical analysis with experimental results in Section <ref>. Section <ref> explores the performance of RA-TPC algorithms from the energy consumption's perspective. Finally, Section <ref> summarises the paper.§ JOINT GOODPUT-ENERGY MODEL In this section, we develop a joint goodput-energy model for a single 802.11 spatial stream and the absence of interfering traffic. It is based on previous studies about goodput and energy consumption of wireless devices. As stated in the introduction, the aim of this model is the isolation of the relevant variables (MCS and TXP) to let us delve in the relationship between goodput and energy consumption optimality in the absence of other effects such as collisions or MIMO.Beyond this primary intent, it is worth noting that these assumptions conform with real-world scenarios in the scope of recent trends in the IEEE 802.11 standard development, namely, the amendments 11ac and 11ad, where device-to-device communications (mainly through beamforming and MU-MIMO) are of paramount importance. §.§ Goodput model We base our study on the work by Qiao et al. <cit.>, which develops a robust goodput model that meets the established requirements. This model analyses the IEEE 802.11a Distributed Coordination Function (DCF) over the assumption of an AWGN (Additive White Gaussian Noise) channel without interfering traffic.Let us briefly introduce the reader to the main concepts, essential to our analysis, of the goodput model by Qiao et al.. Given a packet of length l ready to be sent, a frame retry limit n_max and a set of channel conditions ŝ={s_1, …, s_n_max} and modulations m̂={m_1, …, m_n_max} used during the potential transmission attempts, the expected effective goodput 𝒢 is modelled as the ratio between the expected delivered data payload and the expected transmission time as follows:𝒢(l, ŝ, m̂) = data/𝒟_data = [succ| l, ŝ, m̂]· l/𝒟_data where [succ| l, ŝ, m̂] is the probability of successful transmission conditioned to l, ŝ, m̂, given by Eq. (5) in <cit.>. This model is valid as long as the coherence time is equal or greater than a single retry, i.e., the channel condition s_i is constant.The expected transmission time is defined as follows:𝒟_data = (1 - [succ| l, ŝ, m̂]) ·𝒟_fail| l, ŝ, m̂ + [succ| l, ŝ, m̂] ·𝒟_succ| l, ŝ, m̂ where𝒟_succ| l, ŝ, m̂ =∑_n=1^n_max[n succ| l, ŝ, m̂] ·{∑_i=2^n_max[T_bkoff(i).. + .T_data(l, m_i) + 𝒟_wait(i)] + T_bkoff(1) + T_data(l, m_1) + T_SIFS+ .T_ACK(m'_n) + T_DIFS} is the average duration of a successful transmission and𝒟_fail| l, ŝ, m̂ =∑_i=1^n_max[T_bkoff(i). + .T_data(l, m_i) + 𝒟_wait(i+1)] is the average time wasted during the n_max attempts when the transmission fails.[n succ| l, ŝ, m̂] is the probability of successful transmission at the n-th attempt conditioned to l, ŝ, m̂, and 𝒟_wait(i) is the average waiting time before the i-th attempt. Their expressions are given by Equations (7) and (8) in <cit.>. The transmission time (T_data), ACK time (T_ACK) and average backoff time (T_bkoff) are given by Eq. (1)–(3) in <cit.>. Finally, T_SIFS and T_DIFS are 802.11a parameters, and they can be found also in Table 2 in <cit.>. §.§ Energy consumption model The selected energy model is our previous work of <cit.>, which has been further validated via ad-hoc circuitry and specialised hardware <cit.> and, to the best of our knowledge, stands as the most accurate energy model for 802.11 devices published so far, because it accounts not only the energy consumed by the wireless card, but the consumption of the whole device. While classical models focused on the wireless interface solely, this one demonstrates empirically that the energy consumed by the device itself cannot be neglected as a device-dependent constant contribution. Conversely, devices incur an energy cost derived from the frame processing, which may impact the relationship that we want to evaluate in this paper.The energy model is a multilinear model articulated into three main components:P(τ_i, λ_i) = ρ_id + ∑_i∈{tx,rx}ρ_i τ_i_classical model + ∑_i∈{g,r}γ_xiλ_i where the first two addends correspond to the classical model and the third is the contribution described in <cit.>. These components are the following:* A platform-specific baseline power consumption that accounts for the energy consumed just by the fact of being powered on, but with no network activity. This component is commonly referred to as idle consumption, ρ_id. * A component that accounts for the energy consumed in transmission, which linearly grows with the airtime percentage τ_tx, i.e., P_tx(τ_tx) = ρ_txτ_tx. The slope ρ_tx depends linearly on the radio transmission parameters MCS and TXP. * A component that accounts for the energy consumed in reception, which linearly grows with the airtime percentage τ_rx, i.e., P_rx(τ_rx) = ρ_rxτ_rx. The slope ρ_rx depends linearly on the radio transmission parameter MCS. * A new component, called generation cross-factor or γ_xg, that accounts for a per-frame energy processing toll in transmission, which linearly grows with the traffic rate λ_g generated, i.e., P_xg(λ_g) = γ_xgλ_g. The slope γ_xg depends on the computing characteristics of the device. * A new component, called reception cross-factor or γ_xr, that accounts for a per-frame energy processing toll in reception, which linearly grows with the traffic rate λ_r received, i.e., P_xr(λ_r) = γ_xrλ_r. Likewise, the slope γ_xr depends on the computing characteristics of the device. Therefore, the average power consumed P is a function of five device-dependent parameters (ρ_i, γ_xi) and four traffic-dependent ones (τ_i, λ_i). §.§ Energy efficiency analysis Putting together both models, we are now in a position to build a joint goodput-energy model for 802.11a DCF. Let us consider the average durations (<ref>) and (<ref>). Based on their expressions, we multiply the idle time (𝒟_wait, T_bkoff, T_SIFS, T_DIFS) by ρ_id, the transmission time (T_data) by ρ_tx, and the reception time (T_ACK) by ρ_rx. The resulting expressions are the average energy consumed in a successful transmission ℰ_succ| l, ŝ, m̂ and the average energy wasted when a transmission fails ℰ_fail| l, ŝ, m̂:ℰ_succ| l, ŝ, m̂ =∑_n=1^n_max[n succ| l, ŝ, m̂] ·{∑_i=2^n_max[ρ_idT_bkoff(i).. + .ρ_txT_data(l, m_i) + ρ_id𝒟_wait(i)] + ρ_idT_bkoff(1) + ρ_txT_data(l, m_1) + ρ_idT_SIFS+ .ρ_rxT_ACK(m'_n) + ρ_idT_DIFS} ℰ_fail| l, ŝ, m̂ =∑_i=1^n_max[ρ_idT_bkoff(i). + .ρ_txT_data(l, m_i) + ρ_id𝒟_wait(i+1)] Then, by analogy with (<ref>), the expected energy consumed per frame transmitted, ℰ_data, can be written as follows:ℰ_data = γ_xg + (1 - [succ| l, ŝ, m̂]) ·ℰ_fail| l, ŝ, m̂ + [succ| l, ŝ, m̂] ·ℰ_succ| l, ŝ, m̂ It is noteworthy that the receiving cross-factor does not appear in this expression because ACKs (acknowledgements) are processed in the network card exclusively, and thus its processing toll is negligible.Finally, we define the expected effective energy efficiency μ as the ratio between the expected delivered data payload and the expected energy consumed per frame, which can be expressed in bits per Joule (bpJ):μ(l, ŝ, m̂) = data/ℰ_data § NUMERICAL RESULTS Building on the joint model presented in the previous section, here we explore the relationship between optimal goodput and energy efficiency in 802.11a. More specifically, our objective is to understand the behaviour of the energy efficiency of a single spatial stream as the MCS and TXP change following our model to meet the optimal goodput.§.§ Optimal goodput We note that the main goal of RA, generally, is to maximise the effective goodput that a station can achieve by varying the parameters of the interface. In terms of the model discussed in the previous section, a rate adaptation algorithm would aspire to fit the following curve:max𝒢(l, ŝ, m̂) We provide the numerical results for this goodput maximisation problem in Fig. <ref>, which are in good agreement with those obtained in <cit.>. For the sake of simplicity but without loss of generality we fix l=1500 octets and n_max=7 retries, and assume that the channel conditions and the transmission strategy are constant across retries (ŝ={s_1, …, s_1} and m̂={m_1, …, m_1}).Fig. <ref> illustrates which mode (see Table <ref>) is optimal in terms of goodput, given an SNR level. We next address the question of whether this optimisation is aligned with energy efficiency maximisation. §.§ Extension of the energy parametrisation The next step is to delve into the energy consumption of wireless devices. <cit.> provides real measurements for five devices: three AP-like platforms (Linksys WRT54G, Raspberry Pi and Soekris net4826-48) and two hand-held devices (HTC Legend and Samsung Galaxy Note 10.1). Two of the four parameters needed are constant (ρ_id, γ_xg), and the other two (ρ_tx, ρ_rx) depend on the MCS and the TXP used. However, the characterisation done in <cit.> is performed for a subset of the MCS and TXP available, so we next detail how we extend the model to account for a larger set of operation parameters.A detailed analysis of the numerical figures presented in <cit.> suggests that ρ_rx depends linearly on the MCS, and that ρ_tx depends linearly on the MCS and the TXP (in mW). Based on these observations, we define the following linear models:ρ_tx = α_0 + α_1·MCS + α_2·TXP ρ_rx = β_0 + β_1·MCS The models are fed with the data reported in <cit.>, and the resulting fitting is illustrated in Figs. <ref> and <ref>, while Table <ref> collects the model estimates for each device (with errors between parentheses), as well as the adjusted r-squared. Since these linear models show a very good fit, they support the generation of synthetic data for the different MCS and TXP required. §.§ Energy consumption To compute the energy consumption using the above parametrisation, first we have to define the assumptions for the considered scenario. We assume for simplicity a device-to-device communication, with fixed and reciprocal channel conditions during a sufficient period of time (i.e., low or no mobility). As we have discussed before, our primary goal is to isolate MCS and TXP as variables of interest, but we must not forget that these are also reasonable assumptions in scenarios targeted by recent 802.11 standard developments (11ac, 11ad).For instance, given channel state information from a receiver, the transmitter may decide to increase the TXP in order to increase the receiver's SNR (and thus the expected goodput), or to decrease it if the channel quality is high enough. Although the actual relationship between TXP and SNR depends on the specific channel model (e.g., distance, obstacles, noise), without loss of generality, we choose a noise floor of N=-85 dBm in an office scenario with a link distance of d=18 m in order to explore numerically the whole range of SNR while using reasonable values of TXP. The ITU model for indoor attenuation <cit.> gives a path loss of L≈ 85 dBm. Then, we can use (<ref>) to obtain the expected energy consumed per frame and MCS mode, with TXP being directly related to the SNR level.The results are reported in Fig. <ref>. As the figure illustrates, consumption first falls abruptly as the TXP increases for all modes, which is caused when the SNR reaches a sharp threshold level such that the number of retransmissions changes from 6 to 0 (i.e., no frame is discarded). From this threshold on, the consumption increases with TXP because, although the number of retransmissions is 0, the wireless interface consumes more power. We note that the actual value of the TXP when the consumption drops depends on the specifics of the scenario considered, but the qualitative conclusions hold for a variety of scenarios. §.§ Energy efficiency vs. optimal goodput We can finally merge previous numerical analyses and confront energy efficiency, given by (<ref>), and optimal goodput, given by (<ref>), for all devices and under the aforementioned assumptions. To this aim, we plot in the same figure the energy efficiency for the configuration that maximises goodput given an SNR value vs. the obtained goodput, with the results being depicted in Fig. <ref>. We next discuss the main findings from the figure.First of all, we can see that the energy efficiency grows sub-linearly with the optimal goodput (the optimal goodput for each SNR value) in all cases. We may distinguish three different cases in terms of energy efficiency: high (Samsung Galaxy Note and HTC Legend), medium (Raspberry Pi) and low energy efficiency (Linksys and Soekris). Furthermore, for the case of the Soekris, we note that the “central modes” (namely, 4 and 5) are more efficient in their optimal region than the subsequent ones.Another finding (more relevant perhaps) is that it becomes evident that increasing the goodput does not always improve the energy efficiency: there are more or less drastic leaps, depending on the device, between mode transitions. From the transmitter point of view, in the described scenario, this can be read as follows: we may increase the TXP to increase the SNR, but if the optimal goodput entails a mode transition, the energy efficiency may be affected.As a conclusion, we have demonstrated that optimal goodput and energy efficiency do not go hand in hand, even in a single spatial stream, in 802.11. There is a trade-off in some circumstances that current rate adaptation algorithms cannot take into account, as they are oblivious to the energy consumption characteristic of the device. §.§ Sensitivity to energy parameter scaling We next explore how the different energy parameters affect the energy efficiency vs. optimal goodput relationship. For this purpose, we selected the Raspberry Pi curve from Fig. <ref> (results are analogous with the other devices) and we scale up and down, one at a time, the four energy parameters ρ_id, ρ_tx, ρ_rx, and γ_xg. The scaling up and down is done by multiplying and dividing by 3, respectively, the numerical value of the considered parameter. One of the first results is that the impact of ρ_rx is negligible, a result somehow expected as the cost of receiving the ACK is practically zero. From this point on, we do not consider further this parameter. We show in Fig. <ref> the overall effect of this parameter scaling. The solid line represents the base case with no scaling (same curve as in Fig. <ref>), and in dashed and dotted lines the corresponding parameter was multiplied or divided by a factor of 3, respectively. As expected, larger parameters contribute to lower the overall energy efficiency. However, the impact on the energy efficiency drops between mode transitions is far from being obvious, as in some cases transitions are more subtle while in others they become more abrupt.To delve into these transitions, we illustrate in Fig. <ref> the “drop” in energy efficiency when changing between modes. As it can be seen, the cross-factor γ_xg is the less sensitive parameter of the three, because its overall effect is limited and, more importantly, it scales all the leaps between mode transitions homogeneously. This means that a higher or lower cross-factor, which resides almost entirely in the device and not in the wireless card, does not alter the energy efficiency vs. optimal goodput relationship (note that this parameter results in a constant term in (<ref>)). Thus, the cross-factor is not relevant from the RA-TPC point of view, and energy-aware RA-TPC algorithms can be implemented by leveraging energy parameters local to the wireless card.On the other hand, ρ_id and ρ_tx have a larger overall effect, plus an inhomogeneous and, in general, opposite impact on mode transitions. While a larger ρ_id contributes to larger leaps, for the case of ρ_tx, the larger energy efficiency drops occur with smaller values of that parameter. Still, the reason behind this behaviour is the same for both cases: the wireless card spends more time in idle (and less time transmitting) when a transition to the next mode occurs, which has a higher data rate.This effect is also evident if we compare the Samsung Galaxy Note and the HTC Legend curves in Fig. <ref>. Both devices have ρ_id and ρ_tx in the same order of magnitude, but the HTC Legend has a larger ρ_id and a smaller ρ_tx. The combined outcome is a more dramatic sub-linear behaviour and an increased energy efficiency drop between mode transitions. §.§ Discussion We have seen that the energy efficiency vs. optimal goodput relationship shows a signature “sawtooth” pattern when RA and TPC are considered for a single 802.11 spatial stream. This sawtooth shape presents a growing trend in the central part of each mode, but there are energy efficiency drops between mode transitions, which conceal a trade-off.Parameter scaling has diverse effects on the final consumption signature, but overall, the qualitative behaviour (i.e., the shape) remains the same. The cross-factor produces an homogeneous scaling of the sawtooth. Thus, a first conclusion is that the trade-off depends on the energy parameters local to the wireless card, which means that a properly designed energy-aware RA-TPC algorithm can be device-agnostic.Moreover, an energy-aware RA-TCP algorithm may also be card-agnostic. This is because the inefficiencies are always constrained at mode transitions, which are exactly the points at which RA-TPC algorithms take decisions. Therefore, there is no need of knowing the exact energy parametrisation, nor the instantaneous power consumption of the wireless card, in order to take energy-efficient decisions.An RA-TPC algorithm moves along the sawtooth shapes of Fig. <ref> in two directions, namely, “up” (towards higher throughput) and “down” (towards lower throughput). In this way, an algorithm requires different policies to make a decision: (i) the upwards policy, in which mode transitions take place by increasing MCS and TXP (to achieve more goodput), and (ii) the downwards policy, in which mode transitions take place by decreasing MCS and TXP. * In the upwards direction, a sensitive policy would be to remain in the left side of the leaps, to prevent falling into the efficiency gaps, until the link is good enough to move to a higher MCS with at least the same efficiency. An heuristic for the upwards policy may be the following: whenever an algorithm chooses a higher MCS, it may hold the decision for some time and, if it persists, then trigger the MCS change (however, if this delay is too long, the algorithm will incur in inefficiencies, too). * In the downwards direction, a sensitive policy would be to try to reach the left side of the leaps as soon as possible. However, it should be noted that this downwards policy is much more challenging, as it implies predicting quality drops to trigger early MCS/TXP changes. In summary, our results suggest that one of the key points of an energy-aware RA-TPC algorithm is the management of mode transitions. A good algorithm should be conservative at mode transitions, in the sense that it should prefer a lower MCS and TXP until a higher MCS can be guaranteed. § EXPERIMENTAL VALIDATION This section is devoted to experimentally validate the results from the numerical analysis and, therefore, the resulting conclusions. To this aim, we describe our experimental setup and the validation procedure, first specifying the methodology and then the results achieved.§.§ Experimental setup We deployed the testbed illustrated in Fig. <ref>, which consists of a station (STA) transmitting evenly-spaced maximum-sized UDP packets to an access point (AP). The AP is an x86-based Alix6f2 board with a Mini PCI Qualcomm Atheros AR9220 wireless network adapter, running Voyage Linux with kernel version 3.16.7 and thedriver. The STA is a desktop PC with a Mini PCI Express Qualcomm Atheros QCA9880 wireless network adapter, running Fedora Linux 23 with kernel version 4.2.5 and thedriver. We also installed at the STA a Mini PCI Qualcomm Atheros AR9220 wireless network adapter to monitor the wireless channel. The QCA9880 card is connected to the PC through a x1 PCI Express to Mini PCI Express adapter from Amfeltec. This adapter connects the PCI bus' data channels to the host and provides an ATX port so that the wireless card can be supplied by an external power source. The power supply is a Keithley 2304A DC Power Supply, and it powers the wireless card through an ad-hoc measurement circuit that extracts the voltage and converts the current with a high-precision sensing resistor and amplifier. These signals are measured using a National Instruments PCI-6289 multifunction data acquisition (DAQ) device, which is also connected to the STA. Thanks to this configuration, the STA can simultaneously measure the instant power consumed by the QCA9880 card,[Following the discussion on Section <ref> the device's cross-factor is not involved in the trade-off, thus we will expect to reproduce it by measuring the wireless interface alone.] and the goodput achieved. As Fig. <ref> illustrates, the STA is located in an office space and the AP is placed in a laboratory 15 m away, and transmitted frames have to transverse two thin brick walls. The wireless card uses only one antenna and a practically-empty channel in the 5-GHz band. Throughout the experiments, the STA is constantly backlogged with data to send to the AP, and measures the throughput obtained by counting the number of received acknowledgements (ACKs). §.§ Methodology and results In order to validate our results, our aim is to replicate the qualitative behaviour of Fig. <ref>, in which there are energy efficiency “drops” as the optimal goodput increases. However, in our experimental setting, channel conditions are not controllable, which introduces a notable variability in the results as it affects both the x-axis (goodput) and the y-axis (energy efficiency). To reduce the impact of this variability, we decided to change the variable in the x-axis from the optimal goodput to the transmission power —a variable that is directly configured in the wireless card—. In this way, the qualitative behaviour to replicate is the one illustrated in Fig. <ref>, where we can still identify the performance “drops” causing the loss in energy efficiency.Building on Fig. <ref>, we perform a sweep through all available combinations of MCS (see Table <ref>) and TXP.[The model explores a range between 0 and 30 dBm to get the big picture, but this particular wireless card only allows us to sweep from 0 to 20 dBm.] Furthermore, we also tested two different configurations of the AP's TXP at different times of the day, to confirm that this qualitative behaviour is still present under different channel conditions. For each configuration, we performed 2-second experiments in which we measure the total bytes successfully sent and the energy consumed by the QCA9880 card with sub-microsecond precision, and we compute the energy efficiency achieved for each experiment.The results are shown in Fig. <ref>. Each graph corresponds to a different TXP value configured at the AP, and depicts a single run (note that we performed several runs throughout the day and found no major qualitative differences across them). Each line type represents the STA's mode that achieved the highest goodput for each TXP interval, therefore in some cases low modes do not appear because a higher mode achieved a higher goodput. Despite the inherent experimental difficulties, namely, the low granularity of 1-dBm steps and the random variability of the channel, the experimental results validate the analytical ones, as the qualitative behaviour of both graphs follows the one illustrated in Fig. <ref>. In particular, the performance “drops” of each dominant mode can be clearly observed (especially for the 36, 48 and 54 Mbps MCSs) despite the variability in the results.§ ON THE PERFORMANCE OF RA-TPC ALGORITHMS So far, we have demonstrated through numerical analysis, and validated experimentally, the existence of a trade-off between two competing performance figures, namely, goodput and energy efficiency. This issue arises even for a single spatial stream in absence of interference.Furthermore, we have discussed in Section <ref> some ideas about the kind of mechanisms that energy-aware RA-TPC algorithms may incorporate, to leverage the behaviour that we have identified in our analysis in these so-called mode transitions. In nuce, the algorithms should be conservative during these transitions.During that discussion, we neglected the challenge of estimating channel conditions. In practice, any RA-TPC algorithm has imperfect channel knowledge, and therefore will adapt to changing conditions in a suboptimal way. In this section, we will analyse and compare the performance of several representative existing RA algorithms, which also incorporate TPC, to confirm if the conservativeness in such decisions may have a positive impact on the achieved performance. §.§ Considered RA-TPC algorithms If we take a look at the actual operation of WiFi networks, the Minstrel algorithm <cit.>, which was integrated into the Linux kernel, has become the de facto standard due to its relatively good performance and robustness. However, Minstrel does not consider TPC and, in consequence, there is no TPC in today's WiFi deployments. Moreover, despite some promising proposals have been presented in the literature, there are very few of them implemented, although there are some ongoing efforts such as the work by the authors of Minstrel-Piano <cit.>, who are pushing to release an enhanced version of the latter for the Linux kernel with the goal of promoting it upstream.[<https://github.com/thuehn/Minstrel-Blues>]As stated before, RA is a very prolific research line in the literature, but the main corpus is dedicated to the MCS adjustment without taking into account the TXP <cit.>. There is some work considering TPC, but the motivation is typically the performance degradation due to network densification, and the aim is interference mitigation <cit.> and not energy efficiency. Given that we are interested in assessing RA implementations with TPC support, we consider only open-source algorithms that can be tested using the NS3 Network Simulator. After a thorough analysis of the literature, we consider the following set of algorithms:* Power-controlled Auto Rate Fallback (PARF) <cit.>, which is based on Auto Rate Fallback (ARF) <cit.>, one of the earliest RA schemes for 802.11. ARF rate adaptation is based on the frame loss ratio. It tunes the MCS in a very straightforward and intuitive way. The procedure starts with the lowest possible MCS. Then, if either a timer expires or the number of consecutive successful transmissions reach a threshold, the MCS is increased and the timer is reset. The MCS is decreased if either the first transmission at a new rate fails or two consecutive transmissions fail. PARF builds on ARF and tries to reduce the TXP if there is no loss until a minimum threshold is reached or until transmissions start to fail. If transmission fails persist, the TXP is increased. * Minstrel-Piano (MP) <cit.> is based on Minstrel <cit.>. Minstrel performs per-frame rate adaptation based on throughput. It randomly probes the MCS space and computes an exponential weighted moving average (EWMA) on the transmission probability for each rate, in order to keep a long-term history of the channel state. As the previous algorithm, MP adds TPC without interfering with the normal operation of Minstrel. It incorporates to the TPC the same concepts and techniques than Minstrel uses for the MCS adjustment, i.e., it tries to learn the impact of the TXP on the achieved throughput. * Robust Rate and Power Adaptation Algorithm (RRPAA) and Power, Rate and Carrier-Sense Control (PRCS) <cit.>, which are based on Robust Rate Adaptation Algorithm (RRAA) <cit.>. RRAA consists of two functional blocks, namely, rate adaptation and collisions elimination. It performs rate adaptation based on loss ratio estimation over short windows, and reduces collisions with a RTS-based strategy. The procedure starts at the maximum MCS. The loss ratio for each window of transmissions is available for rate adjustment in the next window. There are two thresholds involved in this adjustment: if the loss ratio is below both of them, the MCS is increased; if it is above, the MCS is decreased; and if it is in between, the MCS remains unchanged. RRPAA and PRCS build on this and try to use the lowest possible TXP without degrading the throughput. For this purpose, they firstly find the best MCS at the maximum TXP and, from there, they jointly adjust the MCS and TXP for each window based on a similar thresholding system. RRPAA and PRCS are very similar and only differ in implementation details. Based on their behaviour, these algorithms can be classified into three distinct classes. First of all, MP is the most aggressive technique, given that it constantly samples the whole MCS/TXP space searching for the best possible configuration. On the opposite end, RRPAA and PRCS do not sample the whole MCS/TXP space. Instead, they are based on a windowed estimation of the loss ratio, which makes the MCS/TXP transitions much lazier. Finally, PARF falls in between, as it changes the MCS/TXP to the next available proactively if a number of transmissions are successful, but it falls back to the previous one if the new one fails. In practice, this may result in some instability during transitions. §.§ Scenario This evaluation is publicly available,[<https://github.com/Enchufa2/ns-3-dev-git>] and builds upon the code provided by Richart et al. in <cit.>.[<https://github.com/mrichart/ns-3-dev-git>]. We assessed the proposed algorithms in the toy scenario depicted in Fig. <ref>. It consists of a single access point (AP) and a single mobile node connected to this AP configured with the 802.11a PHY. The mobile node at the farthest distance at which is able to communicate at the lowest possible rate (6 Mbps) and highest TXP (17 dBm), and then it moves at constant speed towards the AP. The simulation stops when the node is directly in front of the AP and it is able to communicate at the highest possible rate (54 Mbps) and lowest TXP (0 dBm). This way, we sweep through all mode transitions available.For the whole simulation, the AP tries to constantly saturate the channel by sending full-size UDP packets to the node. Every transmission attempt is monitored, as well as every successful transmission. The first part allows us to compute the transmission time, while the latter allows us to compute the reception time (of the ACKs) and the goodput achieved.The simulation model assembles the power model (<ref>) with the parametrisation previously made (see Table <ref>) for all the devices considered in Section <ref>: HTC Legend, Linksys WRT54G, Raspberry Pi, Samsung Galaxy Note 10.1 and Soekris net4826-48. Thus, the total energy consumed is computed for all the devices and each run using the computed transmission time, reception time and idle time. The beacons are ignored and considered as idle time.We set up one simulation for each algorithm (PARF, MP, PRCS, RRPAA) with a fixed seed, and perform 10 independent runs for each simulation. We use boxplots for the results unless otherwise mentioned. §.§ Results We first analyse the goodput achieved per each algorithm, which are depicted in Fig. <ref>. The median of the average goodput across several runs for RRPAA is the highest, followed by PRCS, PARF and MP. PRCS and RRPAA, which are very similar mechanisms, show a higher variability across replications compared to PARF and MP, which have little dispersion.Fig. <ref> shows the energy efficiency achieved per algorithm, computed for all the devices presented in Section <ref>. As expected, the numerical values of the energy efficiency achieved are different across devices, but the relative performance is essentially the same, as in the previous case. Indeed, the efficiency follows the pattern seen in Fig. <ref>: RRPAA results the most energy efficient in our scenario, followed by PRCS, PARF and MP. PRCS and RRPAA exhibit the same variability across replications as in the case of goodput, which is particularly notable for the most efficient devices, i.e., the HTC Legend and the Samsung Galaxy Note. §.§ Discussion In order to shed some light into the reasons behind the differences in performance, Figs. <ref> and <ref> show the behaviour of each algorithm throughout the simulation time for one run, showing the evolution of the MCS and TXP chosen by each algorithm, respectively. Here, we can clearly differentiate that there are two kinds of behaviour: while MP and PARF are constantly sampling other MCSs and TXPs, PRCS and RRPAA are much more conservative in that sense, and tend to keep the same configuration for longer periods of time. MP randomly explores the whole MCS/TXP space above a basic guaranteed value, and this is the explanation for the apparently uniformly greyed zone. Also, this aggressive approach is clearly a disadvantage in the considered toy scenario (deterministic walk, one-to-one, no obstacles), and this is why the achieved goodput in Fig. <ref> is slightly smaller than the one achieved by the others. PARF, on its part, only explores the immediately higher MCS/TXP, which leads to a higher goodput and efficiency.On the other hand, PRCS and RRPAA sampling is much more sparse in time. As a consequence, Figs. <ref> and <ref> are much more different across replications, leading to the high variability shown in Fig. <ref> compared to MP and PARF.In terms of TXP, all the algorithms exhibit a similar aggressiveness, in the sense that they use a high TXP value in general. Indeed, as Fig. <ref> shows, the TXP is the highest possible until the very end of the simulation, when the STA is very close to the AP. This is the cause for the high correlation between Figs. <ref> and <ref>.A noteworthy characteristic of PRCS and RRPAA is that, in general, they delay the MCS change decision, as depicted in Fig. <ref>. Most of the times, they do not even use the whole space of MCS available, unlike MP and PARF. Because of this, they tend to achieve the best goodput and energy efficiency.§.§ Conservativeness at mode transitions Building on the concept of conservativeness developed in Section <ref> (i.e., the tendency to select a lower MCS/TXP in the transition regions), we explore whether there is any correlation of with the energy efficiency achieved by a certain algorithm and this tendency. For that purpose, we first define a proper metric.In the first place, we define the normalised average MCS as the area under the curve in Fig. <ref> normalised by the total simulation time and the maximum MCS:MCS = 1/max(MCS) · t_sim∫_0^t_simMCS(t)dt where t_sim is the simulation time and max(MCS) is 54 Mbps in our case. The same concept can be applied to the TXP:TXP = 1/max(TXP) · t_sim∫_0^t_simTXP(t)dt where max(TXP) is 17 dBm in our case. Both MCS and TXP are unitless scores between 0 and 1, and lower values mean a more conservative algorithm. Therefore, we can define a Conservativeness Index (CI) as the inverse of the product of both scores:CI = 1/MCS·TXP where CI>1.[It must be taken into account that the CI is not suitable for comparing any algorithm. For instance, in an extreme case, an “algorithm” could select 6 Mbps and 0 dBm always, resulting in a very low CI, but a very bad performance at the same time. The CI should only be used for comparing similarly performant algorithms, as it is the case in our study given the results shown in Figs. <ref> and <ref>.]We computed the CI for each device and run, and the final results are depicted in Fig. <ref> as the average CI across different runs vs. the median energy efficiency in Fig. <ref> (note that the dots have been connected by straight lines to facilitate the visualisation). The results in Fig. <ref> show a positive non-linear relationship between the CI of an algorithm and the energy efficiency achieved for all the devices considered. MP is the algorithm with the lowest CI, which is in consonance with its aggressiveness (i.e., frequent jumps between MCS/TXP values, as shown in Figs. <ref> and <ref>), and the goodput achieved was also the lowest, as depicted in Fig. <ref>. On the other hand, PARF, PRCS and RRPAA achieved a similar performance in terms of goodput, but the ones with the most conservative behaviour (PRCS and RRPAA, as it can be seen in Figs. <ref> and <ref>) also achieve both the highest CI and energy efficiency.This result evidences that the performance gaps uncovered by Fig. <ref> under optimal conditions have also an impact in real-world RA-TPC algorithms. Therefore, we confirm that this issue must be taken into account in the design of more energy-efficient rate and transmission power control algorithms.§ CONCLUSIONS In this paper, we have revisited 802.11 rate adaptation and transmission power control by taking energy consumption into account. While some previous studies pointed out that MIMO rate adaptation is not energy efficient, we have demonstrated through numerical analysis that, even for single spatial streams without interfering traffic, energy consumption and throughput performance are different optimisation objectives. Furthermore, we have validated our results via experimentation.Our findings show that this trade-off emerges at certain “mode transitions” when maximising the goodput, suggesting that small goodput degradations may lead to energy efficiency gains. For instance, a station at the edge of a mode transition may decide to reduce the transmission power a little in order to downgrade the modulation coding scheme. Or an opportunity to achieve a better goodput by increasing the transmission power and modulation coding scheme could be delayed if the expected gain is small.We have assessed the performance of four state-of-the-art schemes through simulation, and we have demonstrated that certain conservativeness at mode transitions can make a difference for properly designed energy-aware rate adaptation with transmission power control algorithms.§ ACKNOWLEDGEMENTS This work has been performed in the framework of the H2020-ICT-2014-2 projects 5GNORMA (grant agreement no. 671584) and Flex5Gware (grant agreement no. 671563). The authors would like to acknowledge the contributions of their colleagues. This information reflects the consortium's view, but the consortium is not liable for any use that may be made of any of the information contained therein.§ elsarticle-num | http://arxiv.org/abs/1706.08339v2 | {
"authors": [
"Iñaki Ucar",
"Carlos Donato",
"Pablo Serrano",
"Andres Garcia-Saavedra",
"Arturo Azcorra",
"Albert Banchs"
],
"categories": [
"cs.NI",
"cs.PF",
"C.2.2, C.4"
],
"primary_category": "cs.NI",
"published": "20170626122801",
"title": "On the Energy Efficiency of Rate and Transmission Power Control in 802.11"
} |
[a]Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China [b]Istituto Nazionale di Fisica Nucleare Sezione di Perugia, I-06123 Perugia, Italy [c]State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei 230026, China [d]Department of Nuclear and Particle Physics, University of Geneva, CH-1211, Switzerland [e]Università del Salento - Dipartimento di Matematica e Fisica "E. De Giorgi", I-73100, Lecce, Italy [f]Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Lecce , I-73100 , Lecce, Italy [g]Dipartimento di Fisica e Geologia, Università degli Studi di Perugia, I-06123 Perugia, Italy [h]Istituto Nazionale di Fisica Nucleare Sezione di Bari, I-70125, Bari, Italy [i]University of Chinese Academy of Sciences, Yuquan Road 19, Beijing 100049, China [j]Institute of Modern Physics, Chinese Academy of Sciences, Nanchang Road 59, Lanzhou 730000, China [k]National Space Science Center, Chinese Academy of Sciences, Nanertiao 1, Zhongguancun, Haidian district, Beijing 100190, China [l]Institute of High Energy Physics, Chinese Academy of Sciences, YuquanLu 19B, Beijing 100049, China [m]ASI Science Data Center (ASDC), I-00133 Roma, Italy [n]Dipartimento di Fisica "M.Merlin" dell'Univerisity e del Politecnico di Bari, I-70126, Bari, Italy [o]School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China The DArk Matter Particle Explorer (DAMPE), one of the four scientific space science missionswithin the framework of the Strategic Pioneer Program on Space Science of the Chinese Academyof Sciences, is a general purpose high energy cosmic-ray and gamma-ray observatory, which wassuccessfully launched on December 17th, 2015 from the Jiuquan Satellite Launch Center.The DAMPE scientific objectives include the study of galactic cosmic rays up to ∼ 10 TeV and hundreds of TeVfor electrons/gammas and nuclei respectively, and the search for dark matter signatures in their spectra.In this paper we illustrate the layout of the DAMPE instrument, and discuss the results of beam tests and calibrationsperformed on ground. Finally we present the expected performance in space and give an overviewof the mission key scientific goals. § INTRODUCTIONThe interest in space-borne particle/astroparticle physics experiments is growing. The achievements of the early space-borne particle detectors such as IMP <cit.>, HEAO-3 <cit.>, ACE <cit.> lead to more advanced experiments, namely EGRET <cit.> , AMS-01 <cit.>, PAMELA <cit.>, AGILE <cit.>, Fermi <cit.>, AMS-02 <cit.> and CALET <cit.>. Additionally, there have been many balloon and ground based experiments including BESS <cit.>,IMAX <cit.>, HEAT <cit.>, ATIC <cit.>, CAPRICE <cit.>, CREAM <cit.>, WIZARD <cit.>,Fly's Eye <cit.>, H.E.S.S <cit.>, MAGIC <cit.>, ARGO-YBJ experiment <cit.>, VERITAS <cit.>, Pierre Auger Observatory <cit.>, HAWC <cit.> etc. Our understanding of the high-energy universe has been revolutionized thanks to the successful operation of these experiments.The DArk Matter Particle Explorer (DAMPE <cit.>), initially named TANSUO <cit.>, was successfully launched into a sun-synchronous orbit at the altitude of 500 km on 2015 December 17^ th from the Jiuquan launch base. DAMPEoffers a new opportunity for advancing our knowledge of cosmic rays, dark matter, and gamma-ray astronomy. In this paper a detailed overview of the DAMPE instrument is provided, the expected instrumental performance based on extensive GEANT4 simulations are presented, and the key scientific objectives are outlined and discussed.DAMPE is able to detect electrons/positrons, gamma rays, protons, helium nuclei and other heavy ions in a wide energy range with much improved energy resolution and large acceptance (see Table <ref> for summary of the instrument parameters). The primary observing mode is the sky survey in a sun-synchronous orbit at the altitude of 500 km, and it is expected to cover the full sky at least four times in two years. The main scientific objectives addressed by DAMPE include: (1) understanding the mechanisms of particle acceleration operating in astrophysical sources, and the propagation of cosmic rays in the the Milky Way; (2) probing the nature of dark matter; and (3) studying the gamma-ray emission from Galactic and extragalactic sources.§ THE DAMPE INSTRUMENTFig.<ref> shows a schematic view of the DAMPE detector. It consists of a Plastic Scintillator strip Detector (PSD), a Silicon-Tungsten tracKer-converter (STK), a BGO imaging calorimeter and a NeUtron Detector (NUD). The PSD provides charged-particle background rejection for gamma rays (anti-coincidence detector) and measures the charge of incident particles; the STK measures the charges and the trajectories of charged particles, and allows to reconstruct the directions of incident photons converting into e^+ e^- pairs; the hodoscopic BGO calorimeter, with a total depth of about 32 radiation lengths, allows to measure the energy of incident particles with high resolution and to provide efficient electron/hadron identification; finally, the NUD provides a independent measurement and further improvement of the electron/hadron identification.§.§ The Plastic Scintillation array Detector (PSD) The main purpose of the PSD is to provide charged-particle background rejection for the gamma ray detection and to measure the absolute value of the charge (hereafter Z) of incident high-energy particles in a wide range (i.e., Z ≤ 26). Therefore high detection efficiency, large dynamic range, and good charge resolution are required for charged particle detection of PSD. The main instrumental parameters of the PSD are summarized in Table <ref>.A schematic view of the PSD is shown in Fig. <ref>. The PSD has an active area of 82.5 × 82.5 cm^2, that is larger than the on-axis cross section of other sub-detectors of DAMPE <cit.>. The PSD consists of 82 plastic scintillator (EJ-200 produced by Eljen <cit.>) bars arranged in two planes, each with a double layer configuration. Each bar is 88.4 cm long with a 2.8 cm× 1.0 cm cross section; the signals are readout by two Hamamatsu R4443 Photomultiplier Tubes (PMTs) coupled to the ends of each scintillator bar. The bars in the top plane are perpendicular to those in the bottom plane. The bars of the two layers of a plane are staggered by 0.8 cm, allowing a full coverage of the detector with the active area of scintillators without any gap. As the efficiency of a single layer is ≥ 0.95, the PSD provides an overall efficiency ≥ 0.9975 for charged particles. The segmented structure of the PSD allows to suppress the spurious veto signals due to the“backsplash effect", which can lead to a misidentification of gamma rays as charged particles. This phenomenon was observed in EGRET and was found to be significant for photon energies in the GeV region and above. A similar choice of the segmented design was adopted in the AGILE <cit.> and the Large Area Telescope onboard the Fermi telescope (Fermi-LAT) <cit.>, both equipped with anti-coincidence detectors consisting of plastic scintillator tiles.Since the PSD is used to identify cosmic-ray nuclei from helium to iron (Z=26), a wide dynamic range extending up to ∼ 1400 times the energy deposition of a minimum ionizing particle (MIP) [A singly charged MIP at normal incidence, which is assumed as reference, deposits on average about 2 MeV in a single PSD bar.]is required. To cover such a broad range with good energy resolution, a double dynode readout scheme for each PMT has been implemented. Signals from the dynode with high gain cover the range from 0.1 MIPs to 40 MIPs, while those from the dynode with low gain cover the range from 4 MIPs to 1600 MIPs; the overlap region can be used for cross calibration <cit.>.The dynode signals are coupled to VA160 ASIC chip developed by IDEAS <cit.>. This chip integrates the charge sensitive preamplifier, the shaper and the holding circuit for 32 channels. Four groups of front-end electronics (FEE) chips are placed at all the sides of the PSD, and each FEE processes 82 signal channels from 41 PMTs in each side. With each group of FEE, there is also a high-voltage fan-out board, which supplies the high-voltages to all the 41 PMTs in the same side.The detector plane, the four groups of FEEs, and the high-voltage fan-out boards are mounted together with the mechanical support. To minimize the materials used in the active area, this mechanical support is mainly made by honeycomb boards with Carbon Fiber Reinforced Plastics (CFRP) as the skin (see Fig.<ref>). Due to the large difference of temperature coefficients between the plastic scintillator and the CFRP, in order to avoid the damage due to large temperature variations, each of the detector modules is only fixed on the support with one end, and the other end is only constrained by a U-shape clamp while keeping the moving freedom along the bar direction. In 2014 and 2015, the Engineering Qualification Model (EQM) of DAMPE has been extensively tested on different particle beams, namely high energy gamma-rays (0.5-150 GeV), electrons (0.5-250 GeV), protons (3.5-400 GeV), π^- (3-10 GeV), π^+ (10-100 GeV), muons (150 GeV) and various nuclei produced by fragmentation of Argon (30-75 GeV/n) and Lead (30 GeV/n) in the European Organization for Nuclear Research (CERN).Fig. <ref> shows the energy deposited in the PSD for different species of charged particles with Z=1. We find that the peaks can be well described by Landau distribution due to the limited number of photons collected by the PMTs. Despite their very different mass and energy, the energy deposits for leptons (electrons, muons) and hadrons (pions, protons) are nearly the same.For a singly charged incident particle, the energy resolution is ∼10% which can be regarded as the charge resolution of PSD. As mentioned above, in order to effectively separate gamma rays from charged particles, the PSD should have a high detection efficiency for Z=1 particles. Such a performance was checked with electron beams of different energies. Fig. <ref> shows the spectra of deposited energy of 20 GeV electron beam in both X and Y layers. To minimize the influence of the backsplash effect, only modules within the beam spot area have been considered. By setting the threshold at 1 MeV, which corresponds to about 0.5 MIP, an efficiency higher than 0.994 has been achieved for each layer.The performance of the PSD has been also tested with the relativistic heavy ion beams at CERN. In this test, the primary Argon beam of 40 GeV/n was sent onto a 40 mm polyethylene target, andthe secondary fragments with A/Z=2 were selected by beam magnets, thus allowing to study the PSD response to all the stable nuclei with Z=2÷18. Fig. <ref> shows the reconstructed charge spectra for different ions (Z>2) from one PSD module within the beam spot. In this figure the Helium contribution has been removed for clarity (the He fraction is much higher than that of other ion species). The signals from both sides of each module are used (geometric mean) and the quenching effect has been corrected based on the ion response from the same test.It can be seen that all the elements from Lithium (Z=3) to Argon (Z=18) can be identified clearly. By applying a multi-Gaussian fit to the spectrum, we get the charge resolution of PSD for all ion species with the typical value of 0.21 for Helium and 0.48 for Argon. The charge resolution is expected to be better in space, because of much lower ion rates with respect to the case of beam tests. The results show that the position of the Ar peak in the raw Analog-Digital Conversion (ADC) spectrum for different PSD modules is only ∼20% of the full dynamic range. By simple extrapolation using the Birks-Chou law <cit.>, this validates that the PSD can cover ion species up to Iron (Z=26).§.§ The Silicon-Tungsten tracKer-converter (STK) The DAMPE STK is designed to accomplish the following tasks: precise particle track reconstruction with a resolution better than 80 μm for most of the incident angles, measurement of the electrical charge of incoming cosmic rays, and photon conversion to electron-positron pairs <cit.>. The DAMPE tracker-converter system combines the main features of the previous successful missions including AGILE <cit.>, Fermi-LAT <cit.> and AMS-02 <cit.>. It is composed of six position-sensitivedouble (X and Y) planes of silicon detectors with a total area of about 7 m^2, comparable with the total silicon surface of the AMS-02 tracker. Multiple thin tungsten layers have been inserted in the tracker structure in order to enhance the photon conversion rate while keeping negligible multiple scattering of electron/positron pairs (above ∼ 5GeV). The total thickness of STK corresponds to about one radiation length, mainly due to the tungsten layers.An exploded view of the STK is shown in Fig. <ref>, and a summary of the DAMPE STK instrument parameters is given in Table <ref>. The mechanical structure is made of 7 supporting trays of aluminum honeycomb layers sandwiched between two CFRP face sheets ofthick. The second, third and fourth planes are equipped with 1 mm thick tungsten plates glued into the CFRP sheet inside the tray, which was produced by Composite Design Sàrl <cit.>. The overall structure is light but stable in order to withstand the vibrations and accelerations during the launch. The alignment of each tungsten plate with respect to the 4 corners of the tray has been checked with a X-ray scan at CERN.The STK detector is equipped with a total of 768 single-sided AC-coupled silicon micro-strip detectors (SSD). Four SSDs are assembled together with a wire bonded strip-to-strip connection to form a silicon detector ladder, shown in Fig. <ref>. The total strip length along a ladder is about 37 cm. The ladders are glued on the seven support trays to form the 12 STK silicon layers. Each silicon layer consists of 16 ladders, as shown in Fig. <ref>. The two sides of the five central trays are both equipped with 16 ladders each, while for the top and the bottom plane only one side is equipped with the silicon ladders. All the planes are piled up together to form the full tracker system. The silicon ladders on the bottom surface of each tray are placed orthogonal with respect to the ones of the top surface of the lower tray, in order to measure the X-Y coordinates of the incident particles. The inter-distance between two consecutive silicon layers is .The silicon micro-strip sensors produced by Hamamatsu Photonics <cit.> have the same geometry of the ones used by AGILE <cit.>, but with different thickness, bulk resistivity and backplane metallization.The detector size is of 95 × 95 × 0.32 mm^3 and each SSD is segmented in 768 strips. The strips are 48 μm wide andlong with a pitch of . The bulk resistivity iswith a full depletion voltage ofmaximum. The average total leakage current is ofat , well below the specification of . The SSDs are glued on the flex part of the Tracker Front-end Hybrid (TFH) board to form a ladder, as shown in Fig. <ref>. The TFH serves as mechanical support for the SSDs and for the collection and amplification of the signals output from the strips. The readout is done one every other strip (corresponding to 384 channels per ladder), in order to keep a good performance in terms of spatial resolution, and at the same time reduce the number of readout channels. The signal shaping and amplification is performed by six VA140 ASIC chips (produced by IDEAS <cit.>) mounted on the TFH. The chip design is an updated version of the VA64HDR9A chips used in AMS-02 <cit.>. Each VA140 chip reads 64 channels. The readout and power supply electronics of the Tracker Readout Boards (TRB) have been mounted on the sides of the trays as shown in Fig. <ref>. Each TRB module reads 24 ladders and is made of three electronics boards: the power board, the control board, and the ADC board. The ladders are connected to the ADC board which provides the conversion of the signal from analog to digital, while the voltage to the front-end electronics and the silicon bias voltage are supplied by the power board. The control board is equipped with two field-programmable gate arrays (FPGAs) which handle not only the communication with the DAMPE DAQ system, but also the reduction of the data size, thanks to a zero-suppression and a cluster finding algorithm. More details of the TRB boards can be found . As discussed in the previous section, several test beam campaigns of the DAMPE EQM have been conducted at CERN in 2014 and 2015. Moreover, in order to better characterize the key constituent of the STK, dedicated tests have been conducted on single ladder units at the CERN Super Proton Synchrotron facility (SPS). As in the PSD case, the response of the detector is the same for different singly charged particles and different energies, as shown in Fig. <ref>, while it changes in case of particles with higher charge numbers (Z > 1). The two peaks structure of the signal distribution, shown on the left side of Fig. <ref>, is due to the floating/readout strip configuration. When a particle crosses a silicon sensor close to a readout strip and with an incident angle of 0^∘, i.e. orthogonally with respect to the silicon surface, almost all the released charge is collected by a single readout strip (higher charge peak). On the contrary, when the particle hits a floating strip, only about 65% of the original charge is collected by the two nearby readout strips, which produces the lower charge peak of the ADC distribution. This charge collection loss reduces as the incidence angle increases, and it could be recovered with a dedicated correction as function of particle incident angle and impact position (more details can be found in <cit.>). The right panel of Fig. <ref> shows the cluster charge distribution after such correction is applied.The ions charge identification power of STK was evaluated with a dedicated test conducted on single ladder units at CERN with a lead beam. The particle charge can be identified by looking at the mean value of the signal associated to the track. The signal mean S=√(∑(ADC_i/MIP/N)) is shown in Fig. <ref>. In this formula N corresponds to the number of clusters composing the track, ADC_i to the signal charge in the i-th cluster, and MIP to the cluster charge in ADC counts of a minimum ionizing particle. This value is proportional to the particle charge and allows a straightforward identification of ions up to Oxygen. Due to the non-linearity of the VAs above a signal of 200 fC, the identification of ions above Oxygen with the STK becomes non-trivial and on-going work is under preparation to improve the charge identification power. Moreover, in order to equalize the signal collected by each ladder and to make it independent from the incidence angle and the particle hit position on the ladder, a comprehensive and charge dependent STK signal calibration is in progress. Further improvement of the STK charge resolution is expected in the future. Thanks to a dedicated campaign of extensive cosmic ray data collected on ground, the STK detector has been aligned before launch, in order to correct for displacement and rotation of the SSDs with respect to the nominal position. The alignment procedure will be the subject of a dedicated paper. Here we only report the spatial resolution as a function of incident angle after alignment, shown in Fig. <ref>. As a result of the alignment, the spatial resolution is below 80μm within the angular acceptance of the STK (i.e. incidence angle < 60^∘) and below 60μm for particle incidence angles within 40^∘. This result is in agreement with the spatial resolution measurements obtained in test beam campaigns at CERN SPS on a single ladder <cit.>. §.§ The BGO calorimeter (BGO) The BGO calorimeter onboard DAMPE has three primary purposes: (1) measuring the energy deposition of incident particles; (2) imaging the 3D profile (both longitudinal and transverse) of the shower development, and provide electron/hadron discrimination; (3) providing the level 0 trigger for the DAMPE data acquisition system <cit.>. A summary of the key parameters of the BGO calorimeter is given in Table <ref>. Fig. <ref> shows the layout of the BGO calorimeter. Each crystal is readout by two Hamamatsu R5610A-01 PMTs (see Fig. <ref>), mounted on both ends (named S0 and S1, respectively). The left/right light asymmetry provides a measurement of the position of the energy deposit along the bar. The signals are read out from three different dynodes (dy2, dy5, dy8), thus allowing to cover a very large dynamic range of energy deposition, The PMTs are coupled to the crystals with optical filters, which attenuate the scintillation light produced in the BGO. The filter on S1 has a 5× attenuation factor with respect to the one on S0. The high gain readout channels (dy8) cover the range 2 MeV - 500 MeV (S0 end) and 10 MeV - 2.5 GeV (S1 end); the medium gain channels (dy5) cover the range 80 MeV - 20 GeV (S0 end) and 400 MeV - 100 GeV (S1 end); the low gain channels (dy2) cover the range 3.2 GeV - 800 GeV (S0 end) and 16 GeV - 4000 GeV (S1 end). The signals are sent to VA160 chip (or VATA160 for the bars which generate the trigger see Sec. <ref>) which is composed of a charge sensitive pre-amplifier, a CR-RC shaping amplifier and a sample-hold circuit. A charge signal can be injected into the front end of the preamplifier which is used to calibrate and monitor the performance of the VA160.The ground calibration of BGO has been performed using both the data collected in a beam test campaign at CERN and cosmic ray data collected from ground. The calibration procedure includes the measurement of the pedestals, the evaluation of the calibration constants from the MIP peaks, the evaluation of the dynode ratios, and the measurement of the bar attenuation lengths. The full details of the calibration procedure are provided in Refs. <cit.>. Fig. <ref> summarizes the performance of energy reconstruction of the BGO calorimeter for electrons with different energies up to ∼ 250 GeV. The data shown in the figure was obtained during the beam test campaigns performed at CERN. Details on the energy reconstruction and the electron/proton separation are discussed in Section <ref> and Section <ref>. The linearity of reconstructed energy is better than 1%, as shown in the Fig. <ref>. The energy resolution is better than 1.2% at the energies above 100 GeV (see Fig. <ref>).§.§ The NeUtron Detector (NUD) The main purpose of the NUD is to perform electron/hadron identification using the neutrons produced in hadronic showers initiated in the BGO calorimeter. In fact, for a given initial particle energy, the neutron content of a hadronic shower is expected to be one order of magnitude larger than that of an electromagnetic shower. Once the neutrons are created, they are quickly thermalized in the BGO calorimeter, and the total neutron activity over a few microseconds is measured by NUD. Table <ref> summarizes the key parameters of the NUD.Fig. <ref> shows the detailed structure of NUD. It consists of four 30 cm× 30 cm× 1.0 cm blocks of boron-loaded plastic scintillator (Eljen Technologies EJ-254), with 5% boron concentration by weight which has the natural ^10B abundance of 20% <cit.>. Each scintillator is wrapped with a layer of aluminum film for photon reflection, anchored in aluminum alloy framework by silicone rubber, and readout by a PMT. The space between plastic scintillators and aluminum alloy framework is 1 mm on each side, and is filled with silicone rubber to relieve the vibration during the launch.The scintillators are embedded with wavelength shift fibers for optical transmission in order to reduce the fluorescence attenuation and increase photon collection efficiency, and then the signals are readout by corner-on Hamamatsu R5610A-01 PMTs. The R5610A-01 is a 0.75 inches diameter head-on, 10-dynode PMT with a maximum gain of 2× 10^6, and a spectral response ranging from 300 nm to 650 nm, which is a good match to EJ-254's 425 nm maximum emission wavelength.Neutron captures are the dominant source of photon generation in the NUD after ∼ 2 μs from the initial calorimeter shower. Neutrons entering the boron-loaded scintillator can in fact undergo the capture process ^10 B+n→ ^7 Li+α+γ with a probability inversely proportional to their speed, and a time constant for capture inversely proportional to the ^10B loading. About 600 optical photons are produced in each capture <cit.>. A block diagram of the readout electronics is shown in Fig. <ref>. There are four signal channels provided in one data processing board. Each channel contains a fast pre-amplifier, a gating circuit (GC), a shaping circuit (SC) and a main amplifier with peak holding chip (PHC). The GC and PHC are controlled by the data control unit of the DAMPE satellite. The GC is designed to prevent any early signal entering the SC, and is switched-on 1.6 μ s after the triggering signal produced by BGO. Then the delayed neutron signal could be shaped and amplified to the PHC. After the ADC finishes the acquisition of all four signals, a release signal will be sent to the PHC and GC to shut off the signal channel and wait for the next trigger. The electron and proton data collected during the beam test has been used to study the particle identification power of the NUD. Since protons deposit in the BGO is about 1/3 of their initial energy, we compared 150 GeV electron events with 450 GeV proton events (depositing ≈ 150 GeV in the BGO calorimeter). In Fig. <ref>, the NUD signals of electrons and protons are compared. The electron signals are always less than 30 channels, and in most cases are below 2 channels, while the proton signals are remarkably larger.The PMTs of the NUD and the bottom BGO layer share the same high voltage module to save electric power and reduce payload weight. As a result, the NUD works in the high gain mode during on-orbit operation, which gives a more powerful capability for electron-proton identification. Detailed GEANT4 simulations suggest a proton rejection power for NUD (in its full performance) of a factor of ∼10, assuming an electron detection efficiency of 0.95. Preliminary estimates, based on on-orbit calibration data, show that a rejection power is ∼12.5 for incoming particles with BGO energy deposit above 800 GeV (details will be published elsewhere).§.§ Data Acquisition System and TriggerThe data acquisition system (DAQ) receives the commands from the satellite computer, implements trigger decision logic, collects science and housekeeping data from the detectors, and transfers them to the ground. Fig. <ref> shows the architecture of the DAQ system. The DAMPE DAQ system <cit.> consists of two electronics crates, including the Payload Data Process Unit (PDPU) and the Payload Management Unit (PMU).The DAQ system is implemented with dual modular redundancy. The PMU is the control center of DAMPE and it is equipped with a 16 GB flash memory for data storage. The central processing unit (CPU) board of the PMU receives commands from satellite computer through 1553B bus (1 Mbps). The PMU decodes the commands and distributes them to the PDPU or the FEE of -X/-Y sides directly (see Fig. <ref>). When the PMU receives a trigger signal from the trigger board in the PDPU, it begins to collect science data from the FEE on the -X/-Y sides, while data from the FEE on the +X/+Y sides are collected from the PDPU. All collected data are finally stored in the 16 GB mass memory. The PMU also collects housekeeping data of DAMPE periodically and sends them to the satellite computer. All the science and housekeeping data are finally relayed to ground with the timestamp of 1ms precision. The PMU calibrates its timer with the clock of the Global Positioning System (GPS) spacecrafts with one pulse per second.The PDPU is responsible for collecting the science data from FEE of +X/+Y direction, collecting housekeeping data from FEE, generating global trigger signal for DAMPE and distributing the commands from PMU. The trigger board of the PDPU receives signals from the BGO calorimeter and makes a trigger decision within 1 μs <cit.>. The trigger is sent to the FEE and to the PMU, while at the same time the PDPU prevents further events to be collected until all science data is stored, which is collected by the event process board of the PDPU and sent to the PMU.Only the signals from eight out of fourteen BGO layers are sent to the trigger board. The trigger board implements the trigger decision logic with a flash memory based FPGA chip.Four different triggers have been implemented: Unbiased trigger, MIP trigger, High Energy trigger and Low Energy trigger. They are “OR-ed" to generate the global trigger signal for the detector (see Fig. <ref>). The Unbiased trigger requires signals in the two top BGO layers exceeding a low threshold of ∼0.4MIPs in each hit BGO bar. The MIP trigger aims to select particles crossing all the BGO layers. The High Energy trigger selects events with energy depositions in the top four BGO layers exceeding a high threshold of ∼10MIPs in each hit BGO Bar. The Low Energy trigger is similar to the High Energy one, but with a lower threshold of ∼2MIPs. A periodic signal of 100 Hz is also implemented in the trigger board for pedestal calibration.The Unbiased, MIP and Low Energy triggers are pre-scaled with the ratios of 512:1, 4:1, 8:1, respectively, when the satellite is in the low latitude region (± 20). At high latitudes, the MIP trigger is disabled and the pre-scaler ratios of Unbiased and Low Energy triggers are set to 2048:1 and 64:1, respectively. The expected average rate of global triggers is about 70Hz in flight (the rate of High Energy triggers is 50Hz, the rate of Unbiased triggers is about 2.5Hz). The DAQ systems works in an “event by event” mode, and a 3ms time interval is set to acquire each event, so that the dead time is fixed to 3ms as a consequence. § INSTRUMENT MODELING AND EVENT RECONSTRUCTION§.§ Instrument modeling A full Monte Carlo (MC) simulation has been developed to accurately evaluate the detector response to incident particles. The simulation is central both in the design/optimization phase and in demonstrating the possible achievements in terms of dynamic ranges, resolutions and background rejection power. The simulation procedure mimics the real data taking condition of the instrument during both ground tests and in-flight observations, by using proper input particle fluxes and fully modeling the detector geometry and readout chain.Fig. <ref> shows the flow chart of the data processing for DAMPE, which includes simulation, digitization and reconstruction. The DAMPE simulation is based on the GEANT4 toolkit <cit.>, a software widely used in high energy physics experiments to handle particle generation, propagation and interactions. The information on the DAMPE geometry, including the position and the materials of all the detector elements (both active and passive), is stored in gdml and xml files which are used by GEANT4 to build a detailed model. The whole simulation procedure is implemented in a GAUDI-like software framework <cit.>, which produces collections of energy hits for each sensitive detector element. A digitization algorithm has been developed to convert energy hits into ADC counts, with the same format as real data, including the calibration constants (i.e., pedestal noises, PMT gains). In this way, the MC data can be processed by the same reconstruction algorithms and the simulation can provide an accurate representation of the instrument response for analysis. Also, for the orbit simulation the same trigger conditions as for real data have been implemented to simulate the final data stream.§.§ Event reconstruction§.§.§ Energy reconstruction The first step of the energy reconstruction algorithm is the conversion of the ADC counts into energy based on the calibration constants, once pedestals have been removed, and choosing the signals from the proper readout dynodes (dy8/dy5/dy2). The total deposited energy is then calculated by summing up the energies of all BGO crystal elements. The typical pedestal width is about 8 fC, corresponding to 0.32 MeV (S0) and 1.6 MeV (S1) for dy8, 12.8 MeV (S0) and 64 MeV (S1) for dy5, 512 MeV (S0) and 2560 MeV (S1) for dy2, respectively. On orbit, cosmic-ray proton MIP events will be selected to calibrate the energy response of ADC for each BGO crystal. The resulting ADC distribution of each individual BGO crystal will be fitted with a Landau function convolved with a Gaussian distribution. The most probable value (MPV) corresponds to the MPV in energy units taken from the simulation (≈ 21.7MeV for protons).Thanks to the multi-dynode readout design, the BGO calorimeter enables a measurement ofthe energy of electrons or gamma rays up to at least 10 TeV without saturation. The measurable energies for a single bar range from 0.5 MIPs (∼11 MeV) to 10^5 MIPs (∼2 TeV), covering a dynamic range of 2×10^5. From the simulation we find that, for a 10 TeV electromagnetic shower, typically the maximum energy deposit in one BGO bar does not exceed ∼2 TeV, which is within the linear region of dy2.The energy deposited in the BGO calorimeter underestimates the true energy of incident particles. Electrons and photons can in fact lose a significant fraction of their energy in the dead materials of calorimeter, such as the carbon fibers and rubber used for the support structure. For incident electron and photon energies above hundreds of GeV, the energy leakage should be taken into account. In addition, the energy deposited in the STK and in the PSD cannot be neglected, in particular for low-energy incident particles. The true energy of electrons and photons is evaluated by properly modeling the transversal and longitudinal development of electromagnetic showers in the calorimeter.Two methods are used to calculate the corrected energy starting from a set of reconstructed variables, exploiting their dependence on the deposited energy. In the first case the correction is performed starting from the ratio between the sum of the maximum energies in each layer and the total deposited energy, which was found to be sensitive to the energy loss in dead material of BGO calorimeter. In the second case the correction is performed starting from the depth of the shower maximum obtained by fitting the longitudinal profile with the Gamma-distribution, which shows a good correlation with the energy leakage. The correction parameters for different incident energies and different incidence angles are obtained from the simulations and are checked with beam test data (see Fig. <ref>). The details of these procedures can be found in ref. <cit.>.The energy measurements for cosmic-ray protons and nuclei are much more complicated than that for electrons or gamma rays, as hadronic showers generally are not fully contained in the BGO. Moreover hadronic showers include an electromagnetic and a hadronic component with large event-by-event fluctuations, which brings relatively large uncertainties in the energy deposition. An unfolding algorithm based on the Bayes theorem <cit.> will be implemented to estimate the primary energy spectra of cosmic-ray nuclei. DAMPE can measure hadronic cosmic rays to an energy of ∼100 TeV without significant saturation. For such high energy events, the maximum deposit energy in one BGO bar is typically a few TeV, within the linear region of dy2. We are developing a correction method using the adjacent non-saturated bars for a few events which may exceed the linear region of the readout dynodes.§.§.§ Track reconstructionBGO Track reconstruction Despite its limited spatial resolution, the BGO calorimeter can also be used for the track reconstruction. The track reconstruction procedure starts by searching for the “clusters” of fired bars in each BGO layer. A cluster is built starting from the bar with the maximum energy deposit and associating to it all the neighboring bars on both sides with decreasing energy deposits. The cluster construction is terminated when one of the following conditions is met: (1) the side of BGO is reached; (2) a non-fired bar is found; (3) a bar with increasing energy deposit is found. Finally, we make clusters symmetric about the maximum energy bar.Therefore, if the left (right) tail of the fired bar cluster has more bars than the right (left) tail, the bars in excess are removed. We allow one cluster per layer at most, and then perform a linear fit starting from the positions of the bars in the clusters, and each bar is weighted with the corresponding energy deposit. The fitting result, however, is found to bear some systematic bias for inclined incident particles. To minimize this bias, we rotate the coordinate to align the X axis with the track direction obtained from the first fit. A second fit is then performed in the new coordinate system, and the final result is obtained by converting back into the original coordinate system. The direction found by the BGO track reconstruction (if available) is used as a seed for the STK track reconstruction. STK Track reconstruction The raw data of STK are ADC values as the output of data reduction algorithm on board of the satellite <cit.>. Preliminary clusterization of signal is performed on board of the satellite, where cluster seeds are found from the channels which have a signal-to-noise ratio S/N>3.5 and additional strips with ADC>5. A refined hit reconstruction is then performed offline from ground, as outlined below. The ADC values are grouped into arrays of 384 channels per ladder. Channels which did not pass the on-board data reduction are assigned to zero. The offline clustering algorithm looks for seeds which are defined as local signal maxima with S/N>4, and then form the cluster by collecting all the neighboring strips with S/N>1.5.In order to resolve multi-peak clusters (which can occur for example in photon conversions into e^+e^- pairs, where each peak correspond to its own particle) the cluster reconstruction terminates if a strip signal fulfills the condition S_n / N-S_n-1 /N>5, where S_n and S_n-1 are the signals in the current strip in the cluster and the next strip respectively. The hits in X and Y projections in same tracking plane are then combined in all possible ways to form three-dimensional hits. Since quarter-planes of STK are readout by separate electronic boards, only X-Y hit combinations coming from the same quarter plane are allowed, thus reducing significantly the number of candidate hits.Track reconstruction is done as follows. The direction found in the BGO is projected onto the closest layer of the STK with the corresponding error matrix, either infinite, or the one evaluated from the shower position and angular resolution as a function of energy. If the hit is found within a reasonable window around the projected position, a seed is formed and the track is reconstructed using the Kalman filter. If the resulting track is of insufficient quality (i.e. the χ^2-test or the number of hits in the track does not fulfill the corresponding threshold values), the procedure is repeated with other hits in that layer. If a track is not found afterward, it is repeated with the hits in the second and third closest layer to the calorimeter. If a track is found, the whole procedure is repeated again with the first point of previous track being removed from the list of available points. The same iterations are repeated from beginning until all seed points are exhausted. Finally, the procedure is repeated also with the three furthermost layers of the tracker in the opposite direction (towards calorimeter). Once a set of tracks is formed, the ghost tracks are eliminated by looping over all tracks and removing those with lover quality crossed by the other tracks. The track forks (two tracks starting from the same point) in the direction towards the calorimeter are allowed, while those which point toward the opposite direction, are considered as a track crossings and treated correspondingly. §.§.§ Charge reconstructionThe measurement of the energy spectra of cosmic-ray nuclei (Z=1-26) in the energy range from 5 GeV to 100 TeV is a major goal of DAMPE. The charge of cosmic rays can be measured by both the PSD and the STK.A charged particle crossing a PSD strip loses energy mainly by ionization, with the energy deposition being proportional to Z^2 and to the path length. The first step of charge reconstruction is to find the candidate track, which allows to find the PSD strips crossed by the particle, and to evaluate the path lengths and the positions in which the tracks intersect the strips. Since each PSD strip is readout by two PMTs mounted at each end, two signals per strip are obtained. From each signal an energy deposition value is calculated, correcting for the path length and the position of the track along the strip to account for light attenuation. Since a track can intersect a maximum of four PSD strips, a total of eight energy values per event can be used for charge reconstruction, which are then combined to provide an accurate estimate of Z.The STK, with its 12 layers of silicon strip detectors, can also be used to measure the charge of incident particles, starting from the energy deposition points for the clusters along the track. The energy deposition for a cluster can be deduced from the impact point and incidence angle. The impact point can be estimated by the ADC values of the readout strips in the cluster <cit.>. The charge number can be estimated by combining all those measurements. Furthermore, in case of fragmentation of an incoming nucleus due to interaction with material of the instrument (for example with the tungsten plates), the charge number is expected to change along the path of the track towards the calorimeter. The PSD and STK will be combined to further improve the measurement of Z.§.§.§ Electron/proton identification The measurement of the total spectrum of cosmic ray electrons/positrons is a major goal of DAMPE. Therefore, besides the track and energy reconstruction, a high identification and discrimination power of protons from electron/positrons is required. The basic approach for electron/proton identification is an image-based pattern recognition method, mainly inherited from the one used in the ATIC experiment <cit.>.Since the BGO has a radiation length of 1.12cm and a nuclear interaction length of 22.8cm, showers initiated by electrons (electromagnetic) and protons (hadronic) will behave very differently in the BGO calorimeter. Two of the most important features are the radial and longitudinal development of the shower. MC simulation and beam test data show that electrons and protons can be indeed well separated. In the GeV-TeV energy range, the proton rejection power can in fact reach a level of 10^5, while keeping at least a 90% electron identification efficiency. Electrons and protons depositing the same amount of energy in the BGO calorimeter can be separated by means of the reconstructed 3D images of the showers. An electron/proton rejection power close to 2× 10^3 while keeping a 94% electron identification efficiency has been achieved using BGO only beam test data.For the DAMPE calorimeter, almost all electrons deposit more than 90% of their energy into the calorimeter while protons usually just deposit ∼ 1/3. Since the cosmic ray proton spectrum is approximately proportional E^-2.7, the on-orbit rejection power will be improved of a factor ≈ 7 (i.e., 3^1.7≈ 7). In addition, the High Energy trigger (see Sec.<ref>) has been optimized to suppress the proton events by a factor of ∼ 3. Finally, the NUD can be used to further increase the rejection power by a factor of ∼2.5 at TeV energies. As an independent check, we also adopt the Toolkit for Multivariate Data Analysis (TMVA) and deep learning techniques to perform the e/p identification, which give rather similar rejection powers. § PERFORMANCE AND OPERATION§.§ Expected performance and testsThe expected instrument performance is summarized in Figs. <ref>-<ref> for electrons/photons, and Figs. <ref>-<ref> for protons. These results are based on simulations of DAMPE instrument performance from the event reconstruction and selection algorithms, which includes trigger filter, track reconstruction, geometry constraints, charge reconstruction, particle identification and energy reconstruction. The efficiency of each step has been carefully studied with MC simulations and checked with beam test data and cosmic-ray muon data at ground. The performance parameters (in particular for gamma ray detection efficiency) are expected to improve in the future with improved algorithms, as the event reconstruction and selection algorithms will be further optimized after a better understanding of the on-orbit performance.Fig. <ref> shows the effective area as a function of energy for gamma ray detection at normal incidence and at 30^∘ off-axis angle, respectively. The adopted event selection algorithm for gamma rays is the following. Firstly events with shower well contained in the calorimeter are selected, then a first hadronic background rejection is performed by using information from the BGO only (see Sec. <ref>). Candidate electron/gamma-ray events with a track in the STK are then selected. Finally the PSD is used as an anti-coincidence detector to reject charged particle events. The drop of effective area above 100GeV (shown in Fig. <ref>) is due to the backsplash effect, which has not been taken into account in the present gamma-rays event selection. The same cuts without the PSD anti-coincidence veto can be used to select electrons/positrons. Starting from events with the High Energy trigger, the resulting acceptance for electrons is larger than 0.3m^2 sr above 50GeV, as shown in Fig. <ref>. The energy resolution for electromagnetic showers is shown in Fig. <ref>. The angular resolution (i.e. the corresponding 68% containment angle) for gamma rays converted in the STK is shown in Fig. <ref> for normal and 30^∘ incidence angles, respectively. For hadronic cosmic rays, the acceptance is about 0.1m^2 sr for energies above ∼100GeV, which varies for different nuclei species due to different trigger efficiency. The energy measurement of cosmic ray nuclei is more complicated than that of electrons/photons, because of the energy leakage due to limited nuclear interaction thickness of the calorimeter (∼1.6 nuclear interaction length) and fluctuation of the hardonic shower development. To convert the measured energy spectrum to the primary energy spectrum, an unfolding algorithm could be used to reconstruct the nucleus energy spectrum, by using the MC detector response matrix. Figure <ref> shows the deposit (blue) and reconstructed (red) energy distributions for on-axis incident proton beams with momenta of 5, 10, 150, and 400 GeV/c. The reconstructed procedure allows to recovery the incident beam energy as well.The energy resolution (σ_E/E) of on-axis incident protons (after the unfolding), estimated from the simulation data, is shown by the dotted line in Fig. <ref>. As a comparison, the results for the beam test data at four energies are overplotted. It is shown that the energy resolution for protons varies from ∼10% at several GeV to ∼30% at 100 TeV. Above 10 TeV, the uncertainties on the hadronic interaction model as implemented in Geant4 are expected to be non-negligible. While a detailed treatment is currently being undertaken in the collaboration, we expect these uncertainties to yield uncertainties in the reconstructed spectrum of about 10%. We are also investigating the use of alternative simulation packages that incorporate hadronic interactions at the TeV scale better (e.g. Fluka [http://www.fluka.org/fluka.php]).As discussed above, the verification of the estimated performance was carried out using the data from the beam test campaign, as well as a set of data collected with cosmic-ray muons at sea level. In particular, several cosmic-ray muon tests were performed during different stages of the DAMPE assembly, especially in the environmental testing phase and in the pre-launch preparation of the satellite. In these tests, a proper trigger logic was adopted to select cosmic-ray muons. We were able to collect a large amount of muon events, which has been used to perform a full calibration of the energy response for MIPs and to implement the alignment procedure for the STK. After launch, the spacecraft entered the sky-survey mode immediately, and a dedicated calibration of the detector was performed in the first 15 days, including pedestals, MIP responses (protons), alignments, and timing etc. Comparison between on-orbit data with simulations and ground cosmic-ray data demonstrates the excellent working condition of DAMPE detectors. Details of the on-orbit calibration and performance evaluation will be published elsewhere <cit.>.§.§ OperationSince December 17^ th 2015, DAMPE is orbiting in solar synchronous mode, with each orbit lasting 95 minutes. The trigger configuration and the pre-scaling factors for the on-orbit science operation have been illustrated in Sect. <ref>, and ensure a global trigger rate around 70Hz. The pedestal calibration is performed twice per-orbit, and all data are regularly transmitted to ground.On ground the data are processed by the Ground Support System (GSS) and the Scientific Application System (SAS). Binary raw data (housekeeping and science data) transmitted to ground are first received by three ground stations located in the south, west and north of China at early morning and afternoon of each day respectively, when the satellite passes China's borderline. Then all binary data are automatically transmitted to the GSS located in Beijing, and are tagged as level-0 data. On average, about 12 GB level-0 data are produced per day. Upon arrival of the level-0 data at the GSS, they are immediately processed and several operations are performed, including data merging, overlap skipping and cyclic redundancy check (CRC) which is an error-detecting code based on the protocol CRC-16/CCITT.The level-0 data are daily processed into level-1 data, which includes 13 kinds of completed telemetry source packages, one for science data and 12 for housekeeping data.Daily level-1 data will then be processed by the GSS within 1 hour. The SAS located at the Purple Mountain Observatory of Chinese Academy of Sciences in Nanjing monitors the level-1 data production 24 hours a day continuously. The new level-1 data will be synchronized to the mass storage at the Purple Mountain Observatory immediately. Then 12 housekeeping data packages are parsed and inserted into the housekeeping database, which allows to monitor the conditions of the DAMPE payload and the satellite platform. After processing the housekeeping data, routine checks on key engineering parameters are performed to guarantee the proper data taking conditions.The processing pipeline of science data includes the Raw Data Conversion, Pre-Reconstruction and Reconstruction algorithms implemented in the DAMPE software framework (DAMPESW). The Raw Data Conversion algorithm splits raw science data into about 30 calibration files and 30 observation files, and converts them into ROOT data files <cit.>. During this procedure, key housekeeping data required by science analysis are also stored into in the ROOT data files. Calibration files are used to extract calibration constants which are used in Pre-Reconstruction and Reconstruction algorithms. Reconstructed data from all sub-detectors are then merged to generate level-2 science data products. These two procedures increases the raw science data volume by approximately a factor of five.The processing pipeline of science data is designed to run on a cluster of batch processors. The SAS hosts more than 1400 computing cores at the Purple Mountain Observatory, which can reprocess three years of DAMPE data within two weeks. In addition, INFN and University of Geneva computing resources are also used, which are mainly dedicated to MC data production and could also be used as backup reprocessing sites if needed. § KEY SCIENTIFIC OBJECTIVESDAMPE is a high energy cosmic-ray and gamma-ray observatory with a broad range of scientific objectives. The data sets provided by DAMPE could be used to study cosmic-ray physics, to probe the nature of dark matter, and to reveal the nature of high energy gamma-ray phenomena. The large field of view of DAMPE provides the opportunity to monitor the violent GeV-TeV transients for various purposes.§.§ Understanding the acceleration, propagation and radiation of cosmic rays Cosmic rays impinging the Earth with energies below ∼10^17eV are believed to be mainly produced through energetic astrophysical processes within the Milky Way. Their interactions with interstellar medium, interstellar radiation fields, and Galactic magnetic fields are the main source of the detected Galactic diffuse gamma-ray emissions. Moreover, cosmic rays are the only sample of matter originated from distant regions of the Galaxy that can be directly measured with spaceborne experiments. Therefore, understanding the origin, acceleration, and propagation of cosmic rays is a crucial subject on the understanding of the Universe.With more than three years of operation, DAMPE will be able to observe electrons/positrons or photons from GeV to10TeV, and protons, helium or heavier nuclei from 10 GeV to 100 TeV. The measurement of energy spectra with unprecedented precision and energy coverage at higher energies, together with spatial distribution of these particles are expected to significantly enhance our understanding of the origin of cosmic rays. Below we outline the key scientific outputs regarding cosmic-ray studies potentially achievable with DAMPE. * The proton and helium are the most abundant components of cosmic rays. The standard paradigm for particle acceleration and propagation predicts single power-law spectra up to the so-called “knee” at ∼ 10^15 eV. Surprisingly, the spectra of cosmic-ray nuclei measured by ATIC <cit.>, CREAM <cit.>, PAMELA <cit.> and AMS-02 <cit.> all showed remarkable hardening at the magnetic rigidity of several hundred GV. Such a result triggered various modifications of the standard, simple picture of Galactic cosmic rays. Interesting possibilities include the superposition of injection spectra of the ensemble of sources <cit.>, the effect of local source(s) <cit.>,the complicated acceleration of particles <cit.>, or a non-uniform diffusion coefficient <cit.>. Current spectral measurements are, however, uncertain for energies above TeV/n. DAMPE will be able to clearly measure the spectral changes and precisely determine the high energy spectral indices of various nuclei species, as shown in Fig. <ref> for proton and helium. Furthermore, the DAMPE data will be able to test whether there are additional structures on the high energy cosmic-ray spectra, as may be expected from nearby sources <cit.>.Recently, the proton and helium spectra from the CREAM-III flight have been published and tentative breaks at ∼ 10-20 TeV are displayed <cit.>. With the energy resolution of ∼ 20% at such energies (see Fig.<ref>; which is better than that of CREAM-III) and an expected exposure of ∼ 0.3 m^2 sr yr, DAMPE will reliably test such a possibility. The cosmic-ray spectra up to 100 TeV by DAMPE will overlap with that measured by the ground-based air shower experiments (e.g. <cit.>), which can provide us with a full picture of the cosmic-ray spectra up to above the knee. DAMPE can also measure the Boron-to-Carbon ratio, to about 5 TeV/n, which can effectively constrain the propagation parameters. * Electrons/positrons contribute ∼ 1% of the total amount of cosmic rays. Unlike the nuclei, electrons/positrons lose their energies efficiently during the propagation in the Galaxy. This is particularly true for ∼TeV electrons/positrons which are expected to reach the Earth only if the source is relatively nearby (≲ 1 kpc) and young (≲ 10^5 yr) <cit.>. With an acceptance of ∼ 0.3 m^2 sr at TeV energies (see Fig.<ref>), DAMPE will precisely measure the trans-TeV behavior of the energy spectra of electrons/positrons, and determine the spectral structures e.g. spectral cut-off <cit.>. As a consequence, DAMPE will be able to directly test a long-standing hypothesis that nearby pulsars or SNRs (e.g., Vela) are efficient TeV electron accelerators <cit.> by measuring the spectrum and/or the spatial anisotropy of TeV electrons/positrons (see Fig.<ref> for an illustration). * DAMPE can also measure gamma-rays from Galactic and extra-galactic cosmic ray accelerators such as SNRs, pulsars, quasars <cit.> etc. Although the effective acceptance of DAMPE is smaller than that of Fermi-LAT, DAMPE may play an auxiliary role in deep observations of these sources, especially in connection with ground-based measurements at hundreds of GeV.§.§ Probing the nature of dark matterAs early as the 1930s, it was recognized that some matter in the Universe is invisible <cit.>. The existence of this so-called dark matter was gradually and firmly established since the early 1970s <cit.>. In the standard model of cosmology, the ordinary matter, dark matter and dark energy share 4.9%, 26.6% and 68.5% of today's total energy density of the Universe. Compelling evidence shows that the commonly existing dark matter is non-baryonic; however, the physical nature of the dark matter particle is still poorly known <cit.>. Many theoretical models have been proposed, and the suggested candidates span over a wide range of masses, mechanisms, and interaction strengths <cit.>. Among various candidates of dark matter particles, one of the most attractive models is the weakly interacting massive particle (WIMP), which is widely predicted in extensions of the standard model of particle physics. The annihilation or decay of WIMPs can give electromagnetic signals, primarily in the gamma-ray band, as well as standard model particle products such as electrons/positrons, neutrinos/anti-neutrinos and protons/antiprotons <cit.>.Anomalous peaks or structures in the energy spectra of cosmic rays (in particular for electrons/positrons and antiprotons) and/or gamma rays from particular directions with accumulated dark matter distribution could indicate the existence of dark matter particles. In the past few years, several anomalous excesses had been reported in different cosmic-ray and gamma-ray data sets, including the electron/positron excesses <cit.>, the Galactic center GeV excess <cit.>, the possible excesses in a few dwarf galaxies <cit.>, and the tentative ∼ 130 GeV gamma-ray line <cit.>. Recently, another line-like structure around 43 GeV from a number of galaxy clusters was reported with the Fermi-LAT Pass 8 data <cit.>. These candidate signals are either too weak to be claimed as a firm detection, or can be interpreted with astrophysical models or potential instrument systematics (e.g., <cit.>).With its much improved energy resolution (see Fig. <ref>), DAMPE is suitable for the search of gamma-ray line emission which can be expected in the annihilation channel of γ X, where X=(γ, Z_0, H) or other new neutral particle. The energies of the monochromatic gamma-rays are given by E_γ=m_χ [1-m_X^2/4m_χ^2], where m_χ is the mass of dark matter particle <cit.>. The firm detection of gamma-ray line(s) is believed to be a smoking-gun signature of new physics, because no known astrophysical process is expected to be able to produce such spectral feature(s). The high resolution is also crucial to identify multiple lines with energies close to each other <cit.>. A set of gamma-ray lines would further provide convincing evidence of dark matter particles, and could provide more information of physical properties of dark matter particle, such as their couplings with standard model particles. Theoretically the line emission is typically suppressed due to particle interactions through loop process, other scenarios e.g. the internal bremsstrahlung from dark matter annihilating into a pair of charged particles, might dominate the potential line signal <cit.>. Axions or axion-like particles (ALPs), if produced non-thermally, could be candidate of cold dark matter <cit.>, which produce spectral fine structures due to the photon-ALP oscillation <cit.>. DAMPE will enhance our capability to search for monochromatic and/or sharp spectral structures of gamma-rays in the GeV-TeV range.For illustration purpose, we take into account two types of dark matter density profile, including a contracted Navarro-Frenk-White profile with γ=1.3 (NFWc) <cit.> and an Einasto profile with α=0.17 <cit.>. Following <cit.>, the Regions of Interest (ROIs) have been taken as a 3^∘ (16^∘) circle centered on the Galactic center, respectively. For the Einasto profile, we also mask the galactic plane region with |l| > 6^∘ and |b| < 5^∘. The Galactic diffuse emission model gll_iem_v06.fits <cit.>, the isotropic diffuse model iso_P8R2_SOURCE_V6_v06.txt <cit.>, and 3FGL point sources of Fermi <cit.> have been combined to model the gamma-ray background. The projected sensitivities of DAMPE in 3 years in case of targeted observations towards the Galactic center and in 5 years of sky-survey observations are presented in Fig. <ref>.The electron/positron spectra can also be used to probe dark matter, although the discrimination from local astrophysical sources may not be trivial. In general the contribution from astrophysical sources is expected to be non-universal, and may induce multiple features on the total energy spectra <cit.>. DAMPE will accurately measure the energy spectra of electrons/positrons at trans-TeV energies to resolve these potential fine structures, which can be used to test/constrain dark matter models, in order to consistently explain the electron/positron excesses. With DAMPE's much improved energy resolution, possible new and fine spectral structures on the electron/positron spectra may be revealed as the existence of dark matter particles <cit.>. §.§ Studying high energy behaviors of gamma-ray transients and the diffuse emission DAMPE observes gamma-ray photons in the energy range of 10 GeV to TeV and above with very high energy resolution. Note that with the low energy trigger system the threshold can be as low as ∼ 0.5 GeV, with a reduced sampling rate by a factor of ∼ 64. Compared with Fermi-LAT, DAMPE has a smaller effective area and a higher energy threshold. Therefore for stable GeV sources, DAMPE is not expected to be competitive compared with Fermi-LAT due to limited counting statistics. However, DAMPE may play a complementary, and possibly crucial role in catching bright GeV-TeV transients, as each gamma-ray detector can only cover part of the sky at the same time.The collected gamma-ray data can be used to study the violent physical processes behind activities of Active Galactic Nuclei (for instance Mrk 421, 3C279 and 3C454.3), the Crab flares, and some bright gamma-ray bursts (GRBs) such as GRB 130427A <cit.>. Bright short GRBs with an isotropic GeV γ-ray energy release of ≥ 2× 10^51 erg, if taking place within ∼ 400 Mpc, might be detectable by DAMPE and could also serve as the electromagnetic counterparts <cit.> of advanced LIGO/Virgo gravitational wave events <cit.> or IceCube PeV neutrino events <cit.>. High energy gamma-ray observations can also be used to probe the extragalactic background light, the intergalactic magnetic field, and the fundamental physics such as Lorentz invariance violation and quantum gravity.The hadronic interaction of cosmic ray nuclei with the interstellar medium produces bright diffuse gamma-ray emission, primarily along the Galactic plane. In some regions (e.g. the Galactic center ridge and the Cygnus region) of the Galactic plane, fresh cosmic ray accelerators may light up surrounding materials with gamma-ray emission on top of the diffuse background <cit.>. Thanks to the improved hadron rejection power of DAMPE (>10^5), it is possible to measure the diffuse gamma-ray emission up to TeV energies without significant contamination from cosmic rays. DAMPE has the potential to reliably detect >1 TeV gamma-rays in space. The DAMPE mission will provide a crucial overlap energy range to bridge the space and ground-based gamma-ray measurements, providing effective constraints on the origin of cosmic rays in the Milky Way.§ SUMMARYDAMPE was successfully launched into a sun-synchronous orbit at the altitude of 500 km on December 17^ th 2015. The combination of the wide field of view, the large effective area and acceptance and the excellent energy resolution offers new opportunities for advancing our knowledge of cosmic rays, dark matter and high energy astronomy.Acknowledgments:The DAMPE mission was funded by the strategic priority science and technology projects in space science of the Chinese Academy of Sciences (No. XDA04040000 and No. XDA04040400). In China this work is also supported in part by National Key Research and Development Program of China (No. 2016YFA0400200), the National Basic Research Program (No. 2013CB837000), National Natural Science Foundation of China under grants No. 11525313 (i.e., Funds for Distinguished Young Scholars), No. 11622327 (i.e., Funds for Excellent Young Scholars), No. 11273070, No. 11303096, No. 11303105, No. 11303106, No. 11303107, No. 11673075, U1531126, U1631111 and the 100 Talents program of Chinese Academy of Sciences. In Europe the work is supported by the Swiss National Science Science Foundation, the University of Geneva, the Italian National Institute for Nuclear Physics, and the Italian University and Research Ministry. We also would like to take this opportunity to thank the scientific laboratories and test facilities in China and Europe (in particular CERN for provision of accelerator beams) that assisted the DAMPE team during the qualification phases.IMP M. Garcia-Munoz, G. 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"A. D'Amone",
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"M. Di Santo",
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"T. K. Dong",
"Y. F. Dong",
"Z. X. Dong",
"G. Donvito",
"D. Droz",
"K. K. Duan",
"J. L. Duan",
"M. Duranti",
"D. D'Urso",
"R. R. Fan",
"Y. Z. Fan",
"F. Fang",
"C. Q. Feng",
"L. Feng",
"P. Fusco",
"V. Gallo",
"F. J. Gan",
"W. Q. Gan",
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"S. S. Gao",
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"X. Jin",
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"Chi Wang",
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"published": "20170626160540",
"title": "The DArk Matter Particle Explorer mission"
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K. J. Genestreti,1J. L. Burch,2 P. A. Cassak, 3 R. B. Torbert,4,2R. E. Ergun, 5,6 A. Varsani, 1T. D. Phan, 7 B. L. Giles, 8C. T. Russell, 9 S. Wang, 10M. Akhavan-Tafti, 11 R. C. Allen 12,2 1Space Research Institute, Austrian Academy of Sciences, Graz, Austria. 2Space Science and Engineering Division, Southwest Research Institute, San Antonio, TX, USA. 3West Virginia University, Morgantown, WV, USA. 4Space Science Center, University of New Hampshire, Durham, NH, USA. 5Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, USA. 6Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA. 7Space Science Institute, University of California Berkeley, Berkeley, CA, USA. 8Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD, USA. 9University of California, Los Angeles, CA, USA 10University of Maryland, College Park, MD, USA 11Climate and Space Sciences and Engineering Department, University of Michigan, Ann Arbor, MI, USA. 12Department of Physics and Astronomy, University of Texas San Antonio, San Antonio, TX, USA. * Determined location where J⃗·E⃗'>0 for 11 asymmetric EDRs with different guide fields. * Increasing guide field strength appears to move J⃗·E⃗'>0 from electron-crescent to X-point. * Guide field allows electron streaming at X-point, which takes work by the electric field. We compare case studies of Magnetospheric Multiscale (MMS)-observed magnetopause electron diffusion regions (EDRs) to determine how the rate of work done by the electric field, J⃗·(E⃗+v⃗_e×B⃗)≡J⃗·E⃗' varies with shear angle. We analyze MMS-observed EDR event with a guide field approximately the same size as the magnetosheath reconnecting field, which occurred on 8 December 2015. We find that J⃗·E⃗' was largest and positive near the magnetic field reversal point, though patchy lower-amplitude J⃗·E⃗' also occurred on the magnetosphere-side EDR near the electron-crescent point. The current associated with the large J⃗·E⃗' near the X-point was carried by electrons with a velocity distribution function (VDF) resembling the magnetosheath inflow, shifted in the -v_∥ direction. At the magnetosphere-side EDR, the current was carried by electrons with a crescent-like VDF. We compare this 8 December event to 10 other EDRs with different guide field strengths. The dual-region J⃗·E⃗' was observed in three other moderate-shear EDR events, whereas three high-shear events had a strong positive J⃗·E⃗' near the electron-crescent point and one low-shear event had a strong positive J⃗·E⃗' only near the B_L=0 point. The dual-region J⃗·E⃗'>0 was seen for one of three “intermediate"-shear EDRs with guide fields of ∼0.2–0.3. We propose a physical relationship between the shear angle and mode of energy conversion where (a) a guide field provides an efficient mechanism for carrying a current at the field reversal point (streaming) and (b) a guide field may limit the formation of crescent eVDFs, limiting the current carried near the stagnation point.§ INTRODUCTION Magnetic reconnection is a fundamental process in plasmas. It is a change in the topology of a magnetized plasma boundary coupled with the exchange of energy from magnetic fields to particles. The topological change occurs in the electron diffusion region (EDR), wherein the electrons are demagnetized, i.e., E⃗+v⃗_e×B⃗≠0. The per-volume rate of work done by the electric field on the plasma, which is often expressed in the electron rest frame as J⃗·(E⃗+v⃗_e×B⃗)≡J⃗·E⃗' <cit.>, occurs in a region sometimes called the `dissipation region' in order to distinguish it from the EDR <cit.>. J⃗·E⃗' specifically represents the rate of work done on the plasma by non-ideal electric fields. Because the E⃗+v⃗_e×B⃗≠0 is a defining condition for both the EDR and the J⃗·E⃗' region, the two regions may partially overlap; however, observations <cit.> and simulations <cit.> show that significant J⃗·E⃗' may occur several electron inertial lengths away from the magnetic X-point, where the magnetic topology changes.Reconnection at the low-latitude magnetopause of Earth is typically asymmetric, as the plasma density in the magnetosheath can exceed the magnetospheric plasma density by an order of magnitude <cit.>. This density asymmetry alters the momentum balance equation in the vicinity of the EDR and causes the electron flow stagnation point, where there is no bulk electron motion, to be displaced from the X-point, where the magnetic field in the reconnection plane is a minimum <cit.>. Guide field or component reconnection occurs when the local shear angle between the magnetosheath and magnetospheric magnetic fields is less than 180^∘. The presence of a guide field causes the magnetic field strength at the X-point to be non-zero, which can magnetize electrons near the X-point and reduce the size of the electron gyroradius relative to the size of the current layer <cit.>.Observations of asymmetric and nearly anti-parallel reconnection have showed that field-to-plasma energy conversion and parallel electron heating occur Earthward of the X-point <cit.>. In the central (asymmetric and anti-parallel) EDR, the current associated with J⃗·E⃗' is carried by electrons with broad crescent-shaped velocity distribution functions (VDFs) that separate in the parallel and perpendicular directions near the stagnation point <cit.>. In the outer EDR, where the Hall magnetic field is observed but electron kinetic motion still allows for non-zero E⃗', parallel crescents carry the current associated with J⃗·E⃗' <cit.>. The electrons may be demagnetized <cit.> at the X-point, but the out-of-plane current there, a result of electron cusp motion <cit.>, is generally weak.Little work has been done to determine how and why the location of the J⃗·E⃗' region may change with the magnetic shear angle. <cit.> compared particle-in-cell simulations of reconnection with B_M=0 and B_M=B_L,sh and found J_∥ E_∥ at the X-point to be larger for the guide field case, though a physical explanation for this difference was not discussed. <cit.> showed that electron-crescent VDFs also appeared near the electron stagnation point in a simulation of guide field (B_M∼ B_L,sh) reconnection. The intensity of the crescent relative to the core of the VDF was significantly reduced in intensity as compared to their similar simulation of anti-parallel reconnection <cit.>. According to <cit.>, crescent VDFs should reduce in intensity and eventually disappear as the guide field intensity increases to the point where the magnetic scale length B_L/(∂ B_L/∂ N) exceeds the electron Larmor radius, preventing mixing of electrons by thermal motion between regions with significantly different magnetic field directions.In this study, we analyze the 8 December 2015 (11:20 UT) EDR event of <cit.>. We determine the electron-frame energy conversion rate and analyze the eVDFs associated with the current. We find that three of the four MMS spacecraft observed strong J⃗·E⃗'>0 near the X-point associated with E_∥-accelerated magnetosheath-inflow electrons. The eVDF is structured, with a higher-energy beam-like portion anti-aligned with the parallel electric field and a low-energy crescent-like portion. All of the four spacecraft also observed smaller positive J⃗·E⃗' Earthward of the X-point and strong negative J⃗·E⃗' where the high-energy beam-like portion of the eVDF wraps from the parallel direction into the perpendicular (E⃗×B⃗) direction. The location of the J⃗·E⃗'>0 region for this event is then compared with 10 other EDR events with different guide field strengths. For three high-shear (B_M/B_L,sh≈0) events, the J⃗·E⃗'>0 region was near the electron-crescent point, Earthward of the B_L=0 point. For one of three “intermediate"-shear (B_M/B_L,sh≈0.2) cases, J⃗·E⃗'>0 was observed at both the B_L=0 and electron-crescent points, whereas J⃗·E⃗'>0 only occurred at the electron-crescent points for the remaining two cases. For three moderate-shear (0.5≤ B_M/B_L,sh≤1) events, J⃗·E⃗'>0 was observed at both the B_L=0 and electron-crescent points, similar to the 8 December 2015 event. Lastly, for the single low-shear (B_M/B_L,sh≈ 3) case, J⃗·E⃗'>0 was observed only at the B_L=0 point and no clear electron crescents were observed.We suggest that, based on these observations, the strength of the guide field may be a crucial factor in determining where electric fields convert their energy during asymmetric reconnection. The absolute and relative locations of the X, electron stagnation, and maximum J⃗·E⃗' points depend on a number of additional factors (see our companion study of Cassak et al. [submitted; this Volume]), including but not limited to the degree of asymmetry between the upstream plasma number densities, the strengths of the reconnecting component of the magnetic field, and the ion and electron temperatures. The absolute distances between the X, electron stagnation, and maximum J⃗·E⃗' points will also depend on the distance from its center where the EDR is observed, as well as the path of the spacecraft through the EDR. We do not attempt to control for each of variables individually, as a full investigation of this parameter space is beyond the scope of this study. However, we note that similar features in J⃗·E⃗' were observed for the few (≤4) events within each category of high, moderate, and low shear, despite significant differences in other upstream conditions.In the following section we describe the MMS instrumentation and data analysis techniques used in this study. In section 3 we present our case analysis of the 8 December 2015 EDR event. In section 4 we compare and the 8 December 2015 EDR event against 10 others with differing guide field strengths and upstream parameters. In section 5 we summarize our findings and outline topics that warrant future investigation. Further discussion may also be found in our companion study, Cassak et al. [submitted; this Volume], which presents simulations of three of the events studied here, as well as a discussion of what may govern the J⃗·E⃗'>0 location for 2-d steady-state reconnection.§ INSTRUMENTATION AND DATA This study analyzes burst-mode data from the suite of plasma particle and field instruments onboard MMS <cit.>. The fast plasma investigation (FPI) dual ion and electron spectrometers (DIS and DES, respectively) measure differential directional fluxes for their namesake particle species at 32 energies between ∼10 eV/q and ∼28 keV/q <cit.>. FPI-DIS and DES measure 4π-steradian velocity distribution functions (VDFs) once every accumulation period, 150 ms for the ions and 30 ms for the electrons. The DC magnetic field vector is provided by the fluxgate magnetometers (FGM) at 128 samples per second <cit.>. The AC magnetic field vector is provided by the search coil magnetometers (SCM) at a rate of 8196 samples per second <cit.>, as are the spin-plane <cit.> and axial <cit.> components of the electric field, which are measured by two sets of probes collectively referred to as the electric field double probes (EDP). All of the data used in this study are available through the MMS science data center (https://lasp.colorado.edu/mms/sdc/public/), with the exception of the level 3 (L3) EDP data used during the analysis of the 8 December event, which are available by request.Some of the data are resampled and/or smoothed prior to analysis. We shifted all of the FPI data forward by half of an acquisition period (+0.075 seconds for DIS and +0.015 seconds for DES), such that the times associated with each data point mark the average, rather than the beginning, of the associated acquisition period. It is unnecessary to perform a similar shift for the fields data, since the time stamps are already centered on a significantly smaller measurement period. We have also smoothed the AC electric field data using a sliding overlapping boxcar scheme, where the width of the boxcar (±15 ms) was chosen to match the sample rate of FPI-DES and provide the best possible agreement between E⃗ and -v⃗_e×B⃗. Smoothing the electric field reduces the magnitude of positive and negative oscillations of J⃗·E⃗', but makes the bulk action of the electric field on the plasma more easily discernible.All data is shown in either magnetopause-normal (LMN) or field-aligned (FAC) coordinates. The LMN eigenvector system for the 8 December 2015 event is taken from <cit.>, which was determined using minimum variance and minimization of Faraday residue <cit.>. The coordinate axes, L̂, M̂, and N̂ are constant in time and defined in a GSE basis as [0.3641, –0.1870, 0.9124], [–0.2780, –0.9568, 0.08515], and [0.8889, –0.2226, –0.4003], respectively. L̂ is the direction of maximum magnetic variance and the reconnection outflow. M̂ is the direction of intermediate variance and the guide and Hall magnetic fields. N̂ is the direction of minimum variance, the magnetopause normal, and the reconnection inflow. For FAC, the coordinate axes are calculated for each magnetic field measurement and are defined as v̂_∥, v̂_1, and v̂_2, which are defined as b̂, (b̂×v̂_e)×b̂, and b̂×v̂_e, respectively, where b̂ is the direction of the magnetic field and v̂_e is the direction of the electron bulk velocity.§ ANALYSIS OF THE 8 DECEMBER 2015 (11:20 UT) EDR EVENT An overview of MMS data for the 8 December 2015 event is provided in <cit.>. As a brief review,identified a ∼2-second EDR encounter at 11:20:43–45 UT, which was observed by all four MMS spacecraft. The average spacecraft separation was 15 km or roughly 8 electron inertial lengths d_e,sh, given the upstream magnetosheath density of ∼8 cm^-3. Several seconds before the spacecraft passed through the EDR, effectively moving from the magnetosheath to the magnetosphere, the density in the upstream magnetosheath was approximately 2.5 times the density in the upstream magnetosphere. There was an asymptotic out-of-plane guide magnetic field B_M approximately the same size as the magnetosheath reconnecting field B_L,sh. <cit.> noted that the out-of-plane current J_M was bifurcated, peaking strongly near the B_L=0 point and several tens of d_e Earthward of the B_L=0 point. This bifurcated current differed significantly from the antiparallel reconnection event of <cit.>, on 16 October 2015, which only had a peak in J_M at the electron-crescent point. Additionally, <cit.> found that the electrons were highly anisotropic, where the parallel temperature exceeded the perpendicular temperature, at both the B_L=0 and electron-crescent points. The antiparallel event of 16 October had parallel heating only at (and Earthward of) the electron-crescent point, where the electron-frame field-to-plasma energy conversion rate J⃗·E⃗'>0 was similarly peaked. <cit.> did not calculate J⃗·E⃗' for the 8 December event, which is calculated here and shown in Figure <ref>.We have shifted the data from MMS2–4 in Figure <ref> such that the first large J⃗·E⃗'>0 peaks from the four spacecraft are artificially aligned in time. This organizes some reconnection-related data, primarily near the B_L=0 point, but does not organize all of the data. Vertical dashed lines t_1 and t_3 on Figure <ref>(a–f) mark the two peaks of the bifurcated out-of-plane current J_M as measured by MMS2. The separation of the two peaks are well resolved by the MMS data, as approximately 20 eVDF measurements are made between t_1 and t_2-t_3. Overall, the data in Figure <ref> show that the J⃗·E⃗' region was highly structured and located primarily at and Earthward of the B_L=0 point. Three of the four spacecraft (MMS2–4) observed strong positive J⃗·E⃗'≈10 nW/m^3 near the B_L=0 point. For MMS2, this first J⃗·E⃗'>0 peak was sunward of the B_L=0 point, while for MMS3 and 4 the first J⃗·E⃗'>0 peak was Earthward of the B_L=0 point. At t_1, the electrons were strongly anisotropic and agyrotropic. In asymmetric reconnection, agyrotropy is expected when the Larmor motion of the electrons allows for mixing between the high and low-density inflow regions in the vicinity of the EDR. The large agyrotropy seen here indicates that the considerable guide field of B_M/B_L,sh≈1 is not sufficient to fully magnetize the electrons and prevent this mixing. For comparison, the largest value of √(Q_e) shown here, which has values ranging from 0 (fully gyrotropic) to 1 (fully agyrotropic), is approximately the same as its largest value for the nearly anti-parallel 16 October event of <cit.>.Patchy, lower-amplitude, and mostly positive J⃗·E⃗' was also observed by all four spacecraft between t_1 and t_3, several tens of electron inertial lengths Earthward of the B_L=0 point. (Given a magnetopause normal velocity of –44 km/s and a magnetosheath electron inertial length of d_e,sh=1.9 km <cit.>, the spacecraft effectively move ∼2.3 d_e every 100 milliseconds). Large negative excursions of J⃗·E⃗' were also observed, though, like the patchy positive J⃗·E⃗', these peaks are not well-organized by the time shifting done to organize the J⃗·E⃗'>0 peaks at t_1. The large temperature anisotropy, which was first observed near t_1, extends Earthward of the B_L=0 point, up to and beyond t_3. The strong electron agyrotropy occurs mostly between t_1 and t_3, as does the strong out-of-plane current J_M. As with the out-of-plane current, the normal electric field is bifurcated. For fully anti-parallel reconnection, E_N is expected to have a small shoulder at the X-point and a much stronger peak Earthward of the X-point near the electron stagnation point <cit.>.Figure <ref>(h–p) shows selected cuts of eVDFs measured by MMS2. Panels (h–j) show eVDF cuts taken at t_1, at the center of the first out-of-plane current peak near the B_L=0 point, where J⃗·E⃗' is strong positive. A more complete set of distribution functions was presented in <cit.>. Figure <ref>(h–j) shows that the eVDF associated with the current near the B_L=0 point is highly structured, with a beam-like counter-streaming portion at higher energies that is partially balanced by a lower-energy crescent-like portion of the eVDF at lower energies. The parallel motion at this point is almost entirely in the out-of-plane direction due to the presence of the strong guide field. In the picture of <cit.>, the higher-energy portion of this eVDF should be meandering sheath electrons that have been already entered the EDR, been accelerated by the normal electric field, then meander back to the X-point. The lower-energy portion of the eVDF then should be the newly inflowing sheath electrons that have not yet been accelerated.Panels (n–p) show eVDFs at t_3, the second of the two out-of-plane current peaks, where J⃗·E⃗' is negative. At the second of the two current peaks, between t_2 and t_3, the high-energy beam-like portion of the eVDF persists (panels (k–m)), then wraps from the parallel direction into the v_1 direction (panels (n–p)). The lower-energy portion of the eVDF does not appear, which is consistent with the idea that this lower-energy portion were sheath electrons that had not yet undergone acceleration by the large E_N. The wrapping of this beam-like portion of the eVDF from the parallel to perpendicular directions is similar to the wrapping of crescent-shaped eVDFs in high-shear EDRs <cit.>. Several studies have found EDRs with J⃗·E⃗'<0 <cit.>. This may occur as a result of time-dependent evolution, such as current sheet thinning <cit.>, or as a result of processes that may occur in a steady-state, such as the breaking of super-Alfvenic electron jets in the outer EDR. In the simplest terms, J⃗·E⃗'<0 is a conversion of plasma energy to electromagnetic energy in the reference frame of the electrons. In our companion paper, this J⃗·E⃗'<0 did not appear during a steady-state period of a 2.5-d particle-in-cell simulation of this event, which may imply that the J⃗·E⃗'<0 here was either a result of time-dependent or 3-d processes. The exact cause (and effect) in this particular case, though, is beyond the scope of this current investigation.In summary, the 8 December 2015 (B_M/B_L,sh∼1) EDR event had the following characteristics: * J⃗·E⃗' was strongly positive at or very near the B_L=0 point, where a strong out-of-plane current was carried by counter-streaming electrons moving against the local magnetic field direction (≈M̂, * Patchy positive and negative J⃗·E⃗' Earthward of B_L=0, where the mostly anti-field-aligned, higher-energy portion of the eVDF wrapped from the parallel direction to the perpendicular direction * The electrons were anisotropic over a broad region extending from the B_L=0 point to far Earthward of the J⃗·E⃗' region * The electrons were agyrotropic over a narrow region, roughly coinciding with the J⃗·E⃗' region § ANALYSIS OF J⃗·E⃗' FOR EDRS WITH DIFFERING SHEARS§.§ Overview of event list Here we determine the energy conversion rate J⃗·E⃗' for 10 additional EDRs, all of which have been identified in previous studies. For many of these events, e.g., the high-shear EDR of <cit.> and the low-shear EDR of <cit.>, the energy conversion rate, electron dynamics, and larger-scale context have already been studied extensively. For other events, including some of those identified by <cit.> and <cit.>, the energy conversion rate has not been calculated in any previous study to the knowledge of the authors. The set of events is presented in Table 1. EDR events and upstream conditions, sorted into four categories based on the strength of the guide field.DateB_M/B_L,sh B_L,sh/B_L,sp n_sh/n_sp T_e,sh/T_e,sp T_i,sh/T_i,sp X,Y,Z [R_E] Reference High2015-10-16/13:07 0.1 0.6 160.20.28.3, 8.5, -0.7 <cit.> shear2015-09-19/09:10 0.1 0.615 0.5 0.2 6.4, 7.7, 0.02 <cit.>2015-12-11/12:16 0.150.4 101 0.4 9.3, 1.8, -0.9<cit.> “Intermediate" 2015-12-06/23:380.21400.07 0.28.5, -4.0, -0.6 <cit.> shear 2015-12-08/00:060.20.4 20 0.3 0.49.0, -3.9, -0.6<cit.>2016-01-10/09:130.30.5 6 0.7 0.88.8, -2.4, -0.8<cit.>Moderate2016-11-28/07:360.5 0.5300.40.110.0, 3.1, -3.2shear2015-11-11/12:35 10.8121 0.76.6, -1.7, -0.1 <cit.>2015-12-08/11:20^(a)10.42.5 0.50.410.2, 1.6, -1.0<cit.> 2015-12-14/01:1710.510 0.30.5 10.1, -4.3, -0.8 <cit.> Low shear 2015-09-08/11:015 0.52.50.3 0.3 4.9, 9.2, 0.1 <cit.> (a)Presented in section 3. The EDR events listed in Table 1 were selected from a larger set of reconnection events on the following basis: first, the EDR must have been observed during a full crossing of the magnetopause, such that (a) the upstream conditions could be determined immediately before and after the crossing and (b) the energy conversion rate J⃗·E⃗' could be calculated at both the B_L=0 and electron-crescent points. Second, the path of the spacecraft through the magnetopause, judging by the profile of B_L, should be reasonably simple, e.g., we exclude events where the spacecraft passes through a portion of the EDR, doubles back, then crosses again. Lastly, we excluded events for which we were unable to obtain a stable LMN coordinate system in which the upstream conditions (namely B_M and B_L) had some reasonably constant-in-time asymptotic value. For the most part, this last criterion is a repetition of our previous criteria, as it mostly excluded partial or complex crossings of the magnetopause. This list of events is far from exhaustive and is nearly entirely biased towards the first of the two MMS magnetopause phases due to the current (at the time of writing) availability of phase 1a surveys <cit.>.The events listed in Table 1 have a diverse set of upstream conditions, with the density asymmetry ranging from ∼2.5 to ∼40, magnetic field B_L asymmetry ranging from ∼1 (no asymmetry) to ∼0.4, and asymmetric electron (ion) temperatures ranging from 1 to less than 0.1 (0.7 to 0.1). For this investigation, we are primarily concerned with the strength of the guide field relative to the reconnecting component of the magnetosheath field B_M/B_L,sh, and any potential impact of these additional parameters on the location of the J⃗·E⃗' is not controlled for (see Section 5).As in Table 1, we separate these events into four categories; the three “High-shear" events have B_M/B_L,sh<0.2, the three “Intermediate-shear" events have B_M/B_L,sh≈0.2, the four “Moderate-shear" events have B_M/B_L,sh≈0.5-1, and the single “Low-shear" event has a B_M ∼5 times larger than B_L,sh. The upstream conditions were determined several seconds before and after the EDR crossing, such that they are as close as possible to the upstream conditions during the EDR crossing. §.§ High-shear events Figure <ref> shows J⃗·E⃗' for the three high-shear EDR events of (Fig <ref>(a)–(c)) <cit.>, (Fig <ref>(d)–(e)) <cit.>, and <cit.>, all of which have guide fields approximately 10–15% as large as B_L,sh. The vertical dashed blue line marks the B_L=0 point where, for all three cases, there is no significant J⃗·E⃗'>0. For the 19 September event, there is some J⃗·E⃗'<0 at/near the field reversal point, which is discussed in <cit.>. In all three cases, both the J⃗·E⃗'>0 peaks and the parallel heating of electrons occurred on the magnetospheric side of the B_L=0 point at the electron-crescent point, which is marked in Figure <ref> with a vertical dashed red line. In the case of the 16 October event, which is thought to have been an observation very near the center of an EDR, both perpendicular (Fig <ref>j) and parallel crescents (Fig <ref>k) were observed Earthward of the B_L=0 point, while the electrons at the B_L=0 point were largely isotropic and gyrotropic. The 19 September and 11 December cases were likely outside the central EDR, as only parallel crescents were observed. This is consistent with the Hall deflections of B_M that were seen during these two EDR encounters, which indicated that the spacecraft passed some distance along L from from the point of symmetry.The separation between the J⃗·E⃗' peaks and the B_L=0 points were well-resolved for all three of these events due to the very high time resolution of MMS measurements. For the 16 October event, ∼10 eVDFs were collected between the B_L=0 point and the J⃗·E⃗' peak; ∼20 eVDFs were collected between these points for the 19 September event, and ∼110 eVDFs were collected between these points for the 11 December event. §.§ Moderate-shear events Figure <ref> shows J⃗·E⃗' for three moderate-shear EDR events, which had guide fields 50–100% as large as the magnetosheath B_L. The event shown in the middle column of Figure <ref> was identified by <cit.> and the event shown in the right-most column was studied by <cit.>. The left-most column has not been identified yet as an EDR to the knowledge of the authors. The locations of the J⃗·E⃗'>0 peaks for these three events are qualitatively similar to that of the 8 December (11:20 UT) event, in that J⃗·E⃗'>0 and the parallel heating of electrons occur at both the B_L=0 and electron-crescent points. For all three cases, the current Earthward of the B_L=0 point was carried by electrons with parallel crescent-shaped VDFs. In the case of the 14 December event, highly agyrotropic perpendicular crescent eVDFs were also observed. For the 28 and 11 November and events, the current at the B_L=0 was carried by electrons with VDFs similar to the anisotropic magnetosheath inflow, but shifted in the local -v_∥ direction, against the guide field in +M. For the 14 December event, the current at the B_L=0 point had parallel and perpendicular components and the VDFs were not clearly organized by the local magnetic field coordinates. In all three cases, the electrons are broadly anisotropic around both the B_L=0 and crescent points, though for the 11 November and 14 December cases, it is difficult to determine if this anisotropy is a result of local heating or is an extension of the anisotropy generated in the upstream magnetosheath inflow region.For the 11 November case, as for the 8 December (11:20 UT) case, the crescent-shaped portion of the eVDF is not as intense, relative to the background plasma, as it was for the very high-shear events. This is not the case for the B_M/B_L,sh∼0.5 event of 28 November or for the B_M/B_L,sh∼1 event of 14 December, both of which had pronounced crescent-shaped eVDFs. The electron agyrotropy, as defined by the Swisdak parameter √(Q_e) <cit.>, was nearly twice as large at the B_L=0 point than at the electron-crescent point for all three B_M/B_L,sh=1 events (including the 8 December (11:20 UT) event). For the 28 November event, which had B_M/B_L,sh=0.5, the agyrotropy was equally as strong at both points. For the high-shear events, the agyrotropy was sharply peaked at the electron-crescent points alone. There was no significant difference between the maximum values of √(Q_e) for the high and moderate-shear EDR cases, as the differences between events in a given shear category were comparable to the differences between events in different categories. This may be due to the small sample size and the large spread in upstream parameters within each category. §.§ Low-shear eventFor the low-shear EDR event of <cit.>, which is shown in Figure <ref>, no clear crescent-shaped eVDFs were observed, meaning that we cannot locate the J⃗·E⃗'>0 peak relative to any magnetosphere-side landmark. However, J⃗·E⃗' was sharply positively peaked only at the B_L=0 point. No second peak or secondary structure was observed. This single sharp J⃗·E⃗'>0 peak was seen by both of the two spacecraft that observed the 8 September EDR <cit.>. There was no significant agyrotropy, which is consistent with the lack of crescent-shaped eVDFs. <cit.> pointed out that the electron gyroradius was smaller than the magnetic scale size, which should prohibit crescent formation <cit.>. §.§ “Intermediate”-shear events Figure <ref> shows data from the remaining three EDR events considered in this study, which we have categorized as having “intermediate” magnetic shear given that the events have B_M/B_L,sh∼20–30%, falling between our high (B_M/B_L,sh≤0.1) and moderate-shear categories. Two of the three events exhibit essentially the same characteristics as the high-shear events, with J⃗·E⃗'>0 and parallel electron heating only at the electron-crescent point and no activity at the B_L=0 point. The third intermediate-shear event, which occurred on 8 December 2015 (∼11 hours before the 11:20 UT EDR event of <cit.>), had J⃗·E⃗'>0 both at and Earthward of the B_L=0 point. For this third event, there is clear evidence of parallel heating Earthward of the B_L=0 point at the second and largest J⃗·E⃗'>0 peak, but no significant anisotropy at or near the B_L=0 point. Again, the separation between the B_L=0 point and the Earthward-side J⃗·E⃗'>0 peaks were well resolved for all three events. § SUMMARY AND CONCLUSIONS We analyzed the electron-frame energy conversion rate J⃗·(E⃗+v⃗_e×B⃗)≡J⃗·E⃗' that occurred during the intermediate-shear (B_M/B_L,sh∼1) 8 December 2015 EDR event of <cit.>. We found that the J⃗·E⃗' region was highly structured, with J⃗·E⃗'≈10 nW/m^3 near the field reversal point, lower amplitude and patchy |J⃗·E⃗'|≤5 nW/m^3 Earthward of the X-point, and strong J⃗·E⃗'≈–10 nW/m^3 near the Earthward edge of the J⃗·E⃗' region. The strong positive J⃗·E⃗' was associated with a current carried by a counter-streaming beam-like portion of the eVDF, which was partially balanced by a lower-energy parallel magnetosheath-inflow-like portion of the eVDF. The strong negative J⃗·E⃗' was associated with the turning of this beam into the v̂_1 direction.We calculated J⃗·E⃗' for 10 other previously published EDR events with differing guide field strengths. For three nearly anti-parallel events, J⃗·E⃗'>0 and parallel electron heating were only observed Earthward of the B_L=0 point at the electron-crescent point. Two of three “intermediate”-shear EDRs had these same characteristics. For one of the three “intermediate”-shear EDRs, as well as for three moderate-shear EDRs (not including the 8 December event of <cit.>), J⃗·E⃗'>0 was observed at both the B_L=0 and electron-crescent points. For some but not all of these dual-region J⃗·E⃗'>0 events, the intensity of the crescent-shaped portion of the eVDF was considerably reduced as compared to the anti-parallel events. Lastly, for one low-shear EDR, J⃗·E⃗'>0 was only observed at the B_L=0 point and no crescent-shaped eVDFs were detected. §.§ Interpretation: influence of shear angle on energy conversion From this collection of cases it appears that the introduction of a guide field enhances J⃗·E⃗' at the X-point, which is similar to a result of <cit.>, as discussed in the first section of this paper. The possible mechanism for this switching is easily explained. The introduction of a guide field causes the magnetic field at the X-point to be non-zero, allowing for free streaming of the electrons along the guide field due to the out-of-plane reconnection electric field. This is consistent with the eVDFs near the B_L=0 points from the intermediate-to-low-shear EDR events of 28 November 2016, 8 December 2015, and 8 September 2015, which had guide fields 0.5, 1, and 8 times as large as B_L,sh, respectively. For these events, the current near the X-point and at the J⃗·E⃗' peak was carried by electrons with magnetosheath-inflow-like VDFs shifted in the +v_M direction. There was significant structure to the eVDF near the X-point for the 8 December (11:20 UT) event, which did not appear in the eVDFs for the other two aforementioned events. This structured eVDF may be a result of the larger electric field for this 8 December event, which was roughly 10 and 4.3 times the size of the parallel electric field for the 28 November and 8 September events, respectively. There is also an apparent trend, based on these few events, where increasing the guide field reduces J⃗·E⃗' near the electron-crescent point. In the 28 November event, with B_M∼0.5B_L, strong parallel crescents were observed. In the 8 December (11:20 UT) event, with B_M∼ B_L, weaker parallel and perpendicular crescents were observed <cit.>. No crescents were observed for the 8 September event <cit.>. Increasing the strength of the guide field may reduce the intensity of the crescents, which are a result of finite gyroradius effects <cit.>, by reducing the ratio of the gyroradius to the skin depth. It should be noted, however, that the upstream conditions were not uniform for these events. Table 1 lists some of the dimensionless parameters for asymmetric reconnection. As discussed in Section 1, the displacement of the flow stagnation point from the X-point depends on, among other factors, the density asymmetry n_sh/n_sph and the magnetic field asymmetry B_L,sh/B_L,sph. The density asymmetries ranged from 2.5 to 40, but there is no obvious correlation with the location of the J⃗·E⃗' region with this parameter. For example, both the low-shear event of <cit.> (J⃗·E⃗'>0 at X-point) and the moderate-shear event of <cit.> (dual-region J⃗·E⃗'>0) had density asymmetries of 2.5, on the lowest end of this parameter range, and the locations of J⃗·E⃗'>0 differed significantly. The 28 November event, which had a guide field of ∼0.5 and a density asymmetry of 30, on the highest end of the parameter range, also had dual-region J⃗·E⃗'>0 similar to the event of <cit.>. Similar comparisons can be made with the asymmetries of B_L, T_e, and T_i, where several events with similar parameter values can be found in different shear categories with different J⃗·E⃗'>0 locations. Given that the locations of J⃗·E⃗'>0 are well organized by the strength of the guide field (as compared to organization by any other single parameter), we suggest that the strength of the guide field plays a dominant role in controlling the location of J⃗·E⃗'>0 for asymmetric reconnection.The three “intermediate”-shear events had similar guide field strengths of B_M/B_L,sh∼0.2, yet only one of the three had dual-region J⃗·E⃗'>0. The other two events in this category had J⃗·E⃗'>0 only at the electron-crescent point, similar to the high-shear events. The differences between events in this category may indicate that there are other factors beyond the strength of the guide field that control the location of the J⃗·E⃗'>0 region and/or that our determination of the upstream conditions was inexact. This approximate value of B_M/B_L,sh∼0.2 may also be unique, as it is thought to be in this range that symmetric reconnection transitions between anti-parallel-like and component-like <cit.>. §.§ Future work This study was based on a small set of events. A more comprehensive analysis and characterization of EDR events with varying guide field strengths should be conducted to confirm or refute the interpretation provided in the previous section. Additional low-shear EDRs should be identified and/or included in this analysis. As of yet, to the knowledge of the authors, there has only been one very low-shear EDR event identified in the MMS data. This limited number of events may be explained if (a) the guide field suppresses crescent formation and (b) crescents are being used to identify EDRs. Another question is related to strength of the density asymmetry. The density asymmetries varied between these events and, as discussed previously, separation of the X-point and the stagnation points and crescent formation are both consequences of asymmetric reconnection. This question may be most easily addressed with simulations, where all of the parameters for a reconnection event may be pre-defined.We have largely introduced the differences between J⃗·E⃗' for low and high-shear reconnection in a phenomenological manner, so there are many open questions related to the underlying physics that should create these differences. The mechanism for electron acceleration near the X-point during component reconnection and the parameters that govern the separation between J⃗·E⃗' peaks for intermediate-shear reconnection are both unknown. It is also unknown why, despite having similar upstream conditions, J⃗·E⃗' was an order of magnitude smaller for the 28 November event than for the 8 December and 9 September events. The authors would like to thank everyone who contributed to the success of the MMS mission and those who contributed to the rich scientific heritage on which this mission is based. This work was funded by the Austrian Academy of Sciences FFG project number 847069. Kevin Genestreti would like to thank Rumi Nakamura, Yasuhito Narita, Stephen Fuselier, and Jerry Goldstein for helpful conversations. The MMS data are publicly available at https://lasp.colorado.edu/mms/sdc/public except for the level 3 EDP data, which is currently only available via special request from the instrument team. 30urlstyle[Burch and Phan(2016)]BurchandPhan.2016 Burch, J. L., and T. D. Phan (2016), Magnetic reconnection at the dayside magnetopause: Advances with MMS, Geophys. Res. 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Giles, D. Gershman, and S. Wang (2017a), Electron-scale quadrants of the hall magnetic field observed by the magnetospheric multiscale spacecraft during asymmetric reconnection, Phys. Rev. Lett., 118, 175,101, 10.1103/PhysRevLett.118.175101. [Zenitani et al.(2011)Zenitani, Hesse, Klimas, and Kuznetsova]Zenitani.2011 Zenitani, S., M. Hesse, A. Klimas, and M. Kuznetsova (2011), New Measure of the Dissipation Region in Collisionless Magnetic Reconnection, Physical Review Letters, 106(19), 195003, 10.1103/PhysRevLett.106.195003. | http://arxiv.org/abs/1706.08404v2 | {
"authors": [
"Kevin Genestreti",
"Jim Burch",
"Paul Cassak",
"Roy Torbert",
"Bob Ergun",
"Ali Varsani",
"Tai Phan",
"Barbara Giles",
"Chris Russell",
"Shan Wang",
"Mojtaba Akhavan-Tafti",
"Robert Allen"
],
"categories": [
"physics.space-ph"
],
"primary_category": "physics.space-ph",
"published": "20170626143531",
"title": "The effect of a guide field on local energy conversion during asymmetric magnetic reconnection: MMS observations"
} |
[ [ December 30, 2023 =====================A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges.A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the non-loop edges of H. Hell andgavea complete characterisation of the complexity of deciding whether there is a homomorphism from an input graph G to a fixed graph H. A complete characterisation is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterisation of the complexity of counting homomorphisms from an input graph G to a fixed graph H. In this paper, we give a complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H and we also give a complete characterisation of the complexity of counting compactions from an input graph G to a fixed graph H. In an addendum we use our characterisations to point out a dichotomy for the complexity of the respective approximate counting problems (in the connected case).§ INTRODUCTIONA homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. That is, the function maps every edge of G to an edge of H. Many structures in graphs, such as proper colourings, independent sets, and generalisations of these, can be represented as homomorphisms, so the study of graph homomorphisms has a long history incombinatorics <cit.>.Much of the work on this problem is algorithmic in nature. A very important early work is Hell and 's paper <cit.>, which gives a complete characterisation of the complexity of the following decision problem, parameterised by a fixed graph H: “Given an input graph G, determine whether there is a homomorphism from G to H.” Hell andshowed that this problem can be solved in polynomial time if H has a loop or is loop-free and bipartite. They showed that it is NP-complete otherwise. An important generalisation of the homomorphism decision problem is the list-homomorphism decision problem. Here, in addition to the graph G, the input specifies, for each vertex v of G, a list S_v of permissible vertices of H. The problem is to determine whether there is a homomorphism from G to H that maps each vertex v of G to a vertex in S_v. Feder, Hell and Huang <cit.> gave a complete characterisation of the complexity of this problem.This problem can be solved in polynomial time if H is a so-called bi-arc graph,and it is NP-complete otherwise.More recent work has restricted attention to homomorphisms with certain properties. A function from V(G) to V(H) is surjective if every element of V(H) isthe image of at least one element of V(G). So a homomorphism from G to H is surjective if every vertex of H is “used” by the homomorphism. There is still no complete characterisationof the complexity of determining whether there is a surjective homomorphism from an input graph G to a graph H, despite an impressive collection of results<cit.>. A homomorphism from V(G) to V(H) is a compaction if it uses every vertex of H and also every non-loop edge of H (so it is surjective both on V(H) and onthe non-loop edges in E(H)). Compactions have been studied under the name “homomorphic image” <cit.> and even under the name “surjective homomorphism” <cit.>. Once again, despite much work <cit.>, there is still no characterisation of the complexity of determining whether there is a compaction from an input graph G to a graph H. Dyer and Greenhill <cit.> initiated the algorithmic study of counting homomorphisms. They gave a complete characterisation of the graph homomorphism counting problem, parameterised by a fixed graph H: “Given an input graph G, determine how many homomorphisms there are from G to H.” Dyer and Greenhill showed that this problem can be solved in polynomial time if every component of H is a clique with all loops present or a biclique (complete bipartite graph) with no loops present. Otherwise, the counting problem is -complete. , Serna and Thilikos <cit.> and Hell and <cit.> have shown that the same dichotomy characterisation holds forthe problem of countinglist homomorphisms.The main contribution of this paper is to give complete dichotomy characterisations for the problems of counting compactions and surjective homomorphisms. Our main theorem, Theorem <ref>, shows that the characterisation for compactions is different from the characterisation for counting homomorphisms. If every component of H is (i) a star with no loops present, (ii) a single vertex with a loop, or (iii) a single edge with two loops then counting compactions to H is solvable in polynomial time. Otherwise, it is -complete. We also obtain the same dichotomy for the problem of counting list compactions. Thus, even though the decision problem is still open for compactions,our theorem gives a complete classification of the complexity of the correspondingcounting problem.There is evidence that computational problems involvingsurjective homomorphisms are more difficult than those involving (unrestricted) homomorphisms.For example, suppose that H consists of a 3-vertex clique with no loops together with a single looped vertex. As <cit.> noted, the problem of deciding whether there is a homomorphism from a loop-free input graph G to H is trivial (the answer is yes, since all vertices of G may be mapped to the loop) butthe problem of determining whether there is a surjective homomorphism from a loop-free input graph G to H is NP-complete. (To see this, recallthe NP-hard problem of determining whether a connected loop-free graph G'that is not bipartite is 3-colourable. Given such a graph G', we may determine whether it is 3-colourable by letting G consist of the disjoint union of G' and a loop-freeclique of size 4, and then checking whether there is a surjective homomorphism from G to H.) There is also evidence that counting problems involving surjective homomorphisms are more difficult than those involving unrestricted homomorphisms. In Section <ref>we consider a uniform homomorphism-counting problemwhere all connected components of G arecliques without loops and all connected components of H arecliques with loops, but both G and H are part of the input. It turns out (Theorem <ref>) that in this uniform case, counting homomorphisms is inbut counting surjective homomorphisms is -complete. Despite this evidence,we show (Theorem <ref>) that the problem of counting surjective homomorphisms to a fixed graph H has the same complexity characterisationas the problem of counting all homomorphisms to H: The problem is solvable in polynomial time if every component of H is aclique with loops or abiclique without loops. Otherwise,it is -complete.Once again, our dichotomy characterisation extends to the problem of counting surjective list homomorphisms. Even though the decision problem is still open for surjective homomorphisms, our theorem gives a complete complexity classification of the corresponding counting problem.In Section <ref> we will introduce one more related counting problem — the problem of counting retractions. Informally, if G is a graph containing an induced copy of H then a retraction from G to H is a homomorphism from G to H that maps the induced copy to itself. Retractions are well-studied in combinatorics, often from an algorithmic perspective <cit.>. A complexity classification is not known for the decision problem (determining whether there is a retraction from an input to H). Nevertheless, it is easy to give a complexity characterisation for the corresponding counting problem (Corollary <ref>). This characterisation, together with our main results, implies that a long-standing conjecture of Winkler about the complexity of the decision problems for compactions and retractions is false in the counting setting. See Section <ref> for details.Finally, in an addendum to this work, we address the relaxed versionsof the counting problems where the goal is to approximately count surjective homomorphisms, compactions and retractions. We use our theorems to give a complexity dichotomy in the connected case for all three of these problems. §.§ Notation and Theorem Statements In this paper graphs are undirected and may contain loops. A homomorphism from a graph G toagraph H is a function hV(G) → V(H)such that, for all {u,v}∈ E(G), the image {h(u),h(v)} is in E(H). We use GH to denote the number of homomorphisms from G to H. A homomorphism h is said to “use” a vertex v∈ V(H) if there is a vertex u∈ V(G) such that h(u)=v. It is surjective if it uses every vertex of H. We use GH to denote the number of surjective homomorphisms from G to H. A homomorphism h is said to use an edge {v_1,v_2}∈ E(H) if there is an edge {u_1,u_2}∈ E(G) such that h(u_1)=v_1 and h(u_2)=v_2. It is a compaction if it uses every vertex of H and every non-loop edge of H. We use GH to denote the number of compactions from G to H. H is said to bereflexive if every vertexhas a loop. It is said to be irreflexive if no vertex has a loop. We study the following computational problems[The reason that the input graph G is restricted to be irreflexive in these problems,but that H is not restricted, is that this is the convention in the literature. Since our results will be complexity classifications, parameterised by H, we strengthen the results by avoiding restrictions on H. Different conventions are possible regarding G,but hardness results are typically the most difficult part of the complexity classifications in this area, so restricting G leads to technically-stronger results.], which are parameterised by a graph H. Name: H. Name: H. Name:H.[-0.05cm] Input:Irreflexive graph G. Input: Irreflexive graph G. Input:Irreflexive graph G. [-0.05cm] Output: GH. Output:GH.Output:GH. [-0.05cm] A list homomorphism generalises a homomorphism in the same way that a list colouring of a graph generalises a (proper) colouring. Suppose that G is an irreflexive graph and that H is a graph. Consider a collection of sets ={S_v⊆ V(H) :v∈ V(G)} A list homomorphism from (G,) to H is a homomorphism h from G to H such that, for every vertex v of G, h(v)∈ S_v.The set S_v is referred to as a “list”, specifying the allowable targets of vertex v. We use (G,)H to denote the number of list homomorphisms from (G,) to H, (G,)H to denote the number of surjective list homomorphisms from (G,) to H and (G,)H to denote the number of list homomorphisms from (G,) to Hthat are compactions. We study the following additional computational problems, again parameterised by a graph H. Name: H. Name: H. Name:H. [-0.05cm] Input:Irreflexive graph G Input: Irreflexive graph G Input:Irreflexive graph G [-0.05cm] and a collection of lists and a collection of lists and a collection of lists [-0.05cm] ={S_v⊆ V(H) :v∈ V(G)}. ={S_v⊆ V(H) :v∈ V(G)}. ={S_v⊆ V(H) :v∈ V(G)}. [-0.05cm] Output: (G,)H. Output:(G,)H.Output:(G,)H. [-0.05cm]In order to state our theorems, we define some classes of graphs. A graph H is a clique if, for every pair (u,v) of distinct vertices,E(H) contains the edge {u,v}.(Like other graphs, cliques may contain loops but not all loops need to be present.) H is a biclique if it is bipartite (disregarding any loops) and there is a partition of V(H) into two disjoint sets U and V such that, for every u∈ U and v∈ V, E(H) contains the edge {u,v}. A biclique is a star if |U|=1 or |V|=1 (or both). Note that a star mayhave only one vertex since, for example, we could have |U|=1 and |V|=0. We sometimes use the notation K_a,b to denote an irreflexive biclique whose vertices can be partitioned into U and V with |U|=a and |V|=b. The size of a graph is the number of vertices that it has. We can now state the theorem of Dyer and Greenhill <cit.>, as extended to list homomorphisms by , Serna and Thilikos <cit.> and Hell and <cit.>. Let H be a graph. If every connected component of H isa reflexiveclique or an irreflexivebiclique,then H and H are in .Otherwise, H and H are -complete.We can also state the main results of this paper. The tractability results in Theorem <ref> follow from the fact that the number of compactions from agraph G to a graph H can be expressed as a linear combination of the number of homomorphisms from G to certain subgraphs of H, see Section <ref>. A proof sketch of the intractability result in Theorem <ref> is given at the beginning of Section <ref>. Theorem <ref> is simpler, see Section <ref>. §.§ Reductions and Retractions In the context of two computational problems P_1 and P_2, we write P_1≤P_2 if there exists a polynomial-time Turing reduction from P_1 to P_2. If there exist such reductions in both directions, we write P_1≡P_2.Theorems <ref>, <ref> and <ref> imply the following observation.Let H be a graph. ThenH≡H≡H≡H≤H≡H.In order to see how Observation <ref> contrasts withthe situation concerning decision problems, it is useful to define decisionversions of the computational problems that we study. Thus, H is the problem of determining whether GH=0, given an input G of H. The decision problems H, H and H are defined similarly.It is also useful to define the notion of a retraction. Suppose that H is a graphwith V(H)= {v_1,…,v_c} and that G is an irreflexive graph. We say that a tuple (u_1,…,u_c) of c distinct vertices of G induces a copy of H if, for every1 ≤ a < b ≤ c,{u_a,u_b}∈ E(G) ⟺{v_a,v_b}∈ E(H). A retractionfrom (G;u_1,…,u_c) to H is a homomorphism h from G to H such that, for all i∈[c], h(u_i)=v_i. We use (G;u_1,…,u_c)H to denote the number of retractions from (G;u_1,…,u_c) to H. We briefly considerthe retraction counting and decision problems, which are parameterised by a graph H with V(H)={v_1,…,v_c}.[Once again, some works would allow G to have loops, and would insist that loops are preserved in the induced copy of H. We prefer to stick with the convention that G is irreflexive, but this does not make a difference to the complexity classifications that we describe.] Name: H. Name: H. [-0.05cm] Input:Irreflexive graph G and a tupleInput: Irreflexive graph Gand a tuple [-0.05cm] (u_1,…,u_c) of distinct vertices of G (u_1,…,u_c) of distinct vertices of G[-0.05cm] that induces a copy of H. that induces a copy of H. [-0.05cm] Output: (G;u_1,…,u_c)H. Output:Does (G;u_1,…,u_c)H=0?[-0.05cm] The following observation appears as Proposition 1 of <cit.>. The proposition is stated for more general structures than graphs, but it applies equally to our setting. Let H be a graph. ThenH≤H≤H≤H≤H.We have already mentioned the fact (pointed out by Bodirsky et al.) that if H isan irreflexive 3-vertex cliquetogether with a single looped vertex, thenH is in P, butH is NP-complete. There are no known graphs H separating H, H and H. Moreover, Bodirsky et al. mention a conjecture <cit.>, attributed to Peter Winkler, that, for all graphs H, H and Hare polynomially Turing equivalent.The following observation, together with our theorems, implies Corollary <ref> (below),which shows that the generalisation of Winkler's conjecture to the counting setting is false unless =, since H and H are not polynomially Turing equivalent for all H.Let H be a graph. Then H≤H and H≤H Let V(H)={v_1,…,v_c}. We first reduce H to H. Consider an input to H consisting ofG and (u_1,…,u_c). For each a∈[c], let S_u_a be the set containingthe single vertex v_a. For each v ∈ V(G)∖{u_1,…,u_c}, let S_v = V(H). Let ={S_v:v∈ V(G)}. Then (G;u_1,…,u_c)H = (G,)H.We next reduce H to H. Let E^0 be the set of all non-loop edges of H. Consider an input G to H. Suppose without loss of generality that V(G) is disjoint from V(H)={v_1,…,v_c}.Let G' be the graph with vertex set V(G) ∪ V(H) and edge set E(G) ∪ E^0. Then (v_1,…,v_c) induces a copy of H in G' and GH = (G';v_1,…,v_c)H. Observation <ref> immediately implies the following dichotomy characterisation for the problem of counting retractions. Let H be a graph. If every connected component of H isa reflexiveclique or an irreflexivebiclique, then His in .Otherwise, His -complete. The corollary follows immediately from Observation <ref> and Theorem <ref>.Let H be a graph. ThenH≡H≡H≡H≡H≤H≡H.Furthermore, there is a graph H for which H and H are -complete, but H, H, H, H and H are in . Theorems <ref>, <ref>, <ref> andCorollary <ref> give complexity classifications for all of the problems. The reductions in the corollary follow from three easy observations. * All problems inare trivially inter-reducible.* All-complete problems are inter-reducible. * All problems inare reducible to all -complete problems.The separating graph H can be taken to be any reflexive clique of size at least 3 or any irreflexive biclique that is not a star.§.§ Related Work This section was added after the announcement of our results (<https://arxiv.org/abs/1706.08786v1>), in order todraw attentionto some interesting subsequent work <cit.>.Both our tractability results and our hardness results rely on the fact(see Theorem <ref>) that the number of compactions from G to H can be expressed as a linear combination of the number of homomorphisms from G tocertain subgraphs J of H. A similar statement applies to surjective homomorphisms.As we note in the paper, these kinds of linear combinations have beennoticedin related contexts before, for example in <cit.> and in <cit.>. We usethe linear combination of Theorem <ref>,together with interpolation, to prove hardness. Although it is standard to restrict the input graph G to be irreflexive (and this restriction makes the results stronger) the fact that G is required to be irreflexive causes severe difficulties. In fact, Dell's note about our paper <cit.>shows that,if you weaken the theorem statements by allowing the input G to have loops, then a simpler interpolation based on a veryrecent paperby Curticapean, Dell and Marx <cit.>can be used to make the proofs very elegant!The exact same idea, writtenmore generally, was also discovered by Chen <cit.>. § PRELIMINARIESIt will often be technically convenient to restrict the problems that we study by requiring the input graph G to be connected. In each case, we do this by adding a superscript “C” to the name of the problem. For example, the problem H is defined as follows. H.A connected irreflexive graph G.GH. It is well known and easy to see (See, e.g., <cit.>) that if G is an irreflexive graph withcomponents G_1,…,G_tthen GH = ∏_i∈[t]G_iH. Similarly, given ={S_v⊆ V(H) : v∈ V(G)} let _i={S_v : v∈ V(G_i)}. Then (G,)H= ∏_i∈[t](G_i,_i)H. Thus, Dyer and Greenhill's theorem (Theorem <ref>) can be re-stated in the following convenient form. Let H be a graph.If every connected component of H is a reflexiveclique or an irreflexivebiclique,then H, H, H and Hare all in.Otherwise,H, H, H and H areall -complete. Finally, we introduce some frequently used notation.For every positive integer n, we define [n]= {1,…,n}. A subgraph H' of H is said to be loop-hereditary with respect to H if for every v∈ V(H') that is contained in a loop in E(H), v is also contained in a loop in E(H'). We indicate that two graphs G_1 and G_2 are isomorphic bywritingG_1≅ G_2.Given sets S_1 and S_2, we write S_1 ⊕ S_2 for the disjoint union of S_1 and S_2. Given graphs G_1 and G_2, we write G_1 ⊕ G_2 for the graph (V(G_1)⊕ V(G_2), E(G_1) ⊕ E(G_2)). If V is a set of verticesthen we writeG_1 ⊕ V as shorthand for the graph G_1 ⊕ (V,∅). Similarly, if M is a matching (a set of disjoint edges) with vertex set V, then we write G_1 ⊕ M as shorthand for the graph G_1 ⊕ (V,M).§ COUNTING COMPACTIONS The section is divided into a short subsection on tractable cases and the main subsection on hardness results which also containsthe proof of the final dichotomy result, Theorem <ref>. §.§ Tractability Results The tractability result in Lemma <ref>follows from the fact (see Theorem <ref>) that the number of compactions from G to H can be expressed as a linear combination of the number of homomorphisms from G tocertain subgraphs J of H. While we need the full details of our particular linear expansion toderive our hardness results, the following simpler versionsuffices for tractability. Let H be a graph such that every connected component is an irreflexive star or a reflexive clique of size at most 2. Then H and H are in . First we deal with the case that H is the empty graph. Suppose that H isthe empty graph and let (G,) be an instance of H.IfG is empty then (G,)H=1. Otherwise,(G,)H=0. Thus, if H is empty, then H is in .Obviously, this also implies that H is in . Letbe the set of all non-empty graphs in which every connected component is an irreflexive star or a reflexive clique of size at most 2. We will showthat for every H∈,H is in . To do this, we need the following notation. Given a graph H, let m(H) denote the sum of |V(H)| and the number of non-loop edges of H. We will use induction on m(H).The base case is m(H)=1. In this case, H has only one vertex w. If G is non-empty and has w∈ S_v for every vertex v∈ V(G) then (G,)H=1. Otherwise,(G,)H=0. So H is in .For the inductive step, consider some H∈ with m(H)>1. Let (G,) be an instance of H. If G is empty then (G,)H=0, so suppose that G is non-empty. For every subgraph H' of H let _H' denote the set of lists _H' = {S_v ∩ V(H') : v ∈ V(G)}. Itis easy to see that(G,)H = ∑_H'(G,_H')H', where the sum is over all loop-hereditary subgraphs H' of H. This observation is well known and is implicit, e.g, in the proof of a lemma of Borgs, Chayes, Kahn and <cit.> (in a context without lists or loops).A subgraph H' of H is said to be a proper subgraph of H if eitherV(H') is a strict subset of V(H) or E(H') is a strict subset of E(H) (or both).For every graph H, let Sub^<(H) denote the set ofnon-empty proper subgraphs of H that are loop-hereditary with respect to H.Note that if H∈ and H' ∈ Sub^<(H) then H'∈ and m(H')<m(H). We can refine the summation as follows. (G,)H = (G, )H +∑_H' ∈ Sub^<(H)(G,_H')H'.Since H∈, every component of H isa reflexive clique or an irreflexive biclique, so Theorem <ref> shows that thequantity (G,)H on the left-hand side can be computed in polynomial time. By induction, every term of the form (G,_H')H' can also be computed in polynomial time. Subtracting this from the left-hand side, we obtain (G, )H, as desired. Thus, we have proved that H is in . The problem H is a restriction of H, so it is also in . §.§ Hardness ResultsThis is the key section of this work.In this section, we consider a graph H that has a connected component that is not an irreflexive star or a reflexive clique of size at most 2. The objective is to show that H and H are -hard (this is the hardnesscontent of Theorem <ref>).We start with a brief proof sketch. Theeasy case is when H contains a component that is not a reflexive clique or an irreflexive biclique. In this case, Dyer and Greenhill's Theorem <ref> shows that H is -hard. We obtain the desired hardness by giving (in Theorem <ref>) a polynomial-time Turing reduction from H to H. The result is finished off with a trivial reduction from H to H. The proof of Theorem <ref> is not difficult — given an input G to H, we add isolated vertices and edges to G and recover the desired quantity GH using an oracle for Hand polynomial interpolation. There are small technical issues related to size-1 components in H, and these are dealt with in Lemma <ref>.The more interesting case is when every component of H is a reflexive clique or an irreflexive biclique, but some component is either a reflexive clique of size at least 3 or an irreflexive biclique that is not a star. The first milestone is Lemma <ref>, which shows -hardness in the special case where H is connected. We prove Lemma <ref> in a slightly stronger setting where the input graph G is connected. This allows us, in the remainder of the section, to generalise the connected case to the case in which H is not connected.The main difficulty, then, is Lemma <ref>. The goal is to show that H is -hard whenH is a reflexive clique of size at least 3 or an irreflexive biclique that is not a star.Our main method for solving this problem is a technique (Theorem <ref>) that lets us compute the number of compactions from a connected graph G to a connected graph Husing a weighted sum of homomorphism counts, say GJ_1,…,GJ_k.An important feature is that some of the weights might be negative.Our basic approach will be to find a constituent J_i such that J_i is -hard and to reduce J_i to the problem of computing the weighted sum. Of course, if computing GJ_1 is -hard andcomputing GJ_2 is -hard, it does not follow that computing a weighted sum of these is -hard.In order to solve this problem, in Lemmas <ref>and <ref> we use an argument similar to that of Lovász <cit.>to prove the existence of input instances that help us to distinguish between the problemsJ_1, …, J_k. Theorem <ref>then provides the desired reduction fromachosen J_i to the problem of computing the weighted sum. Theorem <ref> is proved by a more complicated interpolation construction, in whichwe use the instances from Lemma <ref> to modify the input. Having sketched the proof at a high level, we are now ready to begin. We start by workingtowards the proof of Theorem <ref>. The first step is to show that deletingsize-1 components from H does not add any complexity to H. Let H be a graph that has exactly qsize-1 components. Let H' be the graphconstructed from H by removing allsize-1 components. Then H'≤H. Let W={w_1, …, w_q}be the vertices of H that are contained in size-1 components. We can assume q≥ 1, otherwise H'=H. Let G' be an input to H' and note that G' might contain isolated vertices. For any non-negative integer t, let V_t be a set of t isolated vertices, distinct from the vertices of G', and let G_t = G' ⊕ V_t. For all i∈{0,…, t}, we define S^i(G') to be the number of homomorphisms σ from G' to H with the following properties: * σ uses all non-loop edges of H'.* σ(V(G'))∩{w_1, …, w_q} = i,where σ(V(G')) is theimage of V(G') under the map σ.We define N^i(V_t) as the number of homomorphisms τ from V_t to H such that {w_1, …, w_i}⊆τ(V(V_t)).Intuitively, N^i(V_t) is the number of homomorphisms from V_t to H that use at least a set of i arbitrary but fixed vertices of H, as the particular choice of vertices {w_1, …, w_i} is not important when counting homomorphisms from a set of isolated vertices. For any compaction γ V(G_t) → V(H), the restriction γ|_V(G) has to use all non-loop edges in H'. As H' does not have size-1 components, this implies that all vertices other than w_1, …, w_q are used by γ|_V(G). Say, additionally, that γ uses q-i vertices from W, for some i∈{0,…,q}. Then, γ|_V_t has to use the remaining i vertices. Thus, for each fixed t≥ 0, we obtain a linear equation:G_tH_b_t = ∑_i=0^q S^q-i(G')_x_iN^i(V_t)_a_t,i. By choosing q+1 different values for the parameter t we obtain a system of linear equations. Here, we choose t=0,… ,q. Then the system is of the form = for = [ b_0; ⋮; b_q ]= [ a_0,0 … a_0,q; ⋮ ⋱ ⋮; a_q,0 … a_q,q; ]and= [ x_0; ⋮; x_q ].Note, that the vectorcan be computed using q+1 H oracle calls. Further,x_q = S^0(G') = G'H'.Thus, determiningis sufficient for computing the sought-for G'H'. It remains to show that the matrixis of full rank and is therefore invertible.If t<i, we observe that a_t,i = 0 as we cannot use at least i vertices of H when we have fewer than i vertices in the domain. For the diagonal elements with t∈{0,…,q} we have that a_t,t= N^t(V_t)=t! (note that 0!=1). Hence, = [ 0!0⋯0;∗ 1!⋱⋮;⋮⋱⋱0;∗⋯∗ q! ]is a triangular matrix with non-zero diagonal entries, which completes the proof.Let H be a graphwithout any size-1 components. Then H≤H. The proof is by interpolation and is somewhat similar to the proof of Lemma <ref>. Let G be an input to H. We design a graph G_t= G⊕ I_t as an input to the problem H by adding a set I_t of t disjoint new edges to the graph G.We introduce some notation. Let E^0(H) be the set of non-loop edges of H and let r= E^0(H). Let S^k(G) be the number of homomorphisms σ from G to H that use exactly k of the non-loop edges of H (additionally, σ might use any number of loops). Let {e_1, …, e_k} be a set of k arbitrary but fixed non-loop edges from H. We define N^k(I_t) as the number of homomorphisms τ from I_t to H such that {e_1, …, e_k} are amongst the edges used by τ. Note that the particular choice of edges {e_1, …, e_k} is not important when counting homomorphisms from an independent set of edges to H—N^k(I_t) only depends on the numbers k and t.We observe that, for each compaction γ V(G_t) → V(H), the restriction γ|_V(G) uses some set F⊆ E^0(H) of non-loop edges and does not use any other non-loop edges of H. Suppose that F has cardinality F=r-k for some k∈{0,…, r}. Then γ|_V(I_t) uses at least the remaining k fixed non-loop edges of H. As H does not have anysize-1 components, this ensures at the same time that γ is surjective.Therefore, we obtain the following linear equation for a fixed t≥ 0:G_tH_b_t =∑_k=0^r S^r-k(G)_x_kN^k(I_t)_a_t,k. As in the proof of Lemma <ref>, we choose t=0,… ,r to obtain a system of linear equations with= [ b_0; ⋮; b_r ]= [ a_0,0 … a_0,r; ⋮ ⋱ ⋮; a_r,0 … a_r,r; ]and= [ x_0; ⋮; x_r ].We can computeusing a H oracle. Further, ∑_k=0^r x_k = ∑_k=0^r S^r-k(G)= ∑_k=0^r S^k(G) = GH.Thus, determiningis sufficient for computing the sought-for number of homomorphisms GH. Finally, we show thatis invertible. If t<k, we observe that a_t,k = N^k(I_t) = 0, as clearly it is impossible to use more than t edges of H when there are only t edges in I_t. Further, for the diagonal elements it holds that for t∈ [r] we have a_t,t= N^t(I_t)=2^tt! as there are t! possibilities for assigning the edges in I_t to the fixed set of t edges of H and there are 2^t vertex mappings for each such assignment of edges, also N^0(I_0)=1. Hence, = [ 1 0 ⋯ 0; ∗ 2^11! ⋱ ⋮; ⋮ ⋱ ⋱ 0; ∗ ⋯ ∗ 2^rr! ]is a triangular matrix with non-zero diagonal entries and is therefore invertible.Let H be a graph. Then H≤H. Let H' be the graph constructed from H by removing allsize-1 components. By Lemma <ref> we obtain H'≤H. Then Lemma <ref> can be applied to the graph H' and thus we obtain H'≤H'≤H. Finally, it follows from Theorem <ref> that H'≡H, which gives H≡H'≤H'≤H. Theorem <ref> shows that hardness resultsfrom Theorem <ref> will carry over from H to H.We also know somecases where H is tractable from Lemma <ref>.The complexity of H is still unresolved if every component of H is a reflexive clique or an irreflexive biclique, but some reflexive clique has size greater than 2, or some irreflexive biclique is not a star.This is the case described at length at the beginning of the section. Recall that the first stepis to specifya technique (Theorem <ref>) that lets us compute the number of compactions from a connected graph G to a connected graph Husing a weighted sum of homomorphism counts, say GJ_1,…,GJ_k.Towards this end,we introduce some definitions which we will use repeatedly in the remainder of this section. A weighted graph set is a tuple (,λ), whereis a set of non-empty, pairwise non-isomorphic, connected graphs and λ is a function λ→.Let H be a connected graph. By Sub(H) we denote the set of non-empty, loop-hereditary, connected subgraphs of H. Let _H be a set which contains exactly one representative of each isomorphism class of the graphs in Sub(H). Finally, for H' ∈_H, we define μ_H(H') to be the number of graphs in Sub(H) that are isomorphic to H'. Note that for a connected graph H, we have μ_H(H)=1.For each non-empty connected graph H, we define a weight function λ_H which assigns an integer weight to each non-empty connected graph J.* If J is not isomorphic to any graph in _H,then λ_H(J)=0. * If J≅ H,then λ_H(J)=1.* Finally, if J is isomorphic to some graph in _H but J H, we define λ_H(J) inductivelyas follows.λ_H(J)=-∑_H'∈_Hs.t. H' Hμ_H(H')λ_H'(J). Note that λ_H is well-defined as all graphs H'∈_H with H' H are smaller than H either in the sense of having fewer vertices or in the sense of having the same number of vertices but fewer edges. The following theorem is the key to our approach forcomputing the number of compactions from a connected graph G to a connected graph H using a weighted sum of homomorphism counts. Inthe Appendix, we give an illustrative example where we verify the theorem for the case H=K_2,3 andwe give the intuition behind the definitions.Here we go on to give the formal statement and proof. Let H be a non-empty connected graph. Then for every non-empty, irreflexive and connected graph G we have GH = ∑_J∈_Hλ_H(J)GJ.Let H_1, H_2, … be the set of non-empty connected graphs sorted by some fixed ordering that ensures that if H_i is isomorphic to a subgraph of H_j, then i≤ j.We verify the statement of the theorem by induction over the graph index with respect to this ordering. Let G be non-empty, irreflexive and connected.For the base case, H_1 is K_1, which is the graph with one vertex and no edges. In this case, _H_1={K_1} and λ_K_1(K_1)=1. Also GK_1 = GK_1.So the theorem holds in this case. Now assume that the statement holds for all graphs up to index i and consider the graph H_i+1. For ease of notation we set H=H_i+1. We use the fact that every homomorphism from a connected graph G to H_i+1 is a compaction onto some non-empty, loop-hereditary and connected subgraph of H_i+1 and vice versa. Thus, it holds thatGH = ∑_H'∈_Hμ_H(H')·GH'= GH+ ∑_H'∈_Hs.t. H' Hμ_H(H')·GH'.Thus, we can rearrange and use the induction hypothesis to obtainGH = GH - ∑_H'∈_Hs.t. H' Hμ_H(H')·GH'= GH - ∑_H'∈_Hs.t. H' Hμ_H(H') ·∑_J∈_H'λ_H'(J)GJ. Then we change the order of summation and use that λ_H'(J)=0 if J is not isomorphic to any graph in _H' to collect all coefficients that belong to a particular term GJ. We obtainGH =GH - ∑_J∈_Hs.t. J H(∑_H'∈_Hs.t. H' Hμ_H(H')λ_H'(J))GJ= ∑_J∈_Hλ_H(J)GJ.We remark that Theorem <ref> can be generalised to graphs H and G with multiple connected components by looking at all subgraphs of H, rather than just at the connected ones. However, within this work, the version for connected graphs suffices.Let (,λ) be a weighted graph set. The following parameterised problem is not natural in its own right,but it helps us to analyse the complexity of H: (,λ).An irreflexive, connected graph G.Z_,λ(G)=0if G is empty ∑_J∈λ(J)GJ otherwise.Let H be a non-empty connected graph. ThenH≡(_H,λ_H).The corollary follows directly from Theorem <ref>. Corollary <ref> gives us the desired connection betweenweighted graph sets and compactions.We will use this later in the proof of Lemma <ref> to establish the -hardness of H when H is either a reflexive clique of size at least 3 or an irreflexive biclique that is not a star.Our next goal is to prove Theorem <ref>, which states that,for certain weighted graph sets (,λ),determining Z_,λ(G) is at least as hard as computingGJ for some graph J from the setwith λ(J)≠ 0.To this end, we first introduce two lemmas that help us to distinguish betweendifferent graphs J in the interpolationthat we will later use to prove Theorem <ref>.For the following lemmas, we introduce some new notation. For a graph G with distinguished vertex v∈ V(G) and a graph H with distinguished vertex w∈ V(H), the quantity (G,v)(H,w) denotes the number of homomorphisms h from G to H with h(v)=w. Analogously, (G,v)(H,w) denotes the number of injective homomorphisms h from G to H with h(v)=w.If there exists an isomorphism from G to H that maps v onto w, we write (G,v) ≅ (H,w), otherwise we write (G,v)(H,w). In the following lemma, we show that for two such target entities (H_1 ,w_1) and (H_2, w_2) that are non-isomorphic, there exists an input which separates them. To this end, we use an argument very similar to that presented in <cit.> and in the textbook by Hell and Nešetřil <cit.>, which goes back to the works of Lovász <cit.>.Let H_1 and H_2 be connected graphs with distinguished vertices w_1∈ V(H_1) and w_2∈ V(H_2) such that (H_1 ,w_1)(H_2, w_2).Suppose that one of the following cases holds:Case Case 1. H_1 and H_2 are reflexive graphs.Case 2. H_1 and H_2 are irreflexive bipartite graphs, each of which contains at least one edge.Then * There exists a connected irreflexive graph G with distinguished vertex v∈ V(G) for which (G,v)(H_1,w_1)≠(G,v)(H_2,w_2).* In Case 2 we can assume that G contains at least one edge and is bipartite. In order to shorten the proof, we define some notation that depends on which case holds. In Case 1, we say that a tuple (G,v) consisting of a graph G with distinguished vertex v is relevant if G is connected and reflexive. In Case 2, we say that it is relevant if G is connected, irreflexive and bipartite and contains at least one edge. We start with a claim that applies in either case.Claim: There exists a relevant (G,v) such that(G,v)(H_1,w_1)≠(G,v)(H_2,w_2).Proof of the claim:To prove the claim, assume for a contradiction that for all relevant (G,v) we have(G,v)(H_1,w_1) = (G,v)(H_2,w_2).The contradiction will follow from the following subclaim:Subclaim: For every relevant (G,v), (G,v)(H_1,w_1) = (G,v)(H_2,w_2).Proof of the subclaim:The proof of the subclaim is by induction on the number of vertices of G. For the base case of the induction we treat the two cases separately. In Case 1, the base case of the induction is V(G)=1. The relevant (G,v) is the graph consisting of the single (looped) vertex v. For every reflexive graph H and vertex w∈ V(H) we have that (G,v)(H,w) = (G,v)(H,w).Therefore, (<ref>) implies that the subclaim is true for this (G,v). In Case 2, the base case of the induction is V(G)=2. (There are no relevant (G,v) with V(G)<2 since G has to contain an edge.)Consider a relevant (H,w). Since H is irreflexive and the two vertices of G are connected by an edge (so cannot be mapped by a homomorphism to the same vertex of H) we have(G,v)(H,w) = (G,v)(H,w). Once again, (<ref>) implies that the subclaim is true for this (G,v). For the inductive step, suppose that the subclaim holds for all relevant (G,v)in which G has up to k-1 vertices. Consider a relevant (G,v) with V(G)=k. Let Θ be the set of partitions of V(G) — that is, each θ∈Θ is a set {U_1,…, U_j} for some integer j such that the elements of θ are non-empty and pairwise disjoint subsets of V(G) with ⋃_i=1^j U_i = V(G). For θ∈Θ with θ={U_1,…, U_j}, by G|_θ we denote the corresponding quotient graph, i.e. let G|_θ be the graph with vertices {U_1,…, U_j} that has an edge {U_i,U_i'} if and only if there exist v∈ U_i and u∈ U_i' with {v,u}∈ E(G). Therefore, G|_θ might have loops but no multi-edges, see Figure <ref>. Let v_θ denote the vertex of G|_θ which corresponds to the equivalence class of θ that contains the distinguished vertex v. Finally, let τ denote the partition of V(G) into singletons. Then for every relevant (H,w) it holds that(G,v)(H,w) = ∑_θ∈Θ(G|_θ,v_θ)(H,w)=(G|_τ,v_τ)(H,w) +∑_θ∈Θ∖{τ}(G|_θ,v_θ)(H,w)=(G,v)(H,w) +∑_θ∈Θ∖{τ}(G|_θ,v_θ)(H,w),where the third equality follows as G|_τ = G.Now we show that only relevant tuples (G|_θ,v_θ) actually contribute to the sum in (<ref>). First, note that since G is connected, so is G|_θ.In Case 1, every quotient graph G|_θ is reflexive. Therefore, for every θ∈Θ∖{τ}, the tuple (G|_θ,v_θ) is relevant.In Case 2, H is an irreflexive bipartite graph with at least one edge. Therefore, we have (G|_θ,v_θ)(H,w)>0 only if G|_θ is an irreflexive bipartite graph and also, θ is a proper vertex-colouring of G, i.e. every part of θ is an independent set. For such a partition θ, G|_θ has at least one edge if G does. We have now shown that only relevant tuples (G|_θ,v_θ) contribute to the sum in (<ref>).Therefore, let Γ be the set of all partitions θ of V(G) such that (G|_θ,v_θ) is relevant. Then, we can rephrase (<ref>) as follows.(G,v)(H,w) =(G,v)(H,w) +∑_θ∈Γ∖{τ}(G|_θ,v_θ)(H,w).To prove the subclaim, we can set (H,w) in (<ref>) to be (H_1,w_1). Similarly, we can set it to be (H_2,w_2). Then, we can use the induction hypothesis, the subclaim, on all tuples (G|_θ,v_θ) in the sum as all these tuples are relevant and the partitions θ∈Γ∖{τ} have strictly fewer than k parts. Applying (<ref>), we obtain(G,v)(H_1,w_1) = (G,v)(H_2,w_2),which completes the induction and the proof of the subclaim.(End of the proof of the subclaim.) We show next how to use the subclaimto derive a contradiction. In particular, in the subclaim we can set (G,v) to be either (H_1,w_1) or (H_2,w_2). This implies (H_1 ,w_1) ≅ (H_2, w_2), which gives the desired contradiction. Thus, we have shown contrary to (<ref>) that there exists a relevant (G,v) with(G,v)(H_1,w_1)≠(G,v)(H_2,w_2)and therefore we have proved the claim. (End of the proof of the claim.) In Case 2, the claim is identical to the statement of the lemma. However, in Case 1 a relevant tuple (G,v) contains a reflexive graph G, whereas for the statement of the lemma, G has to be irreflexive. This is easily fixed as we can set G^0 to be the graph constructed from G by removing all loops. Using the fact that H_1 and H_2 are reflexive, we obtain for i=1 and i=2 that(G^0,v)(H_i,w_i) = (G,v)(H_i,w_i).Hence, the choice (G^0,v) has all the desired properties. In the following lemma, we generalise the pairwise property from Lemma <ref>. The result and the proof are adapted versions of <cit.>. For ease of notation let [k]2 denote the set of all pairs {i,j} with i,j∈ [k] and i≠ j. Let H_1, …, H_k be connected graphs with distinguished vertices w_1, …, w_k where w_i∈ V(H_i) for all i∈ [k] and, for every pair {i,j}∈[k]2, we have (H_i ,w_i)(H_j, w_j).Suppose that one of the following cases holds:Case Case 1. ∀ i∈ [k], H_i is a reflexive graph.Case 2. ∀ i∈ [k], H_i is an irreflexive bipartite graph that contains at least one edge.Then * There exists a connected irreflexive graph G with a distinguished vertex v∈ V(G) such that, for every {i,j}∈[k]2, it holds that (G,v)(H_i,w_i)≠(G,v)(H_j,w_j).* In Case 2 we can assume that G contains at least one edge and is bipartite.Again, we use the notion of relevant tuples but slightly modify the definition from the one given in the proof of Lemma <ref>. A tuple (G,v) is called relevant if G is a connected irreflexive graph and, in Case 2, if additionally G contains at least one edge and is bipartite. We show that there exists a relevant (G,v) such that for every {i,j}∈[k]2 we have(G,v)(H_i,w_i)≠(G,v)(H_j,w_j). We use induction on k, which is the number of graphs H_1,…, H_k. The base case for k=2 is covered by Lemma <ref>. Now let us assume that the statement holds for k-1 and the inductive step is for k. By the inductive hypothesis there exists a relevant (G,v) such that without loss of generality (possibly by renaming the graphs H_1,…, H_k)(G,v)(H_2,w_2) >… > (G,v)(H_k,w_k).Let i^*∈[k]∖{1} be an index with (G,v)(H_1,w_1) = (G,v)(H_i^*,w_i^*).If no such index exists, we can simply choose the graph G which then fulfils the statement of the lemma. Using the base case, there exists a relevant (G',v') such that (G',v')(H_1,w_1) > (G',v')(H_i^*,w_i^*),possibly renaming (H_1,w_1) and (H_i^*,w_i^*).Let i ∈ [k]. First, we show that for all i∈[k] we have (G',v')(H_i,w_i)≥ 1. This is clearly true for Case 1, where w_i has a loop. In this case, we can always map all vertices of G' to the single vertex w_i. In Case 2, as H_i is connected and contains at least one edge, there is some w∈ V(H_i) such that {w,w_i}∈ E(H_i). Since (G',v') is relevant, G' is connected and bipartite and has at least one edge. Let {A,B} be a partition of V(G') such that v'∈ A and A and B are independent sets of G.There is a homomorphism h from G' to H_i with h(v')=w_i which maps all vertices in A to w_i and all vertices in B to w. Therefore, in both cases we have shown that for all i∈[k] we have (G',v')(H_i,w_i)≥ 1. For a yet to be determined number t we construct a graph G^* from (G,v) and (G',v') by taking the graph G' and t copies of G and identifying the vertex v' with the t copies of v and call the resulting vertex v^*, cf. Figure <ref>. Note that from the fact that (G,v) and (G',v') are relevant, it is straightforward to show that (G^*,v^*) is relevant as well. Then, for any graph H and w∈ V(H) it holds that(G^*,v^*)(H,w) = (G',v')(H,w)·(G,v)(H,w)^t.The goal is to choose t sufficiently large to achieve(G^*,v^*)(H_2,w_2) > … > (G^*,v^*)(H_i^*-1,w_i^*-1)> (G^*,v^*)(H_1,w_1)> (G^*,v^*)(H_i^*,w_i^*) > … > (G^*,v^*)(H_k,w_k).Accordingly, we define a permutation σ of the indices {1,…, k} that inserts index 1 between position i^*-1 and i^*. The domain of σ corresponds to the new indices to which we assign the former indices. To avoid confusion, we give the function table in Table <ref> Formally,σ(i)=i+1 if i≤i^*-21 if i= i^*-1iotherwise.Let M= (G,v)(H_2,w_2). As (G',v')(H_j,w_j)≥ 1 for all j ∈ [k], it is well-defined to setC= max_j∈ [k]∖{i^*-1}(G',v')(H_σ(j+1),w_σ(j+1))/(G',v')(H_σ(j),w_σ(j))and t= ⌈ CM ⌉. Let G^* be as defined above. For ease of notation, for j∈[k-1], we setξ(j)=(G^*,v^*)(H_σ(j),w_σ(j)/(G^*,v^*)(H_σ(j+1),w_σ(j+1).We want to show ξ(j)>1 for all j∈[k-1] to complete the proof.For j=i^*-1 we obtainξ(j)=(G^*,v^*)(H_σ(i^*-1),w_σ(i^*-1))/(G^*,v^*)(H_σ(i^*),w_σ(i^*))=(G^*,v^*)(H_1,w_1)/(G^*,v^*)(H_i^*,w_i^*)=(G',v')(H_1,w_1)/(G',v')(H_i^*,w_i^*)> 1.For j∈[k-1]∖{i^*-1} we haveξ(j)=(G^*,v^*)(H_σ(j),w_σ(j))/(G^*,v^*)(H_σ(j+1),w_σ(j+1))= (G',v')(H_σ(j),w_σ(j))·(G,v)(H_σ(j),w_σ(j))^t/(G',v')(H_σ(j+1),w_σ(j+1))·(G,v)(H_σ(j+1),w_σ(j+1))^t≥1/C((G,v)(H_σ(j),w_σ(j))/(G,v)(H_σ(j+1),w_σ(j+1)))^t.Since (G,v)(H_σ(j),w_σ(j))≥ 1 + (G,v)(H_σ(j+1),w_σ(j+1)) for j∈[k-1]∖{i^*-1} we haveξ(j) ≥1/C(1+1/(G,v)(H_σ(j+1),w_σ(j+1)))^t.Using (1+x)^t≥ 1+tx >tx for t≥ 1, x≥ 0 we obtainξ(j) >t/C·(G,v)(H_σ(j+1),w_σ(j+1)).Finally, we use that for all j∈[k-1]∖{i^*-1} we have(G,v)(H_2,w_2)> (G,v)(H_σ(j+1),w_σ(j+1))and concludeξ(j)> t/C·(G,v)(H_2,w_2)≥t/CM≥ 1.Thus, we have shown ξ(j) >1 as required, which completes the proof.In the following theorem, we use the separating instances that we obtain from Lemma <ref> for interpolation-based reductions to (, λ).Let (,λ) be a weighted graph set for which one of two cases holds:Case Case 1. All graphs inare reflexive.Case 2. All graphs inare irreflexive and bipartite.Then, for all H ∈ with λ(H)≠ 0, it holds that H≤(,λ). If, in Case 2,contains a graph without edges, i.e. a single-vertex graph K_1, let (', λ') be a weighted graph set constructed from (, λ) by removing the K_1 and its corresponding weight λ(K_1). As K_1 is inwe have (',λ')≤(,λ)and K_1≤(,λ).Therefore, for the remainder of this proof, we assume that every graph incontains at least one edge. Let ^≠0={H_1,…,H_k} be the set of graphs inthat are assigned non-zero weights by λ. Note that all graphs in ^≠0 are pairwise non-isomorphic, connected and non-empty by definition of a weighted graph set. Thus, for every pair {i,j}∈[k]2 and every w_i∈ V(H_i), w_j∈ V(H_j) we have (H_i ,w_i)(H_j, w_j). Now, for each graph H_i we collect the vertices which are in the same orbit of the automorphism group of H_i. Formally, for each i∈ [k] and w∈ V(H_i), let [w] be the orbit of w, i.e. the set of vertices w' such that (H_i,w')≅ (H_i,w). Let W be a set which contains exactly one representative from each such orbit. Further, for each i∈[k] set W_i= W∩ V(H_i). Then, for each w,w'∈ W_i with w'≠ w, we have (H_i ,w)(H_i, w'). Let k'=∑_i=1^k W_i and let (H'_1,w'_1), …, (H'_k',w'_k') be an enumeration of {(H_i,w_i) : i∈[k], w_i∈ W_i}. Then we can apply Lemma <ref> to (H'_1,w'_1), …, (H'_k',w'_k') to obtain a connected irreflexive graph J with distinguished u∈ V(J) such that for every i,j∈ [k] and for all w_i∈ W_i, w_j∈ W_j we have (J,u)(H_i,w_i)≠(J,u)(H_j,w_j).Let i∈ [k] and suppose that H_i∈ and λ(H_i) ≠ 0. Let G be a non-empty graph which is an input to the problem H_i. Let v be an arbitrary vertex of G. We use the same construction as in Figure <ref> to design a graph G_t as input to the problem (, λ) by taking t copies of J as well as the graph G and identifying the t copies of vertex u with the vertex v∈ V(G). As both G and J are connected, G_t is as well. Then, using an oracle for (, λ), we can compute Z_,λ(G_t) withZ_,λ(G_t) = ∑_H∈λ(H)G_tH= ∑_i∈ [k]λ(H_i)G_tH_i= ∑_i∈ [k]λ(H_i)∑_w∈ V(H_i)(G,v)(H_i,w)·(J,u)(H_i,w)^tNow we collect the terms which belong to vertices in the same orbit. To this end, for w∈ W and i∈[k] such that w∈ V(H_i), we define λ_w= [w]·λ(H_i), N_w(G)= (G,v)(H_i,w) and N_w(J)=(J,u)(H_i,w). Let W={w_0,…,w_r}. Then, continuing from Equation (<ref>):Z_,λ(G_t) = ∑_i∈ [k]λ(H_i)∑_w∈ V(H_i)(G,v)(H_i,w)·(J,u)(H_i,w)^t=∑_w∈ Wλ_w N_w(G)N_w(J)^t. By choosing r+1 different values for the parameter t — here it is sufficient to choose t=0,… ,r — we obtain a system of linear equations = as follows:= [ Z_,λ(G_0); ⋮; Z_,λ(G_r) ]= [ λ_w_0N_w_0(J)^0 … λ_w_rN_w_r(J)^0; ⋮ ⋱ ⋮; λ_w_0N_w_0(J)^r … λ_w_rN_w_r(J)^r; ]and= [ N_w_0(G);⋮; N_w_r(G) ]The vectorcan be computed using r+1 (, λ) oracle calls. ThenGH_i = ∑_w ∈ W_i[w] N_w(G).Thus, determining x is sufficient for computing the sought-for GH_i. It remains to show that the matrix ∈^(r+1)× (r+1) is of full rank and therefore invertible. This can be easily seen by dividing each column by its first entry. The division is well-defined as for t∈{0…, r} we have λ_w_t≠0 by definition of ^≠0. The columns of the resulting matrix are pairwise different by the choice of (J,u) and as a consequence the resulting matrix is a Vandermonde matrix and therefore invertible. Next, we give a short technical lemma which follows from Definition <ref> and is used inLemma <ref> to show that Theorem <ref> gives hardness results for H. Let H be a connected graph with at least one non-loop edge. Let H^- be the graph obtainedfrom H by deleting exactly one non-loop edge (but keeping all vertices). If H^- is connected,then λ_H(H^-) ≠ 0. As H^- is non-empty and connected, it is a valid input to λ_H and from the definition of λ_H (Definition <ref>) we obtainλ_H(H^-) = -∑_H'∈_Hs.t. H' Hμ_H(H')λ_H'(H^-). Consider a graph H'∈_H with H' H and H' H^-. H' is a non-empty loop-hereditary connected subgraph of H and not isomorphic to H or H^-. Note that H^- is not isomorphic to any graph in _H' which gives λ_H'(H^-) = 0. Furthermore, μ_H(H^-) ≥ 1. Thus, we proceedλ_H(H^-) = -μ_H(H^-)λ_H^-(H^-)≤ -1. We now have most of the tools at hand to classify the complexity of H. Tractability results come from Lemma <ref>. If H has a component that is not a reflexive clique or an irreflexive bicliquethen hardness will be lifted from Dyer and Greenhill's Theorem <ref> via Theorem <ref>. The most difficult case is when all components of H are reflexive cliques or irreflexive bicliques, but some component is not an irreflexivestar or a reflexive clique of size at most 2.If H is connected then hardness will come from the following lemma, whose proof builds on the weighted graph set technology (Corollary <ref>) using Theorem <ref> and Lemma <ref>(using the stronger hardness result of Dyer and Greenhill, Theorem <ref>).The remainder of the section generalises the connected case to the case in which H is not connected. If H is a reflexive clique of size at least 3 then H is -hard. If H is an irreflexive biclique that is not a star then H is -hard.Suppose that H is a reflexive clique of size at least 3 or an irreflexive biclique that is not a star. Recall the definitions of _H, λ_Hand weighted graph sets (Definitions <ref>, <ref> and <ref>).Note that (_H,λ_H) is a weighted graph set. Let H^- be a graphobtained from H by deleting a non-loop edge. Note thatH^- is connected and it is not a reflexive clique or an irreflexive biclique. Thus Theorem <ref> states that H^- is -complete. We will complete the proof of the Lemma by showing H^-≤H. If H is a reflexive graph then the definition of _Hensures that all graphs in _H are reflexive.If H is an irreflexive bipartite graph, then the definition ensures that all graphs in _H are irreflexive and bipartite.Since H^- is connected and therefore λ_H(H^-)≠0 by Lemma <ref>, we can apply Theorem <ref> to the weighted graph set (_H,λ_H) with H^-∈_H to obtain H^-≤(_H,λ_H). By Corollary <ref>, (_H,λ_H)≡H. The lemma follows.We use the following two definitions in Lemmas <ref> and <ref> and in the proof of Theorem <ref>. Let H be a graph. Suppose that every connected component that has more than j vertices is an irreflexive star. Suppose further that some connected component has j vertices and is not an irreflexive star. Let (H) be the set of reflexive components of H with j verticesand let (H) be the set of irreflexive non-star components of H with j vertices. Let L(H) denote the set of loops of a graph H. We define the graph H^0=(V(H),E(H)∖ L(H)).Let H be a graph in which every component is a reflexive clique or an irreflexive biclique. If J∈(H) then J≤H. Let H be a graph in which every component is a reflexive clique or an irreflexive biclique. Let (H)={A_1,…, A_k}.It follows from the definition of (H)that all elements of (H) arereflexive cliques of some size j(the same j for all graphs in (H)). If j≤ 2, the statement of the lemma is trivially true, since Lemma <ref> shows that A_i is in , so the restricted problem A_i is also in .Now assume j≥ 3. Suppose without loss of generality that J=A_1. LetG be a (connected) input to J. For all i∈ [k], let H∖ A_i be the graph constructed from H by deleting the connected component A_i. Using Definition <ref> we define the (irreflexive) graph G'=(H∖ J ⊕ G)^0 as an input to H. Intuitively, to form G' from H we replace the connected component J with the graph G, then we delete all loops. We will prove the following claim.Claim: Let h V(G')→ V(H) be a compaction from G' to H. Then the restriction h|_V(G) is a compaction from G onto an element of (H). Proof of the claim:As h is a homomorphism, it maps each connected component of G' to a connected component of H. As, furthermore, h is a compaction and G' and H have the same number of connected components, it follows that there exist connected components C_1, …, C_k of G' such that for i∈ [k], h|_V(C_i) is a compaction from C_i onto A_i. To prove the claim, we show that G is an element of ={C_1, …, C_k}.In order to use all vertices of a graph in (H),i.e. areflexive size-j clique,a graph inhas to have at least j vertices itself. Therefore and by the construction of G', an element ofcan only be one of the following: * aclique with j vertices,* abiclique with j vertices,* a star with at least j vertices* or the copy of G.Since j≥ 3, it is easy to see that there is no compaction from a star onto aclique with j vertices. In order to compact onto areflexive clique of size j,an element ofalso has to have at least j(j-1)/2 edges. Thus,does not contain anybicliques. Finally, there are only k-1 connected components in G' that arej-vertex cliques other than (possibly) G. Therefore, G has to be an element of , which proves the claim. (End of the proof of the claim.) Using the notation from Definition <ref>, the claim impliesG'H = ∑_i=1^k GA_i·(H∖ A_i)^0H∖ A_i.We can simplify the expression (<ref>) using the fact that all elements of (H) arereflexive size-j cliques: G'H =k·GJ·(H∖ J)^0H∖ J.As (H∖ J)^0H∖ J does not depend on G, it can be computed in constant time. Thus, using a single H oracle call we can compute GJ in polynomial time as required.Let H be a graph in which every component is a reflexive clique or an irreflexive biclique. If (H) is empty but (H) is non-empty, then there exists a component J∈(H) such that J≤H. The proof is similar to that of Lemma <ref>. For completeness, we give the details. By Definition <ref> the elements of (H) are of the form K_a,b with a+b=j for some fixed j. As stars are excluded from (H), we have a,b ≥ 2. Let ^max(H) denote the set of graphs with the maximum number of edges in (H). The elements of ^max(H) are pairwise isomorphic since the number of edges of a K_a,b is a· b=a(j-a) and this function is strictly increasing for a≤ j/2. For concreteness, fix a and b so that each J∈^max(H) is isomorphic to K_a,b. Let ^max(H)={B_1,…, B_k}. Take J=B_1.For all i∈ [k], let H∖ B_i be the graph constructed from H by deleting the connected component B_i.Let G'=(H∖ J ⊕ G)^0 be an input to H.We will prove the following claim. Claim: Let h V(G')→ V(H) be a compaction from G' to H. Then the restriction h|_V(G) is a compaction from G onto an element of ^max(H). Proof of the claim:As h is a homomorphism, it maps each connected component of G' to a connected component of H. As, furthermore, h is a compaction and G' and H have the same number of connected components, it follows that there exist connected components C_1, …, C_k of G' such that for i∈ [k], h|_V(C_i) is a compaction from C_i onto B_i. To prove the claim, we show that G is an element of ={C_1, …, C_k}. In order to compact onto a graph in ^max(H), a graph inhas to have at least j vertices and a· b edges itself. By the construction of G' and the fact that (H) is empty, a connected component in G' with at least j vertices and a· b edges can only be one of the following: * abiclique K_a,b,* a star with at least j vertices and at least a· b edges* or the copy of G.Since a,b≥ 2, it is easy to see that there is no compaction from a star onto a K_a,b. Finally, there are only k-1 connected components in G' that arebicliques of the form K_a,b other than (possibly) G. Therefore, G has to be an element of , which proves the claim. (End of the proof of the claim.) Using the notation from Definition <ref>, the claim impliesG'H = ∑_i=1^k GB_i·(H∖ B_i)^0H∖ B_i.We can simplify the expression (<ref>) using the fact that all elements of ^max(H) are of the form K_a,b:G'H =k·GJ·(H∖ J)^0H∖ J.As (H∖ J)^0H∖ J does not depend on G, it can be computed in constant time. Thus, using a single H oracle call we can compute GJ in polynomial time as required. Finally, we prove the main theorem of this section, which we restate at this point.thm-1 The membership of H inis straightforward. We distinguish between a number of cases depending on the graph H.Case 1: Suppose that every connected component of H is an irreflexive star or a reflexive clique of size at most 2. Then H is inby Lemma <ref>.Case 2:Suppose that H contains a component that is not a reflexive clique or an irreflexive biclique. Then the hardness of H (from Theorem <ref>) together with thereduction H≤H (from Theorem <ref>) impliesthat H is -hard. The hardness of H follows from the trivial reduction from H to H.Case 3: Suppose that the components of H are reflexive cliques or irreflexive bicliques andthat H contains at least one component that is not an irreflexive star or a reflexive clique of size at most 2. Every graph J∈(H) ∪(H) is a reflexive clique of size at least 3 or an irreflexive biclique that is not a star. By Lemma <ref>,J is -complete.Finally, as (H) ∪(H) is non-empty, we can use either Lemma <ref> or Lemma <ref> to obtain the existence of J∈(H) ∪(H) with J≤H. This implies that H is #P-hard. As in Case 2, the hardness of H follows from the trivial reduction from H to H. § COUNTING SURJECTIVE HOMOMORPHISMSThe proof of Theorem <ref> is divided into two sections. The first of these deals with tractable cases and the second deals with hardness results and also contains the proof of the finaltheorem. Taken together,Theorem <ref> and Dyer and Greenhill's Theorem <ref> show that the problem of counting surjective homomorphisms to a fixed graph H has the same complexity characterisation as the problem of counting all homomorphisms to H.Section <ref> shows that this equivalence disappears in the uniform case, where H is part of the input, rather than being a fixed parameter of the problem.Specifically, Theorem <ref> demonstrates a setting in which counting surjective homomorphisms is more difficult than counting all homomorphisms (assuming ≠). §.§ Tractability ResultsLet H be a graph. Then H≤H. Let H be fixed and V(H)=q. Let (G,) be an input instance of H. Let (v_1,…,v_n) be the vertices of G in an arbitrary but fixed order. With respect to this ordering and with respect to a homomorphism from G to H, let us denote by v_i_1 the first vertex of G which is assigned the first new vertex of H (v_i_1 = v_1), v_i_2 the first vertex of G which is assigned the second new vertex of H and so on. Every surjective homomorphism from G to H contains exactly one subsequence =(v_i_1,…,v_i_q) and every homomorphism containing such a subsequence is surjective. The number of subsequences is bounded from above by nq. Let σ→ V(H) be an assignment of the vertices of H to the vertices in . There are q! such assignments. We call ψ=(,σ) a configuration of G and Ψ(G) the set of all configurations for the given G. For every such configuration ψ we create a H instance (G,^ψ) with ^ψ = {S^ψ_v_i⊆ V(H) : i∈ [n]} andS^ψ_v_i=S_v_i∩{σ(v_i_j)}, if i=i_jfor j∈[q]S_v_i∩{σ(v_i_1),…, σ(v_i_j)}, for i_j<i<i_j+1.Intuitively, we use lists to “pin” the vertices into the vertices assigned by σ and to prohibit the remainder of the vertices of G from being mapped to new vertices of H. Then(G,)H= ∑_ψ∈Ψ(G)(G,^ψ)HWe can compute (G,)H by making a H oracle call for every instance (G,^ψ) and adding the results. The number of oracle calls |Ψ(G)| is bounded from above by the polynomial q!nq≤ n^q.Let H be a graph. If every connected component of H isa reflexiveclique or an irreflexivebiclique then H is in . The statement follows directly from Theorem <ref> usingDyer and Greenhill's dichotomy from Theorem <ref>.§.§ Hardness Results The following result and proof are very similar to that of Theorem <ref> and Lemma <ref>, respectively. For completeness, we repeat the proof in detail.Let H be a graph. Then H≤H. Let V(H)=q and G be an input to H. We design a graph G_t= G⊕ W_t as an input to the problem H by adding a set W_t of t new isolated vertices to the graph G. We introduce some additional notation. Let S^k(G) be the number of homomorphisms σ from G to H that use exactly k of the vertices of H. Let {w_1, …, w_k} be a set of k arbitrary but fixed vertices from H. We define N^k(W_t) as the number of homomorphisms τ from W_t to H such that {w_1, …, w_k} are amongst the vertices used by τ. The particular choice of vertices {w_1, …, w_k} is not important when counting homomorphisms from a set of isolated vertices—N^k(W_t) only depends on the numbers k and t.We observe that, for each surjective homomorphism γ V(G_t) → V(H), the restriction γ|_V(G) uses a subset V'⊆ V(H) of vertices and does not use any vertices outside of V'. Suppose that V' has cardinality V'=q-k for some k∈{0,…, q}. Then γ|_W_t uses at least the remaining k fixed vertices of H.Therefore, we obtain the following linear equation for a fixed t≥ 0:G_tH_b_t =∑_k=0^q S^q-k(G)_x_kN^k(W_t)_a_t,k. By choosing q+1 different values for the parameter t we obtain a system of linear equations. Here, we choose t=0,… ,q. Then the system is of the form = for = [ b_0; ⋮; b_q ]= [ a_0,0 … a_0,q; ⋮ ⋱ ⋮; a_q,0 … a_q,q; ]and= [ x_0; ⋮; x_q ].Note, that the vectorcan be computed using q+1 H oracle calls. Further, ∑_k=0^q x_k = ∑_k=0^q S^q-k(G)= ∑_k=0^q S^k(G) = GH.Thus, determiningis sufficient for computing the sought-for GH. It remains to show that the matrixis of full rank and is therefore invertible.For t<k, clearly a_t,k = N^k(W_t) = 0. Further, for the diagonal elements we have a_t,t= N^t(W_t)=t! for t∈{0,…, q}. Hence, = [10⋯0;∗ 1!⋱⋮;⋮⋱⋱0;∗⋯∗ q! ]is a triangular matrix with non-zero diagonal entries, which completes the proof.thm-1 The easiness resultfollows from Corollary <ref> using the trivial reduction H≤H. The hardness result follows from the same trivial reduction, along with Theorem <ref> and the dichotomy for H from Theorem <ref>.§.§ The Uniform Case We have seen from Theorems <ref> and <ref> that the problem of counting homomorphisms to a fixed graph H has the same complexity as the problem of counting surjective homomorphisms to H.Nevertheless, there are scenarios in which counting problems involving surjective homomorphisms are more difficult than those involving unrestricted homomorphisms. To illustrate this point, we consider the followinguniform homomorphism-counting problems. Motivated by terminology from constraint satisfaction, we use “uniform” to indicate that the target graph H is part of the input, rather than being a fixed parameter. Name: . Name: . [-0.05cm] Input:Irreflexive graph G Input: Irreflexive graph G [-0.05cm] whose components are cliques whose components are cliques[-0.05cm] and reflexive graph H and reflexive graph H [-0.05cm] whose components are cliques. whose components are cliques.[-0.05cm] Output: GH. Output:GH. [-0.05cm] The main result of this section is the following theorem.In order to prove Theorem <ref>, we definea counting variant of the subset sum problem. Given a set of integers ={a_1,…,a_n} and an integer b let S(,b),be the number of subsets '⊆ such that the sum of the elements in ' is equal to b.The counting problem is stated as follows. .A set of positive integers ={a_1,…,a_n} and a positive integer b.S(,b).It is well known thatis -complete (see for instance the textbook by Papadimitriou <cit.>). Thus, Theorem <ref> follows immediately from Lemmas <ref> and <ref>.is in . Let G and H be an input instance of . Let k be the number of connected components of G and let a_1,… ,a_k be the number of vertices of these components, respectively. Let H have q connected components with b_1, …, b_q vertices, respectively. Then, as all components arecliquesand H is reflexive, GH = ∏_i=1^k ∑_j=1^q b_j^a_i.Thus, it is easy to compute GH.≤. Let ={a_1, …, a_k}, b be an input instance of . We define N=∑_i=1^k a_i. Now, we design a polynomial time algorithm to determine S(,b) using an oracle for . If N<b, we have S(,b)=0. Now assume N≥ b. We create an input ofas follows. We set G to be an irreflexive graph with a connected component G_i for each i∈[k], where G_i is aclique with a_i vertices. Furthermore, we set H to be a reflexive graph with two connected components H_1 and H_2. Let H_1 be aclique with b vertices and let H_2 be aclique with N-b vertices. By nk we denote the Stirling number of the second kind, i.e. the number of partitions of a set of n elements into k non-empty subsets. By definition, we have nk=0 if n<k.Let h V(G) → V(H) be a homomorphism from G to H and let b' be the number of vertices of G that are mapped to the connected component H_1. Note that h has to map each connected component of G to a connected component of H. By the construction of G, this implies that there exists a subset '⊆ such that the sum of elements in ' is equal to b'. Furthermore, as all connected components of G and H arecliques and H is reflexive, the number of surjective homomorphisms from G to H that assign exactly b' fixed vertices to H_1 is equal to the number of surjective mappings from [b'] to [b], which is b!b'b.Therefore, we can express GH as follows.GH=∑_b'=0^N S(,b')· b!b'b· (N-b)!N-b'N-b,where the factor (N-b)!N-b'N-b corresponds to the number surjective mappings from the remaining N-b' fixed vertices of G to the component H_2. Finally, we use the fact that the summands in (<ref>) are non-zero only if b'≥ b and N-b'≥ N-b, which implies b'=b. Thus,GH = S(,b)· b!bb· (N-b)!N-bN-b=b!(N-b)!· S(,b).§ ADDENDUM: A DICHOTOMY FOR APPROXIMATELY COUNTING HOMOMORPHISMS WITH SURJECTIVITY CONSTRAINTS The following standard definitions are taken from <cit.>. A randomised algorithm gives an (ϵ,δ)-approximation for the value V if the output X of the algorithm satisfies (|X-V| ≤ϵ V) ≥ 1-δ. A fully polynomial randomised approximation scheme (FPRAS) for a problem V is a randomised algorithm which, given an input x and a parameterϵ∈ (0,1), outputs an (ϵ,1/4)-approximation to V(x) in time that is polynomial in 1/ϵ and the size of the input x. The concept of anapproximation-preserving reduction (AP-reduction) between counting problemswas introduced by Dyer, Goldberg, Greenhill and Jerrum <cit.>.We will not need the detailed definition here, but the definition has the property that if there is an AP-reduction from problem A to problem B (written as A B) then this reduction, together with an FPRAS for B, yields an FPRAS for A.The problem , which is the problem of counting the independent sets of a bipartite graph, comes up frequently in approximate counting becauseit is complete with respect to AP-reductions in an intermediatecomplexity class. It is not believed to have an FPRAS. Galanis,Goldberg and Jerrum <cit.> gave a dichotomy for the problem of approximatelycounting homomorphisms in the connected case, in terms of . Let H be a connected graph. If H is a reflexive clique or an irreflexive biclique, then there is an FPRAS for H. Otherwise, H. In this addendum we give a similar dichotomy for approximately counting homomorphisms with surjectivity constraints[When H is not connected, the complexity of approximate counting is open even for counting homomorphisms. Hence we do not address this case here.]. The tractability part of the following theorem follows fromTheorem <ref>, Corollary <ref> and from Lemma <ref>below. The -hardness follows from Theorem <ref> and from the reductions in Lemmas <ref>, <ref> and <ref>.Let H be a connected graph. If H is a reflexive clique or an irreflexive biclique, then there is an FPRAS forH, H and H. Otherwise,each of these problems is -hard under approximation-preserving reductions. Let H be a reflexive clique or an irreflexive biclique. Then there is an FPRAS for H. Let H be a reflexive clique or an irreflexive biclique with q vertices and p edges. Our goal is give an FPRAS for H. First, we show that we can assume without loss of generality that every input G to H has no isolated vertices. To see this, suppose instead thatG is of the form G'⊕ I where I is the set of isolated vertices in G.As H is connected, we have GH=q^IG'H.Thus, an estimate of the number of compactions from G' to H will immediately enable us to approximately count compactions from G to H.From now on we restrict attention to inputs G which have no isolated vertices. We use (G,H) to denote the set of homomorphisms from G to H. Case 1. H is a reflexive clique. Let G be a size-n input to H. Then GH=q^n.If there is acompaction from G to Hthen there is a set U⊆ V(G)with |U| ≤ 2p and a compaction σ from G[U] to H.Each assignment ofthe (at most n-2p) vertices in V(G)∖ U extends σ to acompaction from G to H.Thus, we have GH≥ q^n-2p=GH/q^2p. Using this lower bound, it is straightforward to apply the naive Monte Carlo method (cf. <cit.>). Hence Algorithm <ref> with c=q^2p and =(G,H)gives an (ϵ,δ)-approximation for the number of compactions in H.If there are no compactions in ℋ then the algorithm answers 0. Otherwise, the number of compactions in ℋ is at least ||/c, so the algorithm gives an (ϵ,δ)-approximation.When the algorithm is run with δ=1/4, the running time is at most a polynomial in n and 1/ϵ because m is at most a polynomial in 1/ϵ and the basic tasks (choosing a sample from ℋ, determining whether a sample is a compaction, and computing |ℋ|=q^n) can all be done in poly(n) time.Thus, the algorithm gives an FPRAS for H. Case 2. H is an irreflexive biclique.Let the bipartition of V(H) be (L_H,R_H) where ℓ_H = |L_H| ≤ |R_H| = r_H.We can assume that ℓ_H ≥ 1, otherwise counting compactions to H is trivial. Without loss generality, we can assume that inputs G to H are bipartite (as well as having no isolated vertices). (If G is not bipartite, then GH=0.) Suppose that G is an input to H. Let C_1,…,C_κ be the connected components of G. For each i∈ [κ], let (L_i,R_i) be a fixed bipartition of C_i such that1≤ℓ_i = L_i≤R_i = r_i. Then GH=∏_i=1^κ(ℓ_H^ℓ_ir_H^r_i + ℓ_H^r_ir_H^ℓ_i) ≤2 ∏_i=1^κℓ_H^ℓ_ir_H^r_i. Let Ω be the set of functions ω [κ] →{L_H,R_H}. Given ω∈Ω, we say that a homomorphism from G to H obeys ω if, for each i∈ [κ], the vertices of L_i are assigned to vertices in ω(i).Case 2a. κ≥ p.Let ω be the function in Ω that maps every i∈ [κ] to L_H. Since G has no isolated vertices, each of C_1,…,C_κ has at least 2 vertices, so there is a compaction from G to H which obeys ω.As in Case 1, there is a set U⊆ V(G) of size at most 2p such thatthere is a compactionσ from G[U] to H that obeys the restriction of σ to U. Every assignment of the vertices in V(G)∖ Uthat obeys ω yields an ω-obeying compaction from G to H. Since r_H≥ℓ_H, we obtain the lower boundGH≥1/(r_H)^2p∏_i=1^κℓ_H^ℓ_ir_H^r_i≥GH/2(r_H)^2p.By the same arguments asin Case 1, Algorithm <ref> with c=2(r_H)^2p and =(G,H) gives an (ϵ,δ)-approximation forthe number of compactions in . When the algorithm is run with δ=1/4, the running time is at most a polynomial in |V(G)| and 1/ϵ, so it can be used in an FPRAS for inputs G with κ≥ p.Case 2b. κ < p.For each ω∈Ω, let _ω(G,H) be the set of homomorphisms obeying ω,and let N_ω(G→ H) and N^comp_ω(G→ H) be the number of homomorphisms and compactions obeying ω, respectively.Given a compaction that obeys ω we obtain a lower bound as before:N^comp_ω(G→ H)≥1/(r_H)^2p∏_i=1^κω(i)^ℓ_i(V(H)-ω(i))^r_i= N_ω(G→ H)/(r_H)^2p.Now Algorithm <ref> withc=(r_H)^2p and =_ω(G,H) gives an (ϵ,δ)-approximation for the number of compactions in_ω(G,H). Taking δ = 1/(4 · 2^κ) and summing over the 2^κ< 2^p functions ω∈Ω, we obtain an (ϵ,1/4)-approximation for the number of compactions in (G,H).The running time of each call to Algorithm <ref> is at most a polynomial in |V(G)| and 1/ϵ.Thus, putting the cases together, we get an FPRAS for H. Let H be a graph. Then HH. Let q=|V(H)|. Given any positive integer t, let s_t,q denote the number of surjective functions from [t] to [q]. Clearly, s_t,q≥ q^t - 2^q (q-1)^t, since the range of every non-surjective function from [t] to [q] isa proper subset of [q], and there are most 2^q of these. Also,the number of functions from [t] ontothis subset is at most (q-1)^t.Given any n-vertex input G to the problem H, let t = ⌈log(5 q^n 2^q)/log(q/(q-1) ⌉.Clearly, t=O(n), and t can be computed in time poly(n). Note that(q/q-1)^t ≥ 5 q^n 2^q ≥ 4 q^n2^q + 2^q.Let G_t be the graph constructed from G by addinga set I_t of t isolated vertices that are distinct from the vertices in V(G). We claim thats_t,qGH≤G_tH≤ s_t,qGH + (q^t - s_t,q) q^n.To see this, note that any homomorphism from G to H, together with a surjective homomorphism from the I_t to V(H), constitutes a surjective homomorphism from G_t to H. Any other surjective homomorphism from G_t to H consists of a non-surjective homomorphism from I_t to H (and there are q^t - s_t,q of these) together with some homomorphism from G to H (and there are at most q^n of these). Dividing through by s_t,q and applyingour lower bound for s_t,q and then inequality (<ref>), we haveGH≤G_tH/s_t,q ≤GH + (q^t - s_t,q/s_t,q) q^n≤GH +2^q (q-1)^tq^n/q^t - 2^q (q-1)^t= GH + q^n/q^t/2^q(q-1)^t -1≤GH + 1/4.Given Equation (<ref>), the proof of <cit.> shows that to approximate GH with accuracy ε, we need only use the oracle to obtain an approximation S forG_tH with accuracy ϵ/21. We can then return the floor of S/s_t,q. The only remaining issue is how to compute s_t,q. However, it is easy to do this intime poly(t) = poly(n) sinces_t,q = tq q! = ∑_j=0^q(-1)^q-jqj j^t, where tq is a Stirling number of the second kind.Let H be a connected graph. Then HH. If not explicitly defined otherwise, we use the same notation and observations as in the proof of Lemma <ref>. In addition let p be the number of non-loop edges in H and c_t,p=2^t s_t,p.If G is an input to H of size n, G_t is the graph constructed from G by adding a set of t isolated edges distinct from the edges in G. If H is a graph of size 1 the statement of the lemma clearly holds. If otherwise H is a connected graph of size at least 2, every homomorphism that uses all non-loop edges of H is also surjective and therefore a compaction. Thus, we obtainc_t,pGH≤G_tH≤ c_t,pGH + (2^tp^t - c_t,p) q^n.Dividing through by c_t,p givesGH≤G_tH/c_t,p≤GH + (p^t - s_t,p/s_t,p) q^n.If we choose t = ⌈log(5 q^n 2^p)/log(p/(p-1) ⌉the remainder of this proof is analogous to that of Lemma <ref>. Let H be a graph. Then HH. Let q=V(H) and G be an input to H. Further, let H' be a copy of H and (u_1,…,u_q) be the vertices of H' ordered in such a way that they induce a copy of H. Then GH = (G⊕ H'; u_1,…,u_q)H.§ APPENDIX: DECOMPOSITION OF GK_2,3In thisappendix, we work through a long example to illustrate some of the definitions and ideas from Section <ref>. We do this byverifying the statement of Theorem <ref> for the special case where H=K_2,3.Of course, the theorem is already proved in the earlier sections of this paper, but we work through this example in order to help the reader gain familiarity with the definitions. For H=K_2,3 and a non-empty, irreflexive and connected graph G we want to proveGH = ∑_J∈_Hλ_H(J)GJ.First, we set _H={H_1,…,H_10}, cf. Figure <ref>, as defined in Definition <ref>. Next, we recall the definitions of μ_H and λ_H from Definitions <ref> and <ref>. For J∈_H, μ_H(J) is the number of non-empty connected subgraphs of H that are isomorphic to J. Also, λ_H(J)=1 if J≅ H. If otherwise J is isomorphic to some graph in _H but J H, we haveλ_H(J)=-∑_H'∈_Hs.t. H' Hμ_H(H')λ_H'(J). In order to verify (<ref>), we have to determine λ_H(J) for all J∈ S_H. As λ_H(J) is defined inductively by (<ref>), we first determine λ_H'(J) for all H'∈_H with H' H.We start with the graph H_10 and determine λ_H_10. Clearly, H_10 has only one connected subgraph and we can choose _H_10={H_10}. Recall that λ_H_10(J)=0 for all graphs J that are not isomorphic to any graph in _H_10, i.e. not isomorphic to H_10 in this case. By definition we have μ_H_10(H_10)=1 as well asλ_H_10(H_10)=1,see Table <ref>.This conforms with our intuition as for the single vertex graph H_10, it clearly holds that GH_10 =GH_10.Thus, we have now verified (<ref>) for H=H_10.Using this information, we consider the graph H_9 next and determine μ_H_9 and λ_H_9 for _H_9={H_9,H_10}, see Table <ref>. H_9 contains two connected subgraphs that are isomorphic to H_10, therefore μ_H_9(H_10) = 2. Then, from (<ref>) we obtainλ_H_9(H_10) = -∑_H'∈{H_10}μ_H_9(H')λ_H'(H_10)=-2.Plugging this into (<ref>) for H=H_9, we getGH_9 = ∑_J∈_H_9λ_H_9(J)GJ=GH_9 - 2 GH_10. Now let us verify this expression. Recall that G is connected. The central idea behind our approach is that every homomorphism from G to H_9 is a compaction onto some connected subgraph H' of H_9. Furthermore, μ_H_9(H') tells us how many such subgraphs there are that are isomorphic to H'. Thus,GH_9 =μ_H_9(H_9)·GH_9 + μ_H_9(H_10)·GH_10= GH_9 + 2 GH_10.Rearranging and using the fact that we already know GH_10 =GH_10 from (<ref>):GH_9 = GH_9 - 2 GH_10= GH_9 - 2 GH_10.Thus, we have now proved (<ref>) which in turn proves (<ref>) for H=H_9.Using (<ref>) and (<ref>) we can now go on to find (see Table <ref>) that GH_8= GH_8 - 2 GH_9+ GH_10and so on.This gives the intuition behind the formal definitions of μ_H and λ_H. For completeness, we give the values for all graphs H_1 through H_10 in Tables <ref> through <ref>. From Table <ref> we can conclude that for H=K_2,3 the statement of Theorem <ref> givesGK_2,3 = GK_2,3 - 6 GH_2 + 6 GH_3 + 3 GH_4 + 6 GH_5 - 2 GH_6 - 12GH_7 + 3 GH_8. | http://arxiv.org/abs/1706.08786v5 | {
"authors": [
"Jacob Focke",
"Leslie Ann Goldberg",
"Stanislav Zivny"
],
"categories": [
"cs.CC",
"cs.DM",
"F.2.2; G.2.1"
],
"primary_category": "cs.CC",
"published": "20170627112511",
"title": "The Complexity of Counting Surjective Homomorphisms and Compactions"
} |
[E-mail:][email protected] [Present address: ]Department of Mechanical Engineering, University of Benghazi, Benghazi, Libya Sheffield Fluid Mechanics Group SFMG,Department of Mechanical Engineering,University of Sheffield, Mappin Building,Mappin Street, Sheffield, S1 3JD, United KingdomThe paper investigates shock-induced vortical flows within inhomogeneousmedia of nonuniform thermodynamic properties. Numerical simulations areperformed using an Eularian type mathematical model for compressiblemulti-component flow problems. The model, which accounts for pressurenon-equilibrium and applies different equations of state for individualflow components, shows excellent capabilities for the resolution of interfacesseparating compressible fluids as well as for capturing the baroclinic sourceof vorticity generation. The developed finite volume Godunov typecomputational approach is equipped with an approximate Riemann solver forcalculating fluxes and handles numerically diffused zones at flow componentinterfaces. The computations are performed for various initial conditionsand are compared with available experimental data. The initial conditionspromoting a shock-bubble interaction process include: weak to highplanar shock waves with a Mach number ranging from 1.2 to 3 and isolatedcylindrical bubble inhomogeneities of helium, argon, nitrogen, krypton andsulphur hexafluoride. The numerical results reveal the characteristicfeatures of the evolving flow topology. The impulsively generated flowperturbations are dominated by the reflection and refraction of the shock,the compression and acceleration as well as the vorticity generation withinthe medium. The study is further extended to investigate the influence of the ratio of the heat capacities on the interface deformation. 47.40.Nm, 47.55.-t, 47.11.-j, 47.32.-y Passage of a shock wave through inhomogeneous media and its impacton a gas bubble deformation F. C. G. A. Nicolleau December 30, 2023 ===============================================================================================§ INTRODUCTION Compressible multi-component flows with low to high density ratios between components are involved in various physical phenomena and many industrial applications. Some important examples are inertial confinement fusion(ICF) <cit.>, rapid and efficient mixing of fuel and oxidizer in supersonic combustion, primary fuel atomization in aircraft engines anddroplet breakup. A proper understanding of these flows requires studyingthe evolution and creation of interfaces resulting from the interactionof a shock wave with the environment of inhomogeneous gases. The diverseflow patterns and the dynamical interaction of gas phases at the interfacecould cause several physical processes to occur simultaneously. These includeshock acceleration or refraction, vorticity generation and its transport,and consequently shock-induced turbulence. The mechanism of these processesis related to the strength and pattern of the propagating shock waves duringthe short time of their encounter with the surface's curvatures between flowcomponents and inherently to the difference in the acoustic impedance at thecomponents' interfaces. The recent review paper <cit.>provides an excellent description of various possible phenomena occurringduring the shock bubble interaction process.When, as a result of the passageof a shock wave, an interface between fluid components is impulsivelyaccelerated, the development of a so called Richtmyer-Meshkov instability (RMI) <cit.> can be observed. The instability, which directly results from the amplification of perturbations at the interface, is due to baroclinicvorticity generation as a consequence of the misalignment of the pressuregradient of the shock and the local density gradient across the interface.This is a complex phenomenon constituting a challenging task to investigateeither experimentally or numerically as the derivation of a mathematical model for this problem is not straightforward. The nature of the impulsively generated perturbations at the interface oftwo-component compressible flows has been studied experimentally usingidealised configurations. The interaction of a planar shock wave witha cylinder or a sphere is a typical physical arrangement that has receivedattention. However, the measurement of the entire velocity,density and pressure fields for a large selection of physical scales andinterface geometries remains an enormous experimental challenge. The first key work to monitor the interaction between a plane shock waveand a single gas bubble was presented in <cit.>. The shadowgraphphotography technique was utilised to visualise a wave front geometry and thedeformation of the gas bubble volume. The distortion of a spherical bubbleimpacted by a plane shock wave was later examinedin <cit.> using a high speedrotating camera shadowgraph system and in <cit.> by meansof the high speed schlieren photography with higher time resolutions. Allthese experiments were conducted in horizontal shock tubes characterisedby a Mach number smaller than 1.7.Other laser based shock-tube experimentsin <cit.>covered a wider selection of Mach numbers and provided qualitative andquantitative data for the shock-bubble interaction within the Mach numberrange of 1.1-3.5.A similar geometry was investigated in <cit.>to find a mixing mechanism in a shock-induced instability flow. Although atpresent it is possible to consider experiments with a higher Mach numberby building laboratory facilities based on modern laser technologies,such tests still remain rather difficult and expensive to conduct. Therefore the development of numerical techniques for these types ofapplications seems to be an ideal alternative to provide reasonableresults at a significantly lower cost. A shock-capturing upwind finitedifference numerical method has been utilised to solve the compressibleEuler equations for two species in an axisymmetric two-dimensional caseof planar shock interacting with a bubble <cit.>. The evolution of upstream and downstream complex wave patterns and the appearance of vortex rings were resolved in this study. The experiment in <cit.>, inwhich a shock wave with a Mach number 1.22 hits a helium bubble,has inspired several other authors <cit.> who adopted this experiment to demonstrate theperformance of the numerical techniques they developed. In the majority of thecases the authors used the Euler equations to simulate the experiment andthe interface reconstruction was the major task.For example <cit.> proposed to use the front tracking/ghostfluid method to capturefluid interface minimizing at the same time thesmearing of discontinuous variables.In another development <cit.> the two-dimensional simulationsof the shock-bubble interaction were extended to three spatial dimensionsand high Mach numbers using the volume-of-fluid (VOF) method as the numerical approach. The authors considered fourteen different scenarios, including fourgas pairings by using a numerical algorithm solving the same system of partial differential equations for each of the two constituent species with anadditional numerical scheme for the local interface reconstruction.The 2D VOF method was also used in <cit.>. A viscous approach,but without accounting for turbulence, was adopted in the numericalstudy <cit.> which reproduced different experiments performed in <cit.>. The majority of numerical simulations are based on the mixture Euler equations supplemented by two species conservation equations in order to build a reasonableequation of state parameters at the interface(see example <cit.>). By contrast, the ArbitraryLagrangian Eulerian methods or Front Tracking Methods <cit.>consider multi-material interfaces as genuine sharp non-smeareddiscontinuities. These methods are less flexible when dealing with situationsof large interface deformations and topological changes. The mathematical model advocated by the authors of this paper, is based ona different point of view, and while considering the equations for immisciblefluids, does not require the explicit application of boundary conditions atthe interface. The system of equations can be derived fromthe Baer and Nunziato model <cit.>. However, in contrastto <cit.>, who used the original <cit.> formulationto investigate shock-bubble interaction, the equations presented in this paper are considered in an asymptotic limit of the velocity relaxation time of the model <cit.>. This so called“six-equation model”was derived for the first time in <cit.> and was furtherinvestigated in <cit.>.In the latter reference <cit.> the authors showed that thenon-monotonic behaviour of the sound speed which causes errors in thetransmission of waves across interfaces can be circumventedby restoringthe effects of pressure non-equilibrium in the equation of the volume fractionevolution by using two pressures and their associated pressure relaxationterms.The six-equation model takes advantage of the inherent numericaldiffusion at the interface as the necessary condition for interface capturingand avoids the spurious pressure oscillations that frequently occur at themulti-fluid interfaces. Furthermore, and what is of key importance here,the model can naturally handle complex topological changes. The otherattractive and desired features of this model could be summarised as follows:the ability to simulate the dynamical creation and the evolution of interfaces, the numerical implementation with a single solver for a system of unified conservation equations and the ability to use different equationsof state and hence different heat capacity ratios for individual flow components. This paper investigates the two-dimensional flow of a shock wave encountering circular inhomogeneities. It presents a numerical study of the interactionof weak to high Mach number waves with an inhomogeneous medium containinga gas bubble.The inherent features of such flow composition are density jumps across theinterface. The study concentrates on the early phases of the interactionprocess. The purpose is to consider the influence of both the Atwood numberand the shock wave Mach number on the deformation of the gas bubbles andthe associated production of a vorticity field. The physical behaviour of thegas bubbles is monitored using a newly developed numerical algorithm whichhas been built to solve the six equation model. The considered Atwood numbersare within the range (-0.8 ⩽ A ⩽ 0.7) and the shock celeritycovers the range (1.2⩽Ma⩽ 3). The outline of the paper is as follows: section <ref> gives a briefintroduction to the two-component flow governing equations.Then in section <ref> the numerical procedures to solve the systemare described. The main focus of this paper is section <ref>, whichpresents the results of the computational work.First, two independent experimental investigations describedin <cit.> and <cit.> are used for the interface evolutionvalidation. In the case of <cit.>, a shock wave (Ma = 1.5)interacts with three different air/gas configurations which are air/helium (He),air/nitrogen (N_ 2) and air/krypton (Kr). In the case of <cit.>a shock wave (Ma = 1.5) interacts with a sulphur hexafluoride (SF_6) bubble. Second, the study is extended to account for the differentgas pairings in an attempt to evaluate the effect of the Atwood numberon the complex pattern of the gas bubbles evolution. Third, the effect ofthe Mach number on the interface evolution is investigated for all cases withthe intention to discuss and quantify the production of vorticity resultingfrom the passage of the shock. Finally, the investigation of the influence of the ratio of the heat capacities on the interface deformation is made.The conclusions are drawn in section <ref>.§ MATHEMATICAL MODELA two-component compressible flow model is considered. The model consistsof two separate, identifiable and interpenetrating continua that are inthermodynamic non-equilibrium with each other. In its one-dimensionalmathematical framework the model, first derived in <cit.>,consists of six partial differential equations. It constitutes a reduced form of the more general seven equation model <cit.>. The one dimensionalequations of the model are: a statistical volume fraction equation, twocontinuity equations, one momentum equation and two energy equations.It differs from more popular models which rely on instantaneous pressureequilibrium between the two flow components or phases <cit.>.The original six-equation model can be expressed in one dimensional space,as follows: α_1t+uα_1x = μ(p_1-p_2), α_1ρ_1t+α_1ρ_1ux = 0, α_2ρ_2t+α_2ρ_2ux = 0, ρ ut+[ρ u^2+(α_1p_1+α_2p_2)]x = 0, α_1ρ_1 e_1t+α_1ρ_1 e_1ux+α_1p_1ux = P_i μ(p_1-p_2), α_2ρ_2 e_2t+α_2ρ_2 e_2ux+α_2p_2ux = -P_i μ(p_1-p_2),where α_k, ρ_k, p_k and e_k are respectively the volumefraction, the density, the pressure and the internal energy of thek-th (1 or 2) component of the flow. The volume fractions forboth fluids have to satisfy the saturation restriction ∑α_k = 1and the interfacial pressure P_i is defined asP_i= α_1p_1+ α_2p_2. Additionally, the mixture density ρ,velocity u, pressure p and internal energy e are defined as:ρ = α_1ρ_1+ α_2ρ_2, u = (α_1ρ_1 u_1 + α_2ρ_2 u_2)/ρ, p = α_1 p_1 +α_2 p_2, e = (α_1ρ_1 e_1 + α_2ρ_2 e_2)/ρ.The μ variable represents a homogenization parameter controllingthe rate at which pressure tends towards equilibrium and it depends on the compressibility of each fluids and their interface topology.Its physical meaning was justified using the second law ofthermodynamics <cit.>.Instead of using only mixture thermodynamic variables, the model (<ref>)keeps two distinct pressures. As a result, the thermodynamic non-equilibriumsource term μ(p_1-p_2) exists in the volume fraction evolution equationand the source term P_iμ(p_1-p_2) in the energy conservation equations.On the one hand the presence of the left hand side non-conservative termsα_k p_k ∂ u/∂ x complicates theanalytical and computational treatment of the model (<ref>).The non-conservative terms do not allow the governing equations to be writtenin a divergence form, which is preferred for numerical handling ofproblems involving shocks. The classical Rankine-Hugoniot relations cannot be defined in an unambiguous manner and additional relations or regularisationprocedures must be proposed instead.On the other hand when dealing with the bubble interfacerepresented by the density jump, these non-conservative terms enableaccommodating thermodynamic non-equilibrium effects between the bubbleand its surrounding during the passage of shock waves.The pressure non-equilibrium state can be solved using the instantaneousrelaxation model with the efficient numerical algorithm proposedin <cit.>. This model, while retaining the separate equationsof state and pertinent energy equation on both sides of the interfaces,introduces an additional total mixture energy equation.As a result shockwaves can be correctly transmitted through the heterogeneous media and thevolume fraction positivity in the numerical solution is preserved. This keyequation in <cit.> was derived by combining the two internalenergy equations with mass and momentum equations. The final form of thetotal mixture energy equation can be written as: ρ(Y_1e_1+Y_2e_2+1/2u^2)t + u(ρ(Y_1e_1+Y_2e_2+1/2u^2)+(α_1p_1+α_2p_2))x = 0,where, Y_1 and Y_2 are the mass fractions with general formY_k=α_kρ_k/ρ. The numerical procedures discussed in the next section tacklethe overdetermined system of equations consistingof (<ref>) and (<ref>) and correct the errors resultingfrom the numerical integration of the non-conservative terms:α_k p_k ∂ u/∂ x. The solution aspects of the overdetermined hyperbolic systems have beenconsidered earlier (see e.g. <cit.>).The mixture sound speed in this six-equation model has the desiredmonotonic behaviour as a function of volume and mass fractions andis expressed as:c^2=Y_1c_1^2 + Y_2c_2^2,where, c_1 and c_2 are the speeds of soundof the pure fluids.The model (<ref>) is supplemented by a thermodynamic closure. Theideal gas equation of state, relating the internal energy to the pressurep=p(ρ,e), is used for both flow components experiencing differentthermodynamic states.For a given fluid, the equation of state can be written as a pressure law:p_k = (γ_k-1)ρ_k e_k,where γ_k is the specific heat ratio for the component k of theflow. Similarly, the mixture equation of state takes the form:p = (γ-1)ρ e,where the specific heat ratio of mixed gas γ is calculated from:1/γ-1 = ∑_k α_k/γ_k-1. In this two-component formulation there are no explicit diffusive terms.These terms can be neglected as the diffusive effects do not play a major role in the early stages of bubble-shock interaction. The calculationof the kinematic viscosity of the mixture and estimation of resultingviscous length scales were provided in <cit.>.The viscosities of the considered fluids are (μ∼ 10^-5 Pa.s) and the evolution is studied over short time scales(t ⩽ 10^-3 s).To tackle two dimensional geometries and two component flows, the originalsix-equation model (<ref>) can be extended to the following form:α_1t+uα_1x +vα_1y= μ(p_1-p_2), α_1ρ_1t+α_1ρ_1ux+α_1ρ_1vy = 0, α_2ρ_2t+α_2ρ_2ux+α_2ρ_2vy = 0, ρ ut+[ρ u^2+(α_1p_1+α_2p_2)]x +ρ u vy= 0, ρ vt+ρ u vx+[ρ v^2+(α_1p_1+α_2p_2)]y = 0, α_1ρ_1 e_1t+α_1ρ_1 e_1ux+α_1ρ_1 e_1vy+α_1p_1ux+α_1p_1vy = P_i μ(p_1-p_2), α_2ρ_2 e_2t+α_2ρ_2 e_2ux+α_2ρ_2 e_2vy+α_2p_2ux+α_2p_2vy = -P_i μ(p_1-p_2),andρ Et+u(ρ E+(α_1p_1+α_2p_2))x+v(ρ E+(α_1p_1+α_2p_2))y = 0.where, u and v represent the components of the velocity in the x and y directions, respectively. The total energyof the mixture for two dimensional flows is given byE = Y_1e_1+Y_2e_2+1/2u^2+1/2v^2.§ NUMERICAL METHOD As stated in the previous section, model (<ref>) cannot be writtenin a divergence form and hence the standard numerical methods developed forconservation laws are not applied directly here. In order to solve this systema numerical scheme is constructed that decouples the left hand side ofmodel (<ref>) from the pressure relaxation source terms(the right hand side of the model). The left hand side, which representsthe advection part of the flow equations, is then analysed to determinethe mathematical structure of the system and is rewritten in termsof the primitive variables as follows:Wt+A(W)Wx+B(W)Wy = 0,where the vector of primitive variables W, Jacobian matrices A(W)and B(W) for the extended model (<ref>) are:W = [α_1 ρ_1 ρ_2 u v p_1 p_2] , A(W) = [u 0 0 0 0 0 0 0 u 0ρ_1 0 0 0 0 0 uρ_2 0 0 0 p_1-p_2/ρ0 0 u 0α_1/ρ 1-α_1/ρ0 0 0 0 u 0 00 0 0ρ_1 c_1^20 u 00 0 0ρ_2 c_2^20 0 u ]andB(W) = [v 0 0 0 0 0 0 0 v 0 0ρ_10 0 0 0 v 0ρ_2 0 0 0 0 0 v 0 0 0 p_1-p_2/ρ0 0 0 vα_1/ρ 1-α_1/ρ 0 0 0 0ρ_1 c_1^2v 0 0 0 0 0ρ_2c_2^20 v ]The seven eigenvalues of the Jacobian matrix A(W) are determined to be:u-c, u, u, u, u, u and u+c. Similarly, the eigenvalues ofthe Jacobian matrix B(W) are: v-c, v, v, v, v, v and v+c. §.§ Solution of the hyperbolic partThe above primitive form (<ref>) is hyperbolic but not strictlyhyperbolic. Indeed some eigenvalues, which represent the wave speeds,are real but not distinct. The solution of this hyperbolic problem isobtained using the extended Godunov scheme. To achieve second orderaccuracy the numerical algorithm is equipped with the classical Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL) <cit.>.The splitting scheme has been applied to solve the conservative equationson regular meshes. In each time step of simulation (Δ t),the conservative variables evolve in alternate directions (x,y) during time sub-steps (Δ t/2) which are denoted by (U^n+1/2) and (U^n+1).The time increment for the 2D second order discretisation of the Godunovscheme takes the following sub-steps: U_i,j^n+1/2=U_i,j^n-Δ t/Δ x[F^n(U^*(U_i+1/2,j^-, U_i+1/2,j^+))-F^n(U^*(U_i-1/2,j^-, U_i-1/2,j^+))] and U_i,j^n+1=U_i,j^n+1/2 -Δ t/Δ y[G^n+1/2(U^*(U_i,j+1/2^-, U_i,j+1/2^+))-G^n+1/2(U^*(U_i,j-1/2^-, U_i,j-1/2^+))].The components of the vectorU=[α_1ρ_1, α_2ρ_2, ρ u, ρ v, ρ E]^Tare the conservative variables. U_i,j^n represents the state vectorin a cell (i,j) at time n and U_i,j^n+1 represents the statevector at the next time step. F(U^*) is the flux functionin the x direction F(U)=[α_1ρ_1 u, α_2ρ_2 u, ρ u^2 +p, ρ uv, u(ρ E+p)]^T. G(U^*) is the flux function in the y directionG(U)=[α_1ρ_1 v, α_2ρ_2 v, ρ uv, ρ v^2 +p, v(ρ E+p)]^T.The superscript “*”refers to the state at each cell boundary. Figure <ref> presents a diagram for the fluxconfigurations in two dimensional computations. The plus and minus signsrefer to the conservative variable and flux values at cell boundaries inthe second order scheme. The second order accuracy is achieved by applyingthree major steps: the first step consists in the reconstruction ofthe average local values in each computational cell using extrapolationof piecewise linear approximations, the second step consists inthe determination of the variable values at a middle time step and finallythe last step is the solution of the Riemann problem. Figure <ref>shows the piecewise linear variable variation at the boundaries of eachcell. The flux functions in the Godunov scheme are obtained usingthe approximate Harten, Lax and Van Lear (HLL) Riemannsolver <cit.>.Figure <ref> shows a schematic diagram of the typical wave configuration in the approximate HLL Riemann solver. The wave speeds S^R and S^Lare the boundaries of three characteristics regions: left region (U_L),right region (U_R) and the star region (U^HLL). In this Riemann solver, the second order numerical flux functions in the star region at each cellboundary are written as:F_i+1/2,j = {[ F_i,j ,; S_i+1/2,j^RF_i,j - S_i+1/2,j^LF_i+1,j+S_i+1/2,j^RS_i+1/2,j^L(U_i+1,j-U_i,j)/S_i+1/2,j^R-S_i+1/2,j^L ,; F_i+1,j . ].F_i-1/2,j = {[ F_i-1,j ,; S_i-1/2,j^RF_i-1,j - S_i-1/2,j^LF_i,j+S_i-1/2,j^RS_i-1/2,j^L(U_i,j-U_i-1,j)/S_i-1/2,j^R-S_i-1/2,j^L ,; F_i,j . ].and G_i,j+1/2 = {[ G_i,j ,; S_i,j+1/2^RG_i,j - S_i,j+1/2^LG_i,j+1+S_i,j+1/2^RS_i,j+1/2^L(U_i,j+1-U_i,j)/S_i,j+1/2^R-S_i,j+1/2^L ,; G_i,j+1 . ].G_i,j-1/2 = {[ G_i,j-1 ,; S_i,j-1/2^RG_i,j-1 - S_i,j-1/2^LG_i,j+S_i,j-1/2^RS_i,j-1/2^L(U_i,j-U_i,j-1)/S_i,j-1/2^R-S_i,j-1/2^L ,; G_i,j . ].Similarly, the splitting scheme is applied in the descritization ofnon-conservative equations, i.e. the volume fraction and the two energyequations as follows:α_1i,j^n+1/2=α_1i,j^n-Δ t/Δ x[(uα_1)_i+1/2,j^* -(uα_1)_i-1/2,j^* -α_1i,j( u_i+1/2,j^*-u_i-1/2,j^*)]^n, α_1i,j^n+1=α^n+1/2_1i,j-Δ t/Δ y[(vα_1)_i,j+1/2^* -(vα_1)_i,j-1/2^* -α_1i,j( v_i,j+1/2^*-v_i,j-1/2^*)]^n+1/2, (αρ e )_ki,j^n+1/2=(αρ e )_ki,j^n-Δ t/Δ x[(αρ eu)_ki+1/2,j^* -(αρ eu)_ki-1/2,j^* +(α p)_ki,j( u_i+1/2,j^*-u_i-1/2,j^*)]^n, (αρ e )_ki,j^n+1=(αρ e )_ki,j^n+1/2-Δ t/Δ y[(αρ ev)_ki,j+1/2^* -(αρ ev)_ki,j-1/2^* +(α p)_ki,j( v_i,j+1/2^*-v_i,j-1/2^*)]^n+1/2, The stability of the numerical method is controlled bythe Courant-Friedrichs-Lewy (CFL) number, which imposes a restriction onthe time step Δ t as follows:Δ t = CFL×min(Δ x/S_x,Δ y/S_y),where S_x and S_y are the maximum wave speeds in the x and ydirections respectively.S_x = max⟨ 0, u_i±1/2,j^++ c_i±1/2,j^+, u_i±1/2,j^-+ c_i±1/2,j^-⟩,S_y = max⟨ 0, v_i,j±1/2^++c_i,j±1/2^+, v_i,j±1/2^-+c_i,j±1/2^-⟩. §.§ Solution of the pressure relaxation partIn each time step after the hyperbolic advection part is accomplished,the pressure equilibrium is achieved via the relaxation procedure.The pressure relaxation implies volume variations because ofthe interfacial pressure work. This represents the solution of the sub-problem governed by the followingordinary differential equations (ODE), with the source term representingthe right hand side of the model (<ref>). ∂α_1/∂ t =μ (p_1 - p_2), ∂α_1ρ_1/∂ t =0,∂α_2ρ_2 /∂ t =0, ∂ρ u/∂ t =0,∂ρ v/∂ t =0,∂α_1ρ_1e_1/∂ t = μ P_i (p_1 - p_2), ∂α_2ρ_2e_2/∂ t = -μ P_i (p_1 - p_2).The pressure relaxation is fulfilled instantaneously when the valueof μ in system (<ref>) is assumed to be infinite. To solvethe ODE system (<ref>), an iterative method for the pressurerelaxation for compressible multiphase flow is implemented. This methodis the iterative “procedure 4”described in <cit.>. This step rectifies the calculationof the internal energies to satisfy the second law of thermodynamics.The second amendment comes from solving the extra total energyequation (<ref>), where the mixture pressure is calculatedfrom the mixture equation of state.The overall sequence of the numerical solution steps follows theidea of succession of operators introduced in <cit.>.§ RESULTS AND DISCUSSION This section is divided into four parts: In the first part the generaldescription of the physics and the mechanism of the shock-bubble interaction phenomenon are revisited. In the second part the correctness of the resultsobtained using the developed numerical code is quantified. This is made byvalidating the numerical results for different shock-bubble interactionscenarios against the experimental data reported in <cit.>.In the third and fourth subsections the investigation of the shock-bubbleinteraction problem is extended to a wider selection of physicallyintriguing cases for which experimental data are not available.§.§ General features of shock-bubble interaction problemsThe flow configurations for the studied problems are classified accordingto the value of the Atwood number A = (ρ_b - ρ_s) / (ρ_b + ρ_s), where ρ_b is the density of the bubble and ρ_s is the densityof the surrounding medium. If the density of the bubble is lower thanthe density of the surrounding fluid, the value of the Atwood numberbecomes negative and this case represents heavy/light arrangement.In contrast, if the density of the bubble is higher than the densityof the surrounding fluid, the value of the Atwood number becomes positiveand this case represents light/heavy arrangement. Alternative terminologiesexist and describe these flow configurations as divergent in the case oflight bubble or as convergent in the case of heavy bubble <cit.>, or fast/slow interface or slow/fast interface according to the sound speedof the flow constituents <cit.>.Figure <ref> schematically presents the typical flow configurationduring an early stage of the shock-bubble interaction process. The flow patterns of heavy/light, Fig. <ref>(a), and light/heavy, Fig. <ref>(b),scenarios show the set of wave configurations associated with the interaction and the deformation of the bubble interface. After hitting the upstreaminterface of the bubble from the right hand side, the planar shock wave changes its uniform front, which evolves into two parts. One part does not interactdirectly with the gas filling the bubble while the second one transforms into a transmitted wave interacting with the bubble. In addition a reflected wave,as in the case of a heavy bubble, or a rarefaction wave front, as in the case of a light bubble, does occur and propagates back in the right direction. In the case of a light bubble (i.e. negative A number, Fig. <ref>(a)),the transmitted shock travels through the gas bubble faster than the incidentshock outside the bubble. This is the consequence of the mismatch in flowconstituents acoustic impedances (Z = ρ c) across the interface.The shock front also takes a divergent shape due to the curvature of theinterface and provokes the generation of a set of secondary waves insideand outside the bubble boundary. These secondary waves consist in irregularwaves <cit.>. A precursor (refracted) shockwave propagates downstream outside the bubble, internal reflected shocksare generated inside the bubble and move back upstream as the resultof the interaction of the transmitted shock with the internal surfaceof the bubble. A Mach stem shock wave travels outside the bubble.A triple point is formed outside the bubble owing to the intersectionof the exterior incident shock, the precursor shock and the Mach stem. In the case of a heavy bubble (i.e. positive A number, Fig. <ref>(b)) the scenario is completely different. The difference in the acoustic impedance between the fluids across the interface makes the transmitted shock inside the bubble moving more slowly than the incident shock outside the bubble.The transmitted shock becomes convergent. The interaction of the transmittedshock with the internal surface of the bubble produces a rarefaction wavepropagating backward inside the bubble. To explain the role of acoustic impedance mismatch in the creationof a vorticity field it is convenient to consider the vorticitytransport form of the Euler equations. The momentum equation governingthe evolution of vorticity isDω/D t =(ω·∇)u_stretching -ω(∇·u)_dilatation +1/ρ^2(∇ρ×∇ p)_baroclinicity. This equation contains, in contrast to its 2D counterpart, the termcorresponding to vorticity stretching. High resolutionshock-induced 3D simulations were performed to analyzethe relative importance of stretching, dilatation and baroclinic termsin the vorticity equation atMa=3 and Ma=10 <cit.>.It was found that the stretching term contribution manifests its existence after initial phases of shock bubble interaction. The only term of importance in the early stages of a shock-bubble interaction for which ωis initially equal to zero is the baroclinic torque(∇ρ×∇ p). The misalignment of the local pressureand density gradients leads to the non-zero source termin equation (<ref>). Because of the curved surface of the bubble,different parts of the incident planar shock will strike the bubble surface at different times, so the refracted interior wave will be misaligned with the density gradient. The baroclinic torque is the largest where the pressure gradient is perpendicular to the density gradient. Whereas, at the mostupstream and downstream poles of the gas bubble, the baroclinic torqueis equal to zero, owing to the collinearity of density and pressure gradients.The curvature of the shock wave front (the refracted shock wave) has also been used to build a theory behind the vorticity generation. It originatesfrom the conservation of the tangential velocity and the angular momentum acrossthe shock wave as the compression only affects the motion normal to the shocksurface (for details see <cit.> and also recent works of <cit.>). The rotational motion starting from zero-vorticity initial conditions distorts the flow field and the shape of the bubble.The lighter density fluid will be accelerated faster than the high density fluid. §.§ Validation of shock-single bubble interaction casesThe experimental studies performed in <cit.> and <cit.>constituted validation cases for the numerical approach developedin the present paper. The authors provided sequences of flow structuresresulting from experiments and compared them with numerical simulationsthey also conducted. The mathematical model and numerical techniqueconsidered in the present contribution are fundamentally differentfrom the ones utilised in the reference works of <cit.> and <cit.> and hence the main featuresof their approaches are highlighted.The authors in <cit.> employed a homogenization method known as the discrete equations method (DEM), which was earlierintroduced by <cit.>. In this approach the averaged equationsfor the mixture are not used. Instead the DEM method obtains a well-poseddiscrete equation system from the single-phase Euler equations byconstruction of a numerical scheme which uses a sequence of accuratesingle-phase Riemann solutions. The local interface variablesare determined at each two-phase interface. Then, an averaging procedurewhich enables coupling between the two fluids is applied generatinga set of discrete equations that can be used directly as a numerical scheme.The advantage of such an approach is its natural ability to treat correctly the non-conservative terms. In our approach the solution strategy to handlenon-conservative terms is different. It requires the usage of an additionalconservative equation for the total mixture energy (<ref>). As a resultthe present model enables a correct transmission of shock waves throughthe heterogeneous media. The volume fraction positivity in the numericalsolution is also preserved.The authors in <cit.> adopted the 2D axisymmetrical numericalapproach of <cit.> and solved the mixture Euler equationssupplemented by one species conservation equation to capture the interface.It is assumed in this approach that the gas components are in pressure equilibrium and move with a single velocity. This assumption restrictsthe approach to the cases when the density variations between componentsare moderate.§.§.§ Experiments of Layes and Le Métayer <cit.>These experiments were reproduced numerically for three different casesin which Ma = 1.5 planar shock wave interacts with a helium, nitrogenor krypton cylindrical bubble of a diameter D_o = 0.04 m locatedin a shock tube. The thermodynamic properties of the bubbles and surroundingair are given in Table <ref>. The schematic diagram of the computationaldomain and the initial set-up is shown in Fig. <ref>. The shock tubedimensions are L = 0.3 m and H = 0.08 m. The initial positionof the shock is X_o = 0.05 m. The solid walls are treated as reflectingboundary conditions. The inflow boundary conditions are set to the exactpre-shock region parameters summarised in Table <ref> and the standardzero-order extrapolation is used as the outflow boundary conditions.In all three cases the bubble was initially assumed to be in mechanical andthermal equilibrium with the surrounding air. The shock wave propagates in the air from right to left and impacts the bubble. The Atwood numbers arelisted in Table <ref> and represent three distinct regimesof shock-bubble interactions.The first case corresponds to the interaction of a shock wave witha helium bubble surrounded by ambient air. As the density of heliumis lower than the density of air this case represents a heavy/lightinteraction. The second test considers the interaction of the shockwave with a nitrogen bubble. Owing to the very small density ratio betweennitrogen and air, this case is treated as an equal density problem.Finally the third case with a krypton bubble, which is heavier than air,represents a light/heavy interaction problem.The domain is discretised using a regular Cartesian grid consistingof 2700×720 cells, which corresponds to the resolution of 360 cellsacross the bubble diameter. The CFL number is 0.3. The present resolution has been chosen based on information from numerical tests on meshes withdifferent levels of refinement. Table <ref> summarisesthe computational times for a selection of mesh resolutions and provides the circulation values for air/He and air/Kr pairings at the physicaltime 60 μs. The computational times are normalised by the longest simulation run.The difference between the total circulation values of the coarse meshof 900×240 and the refined mesh of 2700×720 is only 1.5%. Figures <ref> and <ref> show the convergence of the solutionas the effective resolution is increased, by comparing the evolution ofthe pressure and density along the centre line of the domain(y = 0.04 m) at time 410 μs for the air/He and air/Krarrangements respectively. The changes in the thermodynamic values are very small, especially between the two refined meshes. After performing numerical discretization tests, the experimentalshadowgraph frames presented in <cit.>, were reproducednumerically. The computational simulations of the density fieldare presented by means of the idealized schlieren function atthe same instants as in the experiment in <cit.>. Although theshapes of the deformed interfaces are recovered and can be observedclearly for different gas pairings it has to be noted that the accuracyof investigation is a function of experimental reproducibility.It is difficult to maintain the initial parameters of the shockwave Mach number, shape as well as size of the bubble and the gascomposition from one to another experimental realisation <cit.>. For example the tolerance for Ma = 1.5 in the experiment was within the range: 1.45-1.52. In spite of these difficulties the numerical results show a good approximation of the density contourplots obtained in the reference experiment. The positions of the characteristic interface points are recorded against time in Fig. <ref>. This figure also shows the changing positions of the incidentand transmitted shock waves for air/He, air/N_2 and air/Kr configurations alongthe tube, which are measured from the shock initial position X_0, Fig. <ref>. The dynamic evolution of a bubble is observed by tracking points (1) and (2)originally placed on its contour, see Fig. <ref>. Point (1) is associated withthe upstream front position and point (2) is the most downstream interface point.The usage of these tracking points to record numerically calculated spatial positionsfollows directly the experimental convention described in <cit.>. As it was earlier mention reproducible experiments with the same size of bubble were difficultto obtain. For example, the initial diameter of the nitrogen bubble in the experimentaldata is slightly larger than 4 cm, Fig. <ref>(b2). Therefore the data from tworealizations of the same experimental procedures presented in <cit.>,are utilized in the present comparison study. In spite of these difficultiesthe quantitative analysis of the computed positions in Fig. <ref> shows excellentagreement with the experimental findings. The numerical results confirm the validityof the underlying governing equations and numerical method. The first study presented in Fig. <ref>(a1, a2) shows the results forthe helium bubble with lower acoustic impedance than the surrounding air.The difference in densities and therefore the higher sound speedin helium (1007 m/s) than in air (340 m/s) results in a higher speed for thetransmitted shock through the helium bubble than for the incident shock in the air.The waves merge after passing the bubble to form a normal shock wave at around270 μs Fig. <ref>(a1). The early stages of the physical process reproducednumerically confirm the vorticity generation by the baroclinic effects. The rear interfaceof the helium bubble is caught by the front interface and the bubble evolves into a kidneyshape. A penetrating high velocity jet along the flow direction moves through the bubbleforming two symmetric flow configurations, Fig. <ref>. When the bubble deformsthe associated flow field is subsequently split into two rings of vorticity.This characteristic separation further intensifies the deformation of the inhomogeneity.The second study presented in Fig. <ref>(b1, b2) characterizes the interactionof a shock wave encountering a nitrogen bubble with comparable acousticimpedance as the surrounding air. The nitrogen and the air densities havesimilar values and therefore the corresponding Atwood number is close to zero.In such flow regime both the incident and the transmitted shock waves propagatewith a small difference in velocities. After approximately 180 μsthe waves are combined again to form a planar shock wave. The generation andsubsequent development of the vorticity field is negligible in this case.The compression process dominates the flow and the bubble evolution. The shapeof the nitrogen bubble does not change significantly with time after around200 μs when the compression rate stabilizes.The third study included in Fig. <ref>(c1, c2) reveals the numerical resultsfor the krypton bubble. The krypton acoustic impedance is higher than the air acoustic impedance. Such situation makes the transmitted shock throughthe krypton bubble moving more slowly than the incidentshock in the surrounding air. These waves fully converge after 300 μs.This case clearly shows that the vorticity drives the distortion mechanism.The shock passage generates vorticity on the bubble interface owing to misalignmentof the pressure and density gradients across the interface.The vortical flow then distorts the bubble interface together witha penetrating jet that is generated after around 160 μsalong the symmetry line of the bubble which moves upstream towardsthe right hand side. In all these cases the different times at whichshocks leave the tube were recorded. The accelerated shock in thehelium case left the tube after around 480 μs,in the nitrogen case the shock left the tube at around 500 μs and finally, in the krypton case the shock wasdecelerated and left the tube after 510 μs. §.§.§ Experiments of Zhai et al. <cit.> The comparison study with both experimental and numerical results wascarried out for the interaction of a weak Ma=1.2planar shock wave with a sulphur hexafluoride SF_6 bubble immersedin air. The Atwood number for this configuration is A=0.66.The advantage of the considered experiment over the previous investigationsof <cit.> is the application of a high speed schlieren photographywith higher time resolution. This allows precise validations of the locationof the wave front evolution both inside and outside the gas bubble at the veryearly stages of the interaction.As in <cit.> the experiment was performed using a rectangular shocktube, Fig. <ref>, but with different dimensions of the observation windowwhich were 0.07×0.5 m. The bubble initial diameteris D_0=0.03 m and the centre is located at a 0.02 m distance from the shock. The numerical initial conditions are set per analogyto the previous investigation and are summarised in Table <ref>.The 2000×560 computational grid provided a resolutionof 240 cells per bubble diameter. The CFL number was set tobe equal to 0.3. The schlieren images from the present simulationare collected in Fig. <ref>.The time interval between consecutive images is set to10 μs to capture the same sequences of the processas presented earlier in the experiment of <cit.>. Theimages reveal the characteristic moments of the interaction andare in a very good agreement with the experimental and numerical results of <cit.>. The images are numberedusing the same convention as in the reference paper. The transmitted shock wave takes the convergent shape owing tothe difference in acoustic impedance, Fig. <ref> (images 1 to 5).Images 6 to 9, in the same figure, show two parts of the incident shockwave passing the top and the bottom poles of the bubble and moving towardsthe most downstream point of the interface. The transmitted shock startsto converge inside the bubble towards the centre of the downstream interface,Fig. <ref> (images 10 to 13). As a result the formation ofthe penetrating jet can be observed in the following images. The processis driven by a high pressure zone resulting from the shock formation whichconcentrated at the downstream pole, Fig. <ref> (images 14 and 15).This causes an explosion producing a refracted shock wave moving throughthe downstream boundary of the bubble from left to right and a shock wavepropagating inside the bubble.The evolution is recorded using x-t diagrams. Figure <ref>defines the different tracking locations for the consideredshock-bubble interaction reproduced in Fig. <ref>. The positions of the upstream interface (P1), downstream interface (P2), refracted (Rr), reflected (R1) and transmittedshock (Tr) are obtained at the horizontal axis while the incident shock (Ins)wave is measured at the undisturbed locations above the bubble.Figure <ref> refers to the distinct tracking points indicated in Fig. <ref>. The positions of these characteristic points,representing interface and various waves involved, are determinedduring the numerical simulation and their evolution is comparedwith (a) the experimental and (b) the numerical resultsreported in <cit.>. The numerical predictions are in perfectagreement with the experimental data in the first stages of the interaction although a small difference in the position of the most downstream pole ofthe bubble is observed. Table <ref> lists the velocities associated with the different shock waves.V_jet is the velocity of the jet and t_jet refers to the time at whichthe jet starts to form. Slight differences can be noticed between the numericaland experimental results. The main reason for these differences is due to theimpurity of the gases inside and outside the bubble in the experiment which wasacknowledged by <cit.>.§.§ Interface evolution and vorticity productionas a function of Mach and Atwood numbers After these successful validations the numerical procedures are applied to examine additional cases for which experimental datacannot be collected owing to the restrictions set by the physical apparatus.These new computational simulations consider the effect of a wider rangeof Atwood numbers on the shape of the interface as well as the effectof the Mach number on the interface growth and development. The influence of the Atwood and Mach number changes on the baroclinic source of the vorticity field is also investigated. Apart from the gases consideredin the previous section the extra cases include the pairings of air/argon (Ar)with A = 0.13. Therefore the numerical study accounts for the total of five different bubble/air configurations interacting with the waves for which Machnumbers were set to be 1.5, 2, 2.5 and 3. The initial data for thesesimulations are listed in Tables <ref> and <ref>. Figure <ref> shows the mixture density field profiles for different air/gas constitutions captured at the same early stage (60 μs) of the bubble interactions with a shock wave of Ma = 1.5. The form of deformation of the bubble during the penetration of the shock wave is determined by thedensity and acoustic impedance of each constituent. The fastest speed of penetration was observed in the He bubble and this speed was at the sametime faster than the normal incident shock. The situation was different in the cases of N_2 and Ar, where a slight difference between the speeds of transmitted and incident shock waves results in a smaller deformation of the bubble. The case of Kr and SF_6 exemplified a scenario opposite from that for He. In these cases the transmitted shock propagates through the bubbles more slowly than the incident shock outside the bubbles boundary. The transmitted shock in the SF_6 bubble moves also more slowly thanin the Kr bubble. The early stages of the shock-bubble interaction in thelast two cases did not allow for large deformations of the bubble. Figure <ref> illustrates the relation between the velocity ratio(U_transmitted/U_incident) and the Atwood number at time60 μs. It is observable that as the Atwood number goes towards positive values and becomes larger, the transmitted shock propagatesthrough the bubble more slowly. Figure <ref> shows the incident shockposition and the location of the bubbles, filled with different gases, along the domain as a function of time for two different Mach numbers:Ma = 1.5 and Ma = 2.5. It is confirmed inFig. <ref>(a1) and (b1) that the incident shock travels through the domain containing light bubbles faster than in the cases of heavy bubbles.The fact that the transmitted shock could accelerate or deceleratein the bubble environment has consequences at a later time,when the transmitted shock leaves the bubble and eventually combines withthe incident shock. These are manifested by higher or lower wave speeds as compared to the medium containing a bubble with comparable physical properties to the surrounding medium or not containing a bubble at all. The location of the bubbles has been measured by tracking point (1) on the front of thebubble (upstream side, Fig. <ref>). The interpretation ofFigs. <ref>(a2) and (b2) confirms that as the bubble is heavier it moves more slowly. Similarly these figures show the effect of the Mach number on the movement of the gas bubbles.The bubble covers a longer distance with higher Mach numbers. To assist in understanding the changes in the dynamics of the interface,the velocity values associated with point 1 on the upstream pole of thegas bubble were monitored. The values were collected for the Mach numbers1.5 and 2.5, Fig. <ref>.Figure <ref> presents the evolutionary patterns of the interfacesrepresented by the volume fraction contours for various air/gasconfigurations and Mach numbers. The images are all taken at the samephysical time equal to 236 μs after the shock started to interact with the gas bubbles. Figure <ref> can be read either fromleft to right (horizontal images - increase of Mach number)or from top to bottom (vertical images - increase of Atwood number).In the horizontal view one can see the effect of the Mach number on theinterface evolution. The higher the Mach number the more changes in theinterface shape, which is clearly seen in the last row of this horizontalview where the bubble undergoes distortion and consequently is dividedinto three entities with a significant interface evolution. In the caseof the He bubble two symmetric contours can be observed as a result ofthe shock-bubble interaction. At the same time a highspeed penetrating jet develops along the axis of symmetry in the mainflow direction. The formation of the symmetric contours is more pronouncedwith the higher Mach numbers, leading eventually to a faster splitting of the bubble into two entities. A different situation is observed in case of the N_2 bubble. Here, the bubble is experiencing a compression process which intensifies with the higher Mach number. It is also found that the compressionprocess happens at the early stages of the shock-bubble interactions allowing the bubble to stabilize its shape after around 200 μs fromthe start of the interaction. This physical behaviour can be attributedto the fact that there is no penetrating jet or associated vorticity field(as the vorticity values are negligible in this case) which is clearly a direct consequence of the small density ratio of the constituents. Figure <ref>illustrates the rate of compression of the N_2 bubble as a functionof Mach number at the time 510 μs. The compression ratio increases with the Mach number and it is measured by dividing the horizontal diameterof the bubble (D_x) at time 510 μs by the initial diameter(D_o). The Ar, Kr and SF_6 bubbles undergo a similar physical processuntil the moment when the baroclinic source of vorticity comes into play.This leads to greater bubble deformation and its interface distortionis even more apparent with the increasing Mach number. Another distinctfeature of this process is the formation of the penetrating high speed jetalong the bubble axis of symmetry, which moves in the opposite direction to the normal shock wave. The interface changes and jet development are clearer for a higher Atwood number Kr and SF_6 bubbles. The cases withthe higher absolute value of the Atwood number experience a higher rate of the bubble deformation with increasing Mach numbers. In contrast,for the Atwood number close to zero the deformation rate of the interfaceis relatively slow. Figure <ref> shows the development of the vorticity field for all considered bubbles as a function of the Mach number.The snapshots were taken at the same time t = 166 μs.These pictures assist in the interpretation of the interfaceevolution process discussed previously, in which the vorticitycreation plays an important role. To better understand Fig. <ref>,the vorticity field for different gases and Mach numbers is quantifiedby calculating the total circulation generated in the symmetrical halfof the computational domain. The circulation values are listed in Table <ref> fordifferent Atwood and Mach numbers.The time evolutions of these values are presented in in Fig. <ref>.The shock propagates from the right to the left. Therefore in thecase of light bubbles positive (anticlockwise) vortices are generated on the bottom side of the bubble and negative (clockwise) vortices are generated on the top side of the bubble.An opposite scenario is observed for the heavy bubbles, where vortices with positive sign are generated on the top and negative sign onthe bottom of the bubbles. In later times(see especially the SF_6 evolution in Fig. <ref>),a dilation of the vorticity torus induced by its spiral effect canbe observed. Looking at the values of the calculated circulation one canconclude that for the Atwood numbers of relatively high absolute valuesthe vorticity generation rate becomes higher. When the Mach numberis increased the value of the total circulation is also higheras its growth rate during the shock-bubble interaction processbecomes faster. The effect of the vorticity on the N_2 bubbleis negligible owing to the small differences in densities and acousticimpedance between N_2 and the surrounding air. §.§ Influence of the heat capacity ratio (γ)on the bubble compression In addition to the essential effect of the density ratio acrossthe interfaces and the corresponding acoustic impedance difference,there is another fundamental parameter that contributes to the interface evolution. The heat capacity ratio γ influences bubblecompression and deformation.This parameter monitors how compressible the medium is.For example, SF_6 with γ = 1.08 is much more compressible than the other mono and di-atomic gases, such as helium and nitrogendiscussed in the previous section. In most of the literature concernedwith the shock-bubble interaction problem, the analysis and discussionsof this phenomenon is focused on the role of the acoustic impedanceand the pressure misalignment on the interface deformation and vorticityproduction. The effect of γ was not highlighted. A new hypothetical shock-bubble interaction case study is designedto address the role of γ. This case study considersa planar shock of Ma = 1.5 propagating in ambient airand interacting with an air bubble characterised by the same densities,i.e. zero Atwood number, but different values of γ. That is γ=1.2, 1.4 and 1.67.The physical domain, Fig. <ref>, and computational set-up arethe same as in subsection <ref>.Figures <ref>(a) to (c) show the volume fraction contoursat the same physical time 306 μ s from the beginning ofthe interaction for these three values of γ. The bubblesunderwent a compression process as in the case of air/N_2 thatwas shown previously. However, the contour of each bubble hasa slightly different shape. Figure <ref>(d) summarisesthe relative change of these different shapes of the bubbleswith respect to γ. While for the case ofγ = 1.2 the bubble compresses more than the othertwo bubbles, the bubble with the largest gamma stretchesvertically more than the others as shown in Fig. <ref>(d).Figure <ref> quantifies the variation of the size ofthe bubbles along their horizontal and vertical diameters,with respect to time. D_0 is the initial diameter andD_x and D_y represent bubble horizontal and verticaldiameters during the process of the shock-bubble interaction.The numerical results confirmthat the heat capacity ratio γ has an effecton the interface deformation. However, one has to remember that using this parameter separately in the discussion can be misleading. This is because the heat capacity ratio effectsare already indirectly included in the acoustic impedance sincethe calculation of the sound speed of the gases acrossthe interfaces requires γ. The total circulation values recorded for various γ are listed in Table <ref>. Although these values arenegligible owing to small difference in acoustic impedance,they confirm the observations discussedin section <ref>.§ CONCLUSIONS The computations of flows in inhomogeneous media of various physical regimes leading to shock-bubble interactions were performed usinga newly developed numerical code based on an Eulerian multi-component flow model. The numerical approach was validated using available data from shock tube experiments, for which very good qualitative and quantitative agreements were found. The present numerical approach could be applied to design better shock-induced mixing processes. In order tobetter understand the bubble shape changes and describe the mechanism of its interface deformation, the study was extended to include additional cases for which experimental data cannot be collected. These enabled us to account for the effect of the Atwood number and shock wave intensity (various Mach numbers) on the interface evolution and on the vorticity generation within the surrounding medium. The constant Mach numbercomparison showed that the Atwood number increase leads to highervorticity generation and its effect on the interface evolution becomes more pronounced.Similarly the constant Atwood number comparison shows that increasing the Mach number produces a higher circulation which also means a highervorticity generation. Apart from highlighting the cases characterisedby the difference in acoustic impedance the study was extended to account for the influence of the heat capacity ratio γ of the heterogeneousmedia on the interface deformation. The results of this study, which couldpotentially constitute a benchmark test for other numerical simulations,confirm that the baroclinic term in the vorticity transport equationhas a large effect on the interface evolution and the vorticity generation.The 2D simulations can only be used as a platform for the analysis of earlystages of a shock wave-spherical bubble interaction. For longer timeperiods after a planar shock-inhomogeneity interaction, the flow becomes 3Dand vorticity structures are influenced by vortex stretching. | http://arxiv.org/abs/1706.08847v1 | {
"authors": [
"A. F. Nowakowski",
"A. Ballil",
"F. C. G. A Nicolleau"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170627135059",
"title": "Passage of a shock wave through inhomogeneous media and its impact on a gas bubble deformation"
} |
Accepted in "Rendiconti Lincei - Matematica et Applicazioni (2017)"Chapter 41 - Mathematical Biology 0.5cm Continuum mechanics at nanoscale :0.3cmAtool to studytrees' watering and recovery0.3cmHenri Gouin 0.2cm Aix-Marseille Univ, Centrale Marseille, CNRS, M2P2 UMR 7340, 13451 MarseilleFrance 0.2cmE-mail: [email protected], [email protected] 0.4cmIn memory of Professor Giuseppe Grioli0.5cmAbstractThe cohesion-tension theory expounds the crude sap ascent thanks tothe negative pressuregenerated by evaporation of water fromleaves.Nevertheless, trees pose multiplechallenges andseemto live inunphysical conditions: the negative pressure increases cavitation;itis possible to obtain a water equilibrium between connected parts where one is at a positive pressure and the otherone is at negative pressure;no theory isable to satisfactorily account for the refilling ofvessels after embolism events.A theoreticalform of our paper <cit.> inthe Journal of Theoretical Biology is proposedtogether with new results:a continuum mechanicsmodel of the disjoining pressure concept refers to the Derjaguinschool of physical chemistry. A comparison betweenliquidbehaviour both in tight-filledmicrotubes and in liquid thin-films is offered when the pressureis negativein liquid bulks andis positive in liquid thin-filmsand vapour bulks. In embolized xylem microtubes,when the air-vapour pocket pressure is greater than the air-vapour bulk pressure, arefilling flow occursbetween the air-vapourdomains to empty the air-vapour pockets although the liquid-bulk pressure remains negative.The model hasa limit of validitytaking the maximal size of trees into account. These results drop an inkling that the disjoining pressure is an efficient tool to study biological liquids in contact with substrates at a nanoscale range.PACS numbers: 68.65.k; 82.45.Mp; 87.10.+e; 87.15.Kg; 87.15.La———————————————————————————————–§ INTRODUCTION Trees are engines running on water,but unlike animals, plants miss an active pump to move liquids along their vascular system. The crude sap contains diluted salts and ascents from roots to leavesthanks to the water evaporation from leaves; its physical properties are roughly those ofwater. The flow is driven along of xylem microtubes made of dead cells which constitute a watering network.Hydrodynamics, capillarity and osmotic pressure induce the crude sap ascent of a few tens of meters only <cit.>; nevertheless a sequoia of 115.55 meters height is living in California <cit.>. Additively, trees operate a second vascular system - phloem sieve tubes - for the circulation of metabolites through theirliving tissues and elaborated sap flows passing from leaves to roots. Measurements of the pressure within the terminal xylem vessels illustrate an extraordinary consequence of the tree behaviour for moving water: the liquid water is under tension. An experimental checking comes from an apparatus called Scholander pressure chamber(see Fig.<ref>, <cit.>). The pressure difference across plants can easily be of the order of 1 to 10 MPa <cit.>. Although trees do not approach the ultimate tensile strength of liquid water during transpiration <cit.>, multiple types of measurements provide evidence for cavitation of liquid waterin the xylem microtubes and cavitation events have been acoustically detected with ultrasonic transducers pressed against the external surface of trees <cit.>. The porous vessel walls can prevent thebubbles from spreading <cit.> and the principal flow of water during transpiration goes to evaporation through stomata on the underside of leaves. The pores - or bordered pits - connecting adjacent segments in the xylem vessels pass through the vessel walls, and are bifurcated by bordered-pit membranes which are thin physical fluid-transmitters. No vessels are continuous from rootsto petioles, and the water does not leave the vessels in the axial directionbut laterally along a long stretch <cit.>.When wetted on both sides, the bordered-pit membranes allow the liquid-water flow to pass through. In the leaves, the membranes serve as capillary seals; in the stems, the bordered-pit membranes also serve as seals between a gas-filled segment and an adjacent liquid-filled segment avoiding propagation of massive embolisms <cit.>. Consequently, trees seem to live in unphysical conditions <cit.>; to be hydrated, they exploit liquid water in metastable states at negative pressure <cit.>. A classical explanation of the sap ascent phenomenon in tall trees is thecohesion-tension theory propounded in 1893-1895 by Boehm, Dixon and Jolyand Askenasy <cit.>, followed by an analysis of the sap motion propounded by van der Honert <cit.>. According to this theory, the crude sap tightly fills microtubes of dead xylem cells and its transport is due to a gradient of negative pressure producing the traction necessary to lift water against gravity. The decrease innegative pressure is related to the closing ofaperture of microscopic stomata in leaves through which water vapour is lost by transpiration.The considered aperture is about 2 μm - or less at the top of tall trees as suggested in<cit.> - which is the right size to prevent cavitation for nucleus germs of the same order of magnitude. Nonetheless, several objections question the validity of the cohesion-tension theory, and worse, preclude the possibility of refilling embolized xylem tubes. To this goal, we first refer to the textbook by Zimmermann <cit.>. He said:The heartwood is referred to as a wet wood. It may contain liquid under positive pressure while in the sapwood the transpiration stream moves along a gradient of negative pressures. Why is the water of the central wet core not drawn into the sapwood? The heartwood is relatively dry i.e. most tracheids are embolized. It is rather ironic that a wound in the wet wood area, which bleeds liquid for a long period of time, thus appears to have the transpiration stream as a source of water, in spite of the fact that the pressure of the transpiration stream is negative most of the time. It should be quite clear by now that a drop in xylem pressure below a critical level causes cavitations and normally puts the xylem out of function permanently.At great elevation, the value of the negative pressure increases risks of cavitation and consequently, the formation of embolisms may cause a definitive break-down ofcontinuous columns of sap inducing leaf death. Crude sap is a fluid with superficial tensionlower than superficial tension γ of the pure water, which is about 72.5 × 10^3 N.m^-1 at 20^ Celsius <cit.>; if we consider a microscopic air-vapour bubble with a diameter 2R smaller than xylem microtube diameters, the difference between air-vapour pressure ℘_v and liquid sap pressure ℘_l is expressed by Young-Laplace formula ℘_v - ℘_l = 2γ /R; the air-vapour pressure is positive and consequently unstable bubbles will appear when R ≥ - 2γ/℘_l. For a negative pressure ℘_l =- 0.6 MPa in the sap, corresponding to an approximative minimal value of the hydrostatic pressure for embolism reversal in plants of Laurus nobilis <cit.>, we obtain R ≥ 0.24 μm; then, when all the vessels are tight-filled, germs naturally pre-existing in crude water may spontaneously embolize the tracheids.Another objection to the confidence in the cohesion-tension theory was also the experiment which demonstrated that tall trees survived double saw-cuts, made through the cross-sectional area of the trunk to sever all xylem elements, by overlapping them <cit.>. This result, confirmed by several authors does not seem to be in agreement with the possibility of strong negative pressures in the water-tight microtubes <cit.>. Using a xylem pressure probe, Balling & Zimmermann <cit.> showed that, in many circumstances, the apparatus does not measure any water tension <cit.>. However, there are other possibilities for the tree survival and researchers presented some experimental evidences for the local refilling that restores embolized conduits by visualizing the conduits with microscope <cit.>. A negative argument also seems to come from the crude sap recovery in embolized xylem tubes.At high elevation, it does not seem possible to refill a tube full of air-vapour at a positive pressure when liquid-water is at a negative pressure. In xylem, the liquid-watermetastability - due to negative pressures - may persist even in the absence of transpiration. Consequently, refilling processes pose a tough physical challenge to push the liquid-water back intoxylem vessels: once embolized vessels have reached a nearly full state, is the refilling solution still at positive pressurewith some remaining air? The most popular theory of refilling process has been proposed in several papers. Due to the fact that xylem microtubes are generally in contact with numerous living cells <cit.>, it is hypothesized that crude sap is released into the vessel lumen from the adjacent living cells in a manner similar to root exudation <cit.> and it is assumed that the mechanism for water movement into embolized conduits involves the active secretion of solutes by the living cells <cit.>. Nonetheless, a survey across species indicated that the root pressure can reach 0.1-0.2 MPa above atmospheric pressure <cit.> and seems the only logical source of embolized vesselsrepairing at night in smaller species with well-hydrated soil. TheMunch pumping mechanism <cit.> was invoked, but basic challengesstill persist: osmotic pressures measured in sieve tubes do not scale with the height of a plant as one would expect <cit.> and such scenarios have not yet been empirically verified. Hydraulic isolation is also required to permit the local creation of the positive pressures necessary to force the gas into solution and the embolism removal may be concurrent with tree transpiration <cit.>. Additively, refilling in the presence of tension in adjacent vessels requires the induction of an energy-dissipating process that locally pumps liquid into the emptied vessels <cit.> or lowers the water potential in the vessel with the secretion of solutes <cit.>. As a consequence, many authors suggested that alternative mechanisms must be required <cit.>. Nowadays,thedevelopment of techniques allows us to observe phenomena at length scales of a very few number of nanometers. This nanomechanics reveals new behaviors, often surprising and essentially different from those that are usually observed at macroscopic but also at microscopic scales <cit.>. As pointed out in experiments, the density ofwater is found to be changed in narrow pores. The first reliable evidence of this effect was reported by B.V. Derjaguin, V.V. Karasevand E.N. Efremova <cit.> and found after by many others <cit.>, pp. 240-244. In order to evaluate the structure of thin interlayers of waterand other liquids, Green-Kelly and Derjaguin employed a method based on measuring changes in birefringence <cit.> and they found significant anisotropy ofinterlayers. Slightly compressible liquids wetting solid substratespoint out an unexpected behaviour in which liquids do not transmit the pressure to all their connected domains <cit.>: it is possible to obtain an equilibrium between connectedparts where one is at positive pressure - the pressure in a liquid thin-film - and the other is atnegative pressure - the pressure in the liquid bulk. The air-vapour phase in contact with the liquid thin-film is at the same positive pressure as the liquid thin-film. The experiments and model associated with this behaviour fit the disjoining pressure concept<cit.> which is a well adapted tool for a very thin liquid film of thickness h. In cases of Lifshitz' analysis <cit.> and van der Waals' theory <cit.>, behaviours of disjoining pressure Πare respectively as Π∼ h^-3 and Π∼exp ( -h). None of them fitsexperimental results for a film with a thickness ranging over a few nanometers.Since van der Waals, the fluid inhomogeneities in liquid-vapour interfaces have been represented with continuous models by taking a volume energy depending on space density derivativeinto account <cit.>. Nevertheless, the corresponding square-gradient functional is unable to model repulsive force contributions and misses the dominant damped oscillatory packing structure of liquid interlayers near a substrate wall <cit.>. The decay lengths are correct only close to the liquid-vapour critical point where the damped oscillatory structure is subdominant <cit.>.In contrast, fluctuations strongly damp oscillatory structure and it is mainly for this reason that van der Waals' original prediction of a hyperbolic tangent curve in density is so close to simulations and experiments <cit.>. To propose an analytic expression in density-functional theory for liquid film of a very few nanometer thickness near a solid wall, we add a liquid energy-functional at the solid surface and a surface energy-functional at the liquid-vapour interface to the square-gradient functional representing the volume free energy of the fluid. This kind of functional is well-known in the literature <cit.> and the process is simpler than the renormalization group theory <cit.> mainly used near critical points. It was used by Cahn in a phenomenological form<cit.>. An asymptotic expression is obtained in <cit.> with an approximation of hard sphere molecules for liquid-liquid and solid-liquid interactions: in this way, we also took account of the power-law behavior which is dominant in a thin liquid film in contact with a solid<cit.>. The paper is organized as follows:Section 2 expounds that nanofluidic and liquid thin-films concepts arefundamentaltools used in the paper; following Derjaguin's Russian school of physical chemistry, wepropose an experimental overview of the disjoining pressure concept for liquid thin-films at equilibrium. Section 3 is an analytical and numerical study of the disjoining pressure along vertical liquid thin-films. Section 4 studies the liquid motions along vertical liquid thin-films, and Section 5 is a comparison betweenliquid-motions' behaviours both in tight-filledmicrotubes and in liquidthin-films. Section 6 focuses on trees containingvessels considered as machines. From experiments presented inprevious sections, a model of xylemusing liquid thin-films is proposed.Such amodel of xylem allows to explain both the thermodynamicalconsistence ofthe cohesion-tension theory and the conditions of the crude-sap refilling at high elevation. This previous thought experiment is modified to takeaccount of air-vapour pockets:when the air-vapour pocket pressure is greater than the air-vapour bulk pressure, a huge flow occursbetween the two parts filled by air-vapour gas to empty the air-vapour pockets although the liquid-bulk pressure is negative. Finally, the pancake-layer concept, associated with the breaking-down of vertical liquid thin-films, allows to forecast the limit of validity of the model and yields a maximum height for the tallest trees. We present new results concerning models and numerical calculations for comparing filled microtubemotionsand thin-film motions, a new study of laterally transfer of masses between xylem microtubes and an explanation for ultrasounds eventually generated in the watering network. A conclusion ends the article. The apparent incompatibility between the model in <cit.> and the cohesion-tension theory is now solved. Experiments are suggested toverify the accuracy of the sap ascent for tall trees and of the crude-sap's refilling. § THE DISJOINING PRESSURE The disjoining pressure concept is associated with liquid thin-films bordered by vapourbulks and wettingflat solid surfaces. Experiments and analysis are described by Derjaguin et al <cit.>. Atgiven temperature T__0, two experiments allow to understandthe physical meaning of horizontalliquid thin-films at equilibrium. ∙ The first experiment wasdescribed in <cit.> pp. 330–331: a liquid bulk submitted topressure P_l_b contains a microscopic bubble of radius R contiguous to a solid(Fig. <ref>). Thebubble floats upward and approaches a horizontal smooth plate, and a planar liquid thin-film is formed after some time. The liquid thin-film separates the flat part of the bubble which is squeezed ontothe solid surface, from inside. Inside the bubble,the pressure of vapour bulkv_b of density ρ_v_b (mother vapour-bulk)is P_v_b. The film is thin enough for gravity to be neglected thickness-wise and the hydrostatic pressure of the liquid thin-filmis identical to the vapour-bulk pressure inside the bubble. Pressure P_v_b differs frompressure P_l_b of liquid bulk l_b of density ρ_l_b (mother liquid-bulk) <cit.>, page 32. Theanalysis can apply to the bulk pressure P_l_b in the liquid atshort distance away from the surface;bulk pressure P_l_b is not really affected by the gravity because of the microscopic size of the bubble which remains spherical outside the liquid thin-film.The Young-Laplace formula describes the difference between the two bulk pressures: P_v_b- P_l_b = 2γ/R, where γ is the surface tension of the bubble liquid-vapour interface. The liquid thin-film extends from the bulks which createthe pressure difference already estimated in Eq. (<ref>)and named Π(h):Π(h)=P_v_b-P_l_b.Interlayer pressureΠ(h) additional to the mother liquid-bulk pressure is calledthe disjoining pressure of the thin film of thickness h, and curve h ⟶Π(h)- obtained by changing the bubble's radius and thereby film thickness h -is the disjoining pressure isotherm. ∙ The second experimentis associated with theapparatus due to Sheludko<cit.> and described in Fig. <ref>. The film is thin enough such that the gravity effect is neglected across the liquid layer. The hydrostatic pressure in the thin liquid layer included between a solid wall and the vapour bulk differs from the pressure in the contiguous liquid bulk from which the liquid layer extends (this is the reason for which Derjaguin usedmother-bulk term). The forces arising during the thinning of the film of uniform thickness h produce the disjoining pressure which is the additional pressure on the surface of the filmto the pressure within themother liquid-bulk. Clearly, a disjoining pressure could be measured by applying an external pressure to keep the complete layer in equilibrium and verifies Eq. (<ref>). Derjaguin's clever idea was to create an analogy between liquid thin-films and liquid-vapour interfaces of bubbles. Liquid thin-films - allowingto obtain an equilibrium between fluid phases at different pressures - are physically similar tobubbles' flat interfaces.The pressure in the liquid phase is differentfrom the liquid pressure in the liquid thin-film, which is the same asthe pressure in the vapour phase; thereby, the liquiddoes not completely transmitthepressure in all places where itlays.Let us consider the Gibbs free energyof theliquid layer (thermodynamic potential). As pointed out by Derjaguin et al in(<cit.>, Chapter 2), at temperature T_0,the Gibbs free energy per unit area G can be simply expressed as a function ofh:dG(h)/dh = -Π(h),and can be integrated as:G (h) = ∫_h^+∞Π(h) dh,where h=0 is associated with the dry wall in contact with the vapour bulk and h=+∞ is associated with a wall incontact with liquid bulkwhen the value of G is 0. An important property related to theproblem of wettingis associated with thespreading coefficient <cit.>: S = γ__SV - γ__SL-γ__LV,whereγ__SV, γ__SL, γ__LV are respectively thesolid-vapour, solid-liquid and liquid-vapour free energies per unit area of interfaces.The liquid-layerenergy per unit area can be written as E =γ__SL+γ__LV+ G(h).When h=0, we obtain the energy γ__SV of the dry solid wall; when h=+∞, we obtain γ__SL + γ__LV. In complete wetting of liquid onsolid wall, the spreading coefficient is positive. The conditions of stability of a liquid thin-film essentially depend on phases between which the film is sandwiched. In case ofsingle film in equilibrium between the vapour and a solid substrate, the stability condition is classically: ∂Π (h)/∂ h < 0⟺∂^2 G (h)/∂ h^2 > 0 . The coexistence of two film segments with different thicknesses is a phenomenon which can be interpreted withthe equality of chemical potential and superficial tension of the two films. A spectacular case corresponds to the coexistence of a liquid film of thickness h_p and the dry solid wall associated with h=0.The film is the so-calledpancakelayercorresponding to conditionG(0) =G(h_p)+ h_p Π(h_p). Equation (<ref>) expresses that the value of the Legendre transformation of G(h) at h_p is equal to G(0).Liquid films of thickness h > h_p are stable and liquid films of thickness h < h_p are metastable or unstable. Thickness h_p can be obtained by the geometric transformationdrawnin Fig. <ref>.§ EQUILIBRIUM OF VERTICAL LIQUID THIN-FILMS §.§ A functional of energy associated with liquid thin-films The modern understanding of liquid-vapour interfaces begins with papers of van der Waals <cit.>. In current approaches, it is possible to give exact expressions of the free energy in terms of pair-distribution function and direct correlation function <cit.>. In practice, these complex expressions must be approximated to lead to a compromise between accuracy and simplicity.When we are confronted with such complications, the mean-field models are generallyinadequate and the obtained qualitative picture is no more sufficient. The main alternatives aredensity-functional theories which are a lot simpler than the Ornstein-Zernike equationin statistical mechanics since the local density is a functional at each point of the fluid <cit.>. We use this approximation enabling us toanalytically compute the density profiles of simple fluids where we takeaccount of surface effects and repulsive forces by adding density-functionals at boundary surfaces. The energyfunctional of the inhomogeneous fluid in a domain O of boundary ∂ Ois taken in the form:F_o = ∭_O ρ ε dv + ∬_∂ Oϖds . The first integral is associated with a square-gradient approximation when we introduce a specific free energy ε of the fluid at a given temperature T_0, as a function of density ρ andβ= ( grad ρ)^2 and ϖ is a convenient surface energy. Specific free energy ε characterizes together the fluid properties of compressibility and capillarity.In accordancewith kinetic theory,ρ ε =ρ α (ρ)+λ/2 (grad ρ )^2,where term (λ/2) ( grad ρ )^2is addedto the volume free-energy ρ α (ρ) of thecompressible-fluid bulk and coefficient λ =2ρ ε _β^'(ρ, β)( ' denotes the partial derivative) is assumed to be constant at given temperature T_0 <cit.>. Specific free energy α enables to connect continuously liquid and vapour bulks, and thermodynamicalpressure P(ρ)=ρ ^2α _ρ^'(ρ ) is a state equation for liquid-vapour interfaces likevan der Waals' pressure. Near a solid wall, London potentials of liquid-liquid and liquid-solid interactions are{[ φ _ll=- c_ll/r^6 , whenr>σ _land φ _ll=∞whenr≤σ _l,;φ _ls=- c_ls/r^6 , whenr>δand φ _ls=∞ when r≤δ , ].where c_ll and c_ls are two positive constants associated with Hamaker coefficients, σ _l and σ _s denote fluid and solidmolecular-diameters, δ =1/2( σ _l+ σ _s) is the minimal distance between centers of fluid and solid molecules <cit.>. Forces between liquid and solidare short range and can be simply described by adding a special energy at the surface. This energy is the contribution to the solid/fluid interfacial energy which comes from direct contact.For a plane solid wallat a molecular scale, this surface free energy is in the form:ϕ(ρ)=-γ _1ρ+1/2 γ_2 ρ^2.Here ρ denotes the fluid density valueat surface (S); constants γ _1, γ _2 and λ are positive and given by the mean field approximation:γ _1=π c_ls/12δ ^2m_lm_s ρ _sol,γ _2=π c_ll/12δ^2 m_l^2,λ = 2π c_ll/3σ_lm_l^2,where m_l and m_s denotemasses of fluid and solid molecules, ρ _sol is the solid density <cit.>.This is notthe entire interfacial energy: another contributioncomes from the distortions in the density profile near the wall <cit.>. We consider a liquid layer contiguous to its vapour bulk and in contact with a plane solid wall (S); the z-axis is perpendicular to the solid surface.The conditions inthe vapour bulk aregrad ρ =0 andΔρ = 0 where Δ denotes the Laplace operator. Far below from the fluid critical-point,a way to compute the total free energy of the complete liquid-vapourlayer is to add the surface energy ofsolid wall (S) at z=0, the energy ofliquid layer (L)located between z=0 and z=h, the energy of the sharp liquid-vapourinterface of a few Angström thickness assimilated to surface (Σ) at z=h and the energy of the vapour layer located between z=h and z=+∞ <cit.>. The liquid at level z=h is situated at a distance order of two molecular diameters from the vapour bulk and the vapour has a negligible densitywith respect to the liquid density <cit.>.In our model, the two last energies can be expressed with writing a unique energy ψ per unit surface located onmathematical surface (Σ) atz=h: by a calculus like in<cit.>, we can write ψ in the same form than Rel. (<ref>)and expressed in <cit.> like ψ(ρ) = -γ _5ρ+(1/2) γ_4 ρ^2; but vapour density being negligible, γ _5≃ 0 andsurface free energy ψ is reduced toψ (ρ )=γ _4/2 ρ_h ^2,where ρ_h is the liquid density at level z=h andγ_4 is associated with distance d of theorder of the fluid molecular diameter (then d ≃δ and γ_4≃γ_2). The properties of Section 2 can be extended to vertical thin-films when we add the gravitational potential Ω to the energy functional; we obtain a functional F in final formF = ∭_(L)ρ ε dv +∭_(L)ρ Ω dv + ∬_(S)ϕ ds + ∬_(Σ)ψ ds.In case of equilibrium, functional(<ref>) must be stationary andyields equation of equilibriumandboundary conditions<cit.>. §.§ Equation of equilibrium The equation of equilibrium is obtained by using the virtual work principle <cit.>. We denote by δx the variation of Euler position x as defined by Serrin in <cit.>. Forδx null on theboundariesof L, the integrals onS and Σ havenull contributions; the virtual work principle yields:δ(∫_Lρ(ε +Ω )dv)= 0.Takinginto account the relation expressing the variation of a derivative,δ(∂ρ/∂x)=∂δρ/ ∂x-∂ρ/∂x∂δx/ ∂x ,with ∂/∂x =grad^T, where ^T denotes the transposition; we obtain:δβ =2 (∂δρ/∂x-∂ρ/∂x∂δx/∂x) grad ρandδε =ε '_ρ δρ +ε '_β δβ ,as well as eqs. (14.5) and (14.6) in <cit.>, then∫_L( -div^T σ +ρgrad Ω ) δxdv=0,where σ =-p 1-λgrad ρ ⊗grad ρis the generalization of the stress tensor with p=ρ ^2ε _ρ^'-ρ div (λgrad ρ ) <cit.>.Classical methods of the calculus of variations lead tothe equation of equilibrium:div^T σ -ρgradΩ = 0 .Let us consider an isothermal vertical film of liquid bounded respectively by flat solid wall and vapour bulk; i is the upward direction of coordinatex; then,gravitational potential is Ω =𝒢 x , anddiv^T σ - ρ 𝒢 i=0Coordinatez isnormal to thesolid wall. Density derivatives are negligiblein directions other thanz. In liquid-vapour layer (we also callinterlayer),σ= [ [ a_1, 0,0; 0, a_2,0; 0, 0,a_3 ]], with{[ a_1=a_2 =-P+λ/2(dρ/dz)^2+λ ρ d^2ρ/dz^2;a_3=-P-λ/2(dρ/dz)^2+λ ρ d^2ρ/dz^2 ].Equation (<ref>) yields a constant eigenvalue a_3,P+λ/2(dρ/dz)^2-λ ρ d^2ρ/dz^2=P_v_b_x,where P_v_b_x≡ P(ρ_v_b_x) denotes the pressure in vapour bulk v_b_x bounding the liquid layer at level x. In the interlayer, eigenvalues a_1, a_2 depend on distance z to the solid wall. In all parts of the isothermal fluid, Eq. (<ref>) can be written <cit.>:grad (μ -λ Δρ + 𝒢 x ) =0,where μ is the chemical potential which is defined to an unknown additive constant.We note that Eqs (<ref>) and (<ref>)are independent of surface energies (<ref>) and (<ref>). The chemical potential is a function of Pand temperature T; due tothe equation of state, the chemical potential can be also expressed as a function of ρand T.We choose as reference chemical potential μ _o=μ _o(ρ) which is null for bulks of densities ρ _l and ρ _v associated with the phase equilibrium in normal conditions (temperature T_o and atmospheric pressure P_o<cit.>). Due to Maxwell'srule, the volume free energy associated with μ _o is g_o(ρ)-P_owhere P_o≡ P(ρ _l)= P(ρ _v) is the bulk pressureand g_o(ρ)=∫_ρ _v^ρμ _o(ρ) dρ is null for the liquid and vapourbulks of the phase equilibrium. Pressure P is <cit.>:P(ρ)=ρ μ _o(ρ)-g_o(ρ) +P_o .Thanks to Eq. (<ref>), we obtain in all the fluid:μ _o(ρ)-λΔρ + 𝒢 x =μ _o(ρ _b) ,where μ _o(ρ _b) is the chemical potential value of mother liquid-bulkof densityρ _bsuch that μ _o(ρ _b)= μ _o(ρ_v_b); ρ_b and ρ_v_b are the densities of themotherbulks bounding the layerat levelx =0.Weemphasize thatP(ρ _b) and P(ρ_v_b) are unequal asdrop or bubblebulk pressures. Likewise, we define a mother liquid-bulkof density ρ_b_xat levelx such that μ _o(ρ _b_x)= μ _o(ρ_v_b_x) with P(ρ _b_x) ≠P(ρ_v_b_x). Then, λ Δρ=μ_o(ρ)-μ_o(ρ _b_x)withμ_o(ρ _b_x) = μ_o(ρ _b) - 𝒢 xand density derivatives being negligiblein directions other than z, in the interlayer, λ d^2ρ/dz^2 = μ_b_x(ρ),withμ_b_x(ρ) = μ_o(ρ)-μ_o(ρ _b_x) §.§ Boundary conditionsConditionatsolid wall (S) associated with free surface energy(<ref>) yields <cit.>λ(dρ/dn)_|_S+ϕ ^'(ρ)_|_S =0,where n is the external normal direction to the fluid; then, Eq. (<ref>) yieldsλ(dρ/dz)_|_z=0=-γ _1+γ _2 ρ_|_z=0 .The condition at liquid-vapour interface(Σ) associated with the free surface energy (<ref>) yieldsλ(dρ/dz)_|_z=h=-γ _4ρ_|_z=h .In Eq. (<ref>), density derivative dρ/dz is large with respect to the variations of the density in the interlayer and corresponds to the drop of density in the liquid-vapour interface. Consequently, Eq. (<ref>) defines the film thickness inside the liquid-vapour interface bordering the liquid layerat surface z=h considered as a dividing-like surface(<cit.>, chapter 3).§.§ Disjoining pressure of vertical liquid thin-filmsEquation (<ref>) can be extended to the disjoining pressure at level x; we obtain the disjoining pressure value:Π =P_v_b_x-P_b_x ,where P_b_x and P_v_b_x are the pressures in mother-liquid and mother-vapour bulkscorresponding to level x. At a given temperature T, Π is a function of ρ_b_x or equivalently a function of x. Let us denote byg_b_x(ρ) = g_o(ρ)-g_o(ρ_b_x)-μ_o(ρ_b_x)(ρ-ρ_b_x),the primitive ofμ _b_x(ρ) null for ρ _b_x. Consequently,Eq. (<ref>) givesΠ (ρ _b_x) = -g_b_x(ρ_v_b_x) ,and integration of Eq. (<ref>) yieldsλ/2 (dρ/dz)^2=g_b_x(ρ)+Π (ρ _b_x),where dρ/dz = 0 when ρ = ρ_v_b_x.The reference chemical potential linearized near density ρ_l is μ _o(ρ)= (c_l^2/ρ _l)(ρ -ρ_l) where velocity c_l is the isothermal sound-velocity in liquid bulk of density ρ_l at temperature T_o <cit.>. In the liquid part of the liquid-vapour film,Eq. (<ref>) yields:λd^2ρ/dz^2 =c_l^2/ρ _l(ρ -ρ _b)+ 𝒢 x ≡c_l^2/ρ _l(ρ -ρ _b_x) with ρ _b_x = ρ _b-ρ _l/c_l^2 𝒢x .The reference chemical potential linearized near density ρ_v is μ _o(ρ)=c_v^2/ρ _v(ρ -ρ_v) where velocity c_v is the isothermal sound-velocity in vapour bulk of density ρ_v at temperature T_o <cit.>. In the vapour partof the liquid-vapour film, Eq. (<ref>) yields:λd^2ρ/dz^2=c_v^2/ρ _v(ρ -ρ _v_b)+𝒢 x≡c_v^2/ρ _v(ρ -ρ _v_b_x) withρ _v_b_x = ρ _v_b- ρ_v/c_v^2 𝒢 x .Due to Eq. (<ref>), μ _o(ρ) has thesame value for ρ _v_b_x and ρ _b_x; thenc_l^2/ρ _l(ρ _b_x-ρ _l) =μ _o(ρ _b_x)=μ _o(ρ _v_b_x) =c_v^2/ρ _v(ρ _v_b_x-ρ _v), andρ _v_b_x=ρ _v( 1+c_l^2/c_v^2(ρ _b_x-ρ _l)/ρ _l) .In liquid and vapour parts of the interlayer we have, respectivelyg_o(ρ)=c_l^2/2ρ _l(ρ -ρ _l)^2 ( liquid) and g_o(ρ)=c_v^2/2ρ _v(ρ -ρ _v)^2 ( vapour).From Eqs (<ref>)-(<ref>)we deducethe disjoining pressure at level x:Π (ρ _b_x)=c_l^2/ 2ρ _l(ρ _l-ρ _b_x) [ ρ _l+ρ _b_x-ρ _v( 2+c_l^2/c_v^2 (ρ _b_x-ρ _l)/ρ _l) ] .Due to ρ _v( 2+ c_l^2/c_v^2(ρ _b_x-ρ _l)/ρ _l) ≪ρ _l+ρ _b_x, we getΠ (ρ _b_x)≈c_l^2/2ρ _l(ρ _l^2-ρ _b_x^2) . At level x=0, themother liquid-bulk density is closely equal toρ_l(the density of liquid in phase equilibrium). Due toEq. (<ref>), Π can be considered as a function of x:Π{x}≈ρ_l 𝒢 x(1-𝒢 x/2 c_l^2).We denote h_x in place of h for a vertical film, and we consider a film of thicknessh_x at level x; the density profile in the liquid part of the liquid-vapour film is solution of system:{[ λd^2ρ/dz^2=c_l^2/ρ _l (ρ -ρ _b_x), with boundary conditions:; ; λdρ/dz_| _z=0. =-γ _1+γ _2 ρ _| _z=0.andλdρ/dz_| _z=h_x. =-γ _4 ρ _| _z=h_x. . ].Quantities τ and d are τ≡1/d=c_l/√(λρ _l) ,where d is areference length; we introduce coefficient γ _3≡λ τ. The solution of System (<ref>) isρ =ρ _b_x+ρ _1_x e^-τ z+ρ _2_x e^τ z,where the boundary conditions at z=0 and h_x yield the values of ρ _1_x and ρ _2_x:{[ (γ _2+γ _3)ρ _1_x+(γ _2-γ _3)ρ _2_x=γ _1-γ _2ρ _b_x,; ; -e^-h_xτ(γ _3-γ _4)ρ _1_x+e^h_xτ(γ _3+γ _4)ρ _2_x=-γ _4ρ _b_x. ].The liquid density profile is a consequence of solution (<ref>) when z ∈[ 0,h_x].By taking Eq. (<ref>) into accountin Eq. (<ref>) and g_b_x(ρ) inlinearized form in the liquid part of the interlayer, we getΠ (ρ _b_x)=-2 c_l^2/ρ _l ρ _1_x ρ _2_x, and consequently,Π (ρ _b_x)= 2 c_l^2/ρ _l[ (γ _1-γ _2ρ _b_x)(γ _3+γ _4)e^h_xτ+(γ _2-γ _3)γ _4ρ _b_x] × [ (γ _2+γ _3)γ _4ρ _b_x-(γ _1-γ _2ρ _b_x)(γ _3-γ _4)e^-h_xτ] / [ (γ _2+γ _3)(γ _3+γ _4)e^h_xτ+(γ _3-γ _4)(γ _2-γ _3)e^-h_xτ] ^2 .By identification of expressions(<ref>) and (<ref>), we get a relation between h_x and ρ _b_x anda relation betweendisjoining pressure Π (ρ _b_x) andthicknessh_x of the liquid film. For the sake of simplicity, weagain finally denote the disjoining pressure byΠ (h_x) which is a function of h_x at temperature T_o. Only Eq. (<ref>) depends on 𝒢. Due to the fact thatEq. (<ref>) does not depend on 𝒢, itsexpression remains unchanged when we consider h instead of h_x. Due toρ _b_x≃ρ _b≃ρ _l <cit.>, the disjoining pressure reduces to the simplified expressionΠ (h_x)= 2 c_l^2/ρ _l[ (γ _1-γ _2ρ _l)(γ _3+γ _4)e^h_xτ+(γ _2-γ _3)γ _4ρ _l] × [ (γ _2+γ _3)γ _4ρ _l-(γ _1-γ _2ρ _l)(γ _3-γ _4)e^-h_xτ] / [ (γ _2+γ _3)(γ _3+γ _4)e^h_xτ+(γ_3-γ _4)(γ _2-γ _3)e^-h_xτ] ^2 with Π (h_x) ≡Π (h) .Let us notice an important property of anyfluid mixture consisting of liquid-water, vapour-water andair <cit.>. Themixture's total-pressure is the sum ofthe partial pressures of its components, and at equilibrium the partial pressure of air is constant through liquid-air and vapour-air domains. Consequently, results of Section 2 remains unchanged: the disjoining pressure of the mixture is the same as for fluid withoutair when only a liquid thin-filmseparates liquid and vapour bulks <cit.>.§.§ Numerical calculations Mathematica^ TM allows us to drawthegraphs ofΠ(h) defined by Eq. (<ref>) and G(h)defined by Eq. (<ref>) when h ∈ [(1/2) σ_l,ℓ], where ℓ is a few tensÅngstroms length.For a few nanometers, thefilm thickness is not exactly h; we must add an estimated thickness 2 σ_l of liquid part of the liquid-vapour interface bordering the liquid layerand thelayer thickness is approximatively h+ 2 σ_l <cit.>.We considered water at T_o = 20^∘ C.In S.I. units, experimental valuesare<cit.>:ρ_l = 998^-3,c_l = 1.478× 10^3^-1, c_ll=1.4× 10^-77 ^8. ^-2 , σ _l=2.8× 10^-10, m_l=2.99× 10^-26.Silica is deposited in many plant tissuessuch as in bark and wood. We choose σ _s=2.7× 10^-10 ; this value is intermediate between molecular diameter of silicon and diameter of non-spherical molecules of water. We deduce λ = 1.17 × 10^-16 ^-1. ^7. ^-2,d = 2.31 × 10^-10, γ _2= 5.42 × 10^-8 ^-1. ^6. ^-2, γ _3 = 5.06 × 10^-7 ^-1. ^6. ^-2. The superficial tension of water is γ = 72.5 × 10^-3 ^-2.We choose as Young contact-angle θ betweenxylem wall and the liquid-water/vapour interface, the arithmetic average of different Young angles propounded in the literature; this value is θ = 50 degrees <cit.>. Consequently,γ_1 is deduced fromthe solid-liquidsurface energy expressed as ϕ(ρ_s)=-γ _1 ρ_s+ (1/2) γ_2 ρ_s^2. Here ρ_s≃ρ_l denotes the fluid density value at the surface. The vapour density is negligible with respect to the water-liquid density and Young's relation<cit.>yields γcos θ =γ _1 ρ_l- (1/2) γ_2 ρ_l^2 and wegetγ_1 ≈ 75 × 10^-6 ^3. ^-2.In the upper graph of Fig.<ref>, we present the free energy graphG(h_x). Due to h_x>(1/2) σ_l, it is not numerically possible to obtainthe limit pointW corresponding to the dry wall; consequently, point W is obtained by interpolation associated with the concave part of the G-curve. To obtainpancake thickness h_p corresponding to the smallest thickness of the liquid layer, we drawpointP,contact-point of the tangent line issued fromW to G-curve.In the lower graph of Fig. <ref>, we present the disjoining pressure graph Π(h_x). The physical part of the disjoining pressure graph corresponding to ∂Π/∂ h_x <0is associated witha liquid layer of several molecules thickness. The non-physical part corresponding to ∂Π/∂ h_x >0 is alsoobtained by Derjaguin et al <cit.> . The reference lengthd is of the same order as σ_l and is a good length unit for very thin-films. The total pancake thicknessis of one nanometer order corresponding to a good thickness value for a high-energy surface <cit.> ; consequently in the tall trees, at high level, the thickness of the layer is of a few nanometers. The point Pofthe lower graph corresponds to the point P of upper graph. § DYNAMICS OF LIQUID THIN-FILMS ALONG VERTICAL WALLSThe dynamics of liquid thin-films is studied in the isothermal case.When h≪ L, where L is the characteristic length along the wall <cit.>,i) The velocity component along the wall is large with respect to normal velocity components which can be neglected,ii) The velocity value varies orthogonally to the wall and it is possible to neglect velocity spatial derivativesalong the wall with respect tonormal velocity derivatives,iii) The pressure is constant in the direction normal to the wall. It is possible to neglectinertial term when Re≪ L/h, where Re is the Reynolds number of the flow.The fluid is heterogeneous and the liquid stress tensor is not scalar. However, it is possible to adapt the results obtained for viscous flows to motions in liquid thin-films: due to ϵ =h/L≪ 1, we arein the case of long wave approximation. We denote the velocity by v=(u,v,w) where (u,v) are the tangential components to the wall. Due to the fact that e=sup( | w/u| ,| w/v|) ≪ 1, we arein the case oflubrication approximation. The main parts of terms associated with second derivatives of liquid velocity components correspond to ∂ ^2u/∂ z^2 and ∂ ^2v/ ∂ z^2.The density is constant along each stream line (ρ=0⟺ div v=0) and isodensity surfaces contain the trajectories. Then, ∂ u/∂ x,∂ v/∂ y and ∂ w/∂ z have the same order of magnitude and ϵ∼ e.As in <cit.>, we assume that the kinematic viscosity coefficient ν =κ/ρ, where κ is the dynamic viscosity, only depends on the temperature. In motion equation, the viscosity term is like in liquid bulk <cit.>(1/ρ) div σ_v = 2ν[div D + Dgrad { Ln (2 κ)}],where σ_v is the viscous stress tensor, D is the velocitydeformation tensor and D grad{Ln (2 κ)} is negligible with respect to divD. In lubrication and long wave approximations, the liquid nanolayer motion verifies <cit.>:a+grad[ μ _o(ρ )-λ Δρ ]=ν Δv- 𝒢 iwithΔ v≃[ ∂ ^2u/∂ z^2,∂ ^2v/∂ z^2,0 ] ,where a denotes the acceleration vector. The equation corresponds toequation of equilibrium (<ref>) with additional inertial-term a and viscous term ν Δv.In approximation of lubrication, the inertial term can be neglected <cit.>:grad[ μ _o(ρ )-λ Δρ ]=ν Δv- 𝒢 i.Equation ( <ref>) can be separated into tangential and normal components to the solid wall.- The normal componentof Eq. (<ref>) writes in the same form than for equilibrium:∂/∂ z[μ _o(ρ )-λ Δρ] =0,and consequently,μ _o(ρ )-λ Δρ =μ _o(ρ^∗ _b_x),where ρ^∗ _b_x is the dynamicalmother liquid-bulk density at level x (differentfrom ρ _b_x,mother liquid-bulk density at level x and at equilibrium, where quantitiesat equilibrium have corresponding dynamical quantities indicated by ^∗ ).A liquid film of thickness h_x^∗ is associated to density ρ^∗ _b_x. We can write μ _o(ρ _b)-λ Δρ =η (h_x^∗), where function η is such that η (h_x^∗)=μ _o(ρ^∗_b_x).- For motions colinear to the solid wall and to the gravity (direction i and velocity u i), by takingaccount of Eq. (<ref>)the tangential component of Eq. (<ref>) writes:i .grad μ _o(ρ^∗ _b_x)=ν∂ ^2u/∂ z^2 - 𝒢 ,which is equivalent to∂μ _o(ρ^∗ _b_x)/∂ρ^∗ _b_x ∂ρ^∗ _b_x/∂ x=ν ∂ ^2u/∂ z^2 - 𝒢.Generally, the kinematic condition at solid walls is the adherence condition (u_z=0=0). Nevertheless, with water flowing on thin nanolayers <cit.>, there arequalitative observations for slippagewhen the Young contact angle is not zero <cit.>. De Gennes said:the results led us to think about unusual processes which could take place near a wall. They are connected withthickness h of the film when h is of an order of the mean free path <cit.>. Recent papers in nonequilibrium molecular dynamics simulations of three dimensional micro Poiseuille flows in Knudsen regime reconsider microchannels:the results point out that the no-slip condition can be observed for Knudsen flow when the surface is rough and the surface wetting condition substantially influences the velocity profiles <cit.>. In fluid/wall slippage, the condition at solid wall writes:u=L_s∂ u/∂ z atz=0,where L_s is theNavier-length <cit.>. The Navier-lengthmay be as large as a few microns <cit.>. At the liquid-vapour interface, weassume that vapour viscosity stress is negligible; from continuity of the fluid tangential-stress through a liquid-vapour interface, we get∂ u/∂ z=0at z=h_x^∗ .Consequently, Eq. (<ref>) impliesν u=( ∂μ _o(ρ^∗ _b_x)/∂ρ^∗ _b_x ∂ρ^∗ _b_x/∂ x +𝒢) ( 1/2 z^2-h_x^∗ z-L_sh_x^∗) .At level x, the mean spatial velocity u of the liquid in the nanolayer isu=1/h_x^∗∫_o^h_x^∗udzand consequently,ν u=-h_x^∗( h_x^∗/3 +L_s)[ grad μ _o(ρ^∗ _b_x)+𝒢 i ] withu=ui .Let us note that:∂μ _o(ρ^∗ _b_x)/∂ x=∂μ _o(ρ^∗ _b_x)/∂ρ^∗ _b_x∂ρ^∗ _b_x/∂ h_x^∗ ∂ h_x^∗/∂ x≡1/ρ^∗ _b_x∂ P(ρ^∗ _b_x)/∂ρ^∗ _x∂ρ^∗ _b_x/∂ h_x^∗ ∂ h_x^∗/∂ x.Due to the fact the vapour-bulk pressure P^∗_v_b_x is constant alongthe xylem tube, by using relation Π (h_x^∗)=P^∗_v_b_x-P^∗_b_x, we get along the flow motion∂μ _o(ρ^∗ _b_x)/∂ x=-1/ρ^∗ _b_x ∂Π (h_x^∗)/∂ h_x^∗∂ h_x^∗/∂ xand consequently,χ^∗ _b_xu=h_x^∗(h_x^∗/3+L_s) [ grad Π (h_x^∗)- ρ^∗ _b_x 𝒢 i ] ,where χ^∗ _b_x=ρ^∗ _b_xν is the liquid dynamic-viscosity.The mean liquid velocity is driven by variation of the disjoining pressure (andfilm thickness) along the solid wall. Equation (<ref>) differs from classical hydrodynamics; indeed, for a classical liquid thin-films, the Darcy law is u=-K(h)grad ℘,where ℘ is the liquid pressure and K(h) is the permeability coefficient. In Eq. (<ref>), the sign is opposite and the liquid pressure is replaced by the disjoining pressure.We note that χ^∗ _b_x≃χ , where χ is the liquid kinetic viscosity in the liquid bulk at phase equilibrium <cit.>. Moreover h_x^∗/L_s≪ 1, and slippageis strongly different from the adherence conditioncorresponding toL_s=0.Theaveraged mass equation over the liquid depth is∂/∂ t( ∫_0^h_x^∗ρ dz) + div( ∫_0^h_x^∗ρ u dz) =0.Since the variation of density is small in the liquid nanolayer, the equation for the free surface isdh_x^∗/dt+h_x^∗ div u =0.By replacing (<ref>) into (<ref>) weget∂ h_x^∗/∂ t+1/χdiv{ h_x^∗ 2( h_x^∗/3+L_s)[ grad Π (h_x^∗)-ρ^∗ _b_x 𝒢 i]} =0,where ρ^∗ _b_x≃ρ_l. Equation (<ref>) is a non-linear parabolic equation.If ∂Π (h_x^∗)/ ∂ h_x^∗<0 the flow is stable. This result is in accordance with the static criterium of stabilityfor liquid thin-films. When L_s≠ 0, we notice the flow is multiplied by the factor 1+3L_s/h_x^∗. For example, when h_x^∗=3 and L_s=100 which is a Navier length of small magnitude with respect to experiments, the multiplier factor is 10^2; when L_s is 7 μ as considered in <cit.>, the multiplier factor is 10^4, which seems possiblein nanotube observations <cit.>.Infollowing sections, we use previous tools to study watering ofplants and especially trees.§ THE TREE WATERING§.§ Experiments andanalyzesSince the beginning of the cohesion-tensiontheory, many efforts have been done to understandcrude sap motions and to replicate tree functions when vessels are under tension.Synthetic systems simulatingtransport processes have played an important role inmodel testing,and methods creating microfluidic structures to mimic tree vasculature have been developedto capture fundamental aspects of flowsandxylem tension <cit.>.When xylem tubes are completely filled with sap, flowsalong vessels can be compared with flows through capillaries <cit.>.Adjacent xylem wallsare connected byactive bordered-pit membraneswith micropores <cit.>. Themembranes separatetwo volumes of fluid,and generally refer tolipid bilayers that surround living cells or intracellular compartments <cit.>.The micropores are a few tens of microns wide. Due to the meniscus curvatures at micropore apertures, marking off liquid-water bulk from air-vapour atmosphere, the water-bulk pressure is negative inside micropore reservoirs, but, surprisingly,semi-permeable micropores allow flows of liquid-water at negative pressureto be pushed toward air-vapour domains at positive pressure <cit.>. Bubbles spontaneously appear from germ existing in crude sap and cavitation makes some tubes embolized <cit.>. Due to experiments described in Section 2, liquidthin-films must damp embolized xylem walls; consequently,thin-films and microtubes filled of crude sap are in competition. Optical measurements indicateYoung's contact-angles of about 50^∘ for water on the xylem at 20^∘ Celsius <cit.>. This value suggests that xylem wallsare not fully wetting and the capillary spreading cannot really aid the liquid-water refilling but may explain the apparent segregation of liquid-water into droplets <cit.>. The crude sap is not pure water; its liquid-vapour surface tension has a lower value than the surface tension of pure water and it is possible to obtain the same spreading coefficients with less energetic surfaces.Water exitsthe leaves by evaporation through stomata into subsaturated air. Resistance ofstomata sits in the path of vapour diffusion between the interior surfaces of leaves and the atmospherebut many of the tallest trees appear to lack active loading mechanisms<cit.>. When active transpiration occurs, stomata are open and these pumps run.The growth and degrowth of bubbles are rapid within xylem segments, but at night, although the stomata are closed, xylem vesselsdeveloping embolies during the daycan be refilled with liquid-water and the metastability of the liquid-water may persisteven in the absence of transpiration <cit.>. Recent advances in tree hydraulics have demonstrated that, contrary to what was previously believed, embolism and repair may be far from routine in trees. Trees can recover partially or totally from the deleterious effects of water stress until they reach a lethal threshold of cavitation<cit.>.This resultcan be related with the fact that thin films with a thickness greaterthan the pancake-layer's one arestable and the behaviour is different from bubble stability, which is associated with a saddle point <cit.>. §.§Motions in filled microtubesand in thin-films§.§.§GeneralityOne important design requirementis that vapour blockage does not happen in the stems. When the vessel elements are tight-filled with crude sap, liquid motions arePoiseuille flows <cit.>. The flow rate throughcapillary tubes is proportional to the applied pressure gradient, the hydraulic conductivity anddepends on the fourth power of capillary radius <cit.>. To be efficient forsap transportation, the tubes' diameters should be as wide as possible; because of the micronsize of the xylem tubes, this is not the case. Consequently, thetracheary elements' network must be important. But the sap movement is induced bytranspiration across micropores located in tree leaves and the transpiration is bounded bymicropores' sizes; it seems natural to surmise that the diameters of vessels mustnot be too large to generate a sufficient sap movement.When the vessel elements are embolized,thin-films damp the xylem walls and Eq. (<ref>) governs the liquid motion. The diameters of capillary vesselswhich range from 10 to 500 μ and liquid thin-films of some nanometre thickness can be considered as plane interlayers.It is noticeable that trees can avoid having very high energy surfaces: if we replace the flat surfaces of the vessels with corrugated surfaces at molecular scale,it is much easier to obtain the complete wetting requirement, which is otherwise only partial. However, they are still internally wetif crude sap flows through wedge-shaped corrugated pores. The wedge does not have to be perfect on the nanometric scale to significantly enhance the amount of liquid flowing at modest pressures, the wallsbeingendowed with an average surface-energy <cit.>. It is interesting to compareliquid motions both in tight-filled vessels and liquid-water thin-films. An hydraulic Poiseuille flow is rigid due to the liquid incompressibility, the pressure effects are fully propagated in the tube. For a thin layer flow, the flow rate can increase or decrease due to the spatial derivative of h_x^∗ and depends on thelocal disjoining pressure.The tree'sversatility allows it to adapt to the localdisjoining-pressure gradient effects by opening or closing the stomata and the curvature of pit pores, so that the bulk pressurecan be more or less negative and the transport of watercan be differently dispatched through the stem parts.§.§.§ Numerical calculationsThe treachery network of xylem microtubes is extremely developped.For tree hight H = 20, the total area of xylem walls can be estimatedto S = 30 ^2 ≡ 3 × 10^7 ^2 <cit.>. We considerxylem microtubes with diameter 2R = 50 μ≡ 5 × 10^-5. The dynamic viscosity of liquid water at 20^∘ Celsius is χ = 10^-3 ^-1. ^-1. It is experimentally verified that in tightly-filled microtubes, the crude sap velocity usually goes from 1 ^-1≡ 2.8 × 10^-4^-1 to 100 ^-1≡ 2.8 × 10^-2^-1inmaximal transpiration <cit.>. The mean velocity of Poiseuille's flows verifiesu_1 =R^2/8 χ |grad (-℘_b^*_x)|,where ℘_b^*_x is the liquid pressure at level x. Consequently, |grad (-℘_b^*_x)| goes from 3.6 × 10^-2^-1≡ 3.6 × 10^3 ^-1 to 3.6 ^-1≡ 3.6 × 10^5 ^-1. The number of xylem microtubes can be calculated as N=S/(2 π RH); we approximatively obtain N= 10^10 microtubes and the total flow isQ_1 =N π R^4/8 χ |grad (-℘_b^*_x)|.For a velocity u_1 = 1 ^-1≡ 2.8 × 10^-4 ^-1,Q_1= 5.5 × 10^-3^3.^-1≡ 5.5 and for u_1 = 100 ^-1≡ 2.8 × 10^-2 ^-1,Q_1= 5.5 × 10^-1^3. ^-1≡ 550 corresponding tovalues of biological experiments. In embolized microtubes,mean velocity u_2 along thin-films is given by Rel (<ref>). Pressure P_v_b_x≈ P_v being constant and ρ^∗ _b_x≈ρ_l, thengrad( Π (h_x^∗)- ρ^∗ _b_x 𝒢x)≈grad(-P_b^*_x-ρ_l 𝒢 x).From χ u_2 ≈ h_x L_s|grad( Π (h_x^∗)- ρ _b^∗_x 𝒢x)|, together with a layer thickness h_x = 10≡ 10 ^-8, L_s = 7μ≡ 7 × 10^-6, and |grad( Π (h_x^∗)- ρ _b^∗_x 𝒢x)| = 3.6 × 10^5 ^-1, we obtain u_2 =2.5 × 10^-5 ^-1 which is 11 times less than velocity in tight-filled microtubes for a velocityof 1^-1 ≡ 2.8 × 10^-4^-1.For N microtubes, the total flow isQ_2 = N 2π R h_x u_2and for 10^10 microtubes, Q_2 = 4 × 10^-7 ^3.^-1≡ 0.4^3.^-1 which is very small with respect to the sum of Poiseuille's flows. It seems that the liquid thin-films do not affect the watering of trees but in next Section 6,we see it is not the case. For an ascent of 50, we obtain an ascent time of 2 × 10^6≡ 23 days which estimates the tree-recovery time in spring. § EMBOLIZATION AND RECOVERY§.§ A diagram ofvessel elements for tall trees Inphysical conditions of Subsection 3.4 and temperature at 20^∘ Celsius, we consider two vertical adjacent vessels linked by micropore reservoirs with pit membranes dotting their walls. The bordered-pit membranes are of the order of few tens of microns corresponding to the microporous filters in Fig. <ref>. The mothervapour-bulk contains air and the mother liquid-water bulk alsocontains dissolved air. Onevessel - corresponding to subsaturated mother air-vapour bulk - is embolized with apositive pressure;it generates a liquid thin-film which wets the xylem wall. The other vessel is filled with the mother liquid-water bulk at a negative pressure linked to the liquidthin-filmthanks to a micropore reservoir with a bordered-pit membrane.Such a system can be in equilibrium, although the pressure is not the same in the two adjacent vessels.In the same configuration, thevessel elements are now assumed to be weakly out-of-equilibrium.As explained in Section 4, the driving force of the sap ascent comes from the decreasing thickness of the liquid thin-film wetting the walls in the embolized vessels;consequently the negative pressure valueof the mother liquid-water bulk in micropore reservoirsdecreases(its absolute value increases). Additionally, air-vapour pockets can coexist with the mother liquid-water bulk of one of the two vessels. The air-vapour pocketsalso generateliquid thin-films borderingxylem walls (see Fig. <ref>).Due to their curvature,the pressures of air-vapour pockets are generally higher than the mothervapour-bulk pressure in the other vessel. The vapour pockets and their liquid thin-films empty into thevesselwith the lower air-vapour bulk pressure. The analogy proposed in Section 2 between liquid thin-films and liquid-vapour interfaces of bubbles allows to simply understand the directions of motionsbetween air-vapour pockets and the mother air-vapour bulk: for example, when two bubbles are included in a liquid bulk (corresponding to the mother liquid-water bulk atnegative pressure), the smallest bubble with the greatest pressure (i.e. the air-vapour pocket) empties into the largest bubble with the lowest pressure (i.e. the air-vapour bulk of the embolized vessel) <cit.>.Sucheventshappenin particular at night, when - due to the absence of evaporation - the vapour is subsaturated in embolized vessels; the curvature of the air-vapour pockets generates apressure greater than the pressure in the embolized vessels. Conversely, during the day and strong sunlight, the vapour invessels is saturated by evaporation; the air-vapour pressure increases in the embolized vessels and the air-vapour gas mustflowback into the vessels withair-vapour pockets of subsaturated vapour included in the mother liquid-water bulk atnegative pressure. It is not surprising that the heartwood may contain liquid under positive pressure while in the sapwood the transpiration stream moves along a gradient of negative pressure.Embolized vessels creating thin-films may provide a key contribution totree refilling. Crude sap in the heartwood can also fill the vessel elements at negative pressure through the bordered-pit membranes. Consequently, embolized microtubes of xylem fundamentally contribute to the crude-sap ascent and to the refilling of the tree as machine allowing to obtain equilibrium between fluid phases at different pressures and consequently to recover the water from cavitation. §.§ Numerical values of the mass transferIt is interesting to estimate the magnitude ofair-vapour flows between air-vapour pockets and the mother air-vapour bulk. At altitude x, themother bulks are still named (l_b_x) and (v_b), respectively.At equilibrium between the two mother bulks, the verticalliquid thin-film betweenthe mother air-vapour bulk and the xylem wallis also at equilibriumand at the same pressure asthe mother air-vapour bulk. Depending on reservoir altitude x,the mother liquid-bulk can be at negative pressure. The channels crossing from the left tothe right parts of the xylem wall canbe considered assound pipes(Fig. <ref>).We denote by e and2r thelength and diameterofpipes, respectively.The pipe diameters are assumed to be of the same order thandiameters ofpits bordering the xylem walls. The difference of pressure between air-vapour pockets and the mother air-vapour bulk is denoted by P. The flow is assumed to bePoiseuille flow and the mean velocity along the channel is <cit.>:v = r^2/8 χP/e .The velocity is null inthe mother air-vapour bulk and in the air-vapour pockets creating a velocity-pulse between the two extremities of the pipe. On one hand, we assume that the pressure in the air-vapour pocket isgreater than thepressure in the motherair-vapour bulk. The air-vapour pocketand the mother air-vapour bulk are not at equilibrium.The liquid thin-film betweenthe air-vapour pocket and the xylem wall is thinner thanthe liquid thin-film betweenthe mother air-vapour bulk and the xylem wall (see Fig. <ref>). We consider experimental values:e = 50 μ≡ 5× 10^-5 , r = 10 μ≡ 1× 10^-5 , χ = 1.81 × 10^-5 ^-1. ^-1 for air-vapour viscosityand the saturated vapour pressure of wateris2.3 × 10^-2 ≡ 2.3 × 10^3 at20^∘ Celsius <cit.>. In the case of a difference of pressure between the saturated vapour pressure in air-vapour pocket and thesub-saturated vapour pressure in the mother air-vapour bulk P = 10^-2 ≡ 10^3, we obtain v= 14 ^-1 and the crossing time of the pipe is τ= e/v = 3.6 × 10^-6 corresponding to a pulse frequency ω =τ^-1≈ 2.8 × 10^5 ≡ 2 80 associated with ultrasonic vibrations. It is interesting to calcutatethe flow through the pipe,q =π r^4/8 χP/e. We obtain q = 4.3× 10^-9 ^3.s^-1≡ 4.3× 10^-3 ^3.s^-1. Conversely, when themother air-vapour pressure is slightly greater than the air-vapour pocket pressure, the air-vapour bulk can embolize the liquid tight-filledvessel elements with opposite velocityand consequently the same pulse frequency for the same difference of pressure between mother air-vapour-bulk and air-vapour pockets.Let us note that the saturated vapour pressure quickly decreases with the temperature. At0^∘ Celsius, the saturated vapour pressure ofwater is 6 × 10^-4 ≡ 60 Pa and P drastically decreases as the flow through the pipe, and it may be a reason of embolization at low temperature.On the other hand, the crude sap in the tight-filled microtubes can spread in the embolized microtubes when the disjoining pressure is smaller than the disjoining pressure at equilibrium. Equation (<ref>) allows to calculate the mean velocity of the flow:here P represents the difference between the disjoining pressure values at and out from equilibrium. With P = 10^-1 ≡ 10^4and χ = 10^-3 ^-1, we obtain v = 2.5 ^-1 and the crossing time of the pipe is τ= e/v = 2 × 10^-5 corresponding to a pulse frequency ω =τ^-1≈ 5 × 10^4 ≡ 50. We also obtainq = 0.8× 10^-9 ^3.s^-1≡ 0.8× 10^-3 ^3.s^-1.Due to the size of xylem microtubes, a reference diameter of bubbles can be about 2R = 50 μ≡ 5 × 10^-5 corresponding to volume V =6.5× 10^-14 ^3≡ 6.5× 10^-8 ^3 and for both liquid and gas exchanges,the transfers of masses are extremely fast.The fast accelerations of air-vapour gas through micropores generate ultrasounds associated with pulse frequencies and may explain the acoustical measurements obtained in experiments<cit.>. The magnitude of the viscosity of simple wetting fluids increaseswhen they are confined between solid walls,and there is a direct correlation between the air-seeding threshold and the pit pore membranes' diameters <cit.>. However, the acceleration magnitude is so large that it remains very important for viscous fluids and semi-permeable bordered-pit membranes.§.§ Limit of the disjoining pressure model and topmost treesLiquid thin-filmsprimarily contribute toxylem microtubes refilling and consequently therefilling is not possible when the liquid thin-films breakdown. Amazingly, the above studyallows to estimate a maximum of thin-films' altitude.In Subsection 3.5 and in Fig. <ref>, wehave seen that the thickness of the liquid thin-film decreases when its altitude increases and the liquid thin-film disrupts when thicknessreaches the pancake layer thickness-value.The pancake layer of thin-films was presented in Section 2 and a numerical simulation associated with experimental data of xylemat temperature 20^∘ Celsius is presented in Subsection 3.5. Asindicated in Subsection 3.5, the Young contact anglebetween a xylem wall and a liquid-vapour water interface is θ≈ 50^∘. The upper graph in Fig. <ref>presents the free energyG(h) associated with physical values oftrees' xylem walls.The lower graph of Fig. <ref>,presents the disjoining pressure Π(h) and is in accordance with experimental curves obtained in the literature <cit.>. The total pancake thickness h_p is about one nanometer order corresponding to a good thickness value for a high-energy surface <cit.>; consequently in tall trees, at high level, the thickness of the liquid thin-film must be of a few nanometers. Point P on the lower graph corresponds to point P on upper graph.When x__P corresponds to the altitude of the pancake layer, Eq. (<ref>) and 𝒢x/(2 c_l^2) ≪ 1 yields Π(h_p) ≃ ρ_l 𝒢 x__P. From the lower graph in Fig.<ref>, we obtaina maximum thin-film height of120 meters corresponding to 12 atmospheres. At this altitude, we must approximatively add 20 meters corresponding to the ascent of sap due to capillarity and osmotic pressure <cit.> and we obtain 140 meters. This level corresponds to the level order of the topmost trees, as a giant, 128 meter-talleucalyptus or a 135 meter-tall sequoia which were reportedin the past by Flindt <cit.>. Other mechanical or biological constraints may suggest adaptation to height-induced costs<cit.>, but nevertheless our model limits the maximum height of trees. The tallest trees are not the ones with the largest demand for tension; it is rather dry climate shrubs that demand it <cit.>. This observation seems to be in accordance with the possible existence of thin-films in embolized vessels at high elevation.§ CONCLUSIONIn the trees, xylem microtubes are naturally filled with sap up to an altitude of a few ten meters. Above this altitude, whenxylem tubes can be embolized, the molecular forcescreate crude sap thin-films along the walls of xylem associated with micropore pressures versatilely adapted thanks to pit membranes. The disjoining pressure of liquid thin-films is the exhaust valve filling thexylem microtubes and allowingcrude sap to ascend. Consequently, the embolized vessels constitute a necessary network for the watering and the recovery oftall trees (Lampinen and Noponenargued that embolisms were necessary for the ascent of sap <cit.>).The modelexplains aspects of sap movement which the classical cohesion-tension theory was hitherto unable to satisfactorily account for, e.g. the refilling of the vessels in spring, in the morningor after embolism events, as well as the compatibility with thermodynamics' principles. Nevertheless, the xylem tight-filled microtubes under tension are the essential network of tree watering. Simple in vivo observations at the nanometre thickness ofliquidthin-films are not easy to implementand thedirect measurement difficultiesprevent their detection. The progression of MEMS technology<cit.>, and tomography <cit.>, may provide a new route towards this goal. If these biophysical considerations are experimentally verified, they would prove that trees can be an example to use technologies for liquids under tension connected with liquids in contact with solid substratesat nanoscale range. They would provide a context in which nanofluid mechanics points to a rich array ofbiological physics and future technical challenges. It is wondering to observe that the density-functional theory expressed by a rough energetic model with a surface density-functional at the walls enables to obtain a good order ofthe ascent of sap magnitude. The result is obtained without too complex weighted density-functional and without taking account of quantum effectscorresponding to less than an Amgström length scale. These observations seemto prove that thiskind of functionals can be a good tool to study models ofliquids in contact with solids at a small nanoscale range.Moreover there exists no conflictbetween thermodynamics andcohesion-tension theory <cit.>. 0.5cm Acknowledgements: H.G. thanks the Accademia Nazionale dei Lincei, the Istituto Nazionale di Alta Matematica F. Severi (INdAM),and the Gruppo Nazionale per la Fisica Matematica (GNFM), for their nice invitation and support to the conference on"New Frontiers in Continuum Mechanics"hold at the Academy on 21 and 22 June 2016. 1cmReferencesAskenasy Askenasy E., Ueber das Saftsteigen. Verh. Nat. Med. Ver. Heidelb.5(1895) 325–345.batchelor BatchelorG.K.,An Introduction to Fluid Dynamics.Cambridge U. Press (1967).Balling Balling A. - Zimmermann U., Comparative measurements of the xylem pressure of nicotiana plants by means of the pressure bomb and pressure probe. 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"authors": [
"Henri Gouin"
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"categories": [
"cond-mat.soft",
"physics.bio-ph",
"q-bio.TO"
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"primary_category": "cond-mat.soft",
"published": "20170627115225",
"title": "Continuum mechanics at nanoscale. A tool to study trees' watering and recovery"
} |
1]Sylvain BilliardEmail:[1]Univ. Lille, CNRS, UMR 8198 - Evo-Eco-Paleo, F-59000 Lille, France2]Vincent Bansaye Email:[2]Centre de Mathématiques Appliquées, CNRS UMR 7641, Ecole Polytechnique, F-91128 Palaiseau Cedex, France 3]J.-R. Chazottes,Email:[3]Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique, F-91128 Palaiseau Cedex, FranceRejuvenating Functional Responses with Renewal Theory [===================================================== Running title. Stochastic derivation of functional responsesKeywords. stochastic model, ecology, predation, cooperation, inference, emerging properties Corresponding author: Sylvain Billiard: [email protected], Univ. Lille, CNRS, UMR 8198 - Evo-Eco-Paleo, F-59000 Lille, France.Functional responses are widely used to describe interactions and resources exchange between individuals in ecology. The form given to functional responses dramatically affects the dynamics and stability of populations and communities. Despite their importance, functional responses are generally considered with a phenomenological approach, without clear mechanistic justifications from individual traits and behaviors. Here, we develop a bottom-up stochastic framework grounded in Renewal Theory showing how functional responses emerge from the level of the individuals through the decomposition of interactions into different activities. Our framework has many applications for conceptual, theoretical and empirical purposes. First, we show how the mean and variance of classical functional responses are obtained with explicit ecological assumptions, for instance regarding foraging behaviors. Second, we give examples in specific ecological contexts, such as in nuptial-feeding species or size dependent handling times. Finally, we demonstrate how to analyze data with our framework, especially highlighting that observed variability in the number of interactions can be used to infer parameters and compare functional response models. § INTRODUCTION Interactions between individuals affect all ecological processes, and how fast they occur determine resources exchange rates in an ecosystem. Interactions are generally considered at the population (or macroscopic) level and supposed to vary along with species densities following an interaction function, generally called a functional response. The seminal papers introducing functional responses in population ecology followed a reductionistic approach and aimed at giving the underlying mechanisms <cit.>. Yet, population ecologists generally follow a phenomenological approach to justify functional responses <cit.>, and functional responses are rarely derived from the individuals' traits and behaviors. The form given to functional responses is crucial since it can dramatically affect the dynamics and stability of populations and communities. For instance, the stability of predator and prey populations strongly depends on whether predator consumption rates increases linearly (Holling type I functional response) or following a saturating function (Holling Type II and III functional responses) with prey densities <cit.>. It is thus critical to modelize within and between species interactions as accurately as possible in order to have the best predictions and understanding of population and community dynamics, and eventually support wildlife management decisions <cit.>. The form given to functional responses is largely debated for decades, sometimes fiercely (e.g. the long-standing controversy about whether it is best to assume density-dependence or ratio-dependence in predator-prey models <cit.>). Hundreds of functional responses have been proposed in the literature regarding all types of interactions: cooperation <cit.>, plant-pollinators <cit.>, predation (reviewed in <cit.>), competition <cit.>. Strikingly, despite a large variety of possible functional responses, Holling type II or related functional responses are most often used<cit.>, either for predator-prey <cit.> or mutualistic interactions <cit.>. There is however a general agreement that there is much room for improvement. It is for instance difficult to determine which one of alternative functional response fits the best to empirical data because of the poor statistical power of fitting different functions to data <cit.>. Holling Type II functional response is certainly preferred not because it adequately modelizes ecological interactions but rather because it is a saturating function with a single parameter. Many authors argue that bridging the gap between interactions at the level of the individuals (the microscopic scale) and functional responses (the macroscopic level) would be a critical milestone in the field <cit.>, first because it is necessary to mechanistically justify functional responses, and second because evidence of individual traits variation in functional responses accumulate <cit.>. Few studies aimed at making the link between microscopic and macroscopic scales in the context of functional responses. Some authors used a mean-field approximation to derive functional response in consumer-resources relationships<cit.>. In particular, inspired by his famous “Disc experiment”, Hollingshowed that considering mechanisms such as searching times proportional to prey density and constant handling times of prey by predators, gives the well-known Holling's Type II functional response <cit.>. Holling later showed that Holling Type III functional response can be derived by introducing the mechanism of predators learning <cit.>. Other authors used deterministic approaches derived from chemical reactions equations to show which assumptions must be verified for a specific functional response to be valid, e.g. Holling functional responses <cit.>, Beddington-DeAngelis functional response <cit.>, plant-marking pollinators interactions <cit.>. However, these approaches have strong limitations. First, they can be used only to derive simple functional responses. Indeed, approximations of deterministic functional responses can be obtained from a system of ordinary differential equations under assumptions of slow/fast processes. Approximations might generally be difficult to obtain because the number of equations and simplifying assumptions to make increase with the complexity of the system. It makes this approach unlikely to be general enough to embrace the large variety of possible ecological contexts. Second, since only the mean rates of interaction at the population level are considered, variability of traits and behaviors between individuals can not be considered. Third, since these approaches are deterministic, they are only valid in very large populations and the importance of stochasticity can not be taken into account. Deriving stochastic models of functional responses have two purposes. First, it makes explicit the assumptions underlying the average interaction rates, i.e. the mean of functional responses. It has been performed in few studies in specific situations. <cit.> analyzed a Markov-chain model of a predator-prey system adapted from chemical reactions, and showed how Holling functional responses could emerge at the population level. <cit.> and <cit.> used a similar approach to derive a functional response in the specific cases of kleptoparasitism and a Beddington-DeAngelis functional response, respectively. Second, the development of stochastic models are needed for inferring functional responses from empirical data in order to clearly identify the processes and mechanisms underlying the variability of interaction rates, which appears to be large in experiments <cit.>. Three sources of variability are possible: i) environmental or exogenous variability, e.g. temperature or observation errors; ii) inter-individual variation of behavior or traits, e.g. size, color or run speed; iii)endogenous variability due to stochastic fluctuations of the interaction themselves, hereafter called interaction stochasticity.Modelling interactions processes at the microscopic level is necessary to evaluate the part of variance due to interaction stochasticity. <cit.> is the only study to our knowledge which proposed an explicit expression for an approximation of variance due to interactions, in the specific case of competition for resources. They assumed a site-based models where individuals are randomly distributed in patches every generation. They showed how different functional responses for competition can be derived depending on the assumptions about how resources are shared between competitors. A general stochastic framework for the derivation of functional responses is however still lacking. This would help to better justify the choice of the form given to functional responses, in interpreting data and making statistical inference, and evaluating the importance of interaction stochasticity for the dynamics and stability of populations and communities. In addition, since the variance due to interaction stochasticity comes from the interactions process itself, it is a valuable source of informations for inference, and would help for parameters estimation and models selection.In this paper, we propose a general stochastic framework which allows accounting for interaction mechanisms at the level of individuals and the derivation of functional responses at the population or community level. It is based on the modelling of the distribution of the times separating two interactions and on the use of the so-called Renewal Theory, a well-known mathematical stochastic theory (for the reader's convenience, we provide a brief account of this theory in Supp. Mat. <ref>). It is classically used in foraging theory but has never been used, to the best of our knowledge, to derive functional responses and bridge the gap between behavioral ecology and population ecology. We first show how functional responses and their stochastic fluctuations can be approximated under a wide range of possible individual behaviors and interaction types. Second, we show how our model can be used to derive stochastic versions of classical functional responses, such as Holling functional responses. We then show how to derive stochastic functional responses in many different ecological contexts, through two examples: the rate of successful copulations for males in a nuptial-feeding species, and the feeding rate of predators when handling of the prey depends on its size. Finally, we apply our framework for inference through a model comparison framework by reanalyzing a dataset on grey partridges <cit.>. We show in particular that information can be extracted from observed fluctuations in order to estimate model parameters and improve inference, given that variability due to interactions themselves are large enough relatively to other sources of variations. § FUNCTIONAL RESPONSES FROM RENEWAL THEORY: A GENERAL FRAMEWORK Consider a community with three species denoted by e_x, e_y and e_z. We take three species for concreteness, one can of course consider more species, and one can consider genotypes, phenotypes, substrates or resources instead of species. Our goal is to determine the number of times N_Δ(x,y,z) a focal individual of speciese_xsuccessfully interacts with other individuals of a given species e_y (possibly the same species), during a time span Δ, with x, y, z the size or density of each species in the environment.We assume 1 ≪ N_Δ(x,y,z) ≪ x,y,z, i.e. i) the number of interactions is low compared to the size of the populations during Δ, and ii) the number of interactions during Δ is large. In other words, it is assumed that interactions between individuals have a negligible effect on the population sizes and the time span Δ is large enough for many interactions to occur. Under these assumptions, the time betweenthe k-1^th and k^th interactions, denoted by T_k(x,y,z), is a random variable whose distribution is independent of k, but generally depends on x,y,z.The number of interactions N_Δ during Δ is thus a random variable defined as follows (Figure <ref>):N_Δ=kif T_1+⋯+T_k ≤Δ < T_1+⋯+T_k+T_k+1.Approximation of the mean and variance of a functional response.Functional responses, i.e. the expected number of successful interactions per unit of time (in the more specific case of predator-prey interactions the number of prey killed by a predator per unit of time, e.g. <cit.>), can be defined asR(x,y,z)=N_Δ(x,y,z)/Δ.Under the assumptions that the number of interactions is large (i.e. Δ is large) but the environment does not change during this time (i.e. variation in densities x, y and z is negligible), the mean and variance of the functional response R(x,y,z) can be approximated in law according to Renewal Theory byR(x,y,z) 1/(T(x,y,z))+𝒩(0,1/Δ(T(x,y,z))/(T(x,y,z))^3), where 𝒩(0,σ^2) denotes a Normal distribution with mean 0 and variance σ^2 (see Supp. Mat. <ref> for mathematical details). Since all the random variables T_k(x,y,z) have the same distribution, we dropped the index k and simply wrote T(x,y,z) inside the mean and the variance. This formula shows that knowing the expected time between two successful interactions [T(·) ] and its variance [ T(·)] provides the mean of a functional response and a confidence interval. It is important to notice that the stochastic fluctuations of the functional response around its mean are only due to the interactions per se and not to external sources: since we consider interactions between individuals as a stochastic process, it implies stochastic fluctuations. We will call this intrinsic source of variability interaction stochasticity in order to differentiate it with possible others, such as environmental stochasticity or between individuals variability.In order to give an explicit form of a functional response, the next step is to decompose the time T(x,y,z) according to the different events taking place between two interactions. Decomposing the time between two interactions. We suppose that the time between two interactions, namely T(x,y,z), can be decomposed into a sequence of a given number of independent steps, in a definite order. We start with a simple example, and give the general version afterwards. Consider a predator e_x and a prey e_y (the third speciese_z is ignored for simplicity). Assume that a predator has two activities:searching a prey and handling it if its capture is successful (see Fig. <ref>). Suppose it takes a time τ_h to handle a prey and τ_s to find it. The first attempt of the predator to find a prey takes a random time τ_s^(1). The prey is then either caught (`success') or not (`failure'). If the capture is successful, then the time of an interaction is T(x,y)=τ_s^(1)(x,y)+τ_h(x,y). Otherwise, the predator has to make another attempt, which takes a new random time τ^(2)_s. If there is a success, then T(x,y)=τ_s^(1)(x,y)+τ_s^(2)(x,y)+τ_h(x,y), and so on and so forth. The reader can guess that the number of steps is given by the number of trials needed to get the first success. It is well-known that this number is a random variable which follows a geometric law whose parameter is the probability of a success.The decomposition of the time between two interactions can be generalized and applied to many different ecological contexts, for any number of activities. Denote bythe (finite) set of possible activities and by || its cardinality, i.e. the number of necessary activities in order to perform a new successful interaction. In the previous example, we had ={“handling", “searching"} and ||=2.Let τ_a(x,y,z) be the random time needed to carry out a given activity a∈, e.g., a=“searching". Let p_a(x,y,z) be the probability that this activity ends up successfully, and G_a(x,y,z) the random number of attempts the focal individual must perform before succeeding activity a.G_a(x,y,z) is a random variable following a geometric distribution with parameter p_a(·). Then we can write (the notation (x,y,z) is dropped hereafter not to overburden the notations, but the reader should keep in mind that all random variables generally depend on x,y,z):T=∑_a=1^||τ^Σ_awhere τ^Σ_a is the total time spent for activity a to end up successfully, given byτ^Σ_a= ∑_i=1^G_aτ^(i)_a,where, the random variables τ^(i)_a, i=1,2,3,…, are independent and have the same distribution as τ_a (note that G_a appears as a superscript because it is the random number of attempts before succeeding activity a).We can compute the mean and variance of the time span T between two interactions, and we find (see Supp. Mat. <ref>)(T) = ∑_a=1^||(τ_a)/p_aand[4] (T) =∑_a=1^||( (τ_a )/p_a+1-p_a/p_a^2 (τ_a)^2 ). We finally obtain an approximation of the functional responseby plugging Eqs. (<ref>) and (<ref>) into Eq. (<ref>). To achieve this, one needs to know τ_a(x,y,z)and p_a(x,y,z) for all a∈, that is, the time taken by every activity and their probability of success, both generally depending on the species density in the community. These quantities can be determined empirically, or explicitly specified in various ecological contexts. We give examples and applications in the following.§ EXAMPLES OF FUNCTIONAL RESPONSES WITH SEARCHING AND HANDLING TIMES In this section, we show how stochastic versions of classical functional responses can be derived assuming simple processes at the level of the individuals for searching and handling activities.A general form of the functional response with searching and handling times.Our aim is to determine the expectation and variance of the number of interactions between a focal individual e_x and individuals e_y during a time Δ. We want to take into account possible perturbations due to interactions between the focal individual e_x with individuals of a third species e_z. We suppose that the time between two interactions T in a given environment (x,y,z) can be decomposed into a searching time τ_s and a handling time τ_h, that handling always succeeds (p_h=1) and catching an individual e_yhas a probability p_s to succeed. If searching fails, the focal individual starts a new searching phase or stop searching if the total time Δ is reached (Figure <ref>). From Eqs. (<ref>) and (<ref>), we can compute the expectation and variance of the time separating two interactions between the focal e_x individual and e_y individuals: (T) =1/p_s (τ_s)+(τ_h)and(T) =(τ_s)/p_s+1-p_s/p_s^2 (τ_s)^2 + (τ_h) which give an approximation of the stochastic functional response R (Eq. (<ref>)). In order to give explicit expression of the mean and variance of the time between two interactions (T) and (T), Eqs. <ref> and <ref> show that we now need to specify the mean and variance of the time taken to find an individual e_y ((τ_s) and (τ_s)), the probability to effectively interact with an individual once found (p_s), and the time taken for handling the interaction ((τ_h) and (τ_h)). In other words, we need to explicitly specify how individuals move into space and how interactions take place. In the following, we derive functional responses under explicit assumptions regarding foraging and handling times. We first detail a simple case with assumptions which are classically made to obtain a Holling type II functional response, i.e. constant handling time, a single type of prey, and foraging time as a function of prey densities in the environment. Second, we give a generalized model, in a d-dimensions space which can include many different classical functional responses. A simple case: foraging in a 2D space with handling. We assume a focal individual e_x foraging in a 2D space of size L^2, where y individuals e_y are uniformly distributed on a square lattice, such as the distance between the two nearest e_y individuals is L/(√(y)-1). The location of the individual e_x is randomly chosen in the 2D space and we want to calculate the expectation and variance of the distance D(y) to the closest e_y individual. The horizontal u_1 and vertical distances u_2 between e_x and e_y follow uniform distributions in the range [0, δ_2 ] with δ_2=L/2(√(y) -1), which gives ( D(y)) =1/δ_2^2∫_0^δ_2∫_0^δ_2√(u_1^2 + u_2^2) du_1 du_2 = C_ 2^EL/2(√(y)-1), (D(y)^2 )=1/δ_2^2∫_0^δ_2∫_0^δ_2(u_1^2 + u_2^2)du_1 du_2 = C_2^V(L/2 (√(y)-1))^2, where C_2^E and C_2^V are two constants (explicit values are given in Supp. Mat. <ref> ). It is assumed that the focal individual e_x has a perfect knowledge of the spatial distribution of individuals e_y and goes to the nearest e_y individual following a straight line at speed v. Hence, the expectation and variance of the searching time are( τ_s )=( D(y)) / v and( τ_s )=( D(y)) / v^2. Assuming that searching always succeeds (p_s=1) and that handling time is a constant c_y (( τ_h )=0), we finally get from Eqs. <ref> and <ref> the expectation and variance of the functional response R (R) = √(y)-1/c_y (√(y)-1) + C_2^E λ, (R) =1/Δ(C_2^V -C^E_2^2 ) λ^2 (√(y)-1)/( c_y (√(y)-1) + C_2^E λ)^3, with λ=L/2v, the scaled size of the environment (the size of the environment L/2 relative to foraging speed v). Predictions provided by Eqs. <ref> are in very good agreement with individual-based simulations of the present model (Figure <ref>).The functional response given by Eq. <ref> is a saturating function of the density y which looks like a Holling II functional response. Classically, the form of the Holling II functional response is justified by two mechanisms: a searching time depending on the density of the prey, and a constant handling time <cit.>. This is thus not surprising that we recover a functional response close to Holling II here. However, the exact form differs: we obtain a function of √(y) instead of y. This illustrates that being explicit about how individuals forage into the environment, with specific justifications about the mechanisms underlying interactions between individuals, can give rise to alternative functional responses (other alternatives are given in the following section). This also shows that adopting a bottom-up approach allows to estimate how variable is the number of interactions in a given ecological context due to intrinsic interaction stochasticity. In the present case, the variance of the functional response decreases with y. Foraging and handling one or two species in a d-dimensions space.We give here a generalization of the framework with handling and foraging in d-dimensions. The individual e_x now forages for two possible species e_y or e_z, with constant handling times c_y and c_z. We make similar assumptions than in the previous section, but we introduce the parameters α, denoting a possible preference of the species e_x for species e_z, and β denoting a different availability or vulnerability of species e_z (e.g. if β>1, e_z is easier to be detected relatively to e_y, see Supp. Mat. <ref> for details). We will moreover compare two different movements followed by the focal e_x individual: a straight line to the nearest individuals (as in the previous section) or a Brownian motion. We want to calculate the time between two interactions between e_x and e_y which depends on: i) the number of occurrence where e_x interacts with e_z instead of e_y, which follows a geometric distribution with probability of success y/(y+α z); ii) the two first moments of the time taken to reach a given species in a d-dimensions space θ^E_d(w) and θ^V_d(w), respectively (seeSupp. Mat. <ref> for details); iii) the handling times c_y and c_z. Finally, the time between two interactions between a focal individual e_x and individuals e_y are (Supp. Mat. <ref>): (T) =(y+α z/y-1)( θ^E_d(β z)+c_z)+ θ^E_d(y)+c_y, (T)= θ^V_d(y)-θ^E_d(y)^2+z α/y( θ^V_d(β z)-θ^E_d(β z)^2)in the case of a straight movement to the closest patch and (T)=(y+αz/y-1)( θ^E_ BM(β z)+c_z)+θ^E_ BM(y) + c_y, (T)= θ^V_ BM(y)-θ^E_ BM(y)^2+z α/y( θ^V_ BM(β z)-θ^E_ BM(β z)^2)in the case of a Brownian motion.With appropriate assumptions regarding underlying mechanisms, Eqs. <ref>, <ref> and <ref> are general enough to recover some classical functional responses and their variance. As shown in Table 1, assuminga 1D space and direct foraging to the nearest individual yields the equations for Holling Type I, II and III, Beddington-DeAngelis or the Ratio-Dependence functional responses, given further assumptions about handling time and possible interference between species. On a side note, we were not able to find appropriate assumptions to recover the exact equation of the Holling Type III functional responses as defined in the literature: We found a function of the form y^2/(1+y+y^2) instead of the classical form y^2/(1+y^2). It can be due either to a lack of generality of Eq. <ref>, or because there is no realistic biological assumptions which allows to recover the same exact form under our framework, or because Holling Type III is only correct when y is large, i.e. y is negligible relative to y^2. Even if both functions have similar sigmoidal shapes, our results illustrate that deriving functional responses with a bottom-up approach highlights implicit and often hidden assumptions regarding the underlying mechanisms.Interestingly, alternative underlying mechanisms can give rise to similar functional responses, in two ways. First, the exact same equation can emerge for different hypotheses. There are for instance two ways to introduce competition between predators to obtain the ratio-dependence functional response, assuming either that the probability of searching success is p_s=1/x or that the searching time is proportional to the density of individuals e_x, (τ_s(x,y,z))=λ x/y and the probability of success is p_s=1. Second, we can obtain similar form of the functional responses with alternative foraging strategies in a given environment. Fig. <ref> shows that decelerating functional responses can be obtained without handling time (c_y=0) but with foraging in a 2D or 3D space. Sigmoidal functional responses (Holling Type III-like forms) can emerge if foraging follows a Brownian motion in a 1D space. Note that not only forms of functional responses vary with assumptions, but also their range (interactions rates have different scales even though parameter values are identical, see the y-axes on Fig. <ref>). This illustrates the relevancy of the bottom-up approach developed here. Since the same functional responses can be obtained under different hypotheses, deducing underlying mechanisms from an observed form at the macroscopic level is limited; for instance, a sigmoidal functional response does not necessarily mean that a predator is able to learn. Instead, we propose to modelize interactions at the microscopic level (the level of the individuals) in order to get corresponding functional responses.Another merit to adopt a bottom-up approach is that the variance of the functional responses can be estimated (Tab. 1 and Fig.<ref>). The intensity of stochastic fluctuations of the functional responses, depicted by the 95% confidence interval on Fig. <ref>, depends on the supposed underlying mechanisms (e.g. the large fluctuations in the Ratio-Dependence functional response are due to the probability of searching success of the form 1/x), and on the density of species (e.g. fluctuations increase linearly with y when there is no handling). Also note that even if, as shown before, we can recover the same mean functional response with different underlying hypotheses, variance can be different: the ratio-dependence functional response shows different variance whether p_s=1/x and (τ_s(x,y,z))=λ /y orp_s=1 and (τ_s(x,y,z))=λ x/y. This is noteworthy because it shows that, in order to make inference from data, both the mean and the variance of the number of interactions per unit of time can be used to discriminate between concurrent functional responses models. § BUILD THE <INSERT YOUR NAME> FUNCTIONAL RESPONSEIn addition to recover classical functional responses, we argue that our theoretical framework is general enough to be applied to many different ecological contexts. In this section, we give two detailed examples inspired from the empirical literature. This illustrates how anyone with a specific question can derive functional responses and its stochastic fluctuations from basic knowledge about individual traits and behaviors, following a bottom-up approach. Nuptial feeding: how many successful matings for a male? In many species such as insects, spiders or birds, males bring food to females in order to increase their probability to mate <cit.>. Let us imagine that five successive steps must be fulfilled by a male to mate with one female: find a free a gift (e.g. a prey), handle it, find a free female, court it and copulate. Let denote τ_S_G, p_S_G and τ_H_G respectively the time taken and probability to successfully search for a free gift (i.e. not already handled by another male or female), and to handle it;τ_S_F, p_S_F and τ_H_C, p_H_C the time taken and the probability to successfully search for a female, and to court it; finally let p_R be the probability to successfully copulate with it. Applying Eqs. (<ref>) and (<ref>) gives the expectation and the variance of the time taken for a male to successfully mate with a female: (T)=1/p_R( (τ_S_G)/p_S_G +(τ_H_G)+(τ_S_F)/p_S_F+ (τ_C)/p_C)(T)=1-p_R/p_R^2( (τ_S_G)/p_S_G +(τ_H_G)+(τ_S_F)/p_S_F+ (τ_C)/p_C)^2 +1/p_R( 1-p_C/p_C^2 (τ_C)^2+1-p_SG/p_SG^2 (τ_SG)^2+1-p_SF/p_SF^2 (τ_SF)^2 ..+(τ_S_G)/p_S_G +(τ_S_F)/p_S_F).Assuming male-male and male-female competition for finding prey, the probability to successfully find a free gift by a focal male can be supposed as p_S_G=1/(m+f), with m and f the male and female densities, respectively. The form given to p_S_G is arbitrary and, assuming that all females and males have equal chance to find prey, it only reflects that the higher the density of competitor males and females, the more difficult for the focal male to find a gift. Note that the time spent by the focal male to find a free gift also depends on prey density through τ_S_G. Similarly, we assumed that males compete for finding a free female, with equal chance, which gives the probability to successfully find a female to court is p_S_F=1/m. Handling and court times, and the probability to copulate are assumed constant (respectively equal to C_G, C_C and p_R=p). An approximation of the expectation and variance of the number of successful matings by a male can then be obtained using Eq. <ref>. Figure <ref> shows that the variance of the number of successful copulations decreases when the number of males and females decreases, and that the expected number of successful copulations has a non-monotonous variation with the density of females. Such an approach can be useful for studying the evolution of interaction behavior in the context of nuptial-feeding behaviors <cit.>.Trait-dependent interaction times: prey size and functional responses.While most functional responses make the implicit assumption that interaction rates only depend on the number or density of individuals, many investigations suggest that interaction rates might also depend on individual traits and their distribution in a population, especially body mass or size <cit.>. For instance, <cit.> showed that functional responses generally depend on the ratio between predator and prey masses, and <cit.> showed that predators prefer prey with a particular body mass. We show here that our theoretical framework can be used to derive functional responses dependent on quantitative traits.As an illustration, we will focus on the derivation of handling times accounting for body mass. We neglect searching times for the sake of simplicity, but it might be relevant to include the effect of size on the time and probability to find and catch a prey: larger prey can be more easily detected or caught. The effect of prey size on searching time can for instance be taken into account into the spatial scale L introduced before. Let us note s the size of the prey and m(s) its distribution, supposed following a truncated normal distribution defined on [0,+∞], with mean μ_s and standard deviation σ_s. π(s) is defined as the density probability that a predator catches a prey of size s. Finally, we suppose that the expected handling time conditional on the prey size s is (τ_h | s) = t(s). The expectation and variance of handling times are thus(τ_h) =∫_0^∞π(s)t(s)s, (τ_h) =∫_0^∞π(s)t^2(s)s - ((τ_h))^2. We can now be more specific in order to give explicit expression of the functional response. Let P(s) be the preference of the predator for prey with size s, assumed to follow a truncated Gaussian function defined on [0,+∞], with μ_P the most preferred size and σ_P its tolerance. The probability to catch a prey of size s is therefore π(s)=m(s) P(s) ds / ∫ m(u) P(u) du. We can reasonably assume that handling time is a bounded increasing function of prey size s, such as (τ_h|s) = τ_max(1-e^-s). Finally, using Eq. (<ref>), an approximation of the functional response and its variance can be obtained (Fig. <ref>). Figure <ref> shows how the functional response and its variance change as a function of the variance of the body size of the prey μ_s. This illustrates that functional responses not only depend on the average body size, as shown several times <cit.>, but also on the variance of prey body size, possibly giving non-monotonous relationships.§ FUNCTIONAL RESPONSES INFERENCE: A MODEL COMPARISON FRAMEWORKMany works compare different functional responses models with likelihood ratio test or the Akaike's Information Criterion in order to determine which one fits the best <cit.>. However, it is generally difficult to discriminate between alternative functional responses <cit.>. A limit of these approaches is that the variance of the functional responses is directly estimated from the data, while variance depends on interactions themselves, as shown by our framework (Eq. <ref>). Here, we reanalyze a dataset from <cit.> to estimate the feeding rate of grey partridges on seeds in a controlled experiment. <cit.> especially aimed at testing whether vigilant behaviors significantly affect the feeding rate. They compared two models using the AIC method: Holling type II with and without vigilant behaviors. They found no statistical difference between models with or without vigilance, and concluded that vigilant behaviors do not affect feeding rate. Here, we first derive a functional response with vigilant behaviors using our bottom-up approach. Second, we use a likelihood approach to estimate the parameters and test whether vigilance significantly affects grey partridges feeding rates (see Supp. Mat. <ref> for details).We supposed that the time between two successful interactions, i.e. between two eaten seeds, can be decomposed into vigilance bouts with constant duration c_v, occurring with probability p_v, searching bouts τ_s, successful with probability p_s, and handling bouts with constant duration c_h. We fitted and compared models with (p_v ≠ 0) or withoutvigilance (p_v=0) using a maximum likelihood approach. Note that parameters of the model were estimated both from the mean and variance (since mean and variance of the functional responses are explicitly expressed as a function of the ecological parameters). The best model was estimated to be in a 2D space, with a direct movement to the nearest seed, with vigilance (Likelihood-ratio of the models with and without vigilance = 641.3). We estimated the probability of entering a vigilant bout p_v ≃ 0.12, which roughly corresponds to the estimation from direct observations in <cit.>. A comparison of the resulting functional responses is shown on Fig. <ref>.To illustrate the importance of the information included in the variance, we also fitted the models assuming a fixed variance independent of the parameters. In this case, the model without vigilance is the best (using an AIC comparison) because it has a lower number of parameters to estimate. In practice, only using the expectation of the functional response makes difficult to distinguish between different functional responses. This is however made possible using the variance. In our case, the large variability in the functional responses are better explained by the model with vigilance because the probability to enter into a vigilance bout mechanically increases variance in our model (Eq. <ref>).These results must yet be taken with caution. The large variation in the functional response shown in the data (Fig. <ref>) can be due to other sources of variability, observation errors or variability between individuals, not taken into account into the present analyses. Indeed, as shown in <cit.> for a predator fish, most of the variation in the number of prey consumed by unit of time can be due to variation in the predator individual traits: searchingand handling times both decrease with predator body size. This suggests, in the case of the grey partridges treated here, that the large observed variance is not necessarily due to the probability to enter into a vigilance bout, but to between-individuals variability. In order to properly analyze such type of data and improve our ability to infer functional responses parameters, an adapted statistical framework must be developed taking into account both between- and within-individuals variability. Within-individuals variability is taken into accountby our model (Eq. <ref>) but a mixed-effect model is needed to take into account between-individuals variability. Ideally, taking account ofbetween prey variability would also be needed, for instance to take into account the effect of the variability in prey body size. Classically, mixed effects models include between-individuals variability only in the mean part of the model, and not in the variance. In our model, since between-individuals variability would affect both the mean and the variance of the model (Eq. <ref>), a specificmixed-effect statistical framework is needed (development in progress). § DISCUSSIONDealing with identifiability. The rate at which individuals interact, within or between species, is central in most ecological processes: it affects individuals' growth, birth and death, populations' and communities' dynamics, and how energy and matter flow through ecosystems. Predicting ecological dynamics and supporting management decisions depend on our understanding of how individuals interact. It is thus crucial to use functional responses models that fit the best with data and with mechanisms underlying interactions <cit.>. However, functional response models are hardly identifiable: often, different functional responsesfit well to data and inferences show no or few statistical power <cit.>. This identifiability problems of functional responses sometimes even resulted in violent debates <cit.>. Adopting a bottom-up approach, we show in the present paper that such an identifiability is not surprising since, on the one hand, various underlying mechanisms can give rise to similar or identical functional responses (Fig. <ref>). On the other hand, our results show that choosing a functional response with a phenomenological approach does not give insight about the underlying mechanisms.Ignoring the various sources of variability of the interactions rates is certainly responsible in part of the difficulty to infer functional responses from data. It is now recognized that both interspecific and intraspecific variations are important in ecology <cit.>, especially for functional responses <cit.>. For instance, <cit.> and <cit.> respectively showed that functional responses depend on predators and prey body mass. Here we show that, in addition to within and between species variability, a third source of variability should also be considered: interaction stochasticity (Fig.<ref>). Since ecological processes at the level of the individuals are stochastic, the number of interactions per unit of time is random and make functional responses randomly fluctuate. We argue that the variation in the number of interactions per unit of time can have three internal sources: within species variations for both types of interacting individuals (e.g. the predator and the prey), and interaction stochasticity (Fig. <ref>). Inferring functional responses from data would need to take those three sources of variability into account. Our model would allow the development of a statistical framework adapted to this problematic since it follows a bottom-up approach, from the individual to the macroscopic level. Since our model allows to relate the variance of functional responses to ecological parameters and individual traits (Eq. <ref>), variance in data can be used as a source of information to infer parameters (as we showed in reanalyzing <cit.>'s dataset). In many cases, one can expect interaction stochasticity to be the weakest source of variability in real data. As it is illustrated in our reanalysis of the grey partridges dataset, relying on it alone is certainly not relevant for model identification. Developing a statistical framework allowing to disentangle all sources of variability and to extract information from variance is a new challenge.However, one of the main message of our paper is to question the relevancy of making inferences by using phenomenological functional responses: is it really important to know whether an hyperbolic Holling Type II functional response fits better the data than a sigmoidal Ginzburg-Arditi functional response <cit.>? We think it is more relevant to make the link between behavioral ecology, foraging theory, population ecology and community ecology. We argue that our model can help in doing this by i) stopping searching which adhoc functional response fits best to data, and ii) constructing one's own functional response from one's own specific case, using basic knowledge about species and their ecological contexts. Our model is general enough to adapt to many different situations, given it is possible to decompose interactions into a sequence of different activities. Assumptions and time scales. Many situations in natural populations or experiments are obviously expected to depart from our model. We made some choices for the sake of simplicity, without really being necessary. For instance, we supposed that individuals move in straight line or following a Brownian motion, which can be modified accordingly to a specific ecological context. However, two assumptions are inherent to our theoretical framework, and for which departures can be generally found in nature. First, we assume that the interactions can be decomposed into a sequence of activities whose duration are random, independent and identically distributed, which is a fundamental hypothesis of Renewal Theory used here. Yet, individuals are in general able to learn or sets mark in the environment in order to improve their ability to search, or change their preference for a prey or another, which violates the independence in time <cit.>. Individuals can also be clustered in space, which would for instance affect the time taken by a predator to find two succeeding prey <cit.>. Population size of each interacting species can also significantly vary during the considered time frame, because of prey depletion or blooming. Second, we assume that the number of interactions and the considered time frame are large. Yet in general, experiments and observations generally involve a few dozens or hundreds of interacting individuals during a limited time frame. What is the impact of violating these assumptions on the resulting functional response is an open question. Performing stochastic simulations might obviously help in addressing this question. However, we think it can be addressed more generally from a theoretical point of view. Indeed, we think that it mostly relies on how dothe different time scales of the ecological processes involved in the functional response relate with each other. For instance, if individual learning occurs at a slower time scale than it consumes prey, then we can expect that learning little affects the resulting functional response. Similarly, if the time taken by a predator to move from one patch of prey to another is large enough, then it should not affect much the resulting functional response in a short time frame. If prey reproduction is fast enough, then depletion should have a low impact. Some authors proposed for example to take into account prey depletion into account <cit.>, but the approach is still phenomenological, i.e. different time scales are generally not considered. The relative importance of the time scales is central in ecology. It deserves particular attention in the case of functional responses in particular <cit.>, and it is a natural perspective of the current work. The last perspective of this work is about how individuals interactions translates into birth and death, in other words how functional responses are related to numerical responses <cit.>. Since most population and community dynamics models are systems of differential equations, functional responses and numerical responses are generally assumed to play at similar time scales (except when slow/fast processes are explicitly assumed). Some authors showed in specific cases that considering different time scales for functional responses and numerical responses <cit.> can dramatically affect population dynamics and stability.Finally, functional response should be considered as the result of stochastic ecological processes. We highlight in the present paper the phenomenon we called interaction stochasticity, in analogy with demographic stochasticity. It is well-known that demographic stochasticity can affect population stability <cit.>. How interaction stochasticity can affect population and community stability, through its effect in numerical responses, is an open question.§ AUTHOR CONTRIBUTIONSAll authors designed the theoretical framework. SB wrote the first draft of the manuscript, and all authors contributed substantially to revisions. § ACKNOWLEDGEMENTSWe are very grateful to David Baker for sharing its dataset on grey partridges. We want to thank Clotilde Lepers, Elisabeta Vergu and Arnaud Senlis for helpful discussions and Geoffroy Berthelot for providing simulations. SB benefited from the support by the “H.+C.+Y.+M.+V.+D.^†+M. Moens-Sacchettini organization for the advancement of knowledge”. § DATA ACCESSIBILITYData are deposited on a public server (Dryad : doi:10.5061/dryad.c73bm2q) § FUNDING STATEMENT This work was supported by the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement-Ecole Polytechnique-Museum National d'Histoire Naturelle-Fondation X and by the ANR ABIM (ANR-16-CE40-0001). unsrtnat§ FIGURESempty equationsection § SUPPLEMENTARY MATERIALS OF THE PAPER“REJUVENATING FUNCTIONAL RESPONSES WITH RENEWAL THEORY”by S. Billiard, V. Bansaye and J.-R. Chazottes. §.§ Some basics in Renewal Theory For the convenience of the reader, we state and prove the two results we are using in this paper. We refer toe.g. <cit.> or <cit.> for more details. Let T_1, T_2,… be a sequence of independent non negative random variables. We assume that the T_i's are all distributed as a non negative random variable T with distribution F. We assume that F(0)≠ 1 (i.e., the random variables are not identically zero) and (T)<+∞. Finally, let S_n=∑_i=1^n T_i for n≥ 1. The renewal process (N_t)_t≥ 0 associated to T_1, T_2,… is defined byN_t=∑_n=1^+∞1_{S_n≤ t}=sup{n ≥ 1: S_n≤ t}.Since (T≥ 0)=1 and (T=0)<1, we have (T)>0. By the strong law of large numbers, we know that S_n/n converges to (T) with probability one. Therefore we deduce that S_n→+∞ with probability one. In particular, with probability one, S_n<t for only a finite number of n's, showing that (N_t<+∞)=1. It follows that, for all practical purposes, we can put a `max' instead of a `sup' in the definition of N_t.Strong law of large numbers for renewal processes. We haveN_t/t1/(T)with probability one.Proof. By the very definition of N_t we have S_N_t≤ t < S_N_t+1, implying thatS_N_t/N_t≤t/N_t <S_N_t+1/N_t.Now N_t→+∞ with probability one, henceS_N_t/N_t(T)andS_N_t+1/N_t+1N_t+1/N_t(T). To state the second result we use, we have to assume in addition that the variance of T, which we denote by σ^2, is finite.The central limit theorem for renewal processes. We haveN_t -t/(T)/σ√(t/((T))^3)𝒩(0,1),where 𝒩(0,1) is the standard Gaussian distribution (centred at 0 and with variance equal to 1).Proof. Let r_t:=t/(T) + yσ√(t/((T))^3). If it is an integer, let n_t=r_t, if it is not an integer, let n_t=⌈ r_t⌉ +1. Then(N_t -t/(T)/σ√(t/((T))^3) <y)= (N_t<r_t) = (N_t<n_t)= (S_N_t>t) (because {N_t<n}={S_n>t})= ( S_n_t-n_t (T)/σ√(n_t)> t-n_t (T)/σ√(n_t)).We now use the central limit theorem for the sequence T_1, T_2,…, and the fact thatlim_t→∞t-n_t (T)/σ√(n_t)= lim_t→∞t-r_t (T)/σ√(r_t)= lim_t→∞ -y (T)√(t/((T))^3)/√(t/(T)+yσ√(t/((T))^3))=-y.It follows that, if Z is a random variable distributed according to 𝒩(0,1), we havelim_t→+∞(N_t -t/(T)/σ√(t/((T))^3) <y) = (Z>-y)=(Z<y).This ends the proof. Loosely speaking, N_t is approximately Gaussian with mean t/(T) and variance tσ^2/((T))^3.Application to determine the stochastic fluctuations of the functional responses.To apply the two previous theorems in our context, set t=Δ and apply them for each given triplet (x,y,z). More precisely, the functional response, defined as R(x,y,z)=N_Δ(x,y,z)/Δ, is a random variable and we want to determine its fluctuations around its expected value 1/(T(x,y,z)). This can also be achieved by applying Renewal Theory. Colloquially, we get that the distribution of N_Δ(x,y,z)/Δ- 1/(T(x,y,z))is approximately a Gaussian distribution centred at 0 with a variance equal to 1/Δ(T(x,y,z))/((T(x,y,z)))^3,where (T(x,y,z))=(T^2(x,y,z))-((T(x,y,z)))^2 stands for the variance of T(x,y,z). In particular, one can say that standard deviation of (<ref>) is approximately1/√(Δ) √((T(x,y,z)))/(T(x,y,z)))^3/2.The larger the Δ, the better these approximations. Finally, we can condense what precedes by writingR(x,y,z) 1/(T(x,y,z))+𝒩(0,1/Δ(T(x,y,z))/(T(x,y,z))^3), where 𝒩(0,σ^2) is a random variable having a Gaussian distribution centred at 0 and with variance σ^2.§.§ Some probability factsSum of i.i.d random variables X_i a random number of times Y. Let Y be a random variable assuming positive integer values such that E(Y)<+∞. Let X_1,X_2,… be a sequence of independent identically distributed random variables which are also independent of Y, and such that (X_1)<+∞. Then(∑_i=1^Y X_i)=(Y) (X_1).Furthermore, assume that (Y)<+∞ and (X_1)<+∞. Then(∑_i=1^Y X_i )=(Y) (X_1) + ((X_1))^2 (Y).Expectation and variance of the time between two interactions T (Eqs. <ref> and <ref>). Assuming that the probability of success of activity a is p_a, then Y_a the number of times activity a is performed before success follows a geometric distribution with expectation and variance respectively [ Y_a ]=1/p_a and [ (1-p_a)/p_a^2]. Assuming that activity a takes a time X_a=τ_a, with expectation [τ_a ] and variance [τ_a ], which finally gives the expectation and variance of the time between two interactions T in Eqs. <ref> and <ref>. Expectation and variance of the time taken to search and find e_y (Eqs. <ref> and <ref>).We want to calculate T the time taken by a focal individual e_x to interact with an individual e_y given there are individuals e_z in the environment. Y_s is the the number of times the individual e_x forages the environment before encountering e_y, hence e_x interacts Y_s-1 times with e_z before interacting with e_y. Assuming that e_x has a probability y/(y+α z) to choose to searching for individual e_y, the expected number of time e_x interacts with e_z before interacting with e_y is [Y_s -1]=(y+α y)/y-1.Further,in the case of a straight movement to the nearest prey, assuming that the expected time taken to search for individuals e_z is θ^E_d(β z), that handling time of e_z by e_x is a constant c_z, that the expected time to search for individual e_y is θ^E_d(y), and that handling time of e_y by e_x is a constant c_y then the total expected time taken by interaction between e_x and individuals e_z is given by[ T | Y_s ]= (Y_s-1) (θ_d^E (β z) +c_z) + θ_d^E (y) +c_y[ T ]= ( y+α z/y -1 )(θ_d^E (β z) +c_z)+ θ_d^E (y) +c_ywhich corresponds to Eq. <ref>(similar calculations gives Eq. <ref> in the case of a Brownian motion).In order to calculate the variance of the time taken by a focal individual e_x to interact with an individual e_y, we need to calculate the second moment of T. Assuming that the second moment of the time taken to search for individuals e_z is θ^V_d(β z) and to search for individuals e_y it is θ^V_d(y),the second moment of T is given by[T^2 | Y_s ]= (Y_s-1) θ^V_d(β z)+ θ^V_d(y) + 2 (Y_s-1) θ^E_d(β z) θ^E_d(y)+(N-1)(N-2) θ^E_d(β z)^2E[T^2]=( y+α z/y -1 )θ^V_d(β z)+2( y+α z/y -1 )θ^E_d(β z) θ^E_d(y)+( y+α z/y -1 )( y+α z/y -2 )θ^E_d(β z)^2 + θ^V_d(y) which after some calculations gives the variance of T in Eq. <ref>.§.§ Foraging in a d-dimension space The individual e_x forages an environment where two species e_y or e_z are present with densities y and z, respectively. The environment is a d-dimensions space with size L^d. The handling times of the individual e_x with species e_y and e_z are assumed constant, respectively denoted by c_y and c_z. We assume that the focal individual e_x arrives randomly at a given position in space, and that it has a perfect knowledge of the environment. The parameter α denotes a possible preference of the species e_x (if α>1) or rejection (if α <) for resource e_z relatively to species e_y. Because of this possible preference, the individual e_x chooses to interact next with an individual of species e_y with probability y/(y+α z) or with an individual of species e_z with probability α z/(y+α z). Given the species with which individual e_x will interact next, say e_z, we have to compute the time taken for e_x to reach an individual e_z. The parameter β denotes an availability or vulnerability of species e_z relatively to e_y during the foraging process (for instance species e_z is easier to detect if β>1). The expected time to reach an individual e_y isdenoted by θ^E_d(y), and the time taken to reach an individual e_z is denoted by θ^E_d(β z). Two different movements are considered: following a straight line to the nearest patch or following a Brownian motion at speed v. Once the interaction has taken place, foraging starts again from a random position in the environment. There is thus no memory in the foraging process, which is a basic assumption of Renewal Theory used in our framework.§.§.§ Regular repartition in a d-dimensions space, movement to the nearest individualsThe focal e_x individual has a random location in the environment and moves to the nearest individual. We denote D_d(y) the expected distance between the e_x individual and and individual of species e_y given there is a number y individuals in the environment (similarly we denote D_d(β z) the expected distance between e_x and e_z). Assuming that e_y and e_z individuals have a regular distribution in a d-dimensions space of size L^d, the distance between two individuals e_y is δ_d(y)=L/21/y^1/d-1, andthe distance between two individuals e_z is δ_d(β z)=L/21/(β z)^1/d-1 (accounting for β, the vulnerability or availability of e_z). The two first moments of the distance D_d(w) are given by (D_d(w))= 1/δ_d(w)^d∫ ... ∫_0^δ_d(w)√(u_1^2+...+u_d^2)u_1 ...u_d= C^E_d δ_d(w),(D_d(w)^2)=1/δ_d(w)^d∫ ... ∫_0^δ_d(w) (u_1^2+...+u_d^2)u_1... u_d = C^V_d δ_d(w)^2,with C^E_d and C^V_d two constants depending on the dimension, given byC^E_1= 1/2 , C^V_1 = 1/3; C^E_2= √(2)+log(1+√(2) )/3 , C^V_2 = 2/3;C^E_3= 6 √(3)- π+log(3650401+2107560√(3) )/24,C^V_3 =1.Given that the e_xindividual moves at speed v, the two first moments of the time taken by the e_x individual to encounter ane_w individual in a d-dimension space are given byθ^E_d(w)=(D_d(w)/v) =C^E_d δ_d(w)/v,θ^V_d(w) = (D_d(w)^2/v^2)= C^V_d δ_d(w)^2/v^2.Finally, given the focal individual e_x has a preference α for e_z individuals, and given that the handling time of e_z individuals by e_x is c_z, then the total expected searching time τ_s,d(x,y,z) of an e_y individual by an e_x individual in a d-dimensions space is given by(τ_s,d(x,y,z)) =(y+α z/y-1)( θ^E_d(β z)+c_z)+ θ^E_d(y), (τ_s,d(x,y,z))= θ^V_d(y)-θ^E_d(y)^2+z α/y( θ^V_d(β z)-θ^E_d(β z)^2) §.§.§ Regular repartition in a 1-dimension space, Brownian motion We assume as beforethat e_z have weights α and β, and that e_y and e_z are regularly distributed in space, with a distance δ_1(y)=L/(2 (y-1) between individuals y and δ_1(β z)=L/(2 (β z -1) between individuals e_z. Note that we focus here on a 1-dimension space for simplicity. Let us denote H(l,a,b) the random variable representing the time taken by the e_x individual moving following a Brownian motion to first hit the boundary of a segment [a,b], given that its initial location l is random in l ∈[a,b]. The Laplace transformation of H(l,a,b) isℒ(u)=cosh[(b-2l+a)√(u/2)]/cosh[(b-a)√(u/2)].The two first moments of H(l,a,b) are respectively given by ∂ℒ(u)/∂ u|_u=0 and ∂^2 ℒ(u)/∂ u^2|_u=0 <cit.>. Hence, the two first moments of the time taken by an e_x individual to reach either an individual e_w are respectively (with a=0 and b=δ_1(w)): θ_ BM^E(w)=C^E_ BM δ_1(w)^2 / v and θ_ BM^V(w)=C^V_ BM δ_1(w)^4/v^2 with C^E_ BM=1/6, C^V_ BM=1/15. Finally, given a weight α of e_z individuals an handling time of individuals e_z by e_x is c_z, the expected searching time of an e_y individual by an e_x individual in a 1D space with Brownian motion is given by(τ_s, BM(x,y,z)) =(y+αz/y-1)( θ^E_ BM(β z)+c_z)+θ^E_ BM(y),and its variance is(τ_s, BM(x,y,z))= θ^V_ BM(y)-θ^E_ BM(y)^2+z α/y( θ^V_ BM(β z)-θ^E_ BM(β z)^2)§.§ Analysis of Baker et al. (2010)'s data In the <cit.> experiments, the behavior of grey partridges was observed and recorded in controlled conditions, with variable seeds densities. The feeding rate was calculated, i.e. the number of seeds eaten per second per individual. Other behaviors were also recorded such as the duration of vigilance bouts or handling times. The duration of observations bouts was at most of 240 seconds. <cit.> derived a version of Holling Type II functional response with vigilance in a phenomenological manner and fitted it with their data. They showed no statistical difference between models with or without vigilance and concluding that vigilant behaviors do not affect the feeding rates of grey partridges. Here, we derive a functional response with vigilance from the level of the individuals (following our bottom-up approach). We first give the expectation and the variance of the feeding rate, and second we fit our models on <cit.>'s data following a maximum likelihood approach, using both the mean and the variance of the functional response. §.§.§ Derivation of a functional response with vigilanceWe derive the functional response with vigilance by decomposing the time T between two successful interactions, i.e., two eaten seeds, in three steps. First, we suppose that the time between two eaten seeds is decomposed into a time taken to approach a seed and a time to handle it, respectively denoted by τ_a and τ_h. Supposing that the handling time is fixed (τ_h)=c_h, that handling and approaching are always successful, we have(T) =(τ_a)+(τ_h)=(τ_a)+c_h (T) =(τ_a)+(τ_h)=(τ_a). which gives an approximation of the stochastic functional response as given by Eq. (<ref>). Second, we suppose that the time taken to approach a seed is decomposed into times to successfully foraging for a seed, denoted by τ_f which succeeds with probability p_f. We thus have (τ_a)=(τ_f)/p_f (τ_a)=(τ_f)/p_f+1-p_f/p_f^2 (τ_f)^2. Third, we suppose that the time taken by successful foraging can be decomposed into a succession of vigilance bouts, with duration τ_v and searching time denoted by τ_s, such as τ_f=τ_v+τ_v+⋯+τ_s. We denote p_v the probability that the animal enters in a vigilance bout, hence the expected number of vigilance bouts before encountering a seed is 1/(1-p_v)-1. Finally, assuming that the vigilance times are fixed, (τ_v)=c_v and (τ_v)=0, we have (τ_f) =(1/1-p_v-1)(τ_v)+(τ_s)=(1/1-p_v-1)c_v+(τ_s) (τ_f) =(1/1-p_v-1)(τ_v)+p_v/(1-p_v)^2 (τ_v)^2 + (τ_s)=p_v/(1-p_v)^2c_v^2 + (τ_s).The last step consists in determining mean and variance of the searching times, which depends on the movement of the individual and the dimension of space and the density of seeds y, as shown in the main text (see Eqs. (<ref>) and (<ref>)). §.§.§ Likelihood ratio test From the model given before, we can derive an approximation of the expectation and the variance of the functional response as a function of seeds density y and the set of parameters Θ: the time taken by a vigilance bout c_v and the probability to enter in a vigilance bout p_v, the time taken by handling seeds c_y, the probability that foraging succeeds p_f. Our goal is to test whether vigilant behaviors significantly affect the functional response given a data set (D) from the experiments in <cit.>. To answer this question, a commonly used method is to estimate the parameters set Θ of two different models, without (p_v=0) or with (p_v ≠ 0) vigilance bouts, and to compare their likelihood by a likelihood ratio test since both models are nested. The likelihood of the functional response R(Θ,y) given the dataset 𝒟 is defined asℓ( R(Θ,y) | 𝒟) ≡( 𝒟|R(Θ,y))where (E) is the probability of E. An approximation of ( 𝒟|R(Θ,y)) is given by Eq. (<ref>) and its derivation in App. <ref>. The dataset 𝒟 contains the feeding rate as calculated for a given individual in eight treatments with different densities from 5 to 400 seeds.m^-2. We looked for the maximum of logℓ for different models (1D, 2D, or 3D space with movements to the nearest and Brownian motion in a 1D space), with or without vigilance, using the newton method in the “FindMaximum” procedure of Mathematica 10.1 <cit.>. Parameters estimation was constrained to adequate with their definition (for instance probabilities must lie between 0 and 1) and with our information about the protocol of the experiments (the time duration Δ was estimated in our model and supposed to be lower than 240 seconds; the scaled size of the environment λ was supposed between 1 and 20; handling times and vigilance bouts were measured in seconds). The best model was the 2D space with movement to the nearest with vigilance (see main text). To illustrate the importance of the fact that the parameters are estimated not only by fitting the expectation of the functional response but also its variance, we also fitted the models with and without vigilance using nonlinear model fitting(“NonLinearModelFit” in Mathematica 10.1 <cit.>), assuming a fixed variance independent of the parameters. Following this procedure, the model without vigilance had the lowest AIC and thus considered as the best model, in agreement with <cit.>'s results and conclusions. However, this is due only to the fact that it has one less parameter to estimate, hence the AIC chooses the simplest model. As shown in the main text, the AIC of the model fitted with this latter procedure is lower than the model fitted with our framework (using Eq. (<ref>)). The estimation of the parameters are more constrained when both variance and expectation depends on the ecological parameters: the fitting procedure must find a parameters set satisfying the best trade-off between both the expectation and the variance of the functional response. When using the information from the variance of the data to estimate parameters, the different models show largely different log-likelihoods, showing that the difference of the AIC both procedures is not essentially due to the difference in the number of parameters to be estimated. Our results suggest a different conclusion than the one by <cit.>: since the best model is with vigilance, we conclude that it significantly affects the feeding rate of grey partridges in the context of this experiment. Our results and conclusions illustrate the importance to use the information from the variance of the feeding rate, and not only its mean. Especially, our framework makes possible to express this variance as a function of the ecological parameters, because of the bottom-up approach: describing the behaviors and properties at the level of the individuals to make emerge functional responses at the macroscopic level. | http://arxiv.org/abs/1706.08774v2 | {
"authors": [
"S. Billiard",
"V. Bansaye",
"J. -R. Chazottes"
],
"categories": [
"q-bio.PE"
],
"primary_category": "q-bio.PE",
"published": "20170627110257",
"title": "Rejuvenating functional responses with renewal theory"
} |
[email protected] Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.22085, 46071 Valencia, Spain The possible role of the triangle mechanism in the B^- decay intoD^*0π^-π^0η and D^*0π^-π^+π^- is investigated.In this process, the triangle singularity appears from the decay ofB^- into D^*0K^-K^*0 followed by the decay ofK^*0 into π^-K^+ and the fusion of the K^+K^- whichforms the a_0(980) or f_0(980) which finally decay into π^0η orπ^+π^- respectively.The triangle mechanism from the K̅^*KK̅ loop generates apeak around 1420 MeV in the invariant mass of π^-a_0 or π^-f_0,and gives sizable branching fractions (B^-→D^*0π^- a_0;a_0 →π^0η)= 1.66 × 10^-6 and (B^-→D^*0π^- f_0 ; f_0 →π^+π^-)= 2.82 × 10^-6.Triangle singularities in B^-→ D^*0π^-π^0η and B^-→ D^*0π^-π^+π^- E. Oset December 30, 2023 =================================================================§ INTRODUCTIONHadron spectroscopy is a way to investigate Quantum Chromodynamics (QCD), which is the basic theory of the strong interaction. The success of the quark model in the low-lying hadron spectrum gives us an interpretation of the baryons as a composite of three quarks, and the mesons as that of quark and anti-quark <cit.>. Meanwhile, the possibility of non conventional hadrons called exotics, which are not prohibited by QCD, have been intensively studied. One example is the Λ(1405): the quark model predicts a mass at higher energy than the observed peak, and a K̅N (I=0) molecular state seems to give a better description as originally studied in Ref. <cit.> followed by many studies which are summarized in Refs. <cit.>. The spectrum of the low-lying scalar mesons, such as f_0(980) or a_0(980) mesons, is also discussed in this picture <cit.>, while the possible explanation as tetraquark states is also discussed in Refs. <cit.>. These days, in the heavy sector, the XYZ <cit.> and the P_c <cit.> were discovered, which cannot be associated with the states predicted by the quark model. Another sort of non conventional hadrons are the molecular states of other hadrons, which have been often invoked to describe many existing states (see recent review in Ref. <cit.>). Besides ordinary hadrons, molecular states or multiquark states, triangle singularities can generate peaks, but these peaks appear from a simple kinematical effect. These singularities were pointed out by Landau <cit.>, and the Coleman-Norton theorem says that the singularity appears when the process has a classical counterpart <cit.>: in the decay process of a particle 1 into the particles 2 and 3, the particle 1 decays first into particles A and B, followed by the decay of A into the particles 2 and C, and finally the particles BC merge into the particle 3. The particles A, B, and C are the intermediate particles, and the singularity appears if the momenta of these intermediate particles can take on-shell values. A novel way to understand this process is proposed in Ref. <cit.>.For the decay of η(1405) into π^0π^0η via π^0a_0 and π^0π^+π^- via π^0f_0, the triangle mechanism gives a good explanation <cit.>. The K^*K̅K loop generates the triangle singularity in this process, and the anomalously large branching fraction of the isospin-violating π^0f_0 channel reported by BESIII <cit.> is well explained with the mechanism.The peak associated with this singularity can be misidentified with a resonance state. For example, the studies in Refs. <cit.> suggest the possible explanation of Z_c(3900) with the triangle mechanism. Similarly, a peak seen in the π f_0 (980) mass distribution, identified as the "a_1(1420)" meson by the COMPASS colaboration <cit.>, is shown to be a manifestation of the triangle mechanism as studied in Refs. <cit.>. In particular, many XYZ states are discovered as a peak of the invariant mass distribution in the heavy hadron decay. Then, the thorough study on the role of the triangle singularities in the heavy hadron decay is important to clarify the properties of the reported XYZ states. In the B^-→ K^-π^-D_s0^+(2317) (K^-π^-D_s1^+(2460)) process, a peak can be generated by the triangle mechanism around 2800 MeV (2950 MeV) in the π^-D_s0^+ (π^-D_s1^+) invariant mass spectrum, which is driven by the K^*DK (K^*D^*K) loop, and gives a sizable branching fraction into the channel <cit.>. The D_s0^+ and D_s1^+ in the final state are dynamically generated by the DK and D^*K, and have large coupling with these states <cit.>. Because the process of the triangle mechanism contains a fusion of two hadrons, the existence of a hadronic molecular state plays an important role in having a measurable strength. Then, the study of the singularity is also a useful tool to study the hadronic molecular states. Regarding the P_c peak, discovered in the J/ψ p invariant mass distribution of the Λ_b decay <cit.>, the possibility of the interpretation as a triangle singularity was pointed out in Refs. <cit.>. However, in Ref. <cit.> it was noted that if the P_c quantum numbers were 1/2^+ or 3/2^+ the triangle mechanism could provide an interpretation of the narrow experimental peak, but not if the quantum numbers are 3/2^-, 5/2^+, as preferred by experiment.In the present study, we investigate the B^-→ D^*0π^-π^0η and B^-→ D^*0π^-π^+π^- decays via a_0 and f_0 formation. The process of B^-→ D^*0K^-K^*0 followed by the K^*0 decay into π^-K^+ and the merging of the K^+K^- into a_0 or f_0 (see Fig. <ref>) generate a singularity, which would appear around 1418 MeV in the invariant mass of π^-a_0 or π^-f_0, as calculated using Eq. (18) of Ref. <cit.>. In this study, these a_0 and f_0 states appear as the dynamically generated states of ππ, KK̅, ηη, andKK̅, π^0η in the I=0 and I=1 channels, respectively, as studied in Refs. <cit.>. The mechanism proposed here, without the indication of how the K^* K̅ could be formed, and without a quantitative evaluation of the process, was suggested in Ref. <cit.>. We provide here a realistic example of a physical process where this can occur, which also allows us to perform a quantitative calculation of the amplitudes involved.Weak decays of heavy hadrons are turning into a good laboratory to find many triangle singularities. Apart from the work of Ref. <cit.>, the B_c → B_s ππ reaction has been suggested, where B^+_c →K̅^* 0 B^+, K̅^* 0→π^0 K̅^0 and K̅^0 B^+ →π B^0_s. Yet, there are large uncertainties quantizing the K̅^0 B^+ →π B^0_s amplitude.In the present case we rely upon well known KK̅→ππ (KK̅→πη) amplitudes, and the B^- → D^* 0K̅^* 0 K^- vertex can be obtained from experiment. Hence, we are able to quantize the decay rates of the mechanism proposed and we find that the mass distribution of these decay processes shows a peak associated with the triangle singularity, and finally find the branching fractions (B^-→D^*0π^- a_0; a_0 →π^0η)= 1.66 × 10^-6 and (B^-→D^*0π^- f_0; f_0 →π^+π^-)= 2.82 × 10^-6. § FORMALISM We will analyse the effect of triangle singularities in the following decays: B^- → D^* 0π^- ηπ^0 and B^- → D^* 0π^- π^+ π^-. The complete Feynman diagram for these decays, with the triangle mechanism through the a_0 or f_0 mesons, is shown in Fig. <ref>.At first, we evaluate the B^- → D^*π R(R=a_0,f_0). This then produces the triangle diagram shown in Fig. <ref>. The T matrix t_B → D^* π R will have the following form, -i t_B → D^* π R = i ∑_pol. ofK^* ∫d^4 q/(2 π)^4it_B^- → D^* 0 K^* 0 K^-/q^2-m_K^2+i ϵit_K^*K^+ π^-/(P-q)^2-m_K^*^2+i ϵi t_K^+K^- , R/(P-q-k)^2 - m_K^2+iϵ. The amplitude in Eq. (<ref>) is evaluated in the center-of-mass (CM) frame of π R. Now we need to calculate the three vertices, t_B^- → D^* 0 K^* 0 K^-, t_K^*K^+ π^- and t_K^+K^- , R, in Eq. (<ref>).First, we discuss the B^-→ D^*0K^-K^*0 vertex. At the quark level, the Cabibbo-allowed vertex is formedthrough an internal emission of a W boson <cit.> (as can be seen in Fig. <ref>), producing a c u̅ that forms the D^* 0, with the remaining d u̅ quarks hadronizing and producing the K^- and K^* 0 mesons with the selection of the s̅s pair from a created vacuum u̅u+d̅d+s̅s state.Since both D^* 0 and K^* 0 have J^P=1^-, the interaction in the B^- → D^* 0 K^- K^* 0 vertex can proceed via s-wave and we take the amplitude of the form,t_B^- → D^* 0 K^* 0 K^- = Cϵ_μ(K^*) ϵ^μ(D^*).Given that we know that the branching ratio of this decay is (B^- → D^* 0 K^* 0 K^-)=1.5 × 10^-3 <cit.>, we can determine the constant C by calculating the width of this decay, d Γ_B^- → D^* 0 K^* 0 K^-/d M_inv(K^* D^*)=1/(2 π)^3|p⃗_K^-| |p⃗̃⃗_K^*|/4 M_B^2∑∑|t_B^- → D^* 0 K^* 0 K^-|^2, where p⃗_K^- is the momentum of K^- in the B^- rest frame, and p⃗̃⃗_K^* is the momentum of K^*0 in the K^* 0 D^* 0 CM frame. The absolute values of both momenta are given by |p⃗_K^-| = λ^1/2 (M_B^2, m^2_K^-, M^2_inv(K^* D^*))/2 M_B, |p⃗̃⃗_K^*|=λ^1/2 (M^2_inv(K^* D^*), m^2_K^*, m^2_D^*)/2 M_inv(K^* D^*), with λ(x,y,z) the ordinary Källen function.Now, if we square the T matrix in (<ref>) and sum over the polarizations, we get ∑∑ |t_B^- → D^* 0 K^* 0 K^-|^2 = C^2 ∑_polϵ_μ(K^*)ϵ_ν(K^*) ϵ^μ(D^*) ϵ^ν(D^*) = C^2 (2+ (p_K^*· p_D^*)^2/m^2_K^*m^2_D^*) = C^2 (2+ (M_inv^2(K^* D^*)-m^2_K^*-m^2_D^*)^2/4 m^2_K^* m^2_D^*).where we used the fact that (p_K^*+p_D^*)^2=M_inv^2(K^* D^*), i.e., p_K^*· p_D^*= 1/2(M_inv^2(K^* D^*)-m^2_K^*-m^2_D^*).Then, using this last equation in Eq. (<ref>), we getC^2/Γ_B^-=(B^- → D^* 0 K^* 0 K^-)/∫ dM_inv(K^* D^*) 1/(2 π)^3|p⃗_K^-| |p⃗̃⃗_K^*|/4 M_B^2(2+ (M_inv^2(K^* D^*)-m^2_K^*-m^2_D^*)^2/4 m^2_K^* m^2_D^*),where the integral has the limits M_inv(K^* D^*)|_min=m_D^*+m_K^* and M_inv(K^* D^*)|_max=M_B-m_K. Now we calculate the contribution of the vertex K^* 0→π^- K^+. For that we will use the chiral invariant lagrangian with local hidden symmetry given in Refs. <cit.>, which isℒ_VPP = -i g< V^μ[P, ∂_μ P]>,where the VPP subscript refers to the fact that we have a vertex with a vector and two pseudoscalar hadrons. The symbol <...> stands for the trace over the SU(3) flavour matrices, and g=m_V/2 f_π is the coupling of the local hidden gauge, with m_V=800MeV and f_π=93MeV.The SU(3) matrices for the pseudoscalar and vector octet mesons P and V^μ are given byV_μ=[1/√(2)ρ^0_μ + 1/√(2)ω_μρ^+_μK^* +_μ;ρ^-_μ -1/√(2)ρ^0_μ + 1/√(2)ω_μK^* 0_μ;K^* -_μ K̅^* 0_μϕ_μ ], P=[1/√(2)π^0 + 1/√(6)ηπ^+K^+;π^- -1/√(2)π^0 + 1/√(6)ηK^0;K^- K̅^0 -√(2/3)η ].Performing the matrix operations and the trace we getℒ_K^*Kπ =-ig K^* 0^μ(K^- ∂_μπ^+ - π^+ ∂_μ K^- ).So, for the t matrix we get,-i t_K^*K^+ π^- =-i g ϵ^μ_K^*(p_K^+-p_π)_μ ≃ -i g ϵ⃗_K^*· (p⃗̃⃗_π-p⃗̃⃗_K^+),with p⃗̃⃗_K^+ and p⃗̃⃗_π calculated in the CM frame of π R. At the energy where the triangle singularity appears (M_inv(π R)=1418MeV), the momentum of K^* is about 150 MeV/c, which is small enough, compared with the mass of K^*, to omit the zeroth component of the polarization vector in Eq. (<ref>).Finally we only need to calculate t_K^+K^- , R, before we can analyse the triangle diagram. The coupling of R with π^0 η or π^+ π^- proceeds in s-wave. Then, the vertex is written simply as a constant,t_K^+K^- , R = g_K^- K^+ , R.We can now analyse the effect of the triangle singularity on the B^- → D^* π R decay.Substituting Eqs. (<ref>), (<ref>) and (<ref>) for Eq. (<ref>), the decay amplitude t_B^- → D^* 0π^- R iswritten as t_B^- → D^* π R = -i g_K^- K^+, R g C ∑_pol. of K^*∫d^4 q/(2 π)^4 ϵ⃗_D^*·ϵ⃗_K^*/q^2-m_K^2+i ϵϵ⃗_K^*· (p⃗̃⃗_π-p⃗̃⃗_K^+)/(P-q)^2-m^2_K^*+i ϵ1/(P-q-k)^2 - m_K^2+i ϵ , where for t_B^- → D^* 0 K^* 0 K^- we have also the spatial components of the polarization vectors, and p⃗̃⃗_K^+, p⃗̃⃗_K^+ are taken in the CM frame of π R. As we have mentioned below Eq. (<ref>), the momentum of the K^*0 around the triangle peak is small compared with the mass, and we can omit the zeorth component of the polarization vector of the K^*0.Now we only need to calculate the width Γ associated with the diagram in Fig. <ref>. Right away we see that since∑_pol. of K^*ϵ_K^*^i ϵ_K^*^j = δ_ij, Eq. (<ref>) reduces tot_B^- → D^* π R =g_K^- K^+, R g C i∫d^4 q/(2 π)^4 ϵ⃗_D^*· (p⃗̃⃗_K^+-p⃗̃⃗_π)/q^2-m_K^2+i ϵ1/(P-q)^2-m^2_K^*+i ϵ1/(P-q-k)^2 - m_K^2+i ϵ, where p⃗̃⃗_K^+= P⃗-q⃗-k⃗= -(q⃗+k⃗) and p⃗̃⃗_π= k⃗.Defining f(q⃗,k⃗) as a product of the three propagators in Eq. (<ref>), we can use the formula,∫ d^3 q q_i f(q⃗, k⃗) = k_i ∫ d^3 q q⃗·k⃗/|k⃗|^2 f( q⃗, k⃗),which follows from the fact that the k⃗ is the only vector not integrated in the integrand of Eq. (<ref>). Then, Eq. (<ref>) becomest_B^- → D^* π R = -ϵ⃗_D^*·k⃗ g_K^- K^+, R g C t_T,with t_T=i ∫d^4 q/(2 π)^4 (2+ q⃗·k⃗/|k⃗|^2) 1/q^2-m_K^2+i ϵ1/(P-q)^2-m^2_K^*+i ϵ1/(P-q-k)^2 - m_K^2+i ϵ. Squaring and summing over the polarizations of D^*, Eq. (<ref>) becomes∑_ pol|t_B^- → D^* π R|^2 =|k⃗|^2g^2_K^- K^+, R g^2 C^2|t_T|^2, As given in Ref. <cit.>, the analytical integration of t_T in Eq. (<ref>) over q^0 leads to t_T = ∫d^3 q/(2 π)^3(2+ q⃗·k⃗/|k⃗|^2) 1/8 ω^* ωω'1/k^0-ω'-ω^*1/P^0+ω+ω'-k^01/P^0-ω-ω'-k^0 + i ϵ××{2P^0 ω + 2 k^0 ω' -2[ω+ω'][ω+ω'+ω^*] }/P^0-ω^*-ω+iϵ, with ω^*(q⃗)=√(m^2_K^* 0 +|q⃗|^2), ω'(q⃗)=√(m^2_K +|q⃗+k⃗|^2) andω(q⃗)=√(m_K^2 +|q⃗|^2). To regularize the integral in Eq. (<ref>) we use the same cutoff of the meson loop that will be used to calculate t_K^+ K^- →π^ 0 η and t_K^+ K^- →π^+ π^- (Eq. (<ref>)), θ(q_max-|q^*|), where q⃗^ * is the K^- momentum in the R rest frame <cit.>. In in Ref. <cit.> it was found that there is a singularity associated with this type of loop functions when Eq. (18) of Ref. <cit.> is satisfied. From that equation we can determine that the singularity will appear around M_inv(π R) = 1418MeV.To be completely correct in our analysis we have to use the width of K^* 0. We implement that replacing ω^* →ω^* - i Γ_K^*/2 in Eq. (<ref>), which will reduce the singularity to a peak <cit.>.For the three body decay of B^- → D^* 0π^- R in Fig. <ref>, the mass distribution is given byd Γ/d M_inv(π R)= 1/(2 π)^3|p⃗_D^*| |p⃗̃⃗_π|/4 M_B^2∑_ pol.|t_B^- → D^* π R|^2,with |p⃗_D^*| = λ^1/2 (M_B^2, m^2_D^*, M^2_inv(π R))/2 M_B, |p⃗̃⃗_π|=|k⃗|=λ^1/2 (M^2_inv(π R), m^2_π, M^2_R)/2 M_inv(π R) With Eq. (<ref>) and a factor 1/Γ_B^-, the mass distribution of B^- decaying into D^* π R is written as 1/Γ_B^-d Γ/d M_inv(π R) = C^2/Γ_B^-g^2/(2 π)^3|p⃗_D^*| |k⃗|/4 M_B^2|k⃗|^2·| t_T × g_K^- K^+, R|^2, where C^2/Γ_B^- is given in Eq. (<ref>).However, the problem here is that the a_0 and f_0 are not stable particles, but resonances that have a width and decay to π^0 η and π^+ π^-, respectively. To solve this without having to consider R a virtual particle and having a four body decay, we can consider the resonance as a normal particle but we add a mass distribution to the decay width in Eq. (<ref>),d Γ/d M_inv(π R)=1/(2 π)^3∫ d M^2_inv(R)(-1/πImD )|g_K^- K^+, R|^2 |p⃗_D^*| |p⃗̃⃗_π|/4 M_B^2 ∑∑|t̃_B^- , D^* π R|^2,with D= 1/M_inv^2(R) - M_R^2 + i M_R Γ_R,where M_inv(R) stands for M_inv(π^0η) and M_inv(π^+ π^-) for R=a_0 and f_0, respectively, and t̃_B^- , D^* π R= t_B^- → D^* π R/g_K^- K^+, R. What Eq. (<ref>) is accomplishing is a convolution of Eq. (<ref>) with the mass distribution of the R resonance given by its spectral function.Notice also that in the limit of Γ_R → 0, i Im D = -i πδ (M_inv^2(R)- M_R^2) and we recover Eq. (<ref>). Evaluating explicitly the imaginary part of D, Eq. (<ref>) becomes d Γ/d M_inv(π R)= 1/(2 π)^3∫ d M^2_inv(R)1/π|g_K^- K^+, R|^2 |p⃗_D^*| |p⃗̃⃗_π|/4 M_B^2 ∑∑|t̃_B^- , D^* π R|^2 M_R Γ_R/(M_inv^2(R) - M_R^2)^2+(M_R Γ_R)^2. Now, for the case of a_0(980), we only have the decay a_0 →π^0 η (we neglect the small K K̅ decay fraction), and thus,Γ_a_0= 1/8 π|g_a_0 →π^0 η|^2/M^2_inv(π^0η) |q⃗̃⃗_η|,with|q⃗̃⃗_η| = λ^1/2(M_inv^2(π^0η), m_π^2, m_η^2)/2 M_inv(π^0η).Then Eq. (<ref>) becomes d Γ/d M_inv(π a_0)= 1/(2 π)^3 ∫ d M^2_inv(π^0η) |p⃗_D^*| |p⃗̃⃗_π|/4 M_B^2 ∑∑|t̃_B^- , D^* π R|^2M_a_0|g_a_0 →π^0 η|^2 |g_K^- K^+ → a_0|^2/(M_inv^2(π^0η) - M_a_0^2)^2+(M_a_0Γ_a_0)^21/8 π^2|q⃗̃⃗_η|/M^2_inv(π^0η).But since for the resonance we have formally,|g_a_0 →π^0 η|^2 |g_K^- K^+ → a_0|^2/(M_inv^2(π^0η) - M_a_0^2)^2+(M_a_0Γ_a_0)^2 = |t_K^+ K^- →π^0 η|^2,Eq. (<ref>) reduces to d^2 Γ/d M_inv(π a_0) d M_inv(π^0η) = 1/(2 π)^5|p⃗_D^*| |k⃗| |q⃗̃⃗_η|/4 M_B^2 ∑∑|t̃_B^- , D^* π R×t_K^+ K^- →π^0 η|^2, where we approximated (π^0η) as M_R. For the case of f_0(980), f_0 →π^+ π^- is not the only possible decay and as such Γ_f_0 →π^+ π^- will not be the same as the Γ_R in Eq. (<ref>). However, when we put |t_K^+ K^- →π^+ π^-|^2 in the end, we already select the ππ part of the f_0 decay. Thus, for the case of f_0 we just need to substitute, in Eq. (<ref>), t_K^+ K^- →π^0 η→ t_K^+ K^- →π^+ π^-, M_inv(π a_0) → M_inv(π f_0), M_inv(π^0η) → M_inv(π^+π^-), and |q⃗̃⃗_η| → |q⃗̃⃗_π|, with|q⃗̃⃗_π| = λ^1/2(M_inv^2(π^+π^-), m_π^+^2, m_π^-^2)/2 M_inv(π^+π^-).The amplitudes t_K^+ K^- →π^ 0 η and t_K^+ K^- →π^+ π^- themselves are calculated based on the chiral unitary approach, where the a_0 and f_0 appear as dynamically generated states <cit.>. The cutoff parameter q_max which appears for the regularization of the meson loop function in the Bethe-Salpeter equation,t=[1-V G]^-1 V,is determined asq_max=600MeV for the reproduction of the a_0 and f_0 peaks (around 980MeV in invariant mass of π^0 η or π^+ π^-) <cit.>. In Eq. (<ref>), t, V, and G are the meson amplitude, interaction kernel, and meson loop function, respectively.Finally, we can substitute everything we have calculated into Eq. (<ref>) and obtain, 1/Γ_B^-d^2 Γ/d M_inv(π R) d M_inv(R) = g^2/(2 π)^5|p⃗_D^*| |q⃗̃⃗_η| |k⃗|^3/4 M_B^2 | t_T × t_K^+ K^- →π^0 η (π^+ π^-)|^2 C^2/Γ_B^-.§ RESULTS Let us begin by showing in Fig. <ref> the contribution of the triangle loop (defined in Eq. (<ref>)) to the total amplitude. We plot the real and imaginary parts of t_T, as well as the absolute value with (R) fixed at 980 MeV. As can be observed, there is a peak around 1420 MeV, as predicted by Eq. (18) of Ref. <cit.>. In Figs. <ref> and <ref> we plot Eq. (<ref>) for both B^- → D^* 0π^- ηπ^0 and B^- → D^* 0π^- π^+ π^-, respectively, by fixing M_inv(π R)=1418MeV, which is the position of the triangle singularity, and varying M_inv(R). We can see a strong peak around 980MeV and consequently we see that most of the contribution to our width Γ will come from M_inv(R)=M_R. For Fig. <ref> the dispersion is bigger, we have strong contributions for M_inv(π^0 η) ∈ [880,1080]. However, for Fig. <ref> most of the contribution comes from M_inv(π^+ π^-) ∈ [940,1020]. The conclusion is that when we calculate the mass distribution d Γ/d M_inv(π a_0), we can restrict the integral in M_inv(R) to the limits already mentioned.When we integrate over M_inv(R) we obtain d Γ/d M_inv(π R) which we show in Fig. <ref>. We see a clear peak of the distribution around 1420MeV, for f_0 and a_0 production. However, we also see that the distribution stretches up to large values of M_inv(π R) where the phase space of the reaction finishes. This is due to the |k⃗|^3 factor in Eq. (<ref>) that contains a |k⃗| factor from phase space and a |k⃗|^2 factor from the dynamics of the process, as we can see in Eq. (<ref>). Yet, a clear peak in M_inv(π^- R) can be seen for both the B^- → D^* 0π^- f_0 and B^- → D^* 0π^- a_0 reactions.Integrating now d Γ/d M_inv(π a_0) and d Γ/d M_inv(π f_0) over the M_inv(π a_0) (M_inv(π f_0)) masses in Fig. <ref>, we obtain the branching fractions (B^-→ D^*0π^- a_0; a_0 →π^0η) =1.66 × 10^-6 ,(B^-→ D^*0π^- f_0; f_0 →π^+π^-)=2.82 × 10^-6. These numbers are within measurable range.Note that we have assumed all the strength of π^0 η from 880MeV to 1080MeV to be part of the a_0 production, but in an experimental analysis one might associate part of this strength to a background. We note this in order to make proper comparison with these results when the experiment is performed.The shape of t_T in Fig. <ref> requires some extra comment. We see that Im(t_T) peaks around 1420 MeV, where the triangle singularity is expected. However Re(t_T) also has a peak around 1390 MeV. This picture is not standard. Indeed, in Ref. <cit.>, where a triangle singularity is disclosed for the process N (1835) →π N (1535), t_T has the real part peaking at the place of the triangle singularity and Im(t_T) has no peak. In Ref. <cit.>, a triangle singularity develops in the γ p → p π^0 η→π^0 N (1535) process and there Im(t_T) has a peak at the expected energy of the triangle singularity while the Re(t_T) has no peak. Similarly, in the study of N (1700) →πΔ in Ref. <cit.> a triangle singularity develops and here Im(t_T) has a peak but Re(t_T) has not. However, the double peak in the real and imaginary parts of t_T is also present in the study of the B^- → K^- π D_s0^+ reaction in Ref. <cit.>. This latter work has a loop with D^0 K^* 0 K^+, and by taking Γ_K^*→ 0, ϵ→ 0 the peak of Im(t_T) was identified with the triangle singularity while the peak in the Re(t_T) was shown to come from the threshold of D^0 K^* 0. In the present case the situation is similar: The peak of Im(t_T) at about 1420 MeV comes from the triangle singularity while the one just below 1400 MeV comes from the threshold of K^* 0 K^- in the diagram of Fig. <ref>, which appears at 1386 MeV. Yet, by looking at |t_T| in Fig. <ref> and the region of the peak of d Γ/d M_inv in Fig. <ref>, we can see that this latter peak comes mostly from the triangle singularity. § SUMMARY We have performed the calculations for the reactions B^- → D^* 0π^- a_0(980); a_0 →π^0 η and B^- → D^* 0π^- f_0(980); f_0 →π^+ π^-. The starting point is the reaction B^- → D^* 0 K^* 0 K^-, which is a Cabibbo favored process and for which the rates are tabulated in the PDG <cit.> and are relatively large. Then we allow the K^* 0 to decay into π^- K^+ and the K^+ K^- fuse to give the f_0(980) or the a_0(980). Both of them are allowed, since the K^* 0 K^- state does not have a particular isospin. The triangle diagram corresponding to this mechanism develops a triangle singularity at about 1420MeV in the invariant masses of π^- f_0 or π^- a_0, and makes the process studied relatively large, having a prominent peak in those invariant mass distributions around 1420MeV.We evaluate d^2 Γ/d M_inv(π^- a_0) d M_inv(π^0 η), and d^2 Γ/d M_inv(π^- f_0) d M_inv(π^+ π^-) and see clear peaks in the M_inv(π^0 η), M_inv(π^+ π^-) distributions, showing clearly the a_0(980) and f_0(980) shapes. Integrating over M_inv(π^0 η) and M_inv(π^+ π^-) we obtain d Γ/d M_inv(π a_0) and d Γ/d M_inv(π f_0) respectively, and these distributions show a clear peak for M_inv(π a_0), M_inv(π f_0) around 1420MeV. This peak is a consequence of the triangle singularity, and in this sense the work done here should be a warning not to claim a new resonance when this peak is seen in a future experiment. 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C 95, no. 6, 065211 (2017) [arXiv:1702.07220 [hep-ph]]. | http://arxiv.org/abs/1706.08723v2 | {
"authors": [
"R. Pavao",
"S. Sakai",
"E. Oset"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170627083548",
"title": "Triangle singularities in $B^-\\rightarrow D^{*0}π^-π^0η$ and $B^-\\rightarrow D^{*0}π^-π^+π^-$"
} |
Department of Mechanical Engineering, University of Michigan-Dearborn, 48128-1491 MI, [email protected] water modeling of rolling pad instability in liquid metal batteries Oleg Zikanov December 30, 2023 =========================================================================== Magnetohydrodynamically induced interface instability in liquid metal batteries is analyzed. The batteries are represented by a simplified system in the form of a rectangular cell, in which strong vertical electric current flows throughthree horizontal layers: the layer of a heavy metal at the bottom, the layer of a light metal at the top, and the layer of electrolyte in the middle. A new two-dimensional nonlinear model based on the conservative shallow water approximation is derived and utilized in a numerical study. It is found that in the case of small density difference between the electrolyte and one of the metals, the instability closely resembles the rolling pad instability observed earlier in the aluminum reduction cells. When the two electrolyte-metal density differences are comparable, thedynamics of unstable systems is more complex and characterized by interaction between two nearly symmetric or antisymmetric interfacial waves. Shallow water modeling of rolling pad instability in liquid metal batteries Oleg Zikanov December 30, 2023 ===========================================================================§ INTRODUCTION The liquid metal battery is a promising conceptual device for stationary energy storage <cit.>. Recently developed small-scale (about few cm)laboratory prototypes have demonstrated technical feasibility of the concept and its advantages of higher efficiency and longer operational life in comparison to traditional solid-electrode batteries (see <cit.>). Scaling up the concept to large commercially attractive devices has not, however, yet been achieved.As we discuss below, the hydrodynamic effects, in particular that of electromagnetically modified interfacial waves, are expected to be non-negligible factors of this transition. In this paper we consider one version of the battery, which, on the level of simplification sufficient for out purposes, can be viewed as the system illustrated in Fig. <ref>a. It is a cuboid or cylindrical cell filled with three liquid layers: the layer of a heavy metal (Bi, Sb, Zn, or PbSb) at the bottom, the layer of a light metal (Na, Li, Ca, or Mg) at the top, and the thin layer of molten-salt electrolyte sandwiched in the middle. The electrolyte is selected so that it isimmiscible with the metals and conductive to positive ions of the light metal. It has the density intermediate between those of the light and heavy metals, so the entire system is stably stratified by density. The sidewalls are electrically insulated, while the horizontal top and bottom walls are conducting and serve as current collectors. The system is maintained at a temperature above the melting points of all the three materials (200 to 700^∘C depending on the materials used). The energy stored in the battery is the difference between the Gibbs free energies of the light metal in its pure state and the state of alloy with the heavy metal. The battery is charged when the light metal is electrochemically reduced from the alloy and discharged when thealloy is formed. The reactions occur in liquid state in the bottom layer in the presence of strong (∼ 1 A/cm^2) electric currents flowing in the vertical direction.The illustration in Fig. <ref>a lets a fluid dynamicist torecognize immediately that the hydrodynamic instabilities can be a major factor of the system's operation, especially when the transition from small laboratory prototypes to large commercial devices is made. The effect of the instabilities may be positive, when the resulting flow in the bottom layer enhances mixing of reactants. It can also be negative, when the deformation of the interfaces between the layers becomes so strong that it causes rupture of the electrolyte layer, which means short circuit between the metals and disruption of the battery's operation. Several mechanisms of instability have been identified and analyzed so far including the Tayler instability<cit.>, the thermal convection caused by the volumetric Joule heating of the electrolyte <cit.> or by bottom heating <cit.>, and the rolling pad instability <cit.>. We should also mention the electrovortex effect, which may become a significant factor under certain conditions <cit.>. The analysis is far from complete, but it can already be said that all these mechanisms are likely to be active in large-scale batteries. At the same time, the preliminary estimates made in <cit.>for the Tayler instability and in <cit.> for the Joule-heating convection show significant damping effect of the stable density stratification among the layers. It is possible, although not yet proven, that in a wide range of operational parameters, the instabilities are present but do not lead to a disruption. Rather, the instabilities lead to saturated states with weak flows and small amplitudes of interface deformation.In this paper, we study the rolling pad instability. It is conceptually similar to the instability observed and well studied in the Hall-Héroult aluminum reduction cells (see <cit.>). The reduction cell is a shallow and large rectangular bath filled with molten aluminum at the bottom and molten salt electrolyte with alumina dissolved in it at the top. Nearly verticalcurrents of density ∼ 1 A/cm^2 pass through the two layers and cause the desired electrochemical reduction of aluminum from its oxide. The reduction cell is different from the liquid metal battery in many respects: two layers instead of three, turbulent mixing of melts driven by gas bubbles and Lorentz forces, the top wall consisting of many separate carbon anodes, The one important similarity is that in both the systems, the liquid domain consists of horizontal layers with vastly different electric conductivities. The conductivity of molten-salt electrolytes is about four orders of magnitude lower than that ofliquid metals. This means that even a small variation of the local thickness of the electrolyte causes large variation of local electric resistance and, thus, significantly changes the distribution of the electric currents within the system.Lorentz forces are created by the current perturbations interacting with the magnetic field inevitably present within the cell. In unstable systems, the forces modify the flows of the melts in such a way that the interface deformation is enhanced.In the aluminum production, the instability is viewed as a major concern. Itdevelops in a reduction cell when the thickness of the electrolyte is reduced below a certain cell-specific threshold and leads to sloshing waves that grow and, eventually, cause short circuit between the aluminum and the anode. The thickness has to be kept large enough, which results in energy losses to the excessive Joule heating of the electrolyte. The situation improved in the 1980s and 1990s when it was understood that the instability was caused by the interaction between the horizontal components of the current perturbations and the vertical component of the magnetic field generated by the currents flowing in the neighboring cells and electric supply lines. A detailed discussion of the instability mechanism can be found, in<cit.>.Briefly, as illustrated in Fig. <ref>b, in which we should ignore the metal layerfor to consider a reduction cell, the vertical current perturbations j_z form as a result of thevariation of the local thickness of the electrolyte caused by a deformation of interface. The current perturbations close within the highly conducting metal layer resulting in a horizontal component j_. The interaction between this component and the base vertical magnetic field B_0 create the horizontal Lorentz force f that drives the metal in the horizontal direction perpendicular to the plane of the interface deformation. The result is a large-scale interfacial wave structure rotating around the cell - a so-called rolling pad.The mechanism by which the rotating wave grows in amplitude, and, thus, the instability develops is easiest to understand using the linearized shallow water model. In this case, the problem is reduced to a single wave equation (see <cit.>). Application of the Fourier expansion shows that the Lorentz forces modify the natural gravitational standing waves on the interface and introduce coupling between them. In the unstable situation, the eigenvalues of one or several couples of such waves merge to form complex-conjugate pairs, each corresponding to a rotating and exponentially growing wave.The shape of the horizontal cross-section of the cell determines the natural gravitational modes and, thus, strongly affects the threshold of the instability. In particular, cells of square or circular cross-section are degenerate in the sense that theyhave multiple gravitational modes with equal frequencies. In the inviscid limit, such cells are unstable at arbitrarily weak electromagnetic effect. For rectangular cells, the strength of the effect must exceeds a certain not very large threshold. In the real cells, the threshold is increased by the viscosity and interfacial tension, and the situation is further complicated by the background melt flows, gas bubbles, block structure of the anode, and other factors, but the principal instability mechanism remains the same.Fig. <ref>b illustrates the evident fact that a similar instability can develop in a liquid metal battery. The presence of the second metal layer, in which perturbation currents j_ of opposite orientation develop (we discuss this in detail in section <ref>), makes the picture more complicated. The first attempt to analyze the instability was made in <cit.>. A mechanical model based on the approach developed earlier for the aluminum reduction cells in <cit.> was used. In the model, the sloshing motions of the metal layers are imitated by pendulum-like oscillations of two solid metal slabs, which are independently suspended and separatedfrom each other by an electrolyte. The oscillations are modified and coupled to each other by the same physical mechanism as in the battery, i.e. via the Lorentz forces associated with the current perturbations caused by the local changes of the electrolyte thickness. The model is a drastic simplification of a real battery, but provides what can be viewed as a low-mode, nondissipative, linearized analogy of some aspects of the instability. The main result of <cit.> is the clear indication that the rolling pad instability similar to the instability in the reduction cells should be expected in the batteries. The model also suggests existence of an instability caused by the interaction between the current perturbations and the azimuthal magnetic field induced by the base current. The existence of the rolling pad instability in the liquid metal batteries was recently confirmed in the three-dimensional <cit.> and two-dimensional shallow water <cit.> numerical simulations. Batteries of cylindrical <cit.> or rectangular <cit.> shapes were considered. The typical fluid densities of a Mg-Sb battery were chosen, for which the density jump and, thus, the stabilizing effect of the buoyancy force is much larger at the lower than at the upper interface. The situation becomes particularly close to that of a reduction cell in the sense that the instability leads to noticeable deformation of only one interface. As another indication of the closeness, it has been found in<cit.> that the onset of the instability is best determined by thegeneralization of the stability criterion developed for the reduction cells in <cit.> and <cit.>.This paper reports the results of a new computational study. We present a detailed derivation of the nonlinear shallow water model of a liquid metal battery. The model is then applied in a numerical study of the rolling pad instability. The goals are to (i) further analyze the instability appearing at a single interface in the situation of strongly different density jumps and (ii) extend the analysis to the more complex case when the density jumps are comparable to each other, and so both the interfaces are significantly deformed. As we discuss in detail in the model derivation in section <ref>, its validity is limited to shallow cells, in which the horizontal dimensions are much larger than the height. This limitation does not appear especially strict when we consider the main technological requirements to a large commercial battery. It must contain large quantities of both metals, have good vertical mixing in the bottom layer. The electrolyte layer must be thin and have large area in the horizontal plane, so as to maximize the reaction rate and minimize the Joule heat losses. This suggests the optimal shape of the battery in the form of a horizontally large (several m) and shallow (perhaps tens of cm) cell. The additional benefit of such a shape can be avoidance of disruptively strong Tayler and thermal convection instabilities. § PHYSICAL MODEL AND NUMERICAL METHODA battery in the form of a shallow cell shown schematically in Fig. <ref>a is considered. The cell is filled with three liquid layers: metalat the bottom, metalat the top, and electrolytein the middle. The sidewalls are vertical and electrically insulated. The top and bottom walls are horizontal and serve as the electrodes, between which strong electric current of density J flows through the cell. In the following discussion, variables with the superscripts A, B, or E are associated with particular layers. Variables without superscripts either mark universal variables common for all layers (such as the horizontal length scale L) or, for thesake of brevity, stand for the respective variables in each layer (H standing for the layer thicknesses H^A, H^B, and H^E).§.§ Assumptions made in the modelThe model describes the behavior related to the interaction between the magnetic fields and the perturbations of electric current arising in response to the interface deformation. The physical effects leading to the other instabilitiesmentioned in section <ref> are ignored. The justification for this approach is based on the following two arguments. The small-scale, possibly turbulent motions of melts caused by thermal convection and interfacial forcesincrease the dissipative effects, but are unlikely to substantially change the physical mechanisms of the large-scale rolling pad instability. The validity of this argument is confirmed by the results obtained for the aluminum reduction cells (see, e.g. <cit.>) and further supported in the following discussion. Furthermore, as we discuss below, the Tayler instability and the electrovortex flow are either absent or weak in the simplified shallow geometry we consider. The model is based on the following simplifying assumptions. * The cell is shallow in the sense that the thicknesses of all three layers are much smaller than the typical horizontal size of the cell:H^A,H^B,H^E≪ L.This is the strongest of our assumptions, andthe only one that seriously limits the applicability of the model.We further assume that the typical horizontal wavelength λ of the instability is large:H^A,H^B,H^E≪λ∼ L.The error of approximation of the model can be expressed in terms of the shallowness parameterδ≡H/L∼H/λ≪ 1.As we will see in the following discussion,it is partially of the second order (e.g. for the inviscid part of the momentum transport) and partially of the first order (e.g. for the effects of viscosity or some electromagnetic effects).* The metals are immiscible with the electrolyte, so sharp deformable interfaces exist between the layers.* Each layer is assumed to contain aliquid of constant physical properties. This is a simplification, especially in the case of the bottom layer, where a mixture of metalwith the alloy ofandin liquid or, possibly, solid intermetallic state is to be found unless the battery is fully charged. * The system is stably stratified by density:ρ^A<ρ^E<ρ^B. * The electric conductivities of the liquids satisfyσ^E≪σ^A∼σ^B.In real batteries, σ^E is about four orders of magnitude smaller than σ^A and σ^B. * The base electric current flowing through a cell with unperturbed horizontal interfaces is purely vertical and uniform, so its density is J_0=J_0e_z,J_0=const.This means that the electrovortex effect is excluded from consideration.* A simplified model of the magnetic field is used. We limit the analysis to the effect of the vertical magnetic field that is generated in a battery by the currents in external circuits. The field is approximated asB_0=B_0e_z,B_0=const.The other components, in particular the magnetic field generated by the base current J_0 within the cell and the perturbation field b induced by the current perturbations are not considered.This does not allow us to analyze full three-dimensional dynamics of the battery and excludes from consideration the Tayler instability, the effect of b on the rolling pad instability, and the possible second type instability suggested by the mechanical model <cit.>. The rationale of the approach is as follows.Firstly, we leave the simulations of the full three-dimensional dynamics of the system to future studies. The focus of our work is on the rolling pad instability during its low-amplitude stages when it can, as an approximation, be separated from the other magnetohydrodynamic effects. A similar approach was used for analyzing the classical rolling pad instability in the reduction cells <cit.> and in the recent studies of liquid metal batteries <cit.>. Secondly, the Tayler instability is not anticipated in our system, since it is associated with tall cells (as shown, in <cit.>, the typical axial wavelength of the instability is larger than the horizontal dimension of the cell). Thirdly, the general effect of the magnetic field perturbations b on the rolling pad instability, while present, is likely to be insignificant. This is indicated by the results obtained in <cit.> for the reduction cells.Finally, a proper analysis of the instability of the second type predicted in <cit.> would require accurate evaluation of the magnetic fields induced by both the base and perturbation currents, which can only be done in the framework of the three-dimensional model that includes the interior of the cell and the adjacent conductors. This analysis is left to future studies, as well. * As explained in detail in section <ref>, only theelectric current perturbations caused by the deformation of the interfaces are considered.This is justified since, as demonstrated in the three-dimensional simulations <cit.> and confirmed by our results presented below, in an unstable system these perturbations are typically much stronger than the electric currents induced by the melt velocities. One consequence of this assumption is that the current perturbations can be represented as gradients of electric potential functions.Full Ohm's law may need to be applied in a three-dimensional nonlinear analysis of melt flows in a battery with full three-dimensional magnetic field.* The melts are assumed to be at a constant temperature. The thermal convection flows are neglected despite the fact that, as discussed in <cit.>, they are practically unavoidable and likely to be turbulent. This is justified by the estimates (see <cit.>) showing that the stable density stratification between the layers is sufficiently strong to prevent significant interface deformation by the convection in all, but very large batteries. Our assumption is that the convection results in small-scale, possibly turbulent flows but does not critically interfere with the large-scale rolling pad instability.* We now describe the base (unperturbed) state, which is designated in the following discussion by subscript 0. The simplifying assumptions made above imply that the base state has purely vertical current (<ref>) and flat interfaces (see Fig. <ref>a):z=ζ^A_0=H_0^E/2, z=ζ^B_0=-H_0^E/2,where H_0^E is the unperturbed thickness of the electrolyte layer.The Lorentz force J_0 ×B_0=0, and the melt velocities are all zero. This form of the base state is a simplification. In the real battery, J_0 is not uniform and B_0 is a three-dimensional field created by the currents within and without the cell. This and other features of the battery system, for example the thermal convection, inevitably lead to a base state with deformed interfaces and possibly turbulent melt flows. The analysis of the effect of a nontrivial base state on the instability is left to future studies. We note that the model derived below can be applied to systems with background flows and interface deformations after minor modifications (see <cit.> for examples of such analysis in the case of aluminum reduction cells). As a final comment, we note that the model is nonlinear and can be applied to simulation of finite-amplitude as well as small-amplitude perturbations.§.§ Electric current perturbations and Lorentz forcesIn the model, we consider the Lorentz forces that arise as a result of the interface deformations, more specifically, as a result of the local change in the thickness of the electrolyte layerη(x,y,t)=H^E(x,y,t)-H^E_0=ζ^A(x,y,t)-ζ^B(x,y,t)-H^E_0(see Fig. <ref>a). The following derivation is a generalization of the derivation used for the aluminum reduction cells (see <cit.>). The starting point is the approximation based on the relation (<ref>). Due to the much higher electric conductivities of the metals,the distribution of currents in the electrolyte can be accurately approximated as corresponding to equipotential interfaces z=ζ^A(x,y,t) and z=ζ^B(x,y,t).Combined with the shallowness of the electrolyte layer, this allows us to approximate the perturbed current in this layer as vertical (following the shortest path through a resistive medium) and, due to their zero divergence, z-independent:(J_0+j_z^E(x,y,t))e_zand to express it via the Ohm's law asJ_0+j_z^E=σ^E Φ^A-Φ^B/H^E (perturbed state), J_0=σ^E Φ^A_0-Φ^B_0/H^E_0 (base state),where Φ^A and Φ^B are the values of the electric potential at the top and bottom interfaces.We now set Φ^B=Φ^B_0=0 and notice that the total electric resistance of the electrolyte changes by the quantity ∼(η/H^E_0)^2. Either the net current through the cell or the voltage drop has to changewhen η(x,y,t) is not infinitesimal. Following the approach of <cit.>, we assume thatthe voltage drop adjusts as Φ^A=C(t)Φ^E_0 while the total current remains the same (this corresponds, e.g., to the situation when the considered cell is one of many connected in parallel):⟨ J_0+j_z^E⟩≡∫_0^L_y∫_0^L_x(J_0+j_z^E ) dx dy=⟨ J_0⟩.This allows us to determine the current perturbations asj_z^E=σ^E(C(t)Φ^A_0/H^E-Φ^A_0/H_0^E)=J_0(C(t)H_0^E/H^E -1),where the non-dimensional adjustment coefficient is C(t)=⟨ J_0⟩/⟨ J_0H_0^E/H^E ⟩.At infinitesimal perturbations η≪ H_0^E, the coefficient becomes C≈ 1, and the expression (<ref>) can be approximated using the first-order Taylor expansion in η/H_0^E asj_z^E ≈ -J_0η/H_0^E. As the next step, we assume that the perturbation currents completely close within the metal layersand . For the currents in the layer , we integrate ∇·j^A=0from z=ζ^A to the top boundary z=z_t. Using the Leibnitz integration rule, the condition that the vertical perturbation current at z=z_t is zero, and the condition of continuity of normal current at the interface. j_z^E|_z=ζ^A=. j_z^A|_z=ζ^A-. j_x^A|_z=ζ^A∂ζ^A/∂ x-. j_y^A|_z=ζ^A∂ζ^A/∂ ywe obtain∂ J_x^A/∂ x+∂ J_y^A/∂ y=j_z^E,whereJ_x^A=∫_ζ^A^z_t j_x^Adz, J_y^A=∫_ζ^A^z_t j_y^Adzare the vertically integrated horizontalcomponents of the perturbation currentin the layer .The solution for J^A can be found in a straightforward manner if we assume that ∇_×J=0and express (<ref>) asJ^A_=∇_Ψ^A, ∇_^2 Ψ^A = j_z^E,wheremarks the horizontal (x,y) plane. To verify (<ref>), we consider that the three-dimensional current perturbations are curl-free (see assumption 8 in section <ref>). Integration of ∇×j^A=0 from z=ζ^A to z=z_t gives∇_×J^A_=. j_x^A|_z=ζ^A∂ζ^A/∂ y-. j_y^A|_z=ζ^A∂ζ^A/∂ x.The right-hand side of (<ref>) is of the second order in terms of the perturbation amplitude, so it can be neglected when linear stability to infinitesimal perturbations is analyzed. The situation is more complex for perturbations of finite amplitude. The individual terms in the right-hand and left-hand sides of (<ref>) are of the same order in terms of the shallowness parameter (<ref>). This follows from the estimates j^A∼ HJ^A, ζ^A∼ H, and ∂/∂ x∼∂/∂ y∼λ^-1. A closer consideration, however, reveals that the entire right-hand side of (<ref>) is likely to be negligibly small. It is a cross-product of two two-dimensional vector fields: .j^A|_z=ζ^A and ∇_ζ^A. The currents .j^A|_z=ζ^A are flowing in the direction parallel to the local steepest gradient of the perturbation current j_z^E injected into the metal. This direction is, in turn, aligned with the direction of ∇_η. This follows from (<ref>) and is immediately seen in the linearized version (<ref>). If the density differences between the metals and the electrolyte satisfy ρ^E-ρ^A ≪ρ^B-ρ^E, so only the upper interface is significantly deformed (see section <ref>), ∇_η≈∇_ζ^A. The situation is less certain when the density jumps and, thus, the deformations of the two interfaces are comparable (see section <ref>). For this case we can argue that the interfacial waves are nearly anti-symmetrically coupled (see <cit.> and our results in section <ref>), so ∇_η and ∇_ζ^A are nearly parallel. In summary, we see that the vectors .j^A|_z=ζ^A and ∇_ζ^A are perfectly or nearly parallel in the flow. We can neglect the right-hand side of (<ref>) and consider the field of integrated currents J^A as curl-free. While highly plausible and certainly correct in many cases, this approximation may become inaccurate in some situations, in which case the model becomes only valid for infinitesimal perturbations.For the vertically integratedperturbation currents in the layer , we can derive similar relations with -j_z^E in the right-hand side of the Poisson equation or simply observe that theconservation of electric charge requires J^B_=-J^A_. In our case of purely vertical magnetic field, the Lorentz forces are zero in the layerand purely horizontal in the layersand . The vertically integrated forces areF_^A=J^A_×(B_0e_z), F_^A=-F_^B. In the general case of a three-dimensional magnetic field B we would need to use the general three-dimensional expression F=J×B. In addition to the horizontal currents (<ref>), (<ref>), the field of the perturbation currents J would include the vertically integrated vertical components J_z^E=H^E j_z^E,J_z^A=H^A/2j_z^E, J_z^B=H^B/2j_z^E,where the last two expressions are approximations with the error ∼δ^2. §.§ Shallow water approximationThe derivation of the two-dimensional model follows the principal steps used for shallow flows in hydrology. There are differences, in particular the presence of Lorentz forces and immiscible layers of distinct physical properties in our case. Similarly to the approach applied in<cit.> to the aluminum reduction cells, the shallow water equations in the conservative de St. Venant form are derived and used. The main advantage of this approximation over the simpler non-conservative shallow water model is that its error in representing the inviscid flow's dynamics is of the second rather than first order in terms of the shallowness parameter (<ref>). As discussed in <cit.>, thetwo models produce nearly identical solutions in the case of weak interface deformations, but diverge noticeably for the nonlinear states, where the deformations are strong.The derivation starts with the full three-dimensional momentum equations for flows in each layer, with the liquids treated as incompressible, inviscid and having constant and uniform physical properties. The effect of viscosity will be introduced at a later stage. For the sake of generality and possible future use, the equations are derived for the case of a general three-dimensional Lorentz force that would exist in a cell with a general three-dimensional magnetic field. With the purely vertical currents in the electrolyte and three-dimensional currents in the metals, the general force field has the formf^A =f_^A+f_z^Ae_z, f^E =f_^E, f^B =f_^B+f_z^Be_z.The reduction to the actually considered in this paper case of a purely vertical magnetic field, in which the force is two-dimensional, will be done later. The boundary conditions are those of zero velocity at all walls. At the interfaces z=ζ^A(x,y,t) and z=ζ^B(x,y,t), continuity of pressure and the kinematic condition ∂ζ/∂ t +u∂ζ/∂ x+v∂ζ/∂ y= w,where u, v, and w are the velocity components, are required.The incompressibility implies the estimate of the vertical velocityw∼δ u∼δ v.Applying it to the z-momentum equation and dropping all the terms ∼δ^2 we obtain, in each layer:∂ p/∂ z=-ρ g+f_z.Next, we introduce the pressure distribution at the mid-plane of the electrolyte layer z=0p_0(x,y,t)≡ p(x,y,z=0,t)and integrate (<ref>) in z. The resulting pressure distributions satisfying the requirement of continuity at the interfaces are:p^A(x,y,z,t) = p_0+(ρ^A-ρ^E)gζ^A-ρ^Agz+∫_ζ^A^z f_z^Adξ, p^E(x,y,z,t) = p_0-ρ^Egz,p^B(x,y,z,t) = p_0+(ρ^B-ρ^E)gζ^B-ρ^Bgz+∫_ζ^B^z f_z^Bdξ. They are substituted into the horizontal momentum equations written in conservation form for each layer. Together with the incompressibility conditions, the equations are:∂ u/∂ x+∂ v/∂ y+∂ w/∂ z= 0, ∂ u/∂ t+∂/∂ x(u^2)+∂/∂ y(uv)+∂/∂ z(uw) = -1/ρ∂ p/∂ x +1/ρf_x, ∂ v/∂ t+∂/∂ x(uv)+∂/∂ y(v^2)+∂/∂ z(vw) = -1/ρ∂ p/∂ y +1/ρf_y. The approaches of the simple non-conservative shallow water model and the conservative de St. Venant model diverge at this point. In the former, we would just assume that w=0 and all the other variables are z-independent. This would be equivalent to neglecting all but the zero-order terms in the δ-expansions of the flow fields. Instead, we follow theroutine of the second order in the shallowness parameter (<ref>) based on the integration of the equations vertically across each layer. The integration requires the use of the Leibniz rule for the x-, y-, and t-derivatives, the zero-velocity conditions at z=z_b and z=z_t, and the interface conditions (<ref>).We use the velocity fluxesU^A(x,y,t)=∫_ζ^A^z_tu^Adz,U^B(x,y,t)=∫_z_b^ζ^Bu^Bdz,U^E(x,y,t)=∫_ζ^B^ζ^Au^Edz,and the integrated horizontal components of the Lorentz forcesF^A_(x,y,t)=∫_ζ^A^z_tf^A_dz,F_^B(x,y,t)=∫_z_b^ζ^Bf^B_dz,F_^E(x,y,t)=∫_ζ^B^ζ^Af^E_dz.We also need the vertically integrated terms of the horizontal gradients of the pressure fields (<ref>)-(<ref>), of which only the Lorentz force terms are non-trivial. For them, we apply the Leibniz rule and the approximation to the second order in δ to obtain∫_ζ^A^z_t∇_(∫_ζ^A^z f_z^A dξ)dz≈ ∇_(H^AF_z^A/2),∫_z_b^ζ^B∇_(∫_ζ^B^z f_z^B dξ)dz≈- ∇_(H^BF_z^B/2), where F_z^A=∫_ζ^A^z_t f_z^A dz, F_z^B=∫_z_b^ζ^B f_z^B dz.In the integration of the nonlinear terms of the momentum equations, we utilize the second order approximations, such as∫_ζ^A^z_t(u^A)^2dz ≈1/H^A(U^A)^2. The final momentum equations areD U^A/D t= -H^A/ρ^Ap_0 -(1-ρ^E/ρ^A)H^Agζ^A-1/2ρ^A(F_z^AH^A)+F_^A/ρ^A+τ^A, D U^E/D t= -H^E/ρ^Ep_0 +F_^E/ρ^E+τ^E, D U^B/D t= -H^B/ρ^Bp_0 -(1-ρ^E/ρ^B)H^Bgζ^B+1/2ρ^B(F_z^BH^B)+F_^B/ρ^B+τ^B, where τ^A, τ^B, and τ^E are the viscous friction terms, which we introduce here for the first time and discuss in detail below, andthe material derivative in the left-hand side of each equation has the x- and y-componentsD U/D t=∂ U/∂ t+∂/∂ x(U^2/H)+∂/∂ y(UV/H), D V/D t=∂ V/∂ t+∂/∂ x(UV/H)+∂/∂ y(V^2/H). The incompressibility condition is also integrated using the Leibniz rule and the interface and wall boundary conditions. This leads to∂ζ^A/∂ t=∇_·U^A, ∂ζ^B/∂ t= -∇_·U^B, ∂ζ^A/∂ t-∂ζ^B/∂ t= -∇_·U^E. In our model, (<ref>) and (<ref>) are applied to determine the evolution of interfaces. The equation (<ref>) is replaced by the combination of(<ref>)-(<ref>) expressing the global conservation of mass∇_·(U^A+U^B+U^E )=0. The approximation of the viscous friction terms follows the approach commonly applied in hydrology. We retain the horizontal components of the Laplacian and approximate the friction at the horizontal walls and interfaces by finite differences. Thefriction terms in (<ref>)–(<ref>) are: τ^A =μ^A/ρ^A∇_^2U^A+1/ρ^A[-μ^E+μ^A/H^E+H^A(U^A/H^A-U^E/H^E) - 2μ^AU^A/(H^A)^2]τ^B =μ^B/ρ^B∇_^2U^B+1/ρ^B[-μ^B+μ^E/H^B+H^E(U^B/H^B-U^E/H^E) - 2μ^BU^B/(H^B)^2]τ^E =μ^E/ρ^E∇_^2U^E+1/ρ^E[μ^E+μ^A/H^E+H^A(U^A/H^A-U^E/H^E) . +.μ^B+μ^E/H^B+H^E(U^B/H^B-U^E/H^E)].This friction model is decisively simple. In particular, the coefficients μ are the molecular dynamic viscosity coefficients of the respective liquids. No attempt is made to account for the possible small-scale turbulenceby using eddy viscosities or applying the models derived for turbulent open channel flows. The main justification of this approach is that, as expected for large-wavelength instabilities and confirmed by our results in section <ref>, the effect of the viscous friction terms on the evolution of the large-scale modes of the rolling pad instability is limited to minor quantitative changes.In the computational analysis presented in section <ref>, the equations are simplified for the case of a purely vertical magnetic field by setting all the vertical force components F_z and the forces in the electrolyte F_^E to zero and using the expressions (<ref>) for the remaining horizontal components. The perturbation electric currents are determined as described in section <ref>.§.§ Equations in non-dimensional formAs a physical system, even the simplified battery considered in this paper is described by a large number of parameters: horizontal dimensions, thicknesses and liquid properties of the three layers, base electric current and magnetic field. Converting the problem into non-dimensional form reduces this number, but still leaves many non-dimensional groups. Furthermore, since the battery is a real device, each physical parameter should be allowed to vary only within some practically meaningful limits. In such a situation, it is convenient to perform the analysis using dimensional variables. At the same time, as we will see in section <ref>, the non-dimensional form of the problem helps to understand the physics of the instability. Our discussion takes the hybrid approach. The simulation cases are identified usingdimensional parameters, while the discussion is conducted primarily in terms of non-dimensional quantities. The non-dimensionalization is presented in this section.We take the larger horizontal dimension of the cell, say, L=L_x in a rectangular cell, as the length scale, the unperturbed current J_0 as the scale for the density of electric current, the constant vertical component B_0 as the scale of the magnetic field, and the physical properties of one liquid (we take the electrolyte) ρ^E, ν^E=μ_E/ρ_E as the scales of density and kinematic viscosity. As the velocity scale, we use the typical velocity of a large-scale gravitational wave on the upper interface:U_0=[ (ρ^E-ρ^A)g/ρ^A(H_0^A)^-1 + ρ^E(H_0^E)^-1]^1/2.This choice, also practiced for the aluminum reduction cells (see, <cit.>), is consistent with the physical nature of the instability. The typical scales of time and pressure areL/U_0 and ρ^EU_0^2, respectively. The typical scales for the Lorentz force and the electric potential Ψ^A are J_0B_0L and J_0L^2, respectively.Substituting the scaled variables into the equations of sections <ref> and <ref> and performing thenon-dimensionalization procedure, we obtain the final non-dimensional system. It includes the momentum equations written for each of the three layers, the equations of the interface deformations, and the incompressibility condition:D U^A/D t=-H^A/γ_ρ^Ap_0 -(1-1/γ_ρ^A)H^A/^2ζ^A- ϵ/γ_ρ^A(F_z^AH^A/2)+ϵ/γ_ρ^AF_^A+τ^A, D U^E/D t= -H^Ep_0 +ϵF_^E+τ^E, D U^B/D t=-H^B/γ_ρ^Bp_0 -(1-1/γ_ρ^B)H^B/^2ζ^B+ ϵ/γ_ρ^B(F_z^BH^B/2)+ϵ/γ_ρ^AF_+τ^B,∂ζ^A/∂ t=∇_·U^A, ∂ζ^B/∂ t=-∇_·U^B, ∇_·(U^A+U^B+U^E )=0, where all the variables are now non-dimensional and thematerial derivatives in the left-hand sides of (<ref>)-(<ref>) are expressed as in (<ref>)-(<ref>). The non-dimensional friction terms areτ^A =γ_ν^A/∇_^2U^A+1/[-(γ_ρ^A)^-1+γ_ν^A/H^E+H^A(U^A/H^A-U^E/H^E) - 2γ_ν^AU^A/(H^A)^2], τ^B =γ_ν^B/∇_^2U^B+1/[-(γ_ρ^B)^-1+γ_ν^B/H^B+H^E(U^B/H^B-U^E/H^E) - 2γ_ν^BU^B/(H^B)^2]τ^E =1/∇_^2U^E+1/[1+γ_ν^Aγ_ρ^A/H^E+H^A(U^A/H^A-U^E/H^E) + 1+γ_ρ^Bγ_ν^B/H^B+H^E(U^B/H^B-U^E/H^E)]. TheLorentz forces are expressed in terms of the non-dimensional variables as j_z^E = C(t)H_0^E/H^E-1, C(t)=A/⟨H̃_0^E/H^E⟩ J^E = J_z^Ee_z, J_z^E=H^E j_z^E,J_z^A =H^A/2j_z^E, J_z^B = H^B/2j_z^E, ^2Ψ^A = j_z^E,J^A_=Ψ^A,J^B_=-J^A_, F=J×B_0,where A is the non-dimensional area of the horizontal cross-section of the cell.For the system analyzed in this paper, the non-dimensional magnetic field is B_0=e_z, so only the horizontal components of the Lorentz force are non-zero, and the model for the current perturbations reduces to (<ref>), (<ref>), (<ref>).The boundary conditions include the no-slip and perfect electric insulation conditions at the walls:U^A=U^B=U^E=0,∂Ψ^A/∂ n=0at sidewalls. The non-dimensional parameters of the problem are the Froude number≡U_0/(Lg)^1/2,the electromagnetic parameter ϵ≡J_0B_0L/ρ^EU_0^2,the Reynolds number≡U_0L/ν^E,and the ratios of densities and kinematic viscositiesγ_ρ^A≡ρ^A/ρ^E,γ_ρ^B≡ρ^B/ρ^E, γ_ν^A≡ν^A/ν^E,γ_ν^B≡ν^B/ν^E.We should also add theparameters defining the cell's geometry: the non-dimensional unperturbed layer thicknesses H̃_0^A, H̃_0^B, H̃_0^E, and the parameter describing the shape in the horizontal plane, for example, the aspect ratio Γ=L_x/L_y for a rectangular cell. §.§ Numerical methodThe problem is solved numerically using a finite-difference scheme similar to the scheme applied in <cit.> and <cit.> to the shallow water equations for aluminum reduction cells. The time discretization is based on the explicit time-splitting (projection) scheme of the third order. Each time step includes* Solution of the Poisson equation (<ref>) and computation of the electric current perturbations and Lorentz forces, * Advancement of the time-evolution equations (<ref>)-(<ref>) according to the stiffly-stable scheme of <cit.>:f_*=6/11[3f^n-3/2f^n-1+1/3f^n-2+Δ t( 3q^n-3q^n-1+q^n-2)],where f stands for a velocity flux or interface deformation and q for the respective right-hand side of the evolution equation. For the momentum equations, the pressure gradient term is not included into the right-hand side, and f_* is an intermediate velocity flux. For the interface equations, f_* is the value at the next time step.* Solution of the pressure equation∇_·[(H^A/γ_ρ^A+H^B/γ_ρ^B+H^E )∇_p_0 ]=1/∇_·(U_*^A+U_*^B+U_*^E )with zero normal gradients of p_0 at the sidewalls as the boundary condition.* Correction of the velocity fluxes, in each layer, according toU^n+1=U_*-6/11H/γ_ρ∇_p_0and enforcement of the no-slip boundary conditions. The spatial discretization is of the second order of approximation and uses central differences implemented on a staggered grid. The Poisson equation (<ref>) and the non-separable elliptic equation (<ref>) are solved using the multi-grid algorithms of the library Mudpack <cit.>.A grid sensitivity study was carried out. It has been found that the large-scale character of the instability allows us to obtain accurate results on moderately fine grids. For example, for a battery of rectangular cross-section with the aspect ratio Γ=2.0, the grid of 128× 64 points is certainly sufficient, while the grid of 64× 32 points provides the results, which aregenerally correct with only minor quantitative inaccuracy. The analysis consists of multiple simulations, each reproducing the flow's evolution for a certain set of parameters. Each simulation begins with zero melt velocities and the flat interfaces (<ref>) perturbed randomly with the amplitude of 10^-5H^E_0. The simulation continues until the maximum deformation of one of the interface exceeds 0.5H^E_0 or the growing perturbations reach finite-amplitude saturation, or, in the stable case, for not less than 3000 non-dimensional time units L_x/U_0, with U_0 given by (<ref>). § RESULTS The analysis consists of two parts different from each other by the relation between the density jumps across the interfaces Δρ^A=ρ^E-ρ^A and Δρ^B=ρ^B-ρ^E.The simpler case of Δρ^A≪Δρ^B or Δρ^B≪Δρ^A, in which the significant deformation only occurs at the interface with the smaller jump is considered in section <ref>. The main goal is to continue the analysis of <cit.> toward a more accurate evaluation of the extent, to which the predictions developed for the aluminum reduction cells apply to the batteries. The more complex case when Δρ^A∼Δρ^B, so both the interfaces are significantly deformed, is discussed in section <ref>. §.§ Single interface deformation As a typical example of the instability observed at Δρ^A≪Δρ^B, Figs. <ref> and <ref> show the flow in a rectangular cell with the aspect ratio Γ=2.0, L_x=0.75 m, ρ^A=1000 kg/m^3, ρ^B=8000 kg/m^3, ρ^E=1100 kg/m^3, J_0=10^4 A/m^2, B_0=0.001 T, H_0^A=H_0^B=0.1 m, H_0^E=0.005 m, and ν^A=ν^B=ν^E=5× 10^-7 m^2/s.After an initial adjustment, the time signals of the local and integral characteristics show combined periodicity and exponential growth. Analysis of the interface deformations such as those in Figs. <ref>a,b shows that these features are caused by, respectively, rotation and growth of an interfacial wave. At the positive (directed upward) base current J_0, the rotation is counterclockwise if viewed from above. A negative base current changes the sense of rotation, but does not affect the flow in any other way. The solution in Figs. <ref> and <ref> is qualitatively similar to the solutions obtained in <cit.> in terms of the spatial shape, dynamics, and typical length and time scales of the growing perturbations. The amplitude of the deformation is much smaller at the lower than at the upper interface, approximately in proportion to Δρ^A/Δρ^B. The instability can be characterized by two coefficients computed during the phase of exponential growth: the time period of oscillations T and the growth rate γ. Both can be estimated using thetime signals shown in Fig. <ref>. For example, the point signal of the interface deformation in Fig. <ref>a can be used to determine T, while an exponential fit of the appropriate segments of the curves in Fig. <ref>b,c provides γ. For the specific flow in Fig. <ref>, the values are T=0.947 and γ=0.379.The unstable wave illustrated in Figs. <ref> and <ref> grows until the maximum interface deformation exceeds 0.5H_0^E, at which point the simulations are stopped. Such a growth is typically observed in our simulations in strongly unstable systems. In systems with positive but small growth rate γ, the perturbations may saturate nonlinearly at some finite amplitude of the interface deformation, typically six to three times smaller than H_0^E. The evident similarity with the instability in the aluminum reduction cells indicates that the observed instability may be controlled by the same non-dimensional parameters:the horizontal aspect ratio Γ,the instability parameter, which we discuss shortly, and the Reynolds number, which appears since viscosity is included in our model. An expression for the parameter controlling the instability is suggested by the studies of the reduction cells. The simplified models, in which viscosity and other complicated effects are neglected, and modified wave equations for low-amplitude large-scale perturbations are derived, all produce principally the same results (see <cit.>). The parameter is directly proportional to the product J_0B_0 and the square of the horizontal dimension of the cell, and inversely proportional to the product of the thicknesses of the metal and electrolyte layers, density difference Δρ, and gravity acceleration. In our notation, for the instability developing at the upper interface, this becomesΠ≡J_0B_0L_xL_y/Δρ^A H_0^EH_0^Ag.A similar parameter with Δρ^B instead of Δρ^A should be used at Δρ^B≪Δρ^A when the instability develops at the lower interface. The expression (<ref>) is equivalent to the parameter β derived for the reduction cells in the pioneering work of <cit.> and to the parameter, in terms of which the results of <cit.> were recently recast in <cit.>. It can be rewritten in terms of the non-dimensional parameters introduced in section <ref> asΠ=ϵ^2 Γ1/H̃_0^A H̃_0^E1/γ_ρ^A-1. We have carried outextensive testing in order to verify that Π is a suitable control parameter for the single-interface instability in the battery. Firstly, we verified that a significant change of Π leads to a significant change of the wave characteristics γ and T.Furthermore, tests were performed, in which thebattery parameters, such as L_x, B_0, J_0, H_0^A, or H_0^E were varied substantially in such a way that the value of Π remained the same. The results of one of such tests conducted at Γ=2.0 are shown in table <ref>. Two sets of simulations are exhibited: at the normal values of viscosities and at the values reduced by four orders of magnitude, which practically removed the effect of viscosity on the flow. We see in table<ref> that at negligible viscosity the characteristics of the growing wave γ̃ and T̃ are determined by Π with very good accuracy. The computed values are within 1%. An exception is the simulation with Δρ^A increased to 200 kg/m^3, in which γ̃ and T̃ change by about 3%. This effect will be further discussed in section <ref>.The variation is stronger at the realistic ν=5× 10^-7 m^2/s. This can be attributed to the non-negligible effect of the variation of the Reynolds number.We have also performed simulations with Δρ^B≪Δρ^A, i.e. when the significantdeformations are expected only on the lower interface. The results are qualitatively the same as when Δρ^A≪Δρ^B. The only significant differences are the opposite sense of rotation (clockwise at J_0>0 and counterclockwise otherwise), which is evidently caused by the opposite direction of the horizontal current perturbations in the bottom metal layer (see (<ref>) and the illustration in Fig. <ref>b), and the change of T and γ attributed to the change of the typical time scale.Parametric studies were performed to explore the effect of Π and the aspect ratio Γ on the instability. The realistic values ν^A=ν^B=ν^E=5× 10^-7 m^2/s were used. In the study, the same values ρ^A=1000 kg/m^3, ρ^E=1100 kg/m^3, ρ^B=8000 kg/m^3, and H_0^A=H_0^B=0.1 m were used, while the other parameters were varied in the ranges J_0=5× 10^3, 10^4, A/m^2 B_0=10^-3, 2× 10^-3 T, 0.5≤ L_x≤ 1.5 m, and 0.005≤ H_0^E≤ 0.1 m so as to change the value of Π. The results are presented in Fig. <ref>. We find that, as one would expect by analogy with the reduction cells, the behavior is strongly influenced by Γ. In particular, the instability threshold Π_cr such that the battery is stable at all Π<Π_cr but unstable at Π>Π_cr, is a non-monotonic function of Γ. At Γ=1.0, Π_cr is very close to zero, although it cannot be exactly zero due to the stabilizing effect of viscosity and numerical dissipation. In rectangular cells, Π_cr can be small, as, for example, Π_cr(3.0)≈ 0.15 or large as Π_cr(2.0)≈ 3.7 or Π_cr(4.0)≈ 3.5.Even small variations of Γ may change Π_cr quite significantly. An example is the difference between Π_cr(3.0)≈ 0.15 and Π_cr(3.33)≈ 0.67. This and the strong effect of Γ in general can be explained in the same way as, in <cit.> for the aluminum reduction cells. The aspect ratio determines the set of available natural gravitational wave modes and, so, strength of the electromagnetic effect needed to transform two of them into a pair with complex-conjugate eigenvalues. As an illustration of the aspect ratio effect, Fig. <ref> shows the interface deformations computed for the unstable flows at Γ=3.0 (Fig. <ref>a) and Γ=3.33 (Fig. <ref>a). Almost all the battery parameters are the same in the two cases: J_0=10^4 A/m^2, B_0=0.001 T, L_x=1.5 m, ρ^A=1000 kg/m^3, ρ^E=1100 kg/m^3, ρ^B=8000 kg/m^3, H_0^A=H_0^B=0.1 m. The only differences are in the values of L_y (0.5 m or 0.45 m) and the values of H_0^E (0.02548 m or 0.02317 m) that are selected in such a way that the instability parameter is Π=3.0 in both cases. The time signals of the two unstable solutions show strongly different characteristics: T=0.690, γ=0.110 at Γ=3.0 and T=0.335, γ=0.073 at Γ=3.33. The growing interface deformations shown in Fig. <ref> demonstrate that the explanation suggested above is valid. The electromagnetic coupling, which causes the instability, involves different gravitational wave modes in the two cases, with much larger x-wavelength at Γ=3.0 than atΓ=3.33. A more specific description can be obtained with the eigenvalue analysis.The main conclusion of this section is that the single-interface rolling pad instability occurring when one density difference is much smaller than the other is very similar to the instability observed in the aluminum reduction cells. The instability is caused by the same mechanism, develops in essentially the same form of a rotating interfacial wave, and is determined by three non-dimensional parameters: the instability parameter (<ref>) (or its analog based on Δρ^B, H_0^B instead of Δρ^A, H_0^A if the instability occurs on the lower interface), the Reynolds number , and the horizontal aspect ratio Γ.§.§ Double interface deformationThe situation when Δρ^A and Δρ^B are of comparable magnitudes, and, thus, the instability may significantly deform both the interfaces, is considered in this section. The analysis is performed as a parametric study, in which the metal densities are kept constant at ρ^A=1000 kg/m^3 and ρ^B=8000 kg/m^3, while the density of the electrolyte ρ^E varies in the range between 1100and 6000 kg/m^3. This can be compared with the material densities in the currently considered battery schemes (see <cit.>): 500 to 1500 kg/m^3 for ρ^A, 6000 to 10000 kg/m^3 for ρ^B, and 1000 to 3000 kg/m^3 for ρ^E. In the study, we keep constant metal layer thicknesses H^A=H^B=0.1 m and viscosities ν^A=ν^B=ν^E=5× 10^-7 m^2/s. The other parameters vary as 5× 10^3≤ J_0≤ 4× 10^4 A/m^2, 10^-3≤ B_0≤ 3× 10^-2 T, 1≤ L_x≤ 2 m, 0.005 ≤ H^E≤ 0.04 m. The behavior at the aspect ratios Γ=2.0 and 3.0 is explored.As a typical example, Figs. <ref>-<ref> show the instability in a system with ρ^E=3000 kg/m^3, J_0=2× 10^4 A/m^2, B_0=0.002 T, L_x=2.0 m, L_y=1.0 m, H_0^A=H_0^B=0.1 m, H_0^E=0.005 m. The results of the parametric studies are summarized in Fig. <ref>.We see in Figs. <ref>-<ref>that the main features of the instability remain the same as in the case of the single-interface deformation. The time signals combine periodic oscillations with exponential growth. The spatial structure in Fig. <ref> shows large-scale interfacial waves quite similar to the wave in Fig. <ref>. A conclusion can be made that the instability is still of the rolling pad type. At the same time, the interaction between the two waves growing at the lower and upper interfaces results in more complex and diverse dynamics. The waves are always coupled to each other in the sense that they have the same oscillation period and sense of rotation (counterclockwise at Δρ^A<Δρ^B and clockwise at Δρ^A>Δρ^B if J_0>0). Two types of coupling are observed. At substantially different Δρ^A and Δρ^B (at ρ^E≤ 3000 kg/m^3 or ρ^E≥ 6000 kg/m^3 at Γ=2.0), the waves at the upper and lower interfaces are nearly antisymmetric (the sign of δζ^A is opposite to the sign of δζ^B). The antisymmetry is not exact, but with a small phase shift. For example, the time signals of the single-point oscillations in Fig. <ref>a show the time shift Δ t≈ 0.042, which can be compared with the oscillation period of 1.114. The situation changes at close Δρ^A and Δρ^B. In such cases, the two waves become nearly symmetric (with the signs of δζ^A and δζ^B being mostly the same) and, again, with a small phase shift. This type of behavior is illustrated in Fig. <ref>. The observed change of the type of wave coupling is consistent with the results of therecent three-dimensional potential flow analysis of interfacial waves in a three-layer cylindrical cell without electromagnetic forces <cit.>. It has been shown in this work thatthe type of coupling between the waves at the two interfaces changes from antisymmetric to symmetric when the amplitudes of the waves become comparable. While not a proof due to the evident differences between the systems, the consistency can be considered as an indication that the change of coupling observed in our battery is likely to be related to purely hydrodynamic interaction between the waves, rather than to electromagnetic forces. The results of the parametric study presented in Fig. <ref> show that the effect of the relation between Δρ^A and Δρ^B on stability is not very significant in the strongly unstable system with Γ=3.0. The values of the instability growth rate γ decrease as the two density differences become closer to each other, but the instability continues to occur at small values of Π.On the contrary, for the system with Γ=2.0, increase ofΔρ^A has stabilizing effect, which is not only substantial, but also clearly exceeding the effect of the density stratification already incorporated into the expression (<ref>) of the instability parameter. The critical values Π_cr, such that the system is always stable at Π<Π_cr but unstable at Π>Π_cr, can be estimated as Π_cr=3.7, 4.0, 4.5, 6.2, and 21.5 at Δρ^A=100, 500, 1000, 2000, and 3000 kg/m^3, respectively. This growth is in contradiction with the prediction of the mechanical model <cit.>, where the presence of the second pendulum was found to be always destabilizing. We attribute the contradiction to the oversimplified character of the model used in <cit.>. The stabilizing effect of the second layer can be given a qualitative explanation based on the fact that the horizontal perturbation currents in the top and bottom layers are always flowing in the opposite directions (see (<ref>)). This implies the opposite directions of the horizontal Lorentz forces (<ref>) and, thus, that forces in one layer always oppose the motion of a coupled wave system. Whether this happens in the bottom or top layer is determined by the relation between Δρ^A, Δρ^B, H_0^A, H_0^B and, possibly, other parameters.§ CONCLUDING REMARKSWe have presented a new shallow water model and the resultsof its application to electromagnetically coupled waves in a simplified liquid metal battery of rectangular shape.The results primarily concern the linear stages of one particular instability in a simplified situation when the base state has flat interfaces and zero flow, but the model itself has much broader applicability. It is well suited for analysis of more general situations: with background melt flows and interface deformations, spatially complex or even time-dependentdistributions of base electric current and magnetic field, etc.The instability caused by the interaction between the externally generated vertical magnetic field and horizontal electric current perturbations associated with the interface deformations is demonstrated and analyzed. The growing perturbations have the form of rotating large-scale interfacial waves. The instability mechanism is similar to the mechanism of the rolling pad instability known for the aluminum reduction cells. In the case when the density jump at one interface is much smaller than at the other, only one interface is significantly deformed in the course of the perturbation growth, and the similarity to the behavior of reduction cells is quite close. In particular, the instability is controlled by the same non-dimensional groups: the horizontal aspect ratio (<ref>), the instability parameter (<ref>), and the Reynolds number (<ref>). Critically, the instability parameter (<ref>) is proportional to the square of the horizontal size of the cell.In the case when the density jumps at the two interfaces are comparable, both the interfaces are significantly deformed, and the behavior of the system is more complex and quite different from that of a reduction cell. The two interfacial waves can be coupled either symmetrically or antisymmetrically with a small phase shift between the waves. The presence of the second deformable interface may have a stabilizing effect. The model used in this study is idealized and cannot produce a complete and quantitatively accurate picture of a real system. Further work is advised, preferably in the way of experiments performed with large battery cells and three-dimensional numerical models based on accurate representation of the geometry and the complex distributions of the base current and magnetic field. Eigenvalue analysis of the simplified 2D and 3D system would also be interesting.Even at this point, however, the available results can be utilized to make a preliminary assessment of the potential effect of the rolling pad instability on battery design.We have seen above that the unstable perturbations tend to grow strongly resulting in high amplitudes of interface deformation or even rupture of the electrolyte layer. The situation is dissimilar to that predicted for the thermal convection or Tayler instability in the sense that reliable uninterrupted operation of a battery is only possible if the instability is avoided. Our conclusion is that the instability is a limiting factor for the versions of the battery design with small density difference between the metal and the electrolyte. One such version is the Mg-Sb battery <cit.>, in which ρ^A≈ 1577 kg/m^3,ρ^E≈ 1715 kg/m^3, ρ^B≈ 6270 kg/m^3. Taking, as an example, the relatively stable geometry of a rectangular cell with Γ=2.0 and using the criterion Π_cr=3.7 derived in section <ref> we find that at typical J_0=10^4 A/m^2, B_0=0.001 T, a battery with H_0^A=H_0^B=0.1 m, H_0^E=0.005 m becomes unstable at the horizontal size L_x exceeding 0.7077 m. This is larger than in the currently built laboratory prototypes or commercial concepts, but certainly in the range anticipated for future large-scale energy storage facilities <cit.>. We should also mention that, similarly to the practice of the aluminum smelting industry, the instability can be avoided by rearranging the cell's wiring so as to reduce the vertical component of the magnetic field within the cell. Computational simulation tools built on the basis of our model can be utilized to predict the need for such a rearrangement and to assist in its optimization.The instability appears less important for other battery concepts, in which Δρ^A is substantially larger. 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Physics of Fluids 29(5), 054,101 (2017)Weber:2015 Weber, N., Galindo, V., Priede, J., Stefani, F., Weier, T.: The influence of current collectors on Tayler instability and electro-vortex flows in liquid metal batteries. Phys. Fluids 27(1), 014103 (2015)Weber:2014 Weber, N., Galindo, V., Stefani, F., Weier, T.: Current-driven flow instabilities in large-scale liquid metal batteries, and how to tame them. J. Power Sources 265, 166–173 (2014)Xiang:2017 Xiang, L., Zikanov, O.: Subcritical convection in an internally heated layer. Phys. Rev. Fluids 2, 063,501 (2017)Xu:2016 Xu, J., Kjos, O.S., Osen, K.S., Martinez, A.M., Kongstein, O.E., Haarberg, G.M.: Na-Zn liquid metal battery. J. Power Sources 332, 274–280 (2016)Zikanov:2015 Zikanov, O.: Metal pad instabilities in liquid metal batteries. Phys. Rev. E 92(6), 063,021 (2015)Zikanov:2000 Zikanov, O., Thess, A., Davidson, P.A., Ziegler, D.P.: A New Approach to Numerical Simulation of Melt Flows and Interface Instability in Hall – Heroult Cells. Met. Mat. Trans. B 31, 1541–1550 (2000) | http://arxiv.org/abs/1706.08589v2 | {
"authors": [
"Oleg Zikanov"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170626205009",
"title": "Shallow water modeling of rolling pad instability in liquid metal batteries"
} |
On tree-decompositions of one-ended graphsJohannes Carmesin Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom ,Florian Lehner Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, United Kingdom Florian Lehner was supported by the Austrian Science Fund (FWF), grant J 3850-N32 , andRögnvaldur G. Möller Science Institute, University of Iceland, IS-107 Reykjavík, Iceland Rögnvaldur G. Möller acknowledges support from the University of Iceland Research FundDecember 30, 2023 =========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Portuguese Labor force survey, from 4th quarter of 2014 onwards, started geo-referencing the sampling units, namely thedwellings in which the surveys are carried.This opensnew possibilitiesin analysing and estimating unemployment and its spatial distribution across any region. The labor force survey choose, according to an preestablished sampling criteria,a certain number of dwellings across the nation and survey the number of unemployed in these dwellings. Based on this survey, the National Statistical Institute of Portugal presently uses direct estimation methods to estimate the national unemployment figures. Recently, there has been increasedinterestin estimating thesefigures in smaller areas. Direct estimation methods, due to reduced sampling sizes in small areas, tend to produce fairly large sampling variations therefore model based methods, which tend to “borrow strength” from area to area by making use of the areal dependence, shouldbe favored. These model based methods tend use areal counting processes as models and typically introduce spatial dependence through the model parameters bya latent random effect. In this paper, we suggest modeling the spatial distribution of residential buildings across Portugal by a Log Gaussian Cox process and thenumber of unemployed per residential unit as amark attached to these random points. Thus the main focus of the study is to model the spatial intensityfunction ofthis marked point process.Number of unemployed in any region can then be estimated using a proper functional of this marked point process. The principal objective of this point referenced method for unemployment estimation is to get reliable estimates at higherspatial resolutions andat the same timeincorporatein the modeltheauxiliary information available at residential units such as average income or education level of individuals surveyed in these units.§ INTRODUCTION The knowledge and understanding of unemploymentat a regional level has been increasingly used to make political decisions. In Portugal, the official unemployment figures are published quarterly by the National Statistical Institute of Portugal (INE) atthe national level as well asfor NUTS II regions. NUTS is the classification of territorial units for statistics (see figure <ref> for a better understanding of NUTS regions and the 278 counties in mainland Portugal). The calculation of the official numbers is based on a direct method from the sample of the Portuguese Labor Force Survey. This method is an extension of the Horvitz-Thompson estimator (Horvitz and Thompson, 1952), with a correction for non-response and a calibration for the known population totals.Currently, there is a need to obtain reliable estimates at a more disaggregated level, particularly at the NUTS III level. However, due to the relatively small size of these areas, there is insufficient information to obtain estimates with an acceptable level of accuracy using the direct method. Small area estimation methods (Rao, 2003) provide useful tools for such studies. There have been considerable methodological developments to solve small area estimation problems in an unemployment context. See for example, Lopez-Vizcaino et al (2015), which proposes a multinomial model for the estimation of the labor force indicators in a classical context, and Pereira et al (2016) which proposes alternative spatio-temporal models within a Bayesian context. The majority of small area methods are based on generalized linear models applied to areal data by modeling an appropriate counting process. These methods can "borrow strength" from area to area and make use of auxiliary information at regional level, compensating for the small sample sizes in each area due to the designed sampling survey.From 4th quarter of 2014 onwards all the sampling units (dwellings) of the LFS are geo-referenced according to the coordinates of the respective building, which may be composed by of one or more dwellings. Thus, different sampling units may have the same spatial location, causing some difficulty in modeling strategies. We will address this problem by moving from dwellings to residential buildings as our sampling units. This strategy will solve the awkward problem of multiplicity of geo-referenced units at the cost of introducing some approximations at residential unit level with some consequent loss of precision.Geo-referencingresidential units permits using methods and models for point referenced data, increasingthe spatial resolution of inferential methods giving us a much detailedinformation on the intensity of unemployment across space.This approach allows for the representation of the sample survey as a realization of a spatial point process together withthe associated marks, namely the number of unemployed people in each residential unit. Point referencing also permits us to use more precise auxiliaryinformation at residential units, such as the average education level or income of individuals living in the sameresidential unit.For modelling the intensity of residential unitlocations and their associated marks, we suggest a markedLog Gaussian Cox processes (LGCP) as model. The LGCP is a class of flexible models widely used in the context of spatial point processes (Moller and Waagepetersen, 2004, Illian et al, 2008, Baddeley et al, 2016). Typically, in this frame work, the log intensity of the point process is assumed to be a(latent) Gaussian random field. In order to facilitate calculations, often marks are assumed to be independent of point patterns so that marks and spatial patterns can be modeled separately. However, for inference on such models,Illianet al (2012) proposed a flexible framework using INLA (Rue et al, 2009, Martins et al, 2013, Rue et al, 2017), in whichthe spatial pattern of points and marks are allowed to be dependent, assumingindependenceconditional on a common latent spatial Gaussian processes, making these modelsmore flexible.Inference on such models is not straightforward. Due tocomputational problems that emerge inthis framework, Lindgren et al (2011) proposed a more computationally tractable approach based on stochastic partial differential equation (SPDE) models, which permit the transformation of a Gaussian fieldto a Gaussian markov random field andwe follow this method.The structure of the paper is as follows: In sections <ref> and <ref>, we explain the sampling design of the LFS and the consequent data available for the analysis. In section<ref>, we explain our models as well as inferential methods and give the unemployment estimates for NUTS III regions, as well as for the 278 counties in mainland Portugal. Comparison of reported results with thedirect estimation methods is also given in section<ref>. Finally in section <ref>, we give a brief account of possible extensions on models and methods employed. § LABOR FORCE SURVEYS The methodologies proposed in this study are highly dependent on the sampling design of the LFS. Therefore, it is important to understand both how the sampling units are drawn and how the inclusion probabilities are calculated.The LFS is a continuous survey of the population living in private dwellings within the Portuguese national territory. The survey provides an understanding of the socioeconomic situation of these individuals during the week prior to the interview (reference week). The dwellings are the sampling units and the inhabitants living in these dwellings are the observation units.The unemployment figures are published quarterly by INE at both the national and NUTS II level. From one quarter to another, the sample changes. These samples have 6 sub-samples, with the oldest one being replaced for a new one in each quarter. This process is also known as a rotation scheme. In this way, each individual in the sample is surveyed over 6 consecutive quarters, inducing strong temporal dependence between the quarterly surveys. Between 2011 and the 2nd quarter of 2013,sampleswere selected from a sampling base called “master-sample" (MS). From the 3rd quarter of 2013 however, each new sub-sample (in each quarter) was extracted from the national dwellings register (NDR). Contrary to the MS, in NDR all the dwellings are geo-referenced. This transition process from the MS to the NDR was completed in the 3rd quarter of 2014. After the 4th quarter of 2014,alldwellings were extracted from the NDR, and therefore, new methodologies based on point-referenced data canbe used. Since there may be more than one dwelling in each residential unit, particularlyin areas of high population density, multiple dwellings in the survey have the same spatial location.The sampling of the Portuguese LFS is stratified into 30 NUTS III regions. In each region, multi-stage sampling is conducted, where the primary sampling units are areas consisting of one or more cells of the km^2 INSPIRE grid, and the secondary units are dwellings. Every selected dwelling and all its inhabitants are surveyed. For the selection of the primary units, the dwellings are sorted by their coordinates and a fixed interval K_h is calculated by K_h=A_h/n_h, where A_h is the total number of dwellings in strata h and n_h is theselected number of dwellingsin strata h. A randomvariable u_h ∼ U[1,K_h] is generated, to determine the position of the dwelling which determines the first area selected (the area in which the dwelling is included). The position of the other selected areas is fixed and determined by the position of the dwelling that determined the position of the previous area plus the interval K_h. After the selection of the areas, a similar process is followed to select the dwellings. In each selected area j, a random uniform variable u_jh∼ U[1,K_jh], is generated, where K_jh=A_jh/n_jh,A_jh is the total number of dwellings in the area j and strata h andn_jh is the number of dwellings sampled in area j and strata h. u_jh defines the first selected dwelling in area j and strata h. After the selection of the first dwelling in area j, the position of the other dwellings is fixed and determined by the position of the previous selected dwelling plus the interval K_jh. Following this scheme, the selection probability of the area j in strata h isp_jh=A_jh/A_h× s_h,A_jh < K_h,1, otherwise. where s_h is the number of selected areas in the strata h. The selection probability of each dwelling i in area j and strata h is given by p_ijh=p_jh× p_i|jh = p_jh×n_jh/A_jh, Since all the individuals in each selected dwelling are surveyed, their selection probabilities are equal to the respective dwelling, p_kijh=p_ijh for each individual k in dwelling i. The official estimates of the unemployment figures are calculated using a direct method, based on the Horvitz-Thompson estimator (Horvitz and Thompson, 1952). § DATAIn this study, we analyze the quarterly data of the Labor Force Survey (LFS) regarding to the 4th quarter of 2014. The sample size is about 40,000 observations and each individual can be classified into one of the following three categories: employed, unemployed and inactive. Covariate information about the individuals are available, such as gender, age and education level. Within a point process modelling scheme, the choice of dwellings as sampling units, creates problems. Since residential units are geo-referenced, multiple sampling units appear with the same spatial location. The sampling design and the consequent data are not sufficiently detailed to obtain good information on the multiplicity distribution of dwellings in each residential building, therefore we use residential buildings as design units. Thus we aggregate the number of unemployed observed in dwellings with the same spatial location and we denote by Y(s_j),the number of unemployed in the residential building at spatial position s_j. §.§ Covariates One of the great benefits of point referenced models for spatial variation of unemployment is that very detailed covariate information can be given at residential building level, like average education and age of the individuals in each residential building. For the locations we have available only one covariate, the population density, which we will use as offset in the model. The median of the education level in each residential building and the mean age were considered as covariates to model the marks. Although the education level does not constitute a quantitative variable, it was treated as such due to its ordinal meaning (1-primary level, 2-secondary level, 3-higher level). Higher values of this variable in the Lisbon and Pennsula de Setbal regions can beclearly seen(figure <ref>). It is also interesting to see the spatial distribution of the mean age, with more younger people near the country's coastline than in its interior. The proportion of unemployed people registered in the employment centers depends on the number of unemployed and this information is also used as a covariate. Kernel based estimates of the covariates are given in figure <ref>. § A SPATIAL POINT PATTERNS APPROACH We model the spatialpoint process N_1(·)of residential units bya logGaussian Cox process with intensity λ_1(s), withlogλ_1(s|W(s))= α_1+z'_1(s)θ_1+W(s), We assume that for every s, the markY(s)is aPoisson random variablewithprobability mass functionP_Y(s)|W(s)(y)∼Poisson(λ_2(s| W(s))),where logλ_2(s|W(s))=α_2 +z'_2(s)θ_2+ α_3 W(s). Here, W(s) is a latent Gaussian Markov field,z_1(s), z_2(s) are generic auxiliary information which may help in understanding the spatial patterns of points as well as the marks and θ=(α_1,α_2,α_3,θ_1,θ_2) are the model parameters. We assume that the same latent Gaussian process W(s) is acting both on the point patternsand their marks with different scaling and we assume that conditional on W(s),point mass density and the marksare independent. It is natural to expect that in areas with high density of residential buildings, one would expect higher rate of unemployment and therefore independence of points and marks may not be an reasonable assumption. With the conditonional independence assumption, the corresponding marked point process N(s,y) has the following structure: N_1(s,y)|λ (s,y) ∼Poisson(λ(s,y)), withλ(s,y|W(s))=λ_1(s|W(s))P_Y(s)|W(s)(y) whereλ_1(s|W(s)), and P_Y(s)|W(s)(y) defined as in (<ref>),(<ref>) and (<ref>). §.§ Target quantities for inferenceOur objective is to make inference on the number of unemployed in any given region A, based on the sample survey. Thus, our target quantityis a functional of the marked point process, namelythe number of unemployed people in region A, given by N(A) =∑_j=1 ^N_1(A)Y(s_j).Let s =(s_1,...,s_n) be the location of sampling units chosen in the sampling survey,y(s) the number of unemployed in each residential unit and z_1, z_2 the covariates specific to residential units and marks respectively. We denote by=(n,s, y(s), z_1,z_2) the observed data obtained from the sampling survey. Our specific target quantitiesarethe posterior predictive meanand variance of the random variable N(A) given by respectivelyE(N(A)|) = E_(W(s), θ|)[E(N(A)|,W(s),θ)]= ∫_W(s),θE(N(A)|,W(s), θ) p(W(s),θ |) dW(s)dθ, and Var(N(A)|)= Var_(W(s), θ|)[E(N(A)|,W(s), θ)]+ E_(W(s), θ|)[Var(N(A)|,W(s), θ)]. Calculation ofE(N(A)|,W(s), θ)=E(∑_j=1^N_1(A)Y(s_j)|,W(s), θ)andVar(N(A)|,W(s), θ)=Var(∑_j=1^N_1(A)Y(s_j)|,W(s), θ),require certain assumptions.* Conditional on W(s),the point patterns of the Cox process over disjoint regions are independent. Consequently, conditional on W(s), the point patterns over the design pixels I_j arealso independent and we also assume that withineach pixel the intensity function of the Cox process is homogeneous so that λ_1(s)=λ_1(I_j) for every s∈ I_j. * We assume that conditional on W(s),the marks Y(s) are independent of the point patterns so that the conditional intensity function of the marked point process is given by λ(s, y| ,W(s),θ)= λ_1 (s| ,W(s), θ) P_Y(s)|bx,W(s), θ(y),* We assume that conditional on W(s),marksobserved on disjoint sets are independent. * Finally,we assume thatthe marks Y(s_i)are identical for everys_i∈ I_j, that is the number of unemployed in every residentialunit in pixel I_j are identical. Hence, in each pixel we replace E(Y(s_i)) by E(Y(I_j)). To summarize, wehave two major assumptions in this model: The latent Gaussian field W(s) is the only source of dependence in the model. Not only are the point patterns and marks independent conditional on W(s), but the point pattern and marks are independent over disjoint intervals conditional on W(s). Further, within a km^2 design unit pixels, we assume homogeneity of the point patterns as well as marks. Let N(I_j) be thenumber of residential unitsin each pixel I_j. Then with assumptions (a)-(e), E(N(A)|, W(s),θ)) = E(∑_j=1^N_1(A)Y(s_j)|,W(s), θ)= E(∑_I_j∈ A∑_i∈ I_j Y(s_i)|,W(s), θ) = ∑_I_j∈ AE(∑_i=1^N(I_j) Y(s_i)|,W(s), θ))= ∑_I_j∈ AE(N_I_j|W(I_j))E(Y(I_j)|, W(I_j), θ) = ∑_I_j∈ A ||I_j||λ_1(I_j|, W(I_j),θ)λ_2(I_j|,W(I_j),θ)∼ ∫_s∈ Aλ_1(s|,W(s),θ) λ_2(s|,W(s),θ) ds Here, W(I_j) represents the latent gaussian Markov random field approximating the latent Gaussian random field W(s) obtained by the SPDE method. w(I_j) values are obtained from INLA as explainedin section <ref>. (<ref>) follows from the conditional independence andhomogeneity of the point patterns as well as the marks within each km^2 pixels, whereas(<ref>) follows from the approximation of integrals by sums over the design pixels as is explained in <ref> . Thus the km^2 design pixels are the smallest units over which we approximate the point referenced process.We can calculate, with similar arguments Var(N(A)|,W(s), θ )∼ ∫_s∈ Aλ_1(s|,W(s)),θ) λ_2(s|,W(s),θ) ds + ∫_s∈ Aλ_1(s|,W(s),θ) λ_2^2(s|,W(s),θ) dsThemean and the variance of of the predictive distribution given in (<ref>) and (<ref>) can be calculated numerically. INLA package permits the calculation of the intensity function λ_1(s|, W(s),θ) as well as the mean mark λ_2 (s| ,W(s),θ). INLA also simulatesfrom themarginal posterior densities of the latent process as well as the model parameters, thus target quantities(<ref>),(<ref>)can be efficiently calculated within the INLA platform. In the next section, we briefly discuss how these calculations are carried within INLA.§.§ Bayesian inference using INLAConditional on a realization of W(s), a log-Gaussian Cox process is an inhomogeneous Poisson process. It follows that the likelihood for an LGCP is of the form log( F(θ|))=|Ω| - ∫_Ωλ_1(s|,θ) ds + ∑_s_i ∈ Sλ_1(s_i|,θ),where S is the set of observed locations and λ_1(s) is defined in (<ref>).The integral in the likelihood is intractable due the stochastic nature of λ_1(s). To solve this problem we could use the traditional methods to fit a log-Cox process, which consists of dividing the study regions into cells, forming a lattice, and then counting the number of points into each one. These counts are modeled using the Poisson likelihood. See for example Illian et al (2010). However, Simpson et al (2016) consider that this approach can be very inefficient, especially when the intensity of the process is high, the window of observation is too large or when the pattern is rare. They propose the use of an SPDE (Stochastic Partial Differential Equation) approach, introduced by Lindgren et al (2011), to transform a Gaussian field (GF) to a Gaussian Markov random field (GMRF). This methodology uses a computational mesh only for representing the latent Gaussian random field and not for modeling counts. Lindgren et al (2011) assume the following finite element representationW(s) ≈∑_j=1^N w_j ψ_j(s)where N is the number of the mesh nodes, w=(w_1,w_2,...,w_N)^T is a multivariate Gaussian random vector (representing a Gaussian Markov random field, GMRF) and {ψ_j}_j=1^N are the selected basis functions defined for each mesh node: ψ_j is 1 at mesh node j and 0 in all the other mesh nodes. w is chosen in a way that the distribution of W(s) approximates the distribution of the solution to an SPDE. Lindgren et al (2011) showed that the resulting distribution for the weights is w ∼ N(0, Q(τ, k)^-1) where the precision matrix Q(τ, k) is a polynomial in the parameters τ and k. Working directly with the SPDE parameters k and τ can be difficult because they both affect the variance of the field (Yuan et al (2017)). So, we will consider the standard deviation σ and the spatial range ρ which are respectively given by σ=√(1/4π k^2 τ^2)and ρ=√(8)/k. After that approximation, it follows that the integral in (<ref>) can be written as ∫_Ωλ_1(s) ds=∫_Ωexp(W(s)) ds ≈∫_Ωexp(∑_j=1^N w_j ψ_j(s) ) ds This integral can be approximated using standard numerical integration schemes. Simpson et al (2016) suggest to use the follow quadrature rule ∫_Ω f(s) ds ≈∑_i=1^N+nβ_i f(s_i)where {s_i}_i=1^N+n are the locations of mesh nodes and observations, and {β_i}_i=1^N+n are the quadrature weights. Unlike the traditional methodsforinference in LGCP models, this methodology uses each location to model the point pattern, without aggregation. The LGCP modelbelongs to the latent Gaussian models and consequently, the inference can be done within the INLA platform. As we noted, the SPDE methodology requires a triangulation of the study region to represent a Gaussian random field.Here, we used a Delaunay triangulation with 3923 mesh nodes. In real data applications, it is common that the point pattern and its associated marks are dependent. In our case, we expect that the average number of unemployed people per dwellingto be dependent with the intensity of residential buildings, but the signal of that correlation is not obvious. On the one hand, we expect the number of unemployed people to be higher in regions with higher intensity of residential buildings andon the other hand, we expected more opportunities of employment in theseregions.Illian et al (2008) describes twotypes of marked point process models depending on the type of dependence between the point patterns and marks. Here, we consider two versions of conditional dependence: In the first model we assume, as was explained in section <ref>,that there is a common latent Gaussian field that govern the dependence structures of points and marks and conditional on this field, point patterns and marks are independent. In the second alternative model, we assume that there are two independent fields that govern the dependence structures of points patterns and marks. It is also possible to introduce a third coreginalization model (Banerjee et al, 2004, Gelfand et al, 2004) consisting of two dependent latent processes for point patterns and marks by assuming independence of points and marks conditional on these latent processes. Coreginalization models can be inferred within the INLA platform, however this would require joint modeling of two dependent fields and we will not pursue these models in this work. In table <ref>, a comparison of these alternative models is given.Here, we give the details of the model based on a common latent Gaussian model. Let us consider that {s_i}_i=1^N+n are the locations of the mesh nodes and the locations of the sampled residential buildings, and {y(s_i)}_i=1^N+n are the number of unemployed people per residential building. The hierarchical structure of the model considered is given by* Data|Parameter p({s_i,i=1,...,N+n}|λ_1) ≈∏_i=1^N+nPoisson(β_i λ_1(s_i)) p({y(s_i),i=1,...,N+n}| λ_2) ≈∏_i=1^N+nPoisson(λ_2 (s_i))where β_i is defined in (18).* Parameter|Hyperparameters log(λ_1(s_i))=α_1+offset_1(s_i) +W(s_i),log(λ_2 (s_i))=α_2+offset_2(s_i) +Z_2'(s_i)θ +α_3 W(s_i),where W(s) is the GMRF given in (3).* Hyperparameters α_1 ∼ N(0,1000) α_2 ∼ N(0,1000) θ_j ∼ N(0,1000), j=1,...,p α_3 ∼ N(0,1000)We assume that the latent field W belong to the Matern class with ν=1. We further assume that the model parameter of this field has the same prior structure as given below:We followed Simpson et al (2017) and Fuglstad et al (2017) to construct a joint penalising complexity (PC) prior density for the spatial range, ρ, and the marginal standard deviation, σ, which is given by p(ρ,σ)=R S ρ^-2 e^-R ρ^-1 - S σ where R and S are hyperparameters determined by R=-log(α_1) ρ_0 and S=-log(α_2)/σ_0. The practical approach for this in INLA is to require the user to indirectly specify these hyperparameters through P(ρ < ρ_0)=α_1 and P(σ < σ_0)=α_2. Here, we considered ρ_0=400, α_1=0.5,σ_0=1,α_2=0.5. The term_1 (s_i) in (<ref>) represents the log population density. We know their numbers by NUTS III regions so, based on that, we produced a spatial prediction for all domains by way of a Kernel smoothing, using the centroids of the NUTS III regions. The resulting prediction is given in figure <ref>. Lisboa, Porto and Pennsula de Setbal are the regions that stand out most. The _2 (s_i)term in (<ref>) represents the log of the number of people per residential building. We have the information for the residential buildings locations in the sample, but we need to estimate it for the mesh nodes. For this we also used the Kernel smoothing (see figure <ref>).§.§.§ Model selectionTo evaluate the significance of each covariate and random effect in the marks, we considereddifferent models and compared the results of two model selection criteria: deviance information criterion (DIC) and Watanabe-Akaike information criterion (WAIC). DIC, proposed by Spiegehalter et al (2002), is the most commonly used measure of model fit. It is based on a balance between the fit of the model to the data and the corresponding complexity of the model: DIC = D̅ + p_D where D̅ is the posterior mean deviance of the model and p_D is the effective number of parameters. The model with the smallest value of DIC is the one with a better balance between the model adjustment and complexity. However, this criterion can present some problems, which arise in part from not being fully Bayesian.A typical alternative is the WAIC, proposed by Watanabe (2010), which is fully Bayesian in that it uses the entire posterior distribution. It can be considered as an improvement on the DIC for Bayesian models (Gelman et al, 2014).Several alternative spatial random effects were used in modelling the intensities, namely (i) Common random effect W both for points and marks (ii) Random effect W and its scaled version α_3 W for the points and marks respectively (iii) two independent latent processes W and W_2for thepoints and their marks respectively.Table <ref> shows the values of these two criterions for the models considered for the marked point process. In this case, the model with the best performance was the one that took into account the following factors: * the offset term given by the population density to model the intensity of the point process (offset_1) ; * the covariates to model the mark intensity (number of individuals per residential building(nind_2): the median of the education level (edu_2), the mean age (age_2), and the proportion of registered unemployed people (iefp_2). Here, subscripts 1 and 2 indicate that the corresponding covariate is used in modelling intensity λ_1(s) and λ_2(s)respectively; * two independent latent processes W_1 and W_2 used for points and their marks. Here, p_DIC and p_WAIC are the effective number of parameters, as described in Spiegelhalter et al (2002) and Gelman et al (2014), respectively.It is clear from the table that the model that employs all the covariate information and two independent latent processes, one for points other for marks seems to give the best fit with the model that employsall the covariate information and a single common latent process for the points and marks coming second. Here, we chose the model with lower DIC to continue with these analysis. We also considered anegative binomial distribution for the marks as an alternative to the poisson marks, but these models did not bring gain in terms of DIC. To perform the spatial prediction,we created a regular grid of 1km^2 in the domain. A projection from the mesh to the grid was performed and the resultant maps of the posterior mean of the logarithmic transformation of the intensity of the residential units log(λ_1(s))and the logarithmic of the marks mean are shown in figure <ref>. The plot of the logarithmic transformation of the intensity provides a clearer image about the spatial variation of the residential buildings. As we expected, the highest values are concentrated in Lisboa, Porto, and Algarve regions. The intensity is clearly higher near the coast and lower in the interior of the country.The standard deviations of these fields are plotted in figure <ref>.With these estimates, we can conclude that the average number of unemployed people per residential building is higher in the Grande Porto, Pennsula de Setbal and Alentejo Central regions.§.§ Model validationFor model validation purposes, we chose randomly 26 counties and fitted the model excluding the data from these counties. Figure <ref> gives the 95% credible intervals for the predicted values of unemployment from the model together with the observations and predicted values for these 26 counties.Figure <ref> gives the credible intervals together with observations and estimates for the same 26 chosen counties when all the data are used in fitting the model. Figure <ref> gives thePearson residuals versus fitted values for the 278 counties. Figures <ref> gives the 95% credible intervals, observations and estimates for the NUTS III regions. As is expected, the model gives higher precision estimates at NUTS III regions as compared to county level. §.§ Unemployment estimation for NUTS III regions and countiesThe marked point process model explained in the previous section, projects the sampling survey in space. Thispoint process is a thinned version of the true point patterns of the residential units together with theirmarks across Portugal.Let N_1^*(s) representthe true point patterns of the residential units with intensity λ_1^*(s). Thenλ_1^*(s)=λ_1(s)/P(RU(s)),where, P(RU(s)) is the probability that a residential unit at s is included in the survey. P(RU(s)) should be interpreted asthe proportion of the residential units in any infinitesimal area which is included in the sampling survey. Assume also that N_2^*(s) represent the true intensity of thenumber of unemployed observed in residential unit at location s. then the intensity λ_2^*(s)of this counting process is given byλ_2^*(s)=λ_2(s)/P(D(s)|RU(s)),where the probabilityP(D(s)|RU(s))should be interpreted as the proportion of dwellings in a residential unit which are included in the sampling survey.Target quantities <ref> and <ref> depend on the multiplicative intensity λ_1(s)λ_2(s), which is a thinned version of λ_1^*(s)λ_2^*(s) and this relationship is given by λ_1^*(s)λ_2^*(s)=λ_1(s)λ_2(s)/p(s), wherep(s) =P(RU(s))P(D(s)|RU(s))=P(D(s)), since P(D(s)|RU^c(s))=0. Here, p(s) should be interpreted as the proportion of dwellings that are chosen in the sampling survey. As explained in section <ref>these probabilities areestimated using (<ref>).To define the intensity of the full version of the spatial point process, the knowledge of the sampling probabilities p(s) for whole domainis required. Here, we estimate these probabilities using the kernel method, based on the data given by the sampling survey. This method allows us to generate a spatial prediction for the centers of the cells of the grid, derived from the values of the dwellingss locations. We simulated 1000 values of the predictive posterior distribution of λ_1 and λ_2 for each cell I_j to estimate the target quantities, by simulating samples from the posterior distributions of the model parameters and the latent gaussian markov fields used in the model.Figure <ref> gives the predictive multiplicative intensity functionλ_1(s|)λ_2(s|)/p(s),which will form the basis for calculating the unemployment in any region A, expressed in terms of E(N(A)|) as given in <ref>. Figure <ref> shows the estimates at NUTS III level obtained by this model and the estimates obtained by the direct method, as well as the differences. Indeed, the spatial distribution is very similar in both methods, as is expected. Figure <ref> gives the ratio between the standard deviation of estimates obtained from the direct estimation method and square root ofVar_(W(s), θ|)[E(N(A)|,W(s), θ)],with A representing each of the NUTS III regions. As we can see, the ratio is higher than 1 for the most part of regions, indicating clearly that point referenced method produces estimates with higher precision.Coefficients of variation, given by the ratio of the standard deviation to the mean, were also calculated and there is a clear discrepancy in the methods (see figure <ref>). This map remits to the problem of the small area estimation, where the estimates are not reliable for small areas (areas with low population density). Indeed, in these areas we can see that point referenced method we proposehasbetter performance in terms of coefficients of variation. We also calculated the credible intervals for the posterior mean by NUTS III and compared those with the direct estimates (figure <ref>). We note that the direct estimates are inside the intervals for almost all regions. The highest points correspond to Porto and Lisboa. The figure reveals that there is an underestimation in these regions and an overestimation in the others. This behaviour is probably due to the large differences in the intensity inthese regions and consequent smoothing of intensities across the space. We also give estimation results on a higher resolution, namely for 278 counties of mainland Portugal. Figure <ref> gives comparative results from our model and the direct estimation method. Further comparisons on the standard deviations and coefficients of variation clearlyindicate that point referenced methods we employ give estimates with higher precision at higher resolutions, as expected. § DISCUSSION,CONCLUSIONS AND FURTHER WORK In this study, we employed point referenced spatial models to unemployement estimationmaking use ofthenewly availablegeo-referencedquarterly sample surveys. Unlike the direct method and area level models, our approachtake into account specific information about the families including their spatial location at a very detailed geographical level. Therefore, it is natural that both the methodologies produce estimates at NUTS III level with higher precision than the direct method. It is interesting to see that after applying these complex methodologies the results do not seem to be very different than those obtained by the direct method atNUTS III level. Despite this similarity, thesemethodologies provide important advantages in the small area and official statistics context, particularly deliveringreliable estimates also for much smaller areas. As opposed toareal models proposed in the literature, these models take into account ofthe sampling design, which is important in this context. National Statistical Institutes usually require a great deal of consistency between the estimates obtained at different areal resolutions, but this requirement is not easy to satisfy using the standard small area models. Since this new methodology operates independently from the administrative limits of geographical units, it can provide us with the necessary means to meet this requirement both a consistent and timely fashion. Despite the complexity of this methodology, the computational costs are not high due to the availability of the R-INLA package in the software R. In this work, we do not report on the time dynamics of unemployement. At present, there are 14 quarterly sample surveys with geo-referenced sampling units. As these quarterly data further become available,it may be possible to investigate how spatial variation of unemployment changes over time. This can be done byconsidering space-time marked point processes, in which the latentprocess now is a space time Gaussian process.and by addingtime varying covariates in the model such as a linear or quadratic trends functions in time. It is possible to infer on such models within the INLA platform. Recently, INE has started on a much more ambitiousgeo-referencingmethods by identifying and geo-referencing all the residential units in Portugal. Methods and models that are adequate for these new realitieswill be discussed elsewhere as new data become available.We expect that this methodology will be even more beneficial when time-dynamics will be incorporated, as it is more intuitive that the current results today would act as the prior for the next time point and, additionally, there is a benefit of doing some smoothing in time. § ACKNOWLEDGEMENTSThis work was supported by the project UID/MAT/00006/2013 and the PhD scholarship SFRH/BD/92728/2013 from Fundao para a Cincia e Tecnologia. Instituto Nacional de Estatstica and Centro de Estatstica e Aplicaes da Universidade de Lisboa are the reception institutions. We would like to thank professor Antnia Turkman, Elias Krainski and Paula Pereira for their help.§ NOTE This study is the responsibility of the authors and does not reflect the official opinions of Instituto Nacional de Estatstica. [Baddeley et al (2016)]tex1 Baddeley, A., Rubak, E., Turner, R. (2016) Spatial Point Patterns - Methodology and Applications with R. Chapman and Hall/CRC.[Banerjee et al (2004)]tex2 Banerjee, S., Carlin, B. P., Gelfand, A. E. (2004) Hierarchical Modeling and Analysis for Spatial Data. Chapman and Hall/CRC. [Gelfand et al (2004)]tex3 Gelfand, A. E., Schmidt, A.M., Banerjee, S. and Sirmans, C.F. (2004) Nonstationary multivariate process modeling through spatially varying coreginalization. Test, 13, 263-312. [Blangiardo et al (2015)]tex4 Blangiardo, M., Cameletti, M. (2015) Spatial and Spatio-temporal Bayesian Models with R-INLA. Wiley.[Bonneu (2007)]tex5 Bonneu, F. (2007) Exploring and modeling fire department emergencies with a spatio-temporal marked point process. Case Studies in Business, Industry and Government Statistics, 1, 139-152. [Fuglstad et al (2017)]tex6 Fuglstad, G.-A., Simpson, D., Lindgren, F., and Rue, H. (2017) Constructing Priors that Penalize the Complexity of Gaussian Random Fields. arXiv:1503.00256[Gelman et al (2014)]tex7 Gelman, A., Hwang, J., and Vehtari, A. (2014) Understanding predictive information criteria for Bayesian models. Statistics and Computing, 24, 997-1016.[Gneiting et al (2007)]tex8 Gneiting, T., Raftery, A.E. (2007) Strictly proper scoring rules, prediction, and estimation. Journal American Statistical Associaton. 102, 359-378. [Illian et al (2008)]tex9 Illian, J., Penttinen, A., Stoyan, H.,Stoyan, D. (2008) Statistical analysis and modelling of spatial point patterns. Wiley. Statistics in Practice.[Illian et al.(2012b)Illian, Sørbye, Rue, and Hendrichsen]art588 Illian, J. B., Sørbye, S. H., Rue, H. and Hendrichsen, D. K. (2012) Using INLA to fit a complex point process model with temporally varying effects – a case study. Journal of Environmental Statistics, 30 . [Illian et al (2012)]tex11 Illian, J. B., Sorbye, S. H., Rue, H., (2012)A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA). The Annals of Applied Statistics, 6, 1499-1530.[INE (2014)]tex12 INE (2014) Documento metodolgico do Inqurito ao Emprego.[Krainski et al (2016)]tex13 Krainski, E., Lindgren, F., Simpson, D., Rue, H. (2016) The R-INLA tutorial on SPDE models. http://www.math.ntnu.no/inla/r-inla.org/tutorials/spde/spde-tutorial.pdf [Lindgren et al (2011)]tex14 Lindgren, F., Rue, H., Lindstrom, J. (2011) An explicit link between Gaussian fields and Gaussian Markov random fields: the SPDE approach (with discussion). Journal of Royal Statistical Society Series B, 73, 423-498.[Lopez-Vizcaino et al (2015)]tex15 Lopez-Vizcaino, E., Lombardia, M. J., Morales, D. (2015) Small area estimation of labour force indicators under a multinomial model with correlated time and area effects. Journal of Royal Statistical Society Series A, 178, 535-565.[Martins et al (2013)]tex23 Martins, T., G., Simpson, D., Lindgren, F., Rue, H. (2013) Bayesian computing with INLA: New features. Computational Statistics and Data Analysis, 67, 68-83. [Pereira (2014)]tex16 Pereira, P. (2014) Mtodos Proababilsticos e Estatsticos na Gesto de Fogos Florestais. PhD thesis, FCUL.[Pereira et al (2016)]tex17 Pereira, S., Turkman, F., Correia, L. (2016) Spatio-temporal analysis of regional unemployment rates: A comparison of model based approaches. To appear in Revstat. https://arxiv.org/abs/1704.05767[Rao (2003)]tex18Rao, J.N.K. (2003) Small Area Estimation. New York: Wiley. [Rue et al (2009)]tex24 Rue, H., Martino, S., Chopin, N. (2009) Approximate Bayesian Inference for Latent Gaussian Models Using Integrated Nested Laplace Approximations (with discussion). Journal of the Royal Statistical Society Series B, 71, 319-392. [Rue et al (2017)]tex25 Rue, H., Riebler, A., Sorbye, S. H.,Illian, J. B., Simpson, D. P., Lindgren, F. K. (2017) Bayesian computing with INLA: A review. Annual Reviews of Statistics and Its Applications, 4, 395-421.[Schoenberg (2004)]tex19 Schoenberg, F. P. (2004) Testing separability in spatial-temporal marked point processes. Biometrics, 60, 471-481.[Simpson et al.(2016)Simpson, Illian, Lindgren, Sørbye, and Rue]art583Simpson, D. , Illian, J. ,Lindgren, F. , Sørbye, S. , and Rue, H. . (2016) Going off grid: Computational efficient inference for log-Gaussian Cox processes. Biometrika, 1030,0 1-22, 2016. (doi: 10.1093/biomet/asv064).[Simpson et al.(2017)Simpson, Rue, Riebler, Martins, and Sørbye]art631Simpson, D. P., Rue, H. , Riebler, A. , Martins, T. G. , and Sørbye, S. H. (2017) Penalising model component complexity: A principled, practical approach to constructing priors (with discussion). Statistical Science, 320 ,0 1-28. [Spieghalter et al (2002)]tex21 Spiegelhalter, D. J., Best, N.G., Carlin, B.R., van der Linde, A. (2002) Bayesian measures of model complexity and fit (with discussion). Journal of Royal Statistical Society Series B, 64, 583-639.[Yuan et al (2017)]tex22Y. Yuan, F. E. Bachl, F. Lindgren, D. L. Brochers, J. B. Illian, S. T. Buckland, H. Rue, T. Gerrodette. (2017)Point process models for spatio-temporal distance sampling data. Annals of Applied Statistics. | http://arxiv.org/abs/1706.08320v1 | {
"authors": [
"Soraia Pereira",
"Kamil Feridun Turkman",
"Luis Correia",
"Haavard Rue"
],
"categories": [
"stat.AP"
],
"primary_category": "stat.AP",
"published": "20170626110445",
"title": "Unemployment estimation: Spatial point referenced methods and models"
} |
[ Daniel Selsamstanford Percy Liangstanford David L. Dillstanford stanfordStanford University, Stanford, CADaniel [email protected] Software, Engineering, Verification, Verify, Errors, Bugs, Formal, Math, Mathematics, Certify, Systems0.3in ] Noisy data, non-convex objectives, model misspecification, and numerical instability can all cause undesired behaviors in machine learning systems.As a result, detecting actual implementation errors can be extremely difficult.We demonstrate a methodology in which developers use an interactive proof assistant to both implement their system and to state a formal theorem defining what it means for their system to be correct.The process of proving this theorem interactively in the proof assistant exposes all implementation errors since any error in the program would cause the proof to fail.As a case study, we implement a new system, Certigrad, for optimizing over stochastic computation graphs, and we generate a formal (i.e. machine-checkable) proof that the gradients sampled by the system are unbiased estimates of the true mathematical gradients.We train a variational autoencoder using Certigrad and find the performance comparable to training the same model in TensorFlow. § INTRODUCTION Machine learning systems are difficult to engineer for many fundamental reasons. First and foremost, implementation errors can be extremely difficult to detect—let alone to localize and address—since there are many other potential causes of undesired behavior in a machine learning system. For example, an implementation error may lead to incorrect gradients and so cause a learning algorithm to stall, but such a symptom may also be caused by noise in the training data, a poor choice of model, an unfavorable optimization landscape, an inadequate search strategy, or numerical instability. These other issues are so common that it is often assumed that any undesired behavior is caused by one of them. As a result, actual implementation errors can persist indefinitely without detection.[Theano <cit.> has been under development for almost a decade and yet there is a recent GitHub issue (https://github.com/Theano/Theano/issues/4770) reporting a model for which the loss continually diverges in the middle of training. Only after various experiments and comparing the behavior to other systems did the team agree that it is most likely an implementation error. As of this writing, neither the cause of this error nor the set of models it affects have been determined.] Errors are even more difficult to detect in stochastic programs, since some errors may only distort the distributions of random variables and may require writing custom statistical tests to detect. Machine learning systems are also difficult to engineer because it can require substantial expertise in mathematics (e.g. linear algebra, statistics, multivariate analysis, measure theory, differential geometry, topology) to even understand what a machine learning algorithm is supposed to do and why it is thought to do it correctly. Even simple algorithms such as gradient descent can have intricate justifications, and there can be a large gap between the mechanics of an implementation—especially a highly-optimized one—and its intended mathematical semantics. In this paper, we demonstrate a practical methodology for building machine learning systems that addresses these challenges by enabling developers to find and eliminate implementation errors systematically without recourse to empirical testing.Our approach makes use of a tool called an interactive proof assistant <cit.>, which consists of (a) a programming language, (b) a language to state mathematical theorems, and (c) a set of tools for constructing formal proofs of such theorems.Note: we use the term formal proof to mean a proof that is in a formal system and so can be checked by a machine.In our approach, developers use the theorem language (b) to state a formal mathematical theorem that defines what it means for their implementation to be error-free in terms of the underlying mathematics (e.g.multivariate analysis). Upon implementing the system using the programming language (a), developers use the proof tools (c) to construct a formal proof of the theorem stating that their implementation is correct. The first draft of any implementation will often have errors, and the process of interactive proving will expose these errors systematically by yielding impossible proof obligations.Once all implementation errors have been fixed, the developers will be able to complete the formal proof and be certain that the implementation has no errors with respect to its specification.Moreover, the proof assistant can check the formal proof automatically so no human needs to understand why the proof is correct in order to trust that it is. Figure <ref> illustrates this process. Proving correctness of machine learning systems requires building on the tools and insights from two distinct fields: program verification <cit.>, which has aimed to prove properties of computer programs, and formal mathematics <cit.>, which has aimed to formally represent and generate machine-checkable proofs of mathematical theorems.Both of these fields make use of interactive proof assistants, but the tools, libraries and design patterns developed by the two fields focus on different problems and have remained largely incompatible. While the methodology we have outlined will be familiar to the program verification community, and while reasoning formally about the mathematics that underlies machine learning will be familiar to the formal mathematics community, proving such sophisticated mathematical properties of large (stochastic) software systems is a new goal and poses many new challenges.To explore these challenges and to demonstrate the practicality of our approach, we implemented a new machine learning system, Certigrad, for optimizing over stochastic computation graphs <cit.>.Stochastic computation graphs extend the computation graphs that underly systems like TensorFlow <cit.> and Theano <cit.> by allowing nodes to represent random variables and by defining the loss function for a graph to be the expected value of the sum of the leaf nodes over the stochastic choices. See Figure <ref> for an example of a stochastic computation graph. We implement our system in the Lean Theorem Prover <cit.>, a new interactive proof assistant still under active development for which the integration of programming and mathematical reasoning is an ongoing design goal. We formally state and prove functional correctness for the stochastic backpropagation algorithm: that the sampled gradients are indeed unbiased estimates of the gradients of the loss function with respect to the parameters.We note that provable correctness need not come at the expense of computational efficiency: proofs need only be checked once during development and they introduce no runtime overhead.Although the algorithms we verify in this work lack many optimizations, most of the running time when training machine learning systems is spent multiplying matrices, and we are able to achieve competitive performance simply by linking with an optimized library for matrix operations (we used Eigen <cit.>).[Note that the validity of our theorem becomes contingent on Eigen's matrix operations being functionally equivalent to the versions we formally proved correct.]To demonstrate practical feasibility empirically, we trained an Auto-Encoding Variational Bayes (AEVB) model <cit.> on MNIST using ADAM <cit.> and found the performance comparable to training the same model in TensorFlow.We summarize our contributions: * We present the first application of formal (i.e. machine-checkable) proof techniques to developing machine learning systems. * We describe a methodology that can detect implementation errors systematically in machine learning systems. * We demonstrate that our approach is practical by developing a performant implementation of a sophisticated machine learning system along with a machine-checkable proof of correctness. § MOTIVATION When developing machine learning systems, many program optimizations involve extensive algebraic derivations to put mathematical expressions in closed-form make a broader statement: many program optimizations are based on mathematical equivalences, e.g., closed form, but the mathematical connection is often undocumented and implicit. For example, suppose you want to compute the following quantity efficiently: ∫_x 𝒩(x ; μ, Diag(σ^2)) log𝒩(x ; 0, I_n × n). You expand the density functions, grind through the algebra by hand and eventually derive the following closed form expression:-1/2[∑_i=1^n ( σ_i^2 - μ_i^2 ) + n log 2 π] You implement a procedure to compute this quantity and include it as part of a larger program, but when you run your first experiment, your plots are not as encouraging as you hoped. After ruling out many other possible explanations, you eventually decide to scrutinize this procedure more closely. You implement a naïve Monte Carlo estimator for the quantity above, compare it against your procedure on a few random inputs and find that its estimates are systematically biased. What do you do now?If you re-check your algebra carefully, you might notice that the sign of μ_i^2 is wrong, but wouldn't it be easier if the compiler checked your algebra for you and found the erroneous step? Or better yet, if it did the algebra for you in the first place and could guarantee the result was error-free?also, people might not appreciate the pervasiveness of the problem from this one example; they might think it's just localized to math, and they'll assume that TensorFlow people will get it right, and they just have to write models; I guess this is something we talked about - who is the audience? I think this kind of thing occurs more for ML people developing new algorithms rather than people just using TensorFlow to solve some applied problem; might be worth adding one sentence explaining the regime we're interested in Actually, this example comes from AEVB where people needed to do this derivation to find their model.§ BACKGROUND: THE LEAN THEOREM PROVERTo develop software systems with no implementation errors, we need a way to write computer programs, mathematical theorems, and mathematical proofs all in the same language. All three capabilities are provided by the new interactive proof assistant Lean <cit.>.Lean is an implementation of a logical system known as the Calculus of Inductive Constructions <cit.>, and its design is inspired by the better-known Coq Proof Assistant <cit.>. Our development makes use of certain features that are unique to Lean, but most of what we present is equally applicable to Coq, and to a lesser extent, other interactive theorem provers such as Isabelle/HOL <cit.>.To explain and motivate the relevant features of Lean, we will walk through applying our methodology to a toy problem: writing a program to compute the gradient of the softplus function. We can write standard functional programs in Lean, such as softplus:def splus (x : ℝ) : ℝ := log (1 + exp x) We can also represent more abstract operations such as integrals and gradients:∫ (f : ℝ → ℝ) : ℝ∇ (f : ℝ → ℝ) (θ : ℝ) : ℝHere the intended meaning of ∫ f is the integral of the function f over all of ℝ, while the intended meaning of ∇ f θ is the gradient (i.e. the derivative) of the function f at the point θ. Figure <ref> shows how to represent common idioms of informal mathematics in our formal representation; note that whereas some of the informal examples are too ambiguous to interpret without additional information, the Lean representation is always unambiguous.We can represent mathematical theorems in Lean as well. For example, we can use the following predicate to state that a particular function f is differentiable at a point θ:is_diff (f : ℝ → ℝ) (θ : ℝ) : PropThe fact that the return type of is_diff is Prop indicates that it is not a computer program to be executed but rather that it represents a mathematical theorem.We can also state and assume basic properties about the gradient, such as linearity:∀ (f g : ℝ → ℝ) (θ : ℝ), is_diff f θ ∧ is_diff g θ → ∇ (f + g) θ = ∇ f θ + ∇ g θ Returning to our running example, we can state the theorem that a particular function f computes the gradient of the softplus function:def gsplus_spec (f : ℝ → ℝ) : Prop := ∀ x, f x = ∇ splus xSuppose we try to write a program to compute the gradient of the softplus function as follows:def gsplus (x : ℝ) : ℝ := 1 / (1 + exp x)The application gsplus_spec gsplus represents the proposition that our implementation gsplus is correct, i.e. that it indeed computes the gradient of the softplus function for all inputs.We can try to formally prove theorems in Lean interactively:theorem gsplus_correct : gsplus_spec gsplus := @lean:@ ⊢ gsplus_spec gsplus !user:! expand_def gsplus_spec, @lean:@ ⊢ ∀ x, gsplus x = ∇ splus x !user:! introduce x, @lean:@ x : ℝ ⊢ gsplus x = ∇ splus x !user:! expand_defs [gsplus, splus], @lean:@ x : ℝ ⊢ 1 / (1 + exp x) = ∇ (λ x, log (1 + exp x)) x !user:! simplify_grad, @lean:@ x : ℝ ⊢ 1 / (1 + exp x) = exp x / (1 + exp x) The lines beginning with @lean@ show the current state of the proof as displayed by Lean, which at any time consists of a collection of goals of the form assumptions ⊢ conclusion. Every line beginning with !user! invokes a tactic, which is a command that modifies the proof state in some way such that Lean can automatically construct proofs of the original goals given proofs of the new ones. Here the simplify_grad tactic rewrites exhaustively with known gradient rules—in this case it uses the rules for log, exp, addition, constants, and the identity function.The final goal is clearly not provable, which means we have found an implementation error in gsplus. Luckily the goal tells us exactly what gsplus x needs to return: gsplus x = exp x / (1 + exp x). Once we fix the implementation of gsplus, the proof script that failed before now succeeds and generates a machine-checkable proof that the revised gsplus is bug-free. Note that we need not have even attempted to implement gsplus before starting the proof, since the process itself revealed what the program needs to compute. We will revisit this phenomenon in <ref>.In the process of proving the theorem, Lean constructs a formal proof certificate that can be automatically verified by a small stand-alone executable, whose soundness is based on a well-established meta-theoretic argument embedding the core logic of Lean into set theory, and whose implementation has been heavily scrutinized by many developers.Thus no human needs to be able to understand why a proof is correct in order to trust that it is.[This appealing property can be lost when an axiom is assumed that is not true. We discuss this issue further in <ref>.]Although we cannot execute functions such as gsplus directly in the core logic of Lean (since a real number is an infinite object that cannot be stored in a computer), we can execute the floating-point approximation inside Lean's virtual machine:vm_eval gsplus π – answer: 0.958576 The writing style wavers between general ('we can prove theorems in Lean') and specific ('we fix this bug'), which is awkward; I'd just specialize to the current situation since it reads better § CASE STUDY: CERTIFIED STOCHASTIC COMPUTATION GRAPHS Stochastic computation graphs are directed acyclic graphs in which each node represents a specific computational operation that may be deterministic or stochastic <cit.>. The loss function for a graph is defined to be the expected value of the sum of the leaf nodes over the stochastic choices. Figure <ref> shows the stochastic computation graph for a simple variational autoencoder.Using our methodology, we developed a system, Certigrad, which allows users to construct arbitrary stochastic computation graphs out of the primitives that we provide. The main purpose of the system is to take a program describing a stochastic computation graph and to run a randomized algorithm (stochastic backpropagation) that, in expectation, provably generates unbiased samples of the gradients of the loss function with respect to the parameters. §.§ Overview of Certigrad We now briefly describe the components of Certigrad, some of which have no analogues in traditional software systems.[The complete development can be found at <www.github.com/dselsam/certigrad>.]Mathematics libraries. There is a type that represents tensors of a particular shape, along with basic functions (e.g.exp, log) and operations (e.g. the gradient, the integral). There are assumptions about tensors (e.g.gradient rules for exp and log), and facts that are proved in terms of those assumptions (e.g. the gradient rule for softplus). There is also a type that represents probability distributions over vectors of tensors, that can be reasoned about mathematically and that can also be executed procedurally using a pseudo-random number generator.Implementation. There is a data structure that represents stochastic computation graphs, as well as an implementation of stochastic backpropagation. There are also functions that optimize stochastic computation graphs in various ways (e.g. by integrating out parts of the objective function), as well as basic utilities for training models (e.g. stochastic gradient descent).Specification. There is a collection of theorem statements that collectively define what it means for the implementation to be correct. For Certigrad, there is one main theorem that states that the stochastic backpropagation procedure yields unbiased estimates of the true mathematical gradients. There are also other theorems that state that individual graph optimizations are sound.Proof. There are many helper lemmas to decompose the proofs into more manageable chunks, and there are tactic scripts to generate machine-checkable proofs for each of the lemmas and theorems appearing in the system. There are also tactic programs to automate certain types of reasoning, such as computing gradients or proving that functions are continuous.Optimized libraries. While the stochastic backpropagation function is written in Lean and proved correct, we execute the primitive tensor operations with the Eigen library for linear algebra. There is a small amount of C++ code to wrap Eigen operations for use inside Lean's virtual machine. doesn't emphasize the general story which is that there are general libraries; the glue code isn't that importantThe rest of this section describes the steps we took to develop Certigrad, which include sketching the high-level architecture, designing the mathematics libraries, stating the main correctness theorem and constructing the formal proof. Though many details are specific to Certigrad, this case study is designed to illustrate our methodology and we expect other projects will follow a similar process. Note: Certigrad supports arbitrarily-shaped tensors, but doing so introduces more notational complexity than conceptual difficulty and so we simplify the presentation that follows by assuming that all values are scalars. §.§ Informal specification The first step of applying our methodology is to write down informally what the system is required to do. Suppose g is a stochastic computation graph with n nodes and (to simplify the notation) that it only takes a single parameter θ. Then g, θ together define a distribution over the values at the n nodes (X_1, …, X_n). Let cost(g, X_1:n) be the function that sums the values of the leaf nodes.Our primary goal is to write a (stochastic) backpropagation algorithm bprop such that for any graph g,𝔼_g, θ[ bprop(g, θ, X_1:n)] = ∇_θ( 𝔼_g, θ[ cost(g, X_1:n) ] )While this equation may seem sufficient to communicate the specification to a human with a mathematical background, more precision is needed to communicate it to a computer.The next step is to formalize the background mathematics, such as real numbers (tensors) and probability distributions, so that we can state a formal analogue of Equation <ref> that the computer can understand.Although we believe it will be possible to develop standard libraries of mathematics that future developers can use off-the-shelf, we needed to develop the mathematics libraries for Certigrad from scratch. §.§ Designing the mathematics librariesWhereas in traditional formal mathematics the goal is to construct mathematics from first principles <cit.>, we need not concern ourselves with foundational issues and can simply assume that standard mathematical properties hold. For example, we can assume that there is a type ℝ of real numbers without needing to construct them (e.g. from Cauchy sequences), and likewise can assume there is an integration operator on the reals ∫ (f : ℝ → ℝ) : ℝ that satisfies the well-known properties without needing to construct it either (e.g. from Riemann sums).Note that axioms must be chosen with great care since even a single false axiom (perhaps caused by a single missing precondition) can in principle allow proving any false theorem and so would invalidate the property that all formal proofs can be trusted without inspection.[For example, the seemingly harmless axiom ∀ x, x/x = 1 without the precondition x ≠ 0 can be used to prove the absurdity (0 = 0 * 1 = 0 * (0 / 0) = (0 * 0) / 0 = 0 / 0 = 1). If a system assumes this axiom, then a formal proof of correctness could not be trusted without inspection since the proof may exploit this contradiction.]However, there are many preconditions that appear in mathematical theorems, such as integrability, that are almost always satisfied in machine learning contexts and which most developers ignore.Using axioms that omit such preconditions will necessarily lead to proving theorems that are themselves missing the corresponding preconditions, but in practice a non-adversarial developer is extremely unlikely to accidentally construct vacuous proofs by exploiting these axioms. For the first draft of our system, we purposely omitted integrability preconditions in our axioms to simplify the development. Only later did we make our axioms sound and propagate the additional preconditions throughout the system so that we could fully trust our formal proofs.Despite the convenience of axiomatizing the mathematics, designing the libraries was still challenging for two reasons. First, there were many different ways to formally represent the mathematical objects in question, and we needed to experiment to understand the tradeoffs between the different representations. Second, we needed to extend several traditional mathematical concepts to support reasoning about executable computer programs. The rest of this subsection illustrates these challenges by considering the problem we faced of designing a representation of probability distributions for Certigrad.Representing probability distributions. Our challenge is to devise a sufficiently abstract representation of probability distributions that satisfies the following desiderata: we can reason about the probability density functions of continuous random variables, we have a way to reason about arbitrary deterministic functions applied to random variables, we can execute a distribution procedurally using a pseudo-random number generator (RNG), the mathematical and procedural representations of a distribution are guaranteed to correspond, and the mathematics will be recognizable to somebody familiar with the informal math behind stochastic computation graphs.We first define types to represent the mathematical and procedural notions of probability distribution. For mathematics, we define a Func n to be a functional that takes a real-valued function on ℝ^n to a scalar:def Func (n : ℕ) : Type := ∀ (f : ℝ^n → ℝ), ℝThe intended semantics is that if p : Func n represents a distribution on ℝ^n, then p f is the expected value of f over p, i.e. 𝔼_x ∼ p [ f(x) ].For sampling, we define an Prog n to be a procedure that takes an RNG and returns a vector in ℝ^n along with an updated RNG:def Prog (n : ℕ) : Type := RNG → ℝ^n × RNG We also assume that there are primitive (continuous) distributions (PrimDist := Func 1 × Prog 1) that consist of a probability density function and a corresponding sampling procedure. In principle, we could construct all distributions from uniform variates, but for expediency, we treat other well-understood distributions as primitive, such as the Gaussian (gauss μ σ : PrimDist).Finally, we define a type of distributions (Dist n) that abstractly represents programs that may mix sampling from primitive distributions with arbitrary deterministic computations. A Dist n can be denoted to a Func n (with the function E) to reason about mathematically, and to an Prog n (with the function run) to execute with an RNG. E and run need to be introduced more deliberately; saying 'the function E' presumes the reader already knows about it, but the reader's seeing it for the first time, which is jarring; defining their types would also be helpfulFor readers familiar with functional programming, our construction is similar to a monad.We allow three ways of constructing a Dist n, corresponding to sampling from a primitive distribution (sample), returning a value deterministically (det), and composing two distributions (compose):sample ((pdf, prog) : PrimDist) : Dist 1 det (xs : ℝ^n) : Dist n compose (d_1 : Dist m) (d_2 : ℝ^m → Dist n) : Dist nThe mathematical semantics of all three constructors are straightforward:E (sample (pdf, prog)) f = ∫ (λ x, pdf x * f x) E (det xs) f = f xs E (compose d_1 d_2) f = E d_1 (λ x, (E (d_2 x) f))as are the procedural semantics:run (sample (pdf, prog)) rng = prog rng run (det xs) rng = (xs, rng) run (compose d_1 d_2) rng = let (x, rng') := run d_1 rng in run (d_2 x) rng' We have defined E and run to correspond; we consider a stochastic program correct if we can prove the relevant theorems about its Func denotation, and we sample from it by passing an RNG to its Prog denotation. §.§ Formal specification With the background mathematics in place, the next step is to write down the formal specification itself. First, we design types for every other object and function appearing in the informal description. To start, we need a type SCG n to represent stochastic computation graphs on n nodes, and a function SCG.to_dist that takes an SCG n and a scalar parameter θ to a distribution over n real numbers (Dist n). We also need a function cost that takes a graph and the values at each of its nodes and sums the values at the leaf nodes. Figure <ref> provides the full types of all objects that will appear in the specification.Now we can write down a type-correct analogue of the informal specification presented in Equation <ref>: def bprop_spec (bprop : ∀ n, SCG n → ℝ → ℝ^n → ℝ) : Prop := ∀ (n : ℕ) (g : SCG n) (θ : ℝ), E (SCG.to_dist g θ) (λ xs, bprop g θ xs) = ∇ (λ θ, E (SCG.to_dist g θ) (λ xs, cost g xs)) θGiven the mathematics libraries, implementing the other objects and functions appearing in the specification such as SCG n and SCG.to_dist is straightforward functional programming. §.§ Interactive proof While conventional wisdom is that one would write their program before trying to prove it correct, the interactive proof process provides so much helpful information about what the system needs to do that we began working on the proof immediately after drafting the specification.We split the proof into two steps.First, we implemented the simplest possible function that satisfied the specification (that only computed the gradient for a single parameter at a time and did not memoize at all) and proved that correct. Second, we implemented a more performant version (that computed the gradient for multiple parameters simultaneously using memoization) and proved it equivalent to the first one.For the first step, we started with a placeholder implementation that immediately returned zero and let the interactive proof process guide the implementation. Whenever the proof seemed to require induction on a particular data structure, we extended the program to recurse on that data structure; whenever the proof showed that a branch of the program needed to return a value with a given expectation, we worked backwards from that to determine what value to return. Proving the first step also exposed errors in our specification in the form of missing preconditions. For the specification to hold, we needed to make additional assumptions about the graph, e.g. that the identifier for each node in the graph is unique, and that each leaf node is a scalar (WellFormed g). We also needed to assume a generalization of the differentiability requirement mentioned in <cit.>, that a subset of the nodes determined by the structure of the graph must be differentiable no matter the result of any stochastic choices (GradsExist g θ).For the second step, we wrote the memoizing implementation before starting the proof and used the process of proving to test and debug it. Although the code for memoizing was simple and short, we still managed to make two implementation errors, one conceptual and one syntactic. Luckily the process of proving necessarily exposes all implementation errors, and in this case made it clear how to fix both of them.We completed the main proof of correctness before proving most of the lemmas that the proof depends on, but the lemmas turned out to be true (except for a few missing preconditions) and so proving them did not expose any additional implementation errors.We also completed the main proof while our axioms were still unsound (see <ref>). When we made our axioms sound and propagated the changes we found that our specification required two additional preconditions: that all functions that are integrated over in the theorem statement are indeed integrable (IntegralsExist g θ), and that the many preconditions needed for pushing the gradient over each integral in the expected loss are satisfied (CanDiffUnderInts g θ). However, tracking these additional preconditions did not lead to any changes in our actual implementation.Figure <ref> shows the final specification. §.§ Optimizations We can also use our methodology to verify optimizations that involve mathematical reasoning.When developing machine learning models, one often starts with an easy-to-understand model that induces a gradient estimator with unacceptably high variance, and does informal mathematics by hand to derive a new model that has the same objective function but that induces a better gradient estimator. In our approach, the user can write both models and use the process of interactive proving toconfirm that they induce the same objective function. Common transformations can be written once and proved correct so that users need only write the first model and the second can be derived and proved equivalent automatically.As part of Certigrad, we wrote a program optimization that integrates out the KL-divergence of the multivariate isotropic Gaussian distribution and we proved once and for all that the optimization is sound. We also verified an optimization that reparameterizes a model so that random variables do not depend on parameters (and so need not be backpropagated through). Specifically, the optimization replaces a node that samples from 𝒩(μ, Diag(σ^2)) with a graph of three nodes that first samples from 𝒩(0, I_n × n) and then scales and shifts the result according to σ and μ respectively. We applied these two transformations in sequence to a naïve variational-autoencoder to yield the Auto-Encoding Variational Bayes (AEVB) estimator <cit.>. §.§ Verifying backpropagation for specific models Even though we proved that bprop satisfies its formal specification (bprop_spec), we cannot be sure that it will compute the correct gradients for a particular model unless we prove that the model satisfies the preconditions of the specification. Although some of the preconditions are technically undecidable, in practice most machine learning models will satisfy them all for simple reasons. We wrote a (heuristic) tactic program to prove that specific models satisfy all the preconditions and used it to verify that bprop computes the correct gradients for the AEVB model derived in <ref>. §.§ Running the systemWe have proved that our system is correct in an idealized mathematical context with infinite-precision real numbers. To actually execute the system we need to replace all real numbers in the program with floating-point numbers. Although doing so technically invalidates the specification and can introduce numerical instability in some cases, this class of errors is well understood <cit.>, could be ruled out as well in principle <cit.> and is conceptually distinct from the algorithmic and mathematical errors that our methodology is designed to eliminate.To improve performance, we also replace all tensors with an optimized tensor library (Eigen).This approximation could introduce errors into our system if for whatever reason the Eigen methods we use are not functionally equivalent to ones we formally reason about; of course developers could achieve even higher assurance by verifying their optimized tensor code as well. §.§ Experiments Certigrad is efficient.As an experiment, we trained an AEVB model with a 2-layer encoding network and a 2-layer decoding network on MNIST using the optimization procedure ADAM <cit.>, and compared both the expected loss and the running time of our system at each epoch against the same model and optimization procedure in TensorFlow, both running on 2 CPU cores. We found that the expected losses decrease at the same rate, and that Certigrad takes only 7% longer per epoch (Figure <ref>).§ DISCUSSION Our primary motivation is to develop bug-free machine learning systems, but our approach may provide significant benefits even when building systems that need not be perfect.Perhaps the greatest burden software developers must bear is needing to fully understand how and why their system works, and we found that by formally specifying the system requirements we were able to relegate much of this burden to the computer.Not only were we able to synthesize some fragments of the system (<ref>), we were able to achieve extremely high confidence that our system was bug-free without needing to think about how all the pieces of the system fit together.In our approach, the computer—not the human—is responsible for ensuring that all the local properties that the developer establishes imply that the overall system is correct. Although using our methodology to develop Certigrad imposed many new requirements and increased the overall workload substantially, we found that on the whole it made the development process less cognitively demanding.There are many ways that our methodology can be adopted incrementally. For example, specifications need not cover functional correctness, not all theorems need to be proved, unsound axioms can be used that omit certain preconditions, and more traditional code can be wrapped and axiomatized (as we did with Eigen).When developing Certigrad we pursued the ideal of a complete, machine-checkable proof of functional correctness, and achieved an extremely high level of confidence that the system was correct. However, we realized many of the benefits of our methodology—including partial synthesis and reduced cognitive demand—early in the process before proving most of the lemmas. Although we could not be certain that we had found all of the bugs before we made our axioms sound and filled in the gaps in the formal proofs, in hindsight we had eliminated all bugs early in the process as well. While a pure version of our methodology may already be cost-effective for high-assurance applications, we expect that pragmatic use of our methodology could yield many of the benefits for relatively little cost and could be useful for developing a wide range of machine learning systems to varying standards of correctness. § ACKNOWLEDGMENTSWe thank Jacob Steinhardt, Alexander Ratner, Cristina White, William Hamilton, Nathaniel Thomas, and Vatsal Sharan for providing valuable feedback on early drafts. We also thank Leonardo de Moura, Tatsu Hashimoto, and Joseph Helfer for helpful discussions. This work was supported by Future of Life Institute grant 2016-158712. icml2017 | http://arxiv.org/abs/1706.08605v1 | {
"authors": [
"Daniel Selsam",
"Percy Liang",
"David L. Dill"
],
"categories": [
"cs.SE",
"cs.AI"
],
"primary_category": "cs.SE",
"published": "20170626213002",
"title": "Developing Bug-Free Machine Learning Systems With Formal Mathematics"
} |
Photometry of IP V2069 Cyg KozhevnikovAstronomical Observatory, Ural Federal University, Lenin Av. 51, Ekaterinburg620083, Russia e-mail: [email protected] To obtain the spin period of the white dwarf in the intermediate polar V2069 Cyg with high precision, we fulfilled extensive photometry. Observations were performed within 32 nights, which have a total duration of 119 hours and cover 15 months. We found the spin period of the white dwarf, which is equal to 743.406 50±0.000 48 s. Using our precise spin period, we derived the oscillation ephemeris with a long validity of 36 years. This ephemeris and the precise spin period can be used for future investigations of spin period changes of the white dwarf in V2069 Cyg. In addition, for the first time we detected the sideband oscillation with a period of 764.5125±0.0049 s. The spin and sideband oscillations revealed unstable amplitudes both in a time-scale of days and in a time-scale of years. On average, the semi-amplitude of the spin oscillation varied from 17 mmag in 2014 to 25 mmag in 2015. The semi-amplitude of the sideband oscillation varied from 12 mmag in 2014 to an undetectable level of less than 7 mmag in 2015. In a time-scale of years, the optical spin pulse profile revealed essential changes from an asymmetric double-peaked shape in 2014 to a quasi-sinusoidal shape in 2015. Such drastic changes of the optical spin pulse profile seem untypical of most intermediate polars and, therefore, are of great interest. The pulse profile of the sideband oscillation was quasi-sinusoidal. Moreover, we note that V2069 Cyg possesses strong flickering with a peak-to-peak amplitude of 0.4–0.6 mag. § INTRODUCTION Intermediate polars (IPs), a sub-class of cataclysmic variables (CVs), are interacting binary stars, in which accretion occurs onto a magnetic white dwarf. The magnetic white dwarf spins non-synchronously with the orbital period of the system and therefore produces rapid coherent oscillations with the spin period. The spin oscillation can be observed both in optical light and in X-rays. In optical light, the spin period often appears together with the beat period, 1/P_ beat = 1/P_ spin - 1/P_ orb. This oscillation is named the orbital sideband. Normally, the orbital sideband is produced due to the reprocessing of X-rays at some part of the system that rotates with the orbital period. This part can be the secondary star or hot spot in the accretion disc. In rare cases, the orbital sideband can be produced due to disc-overflow accretion. Then, the orbital sideband can be observed in X-rays <cit.>. Other orbital sidebands such as ω-2Ω and ω+Ω, where ω=1/P_ spin andΩ=1/P_ orb,can be additionally produced from the amplitude modulation with the orbital period<cit.>. A review of IPs is presented in <cit.>.Due to a large white dwarf moment of inertia, short-period oscillations seen in IPs show a high degree of coherence. This high degree of coherence suggests that the spin period of the white dwarf can be measured with very high precision if the observational coverage is long. The precise knowledge of the spin period is important for several reasons. Firstly, at a practical level, a precise spin ephemeris allows us to phase new X-ray data with optical data (e.g., the IP Home Page, https://asd.gsfc.nasa.gov/Koji.Mukai/iphome/iphome.html).Secondly, the precise spin period and oscillation ephemeris make it possible to perform an observational test of spin equilibrium from direct measurements of the spin period or from pulse-arrival time variations. This is an important task because many theoretical works assume that IPs are in spin equilibrium <cit.>. This, however, is questionable, because only one IP, namelyFO Aqr, really proves the spin equilibrium due to alternating spin-up and spin-down <cit.>. Other rare IPs with known spin rates reveal continuous spin-up or spin-down<cit.>. Such data are equally important because they allow us to understand angular momentum flows within the binary <cit.>. Moreover, the oscillation ephemeris allows one to see if any orbital variations were present in the pulse arrival time.<cit.> identified the X-ray source RX J2123.7+4217 with a new CV. This CV was subsequently called V2069 Cyg. Although, by virtue of the hard X-ray spectrum,supposed that this CV belongs to the IP class, their brief photometric observations revealed no short-periodic oscillations typical of IPs. Only 14 years later <cit.> proved that this assumption is correct due to detection of the oscillation with a period of 743.2± 0.4 s, which was observed both in X-rays and in optical light. Shortly afterwards this oscillation was independently confirmed both in X-rays (743.2±0.9 s, ; 743.1±0.6 s, ) and in optical light (743.38±0.25 s, ). Thus, because all short periods measured by different authors are compatible with each other, the IP nature of V2069 Cyg seems undoubted. However, due to insufficient observational coverage, all these periods are of low precision and are not suitable to investigate changes of the spin period of the white dwarf in future or to phase different observations. Indeed, if we imagine an oscillation ephemeris with the spin period, which is measured by , then the formal validity of this ephemeris will be only 26 days.To measure the spin period with high precision and uncover other properties of V2069 Cyg, we performed extensive photometric observations within 32 nights, which have a total duration of 119 hours and cover 15 months. In this paper we present the results obtained from these observations. § OBSERVATIONSIn the observations of V2069 Cyg we used a multi-channel photometer with photomultiplier tubes. This photometer allows us to make brightness measurements of two stars and the sky background simultaneously. The design of the photometer and its noise analysis is presented in <cit.>. The photometer is attached to the 70-cm telescope at Kourovka observatory, Ural Federal University. Advantages of this photometer in observations of IPs were proved in our old observations of V709 Cas where we discovered optical oscillations for the first time <cit.>. Later we incorporated a CCD guiding system into the photometer. Due to this guiding system, the photometer and telescope can operate automatically under computer control. This facilitates the obtaining of long continuous light curves. In addition, the precise automatic guiding improves the accuracy of brightness measurements. Such continuous light curves, which are obtained during a few tens of nights and are spread over a year, allow us to achieve very precise oscillation periods of IPs <cit.>. Because V2069 Cyg is a faint star of 16 mag and is invisible by eye, to centre this star in the photometer diaphragm (16 arcsec), we used a nearby reference star and computer-controlled step motors of the telescope. The diaphragm for the comparison star was the same. However, to reduce the photon noise caused by the sky background, we measured the sky background through a diaphragm of 30 arcsec.Photometric data of V2069 Cyg were obtained in white light (approximately 3000–8000 Å). The time resolution was equal to 4 s. Although such a time resolution seems excessively short for the expected oscillation periods, it allows us to fill gaps in observations more accurately and decreases the errors of the periods.The photometric observations of V2069 Cyg were obtainedin 2014 August–November over 16 nights and in 2015 August–November also over 16 nights. A journal of the observations is presented in Table <ref>. This table contains BJD_ TDB, which is theBarycentric Julian Date in the Barycentric Dynamical Time (TDB) standard. TDB is a uniform time and, therefore, is preferential. We calculated BJD_ TDB by using the online-calculator (http://astroutils.astronomy.ohio-state.edu/time/), which is described in <cit.>, and checked these calculations by using the BARYCEN routine in the 'aitlib' IDL library of the University of Tübingen (http://astro.uni-tuebingen.de/software/idl/aitlib/). One can change our BJD_ TDB into BJD_ UTC, the Barycentric Julian Date in the Coordinated Universal Time (UTC) standard, by subtracting67 s in 2004 and 68 s in 2015 <cit.>. The comparison star is USNO-A2.0 1275-15564230. It has B=14.9 mag and B-R=0.8 mag. Its colour index is similar to the colour index of V2069 Cyg, B-R=0.5 mag. This reduces the influence of differential extinction. The obtained differential light curves possess the significant photon noise because of the low brightness of V2069 Cyg and high sky background. Fig. <ref> presents four longest differential light curves obtained in 2014 and four longest differential light curves obtained in 2015, with magnitudes averaged over 40-s time intervals. The number of points in each of these light curves is in the range 351–685. According to the pulse counts of the two stars and sky background, the photon noise of these light curves (rms) is 0.02–0.04 mag.§ ANALYSIS AND RESULTS Because our multichannel photometer allows us to obtain evenly spaced data, we mainly use the classical Fourier analysis, which seems preferential in comparison with numerous methods appropriate to unevenly spaced data <cit.>. For the analysis of periodic oscillations, using a fast Fourier transform (FFT) algorithm, we calculate individual amplitude spectra, average power spectra and power spectra of the data incorporated into common time series. Before applying a FFT routine, we eliminate low-frequency trends from individual light curves by subtraction of a first- or second-order polynomial fit. This is a usual procedure in Fourier analysis and prevents discontinuity of data. This procedure does not affect high frequencies. In our previous works <cit.> one can find details of our methods of analysis.The longest differential light curves of V2069 Cyg presented in Fig. <ref> show obvious flickering. Although flickering is typical of all types of CVs, the flickering power in V2069 Cyg seems noticeably stronger than the flickering power, which we observed in other IPs using the same technique. From Fig. <ref>, we estimate the flickering peak-to-peak amplitude equal to 0.4–0.6 mag whereas V709 Cas, V515 And and V647 Cyg revealedtheir flickering peak-to-peak amplitudes of less than 0.4 mag <cit.>.Spin oscillations of many IPs are directly visible in the light curves <cit.>. In contrast, the oscillation of V2069 Cyg with a period of about 743 s, which is detected in X-rays and in optical light and obviously corresponds to the spin period <cit.>, is inconspicuous in the light curves (Fig. <ref>). In addition to the large flickering power, the low amplitude of the spin oscillation can be the natural reason for this invisibility. To detect this oscillation, at first we calculated amplitude spectra of individual light curves. The amplitude spectra of four longest individual light curves of 2014 and four longest individual light curves of 2015 are presented in Fig. <ref>. As seen, these amplitude spectra clearly show peaks corresponding to the spin period only in 2015. Probably, in 2014 the spin oscillation had lesser amplitude and, therefore, was difficult to detect. One can also note that the amplitude spectra of 2014 show occasional peaks of the first harmonic of the spin period, whereas these peaks are inconspicuous in the amplitude spectra of 2015.Fig. <ref> presents the average power spectra calculated by the weighted averaging of the power spectra of the individual light curves, which are longer than 3 h. As seen, the spin oscillation is detectable both in the data of 2014 and in the data of 2015, but the amplitude of the spin oscillation in 2014 is somewhat lesser than in 2015. In addition, the peak of the spin oscillation in 2014 is accompanied by an additional peak on the left. Obviously, this additional peak is caused by the sideband oscillation. The intermittent presence of this peak in the amplitude spectra of 2014, which strongly changes its amplitude from night to night (Fig. <ref>), makes the spin oscillation difficult to detect in the amplitude spectra of the individual light curves of 2014. This masking effect becomes stronger because the spin oscillation in itself changes its amplitude from night to night. This is also seen in Fig. <ref>. Moreover, from Fig. <ref>,we conclude that in 2015 the sideband oscillation is not detectableat all and, therefore, does not interfere with detection of the spin oscillation. One can also note the presence of the first harmonic of the spin oscillation in 2014.This means that the spin pulse profile is changeable from year to year, namely this profile is non-sinusoidal in 2014 and quasi-sinusoidal in 2015. This is also consistent with the occasional peaks of the first harmonic seen in the individual amplitude spectra of 2014 (Fig. <ref>). The average power spectra shown in Fig. <ref> presents the overview of the periodic oscillationsin V2069 Cyg and of the behaviour of their amplitudes. These power spectra, however, do not allow us to find precise oscillation periods because of their low frequency resolution.Therefore we analysed data incorporated into common time series, in which the gaps due to daylight and poor weather are filled with zeros. Such power spectra possess much higher frequency resolution. Fig. <ref> shows the power spectra of two common time series containing the data of 2014 and the data of 2015 near the frequency of the spin oscillation.As seen, the spin oscillation displays the principal peaks and one-day aliases, which are apparent from the comparison of these power spectra and the window functions shown in the insets. Most of small peaks visible in the immediate proximity of the principal peaks and one-day aliases also coincide in frequency with the corresponding small peaks of the window functions. This means that the spin oscillation is entirely coherent both during 2014 and during 2015. Using a Gaussian function fit to upper parts of the principal peaks, we found the precise values of the spin period. These values are 743.4060±0.0034 and 743.4033±0.0036 s in 2014 and 2015, respectively. The errors are found according to <cit.>. These values of the spin period are compatible with each other because they differ by only 0.5σ. This compatibility also confirms the conclusion made in our previous works that the errors found by the method ofare true rms errors <cit.>. The semi-amplitudes of the spin oscillation found from the power spectra shown in Fig. <ref> are equal to 18 and 25 mmag in 2014 and 2015, respectively. These semi-amplitudes are compatible with the heights of the peaks visible in the average power spectra (Fig. <ref>).As mentioned, the average power spectrum shown in the upper frame of Fig. <ref> suggests the presence of the sideband oscillation in 2014. Obviously, additional small peaks in the upper frame of Fig. <ref> on the left also belong to the sideband oscillation. These additional peaks, however, show no clear picture conforming to the window function because the sideband oscillation is affected by the spin oscillation, which has close frequency and higher amplitude. To remove the effect of the spin oscillation, we subtracted the spin oscillation from the data. The obtained power spectrum of the data of 2014 (the upper frame of Fig. <ref>) clearly proves detection ofthe sideband oscillation due to the presence of the principal peak andone-day aliases conforming to the window function and showing that this oscillation is coherent. The sideband period and semi-amplitude found from this power spectrum are equal to 764.5125±0.0049 s and 12 mmag, respectively.However, the power spectrum of the pre-whitened data of 2015 (the lower frame of Fig. <ref>) reveals the complete absence of the sideband oscillation. The semi-amplitude of the maximum noise peaks in this power spectrum is about 7 mmag. Hence, the semi-amplitude of the undetected sideband oscillation in 2015 is less than 7 mmag. The absence of signs of the sideband oscillation in the power spectrum of the common time series of 2015 conforms to the absence of this oscillation in the average power spectrum shown in the lower frame of Fig. <ref>.Although the individual amplitude spectra (Fig. <ref>) and average power spectra (Fig. <ref>) suggest that the noticeable first harmonic of the spin oscillation is present only in the data of 2014, convincing evidence of this follows from the power spectra of the common time series, which are presented in Fig. <ref>. As seen, in the data of 2014, the first harmonic of the spin oscillation reveal a distinct picture of the principal peak and one-day aliases corresponding to the window function whereas, in the data of 2015, this picture is nearly inconspicuous among noise peaks. The principal peak visible in the upper frame of Fig. <ref> corresponds to a period of371.7032±0.0015 s, which strictly coincides with the first harmonic of the spin oscillation. The semi-amplitude of this harmonic is equal to 7 mmag. The maximum semi-amplitude of the noise peaks visible in the lower frame of Fig. <ref> is equal to5 mmag.Fig. <ref> presents the segment of the power spectrum of the common time series, which contains both the data of 2014 and the data of 2015, in the vicinity of the spin oscillation. Obviously, this power spectrum gives the most precise spin period due to the highest frequency resolution. The period and semi-amplitude of the spin oscillation found from this power spectrum are equal to 743.406 50±0.000 48 s and 20 mmag, respectively. As seen in Fig. <ref>, the difference of heights of the principal peak and nearest aliases, which are caused by the large gap between the data of 2014 and 2015, is small. None the less, due to the quite high signal-to-noise ratios for the spin oscillation in the power spectra, the aliasing problem is absent. This is evident from the comparison of the spin period derived from all data and the spin periods obtained from the data 2014 and from the data of 2015 taken separately. Indeed, the deviations of the periods are less than 0.9σ when we regard the largest peak as the principal peak. However, if we suppose that the nearest alias is the true principal peak, then the deviations turn out 4–5σ. This proves the absence of the aliasing problem in the power spectrum of all data of V2069 Cyg. Unfortunately, using all data, we cannot improve the precision of the sideband period because the sideband oscillation is not detected in 2015.Final information about the periods and amplitudes of the spin oscillation is presented in Table <ref>. The precise semi-amplitudes and their rms errors were determined from a sine wave fitted to folded light curves. Note that these semi-amplitudes are very close to the semi-amplitudes found from the power spectra. In addition, in the fourth column we give the rms errors derived by the method of <cit.>. The error of the spin period found from all data is much lower than the other errors. Therefore, we found the deviations of the other periods and expressed them in units of their rms errors. This is given in the fifth column. As seen, these deviations are not excessively small and obey a rule of 3σ. This confirms our previous conclusion that the errors calculated according to <cit.> are true rms errors <cit.>. Knowing the precise periods of the observed oscillations, we can obtain the average pulse profiles from the folded light curves. However, because two observed oscillations have close periods and comparable amplitudes and, in addition, the spin oscillation possesses the noticeable first harmonic, these oscillations can affect each other. To find out what kind of pre-whitening of the data is necessary to obtain unaffected pulse profiles, we performed numerical experiments with artificial time series. These three time series consisted of sine waves with the period of the fundamental harmonic of the spin oscillation, with the period of the first harmonic of the spin oscillation, with the sideband period and with the gaps according to the observations in 2014. We learned that the time series containing the sine wave with the period of the fundamental harmonic of the spin oscillation and folding with the sideband period shows a roughly sinusoidal pulse profile with amplitude of 7 per cent of the amplitude of the initial sine wave. The time series containing the sine wave with the sideband period and folding with the spin period shows the same result. The time series containing the sine wave with the period of the first harmonic of the spin oscillation and folding with the sideband period shows a double-humped pulse profile with amplitude of 3 per cent of the amplitude of the initial sine wave.From the comparison of the amplitudes of the real oscillations and the amplitudes of the folded artificial time series we concluded that only the data, which are folded with the sideband period, require pre-whitening with the fundamental harmonic of the spin oscillation. Indeed, the semi-amplitude of the spin oscillation in 2014 is 17 mmag and, depending on the phase, can give an addition to the pulse profile of the sideband oscillation, which can reach ±1.2 mmag. This amounts 10 per cent of the sideband semi-amplitude and can be appreciable against noise.The first harmonic of the spin oscillation cannot affect the sideband pulse profile because the semi-amplitude of the first harmonic is 7 mmag and can give only a small addition of ±0.2 mmag to the sideband pulse profile. Also the sideband oscillation cannot noticeably affect the spin oscillation because it can give an addition of ±0.8 mmag, which is only 5 per cent of the semi-amplitude of the spin oscillation. Moreover, as seen in Fig. <ref>, the sideband oscillation has no high-frequency harmonic and, therefore, cannot change a characteristic shape of the spin pulse profile.Fig. <ref> presents the light curves of V2069 Cyg folded with the spin and sideband periods. In the cases of the sideband period, the light curves were pre-whitened with the fundamental harmonic of the spin oscillation. Other kinds of pre-whitening are not necessary (see text above).As seen, the spin oscillation (on the left) reveals an unstable pulse profile, which varies from an asymmetric shape in 2014 to a quasi-sinusoidal shape in 2015. In addition, the spin pulse profile in 2014 shows a small remarkable hump in phases 0.1–0.3. Although a weak sign of this hump is visible in 2015, we cannot consider it statistically significant. Indeed, in 2015 this hump consists of one point, which deviates from the smooth profile only by 1.5σ, whereas in 2014 this hump consists of five consecutive points, which deviate from the smooth profile by 1–2σ. Therefore, we can characterise the spin pulse profile as double-humped in 2014 and as quasi-sinusoidal in 2015.Such characterisation is consistent with the presence of the first harmonic of the spin oscillation in the amplitudeand power spectra of 2014 and with the absence of this harmonic in the amplitude and power spectra of 2015 (Figs. <ref>, <ref> and <ref>).Moreover, the spin pulse profile has changeable amplitude, which is also consistent with the oscillation amplitudes obtained from the power spectra.The sideband oscillation (on the right) has a quasi-sinusoidal pulse profile in 2014 and is unseen in 2015. This also conforms to the amplitude and power spectra.The high precision of the spin period makes it possible to derive an oscillation ephemeris with a long validity. To derive this ephemeris, in addition to the spin period we need in the oscillation phase. Obviously, due to a rather large noise level, the individual light curves do not allow us to find oscillation phases directly. Moreover, phases of the spin oscillation obtained fromindividual light curves turn out to be shifted due to influence ofthe sideband oscillation when the length of these light curves are not equal to the orbital period <cit.>. Therefore, we found the time of maximum from the folding of all data, in which the effect of the sideband oscillation is inconspicuous. This time was referred to the middle of the observations. In addition, we used the data subdivided into four groups (see Table <ref>) for verification. To find the times of maxima, we might apply a Gaussian function fitted to upper parts of the maxima visible in the folded light curves.However, as seen in Fig. <ref>, thespin pulse profile is changeable and, therefore,such a method can introduce a systematic error depending on the pulse profile.Therefore we preferred to find times of maxima by using a sine wave fit. Comparing these two methods, we found out that the corresponding times of maxima are not very different and consistent with each other within an accuracy of 13 per cent.Finally, for the spin oscillation, we obtained the following ephemeris: BJD_ TDB ( max)= 245 7116.799 10(7)+0.008 604 242(6)E. Using this ephemeris, we obtained the (O–C) values and numbers of the oscillation cycles for the four groups of data and presented them in Table <ref>. The (O–C) values (Fig. <ref>) reveal no significant slope and displacement along the vertical axis. Indeed, they obey the relation: (O-C)= - 0.000 008(67) - 0.000 000 0004(33)E. Because all quantities in this relation are less than their rms errors, the ephemeris demands no correction. From the rms error of the spin period, we found that the formal validity of this ephemeris is equal to 36 years (a 1σ confidence level).Although this formal validity seems large, it is noticeably less than the formal validities of the spin ephemerides, which we obtained for V455 And, V647 Aur and MU Cam (85–100 years) by applying roughly the same observational coverage <cit.>. The reason consists in noticeably less amplitude of the spin oscillation in V2069 Cyg. Obviously, the higher relative noise level in the power spectra results in lower precision of the spin period.The suggestion that the oscillation with a period of 764.5125±0.0049 s is the orbital sideband seems quite obvious. None the less, we can check this suggestion by using the orbital period found from spectroscopic observations, P_ orb = 7.480 39±0.000 05 h <cit.>. Indeed, the orbital period calculated from the sideband and spin periods detected by us is equal to 7.4801±0.0017 h. This period coincides with the spectroscopic period with high precision, where the difference is less than 0.2σ. Thus, we have no doubt that the 764.5125 s period is the orbital sideband. Unfortunately, using two periods found by us, we cannot define P_ orb more precisely than it was obtained by <cit.> because the sideband period is not detected in 2015. As mentioned, this results in much lower precision of the sideband period in comparison with the precision of the spin period.The orbital variability of a CV can be difficult to find due to rising of the noise level at low frequencies. This increased noise level is caused by random changes of the star brightness and flickering. In the case of V2069 Cyg, this difficulty strengthens due to the unusually long orbital period. As mentioned, to detect and analyse the high-frequency oscillations, we removed the low-frequency trends from the light curves by subtraction of a first- or second-order polynomial fit. However, to search for the orbital variability, this procedure cannot be applied because most individual light curves are shorter than the long orbital period of V2069 Cyg, and low frequencies corresponding to the orbital variability will be removed. Therefore, to search for the orbital variability, we removed only nightly averages from the individual light curves. To make sure that the orbital variability of V2069 Cyg was not artificially removed due to subtraction of the nightly averages, we performed numerical experiments with artificial time series. We found out that, at least in 2014, where the individual light curves are mostly longer, this subtraction diminishes the amplitude of the orbital variability only by 20 per cent. The low-frequency parts of the obtained power spectra are shown in Fig. <ref>. As seen, they reveal no signs of the orbital period. The semi-amplitude of the maximum noise peaks in the upper part of Fig. <ref> is roughly 30 mmag. Therefore, we might detect the orbital variability if it had the same or larger semi-amplitude. Hence, the undetectability of the orbital period results from low orbital inclination, which must be less than 50^∘ <cit.>.§ DISCUSSIONWe obtained photometric observations of V2069 Cyg with a total duration of 119 h in 2014 and 2015. Analysing these extensive data, we clearly detected two coherent oscillations with periods of 743.406 50±0.000 48 s and of 764.5125±0.0049 s. The shorter period is consistent with the X-ray periods found by <cit.>, <cit.> and <cit.>.argued that V2069 Cyg is a pure disc accretor. Hence, these X-ray periods and our shorter period conform to the spin period of the white dwarf. Although this spin period was already detected in optical light <cit.>, our measurement of the spin period should be considered as a new result because its precision is three orders of magnitude higher than the precisions of all other measurements of the spin period inV2069 Cyg. The longer period detected by us is the orbital sideband because it conforms to the formula: 1/P_ beat = 1/P_ spin - 1/P_ orb, where P_ orb is the orbital period found by <cit.>. We detected the sideband oscillation in V2069 Cyg for the first time.The semi-amplitude of the spin oscillation detected by us is quite low and changeable both in a time-scale of days and in a time-scale of years. On average, it is equal to 20 mmag.In addition, we noticed that V2069 Cyg possesses flickering, which is noticeably stronger than flickering, which we observed in other IPs. Therefore, the spin oscillation is difficult to recognise directly in the light curve or to detect in the power spectrum when the light curve is short. This explains why <cit.> could not find this oscillation in their first photometric observations in 1992. The amplitude of the sideband oscillation is even less than the amplitude of the spin oscillation and is also changeable. We could detect the sideband oscillation only in 2014 when its semi-amplitude was equal to 12 mmag. In 2015 the sideband oscillation completely disappeared.In addition to changeable amplitude, the spin pulse of V2069 Cyg reveals changes of its profile. This profile varies from an asymmetric shape in 2014 to a quasi-sinusoidal shape in 2015. Moreover, the spin pulse profile observed in 2014 shows the additional small hump in phases 0.1–0.3 (the left part of Fig. <ref>). Examining figure 6 in <cit.>, we found out that in 2009 the spin pulse profile of V2069 Cyg was also double-peaked, where, in contrast with the spin pulse profile observed in 2014, two humps were equally pronounced. The reality of this pulse profile is confirmed by the power spectrum presented in figure 4 in , which reveals the very strong first harmonic of the spin oscillation. Thus, between 2009 and 2015 the optical spin pulse profile of V2069 Cyg extremely varied from a pronounced double-peaked shape to a quasi-sinusoidal shape. Such drastic changes of the optical spin pulse profile seem very interesting and uncommon among other IPs.To account for origin of optical spin pulses in IPs, three possibilities can be considered. The optical spin pulse can be produced by changes of the direct visibility of the hot pole caps, by changes of the visibility of accretion curtains located between the inner disc and the white dwarf and through reprocessing of X-rays in the axisymmetric parts of the accretion disc <cit.>. All three possibilities cannot consistently explain drastic changes of the spin pulse profile of V2069 Cyg. Indeed, in the first case, we must suppose temporary invisibility of one of the poles to the observer when the double-peaked pulse profile turns into the quasi-sinusoidal pulse profile. However, the geometry of the system cannot change and thus hide one of the poles. In the second case, depending on the sizes and shapes of the accretion curtains, both single-peaked, roughly sinusoidal and double-peaked spin pulse profiles can be generated, and these sizes and shapes depend on the magnetic field strength <cit.>. Again, to account for drastic changes of the spin pulse profile, we must suppose significant changes of the magnetic field strength, which seem impossible. The third possibility seems less contradictive because changes of the spin pulse profile can originate from changes of the accretion disc structure, which seem probable. However, the reprocessing of X-rays in the axisymmetric parts of the disc demands a sufficient degree of asymmetry, e.g., between the front and the back of the disc and can happen only in a highly inclined system, which shows eclipses (e.g., DQ Her: ; ). Although we observed no eclipses in V2069 Cyg, none the less, we can suppose that the orbital inclination is yet sufficient to produce the spin pulse from reprocessing of X-rays in the axisymmetric parts of the disc.Two ways are conceivable to explain origin of the optical orbital sideband in IPs. The first way consists in the reprocessing of X-rays at some structure of the system that rotates with the orbital period. The second way consists in alternation of the accretion flow between two poles of the white dwarf with the sideband frequency. The second way seems inappropriate to V2069 Cyg because this process happens in the cases stream-fed and disc-overflow accretion and because <cit.> argued that V2069 Cyg is a pure disc accretor. Pulse profiles and amplitudes of the sideband oscillation often show significant variability.According to the first way, which is the canonical interpretation of the optical orbital sideband, reasons for such variability can consist in changes of the structure of the accretion disc, asymmetric parts of which reprocess X-rays with the sideband frequency. This variability, however, is not accompanied by noticeable changes of the star brightness <cit.>. Then, we can account for the disappearance of the orbital sideband in 2015 by the diminishing of its amplitude to undetectable level due to changes of the structure of the accretion disc.This explanation is also consistent with the simultaneous change of the pulse profile of the spin oscillation in V2069 Cyg because changes of the asymmetric parts of the disc can be accompanied by changes of its axisymmetric parts.Our precise spin period of V2069 Cyg makes it possible to investigate its behaviour in future observations. This can be made by using our oscillation ephemeris, which has a formal validity of 36 years, and pulse-arrival times. Then, the alternating increase and decrease of (O–C) values can indicate spin-up and spin-down and thus indicate spin equilibrium. Of course, this is a difficult task, which demands to perform annual observations during a decade or more (see, e.g., figure 2 in ). In addition, one can use direct measurements of the spin period obtained a decade later.This way seems less laborious. As seen in Table <ref>, photometric observations, which are obtained during roughly 50 hours and cover a few months, give an rms error of the period of about 0.004 s. Then, performing observations during one observing season ten years later, one can detect a spin period change of 0.02 s with a 5σ confidence level. This period change corresponds to dP/ dt=6 × 10^-11.As seen in table 1 in <cit.>, such a detection threshold is close to the dP/ dt measured in most IPs and, therefore, seems insufficient. Performing observations during two observing seasons ten years later, one can achieve a detection threshold of 6 times less. Probably, large dP/ dt averaged over a sufficiently large time span can imply the absence of spin equilibrium.§ CONCLUSIONS We obtained extensive photometric observations of V2069 Cyg over 32 nights with a total duration of 119 h in 2014 and 2015. Performing comprehensive analysis of these data, we obtained the following results:* Due to the large observational coverage, we measured the spin period of the white dwarf with high precision. The spin period is equal to 743.406 50±0.000 48 s. * The semi-amplitude of the spin oscillation was unstable both in a time-scale of days and in a time-scale of years.On average, it varied from 17 mmag in 2014 to 25 mmag in 2015.* During our observations the spin pulse profile revealed strong changes. In 2014 the spin pulse profile showed an asymmetric double-peaked shape whereas in 2015 it became quasi-sinusoidal. Such drastic changes of the optical spin pulse profile seem untypical of most IPs and, therefore, are very interesting.* For the first time we detected the sideband oscillation of V2069 Cyg with a period of 764.5125±0.0049 s.* The semi-amplitude of the sideband oscillation was also unstable. On average, it varied from 12 mmag in 2014 to an undetectable level of less than 7 mmag in 2015.* The pulse profile of the sideband oscillation was quasi-sinusoidal.* The high precision of the spin period allowed us to obtain the oscillation ephemeris with a formal validity of 36 years. This ephemeris and the precise spin period can be used for future investigations of spin period changes of the white dwarf in V2069 Cyg.* We note that V2069 Cyg possesses strong flickering with a peak-to-peak amplitude of 0.4–0.6 mag.§ ACKNOWLEDGMENTS This work was supported in part by the Ministry of Education and Science (the basic part of the State assignment, RK NoAAAA-A17-117030310283-7) and by the Act No211 of the Government of the Russian Federation, agreement No02.A03.21.0006. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research also made use of the NASA Astrophysics Data System (ADS).[Bernardini et al.2012]bernardini12 Bernardini, F., de Martino, D., Falanga, M., et al.: 542, A22 (2012)[Butters et al.2011]butters11 Butters, O. W., Norton, A. J., Mukai, K., Tomsick, J. A.: 526, A77 (2011) [de Martino et al.2009]demartino09 de Martino, D., Bonnet-Bidaud, J. M., Falanga, M., Mouchet, M., Motch, C.: ATel. 2089, 1 (2009)[Eastman et al.2010]eastman10 Eastman, J.,Siverd, R., Gaudi, B. S.: 122, 935 (2010)[Hellier1995]hellier95 Hellier, C.:In: Buckley, D. A. H., Warner, B. (eds.) Cape Workshop on Magnetic Cataclysmic Variables. Publ. Astron. Soc. Pac. Conf. Series, vol. 85, p. 185 (1995)[King and Wynn1999]king99 King, A. R., Wynn, G. A.: 310, 203 (1999)[Kozhevnikov2001]kozhevnikov01 Kozhevnikov, V. P.: 366, 891 (2001)[Kozhevnikov2012]kozhevnikov12 Kozhevnikov, V. P.: 422, 1518 (2012)[Kozhevnikov2014]kozhevnikov14 Kozhevnikov, V. P.: 443, 2444 (2014)[Kozhevnikov2016]kozhevnikov16 Kozhevnikov, V. P.: 361, 273 (2016) doi:10.1007/s10509-016-2859-0 [Kozhevnikov and Zakharova2000]kozhevnikoviz Kozhevnikov, V.P., Zakharova, P.E.: In: Garzon, F., Eiroa, C., de Winter, D., Mahoney, T. J. (eds.) Disks, Planetesimals andPlanets. Publ. Astron. Soc. Pac. Conf.Series, vol. 219, p. 381 (2000)[Kruszewski and Semeniuk1998]kruszewski98 Kruszewski, A., Semeniuk, I.: Acta Astron.48, 757 (1998)[la Dous1994]ladous94 la Dous, C.: Space Sci. Rev.67, 1 (1994)[Motch et al.1996]motch96 Motch, C., Haberl, F., Guillout, P., Pakull, M., Reinsch, K., Krautter, J.: 307, 459 (1996) [Nasiroglu et al.2012]nasiroglu12 Nasiroglu, I., Słowikowska, A., Kanbach, G., Haberl, F.: 420, 3350 (2012)[Norton et al.1999]norton99 Norton, A. J., Beardmore, A. P., Allan, A., Hellier, C.: 347, 203 (1999)[Norton et al.2004]norton04 Norton, A. J., Wynn, G. A., Somerscales, R. V.: 614, 349 (2004)[Patterson1994]patterson94 Patterson, J.: 106, 209 (1994)[Patterson and Steiner1983]patterson83 Patterson, J., Steiner, J. E.: 264, L61 (1983)[Patterson et al.1998]patterson98 Patterson, J., Kemp, J., Richman, H. R., et al.: 110, 415 (1998)[Petterson1980]petterson80 Petterson, J. A.: 241, 247 (1980)[Schwarzenberg-Czerny1991]schwarzenberg91 Schwarzenberg-Czerny, A.: 253, 198 (1991)[Schwarzenberg-Czerny1998]schwarzenberg98 Schwarzenberg-Czerny, A.: Baltic Astron. 7, 43 (1998)[Thorstensen and Taylor2001]thorstensen01 Thorstensen, J. R, Taylor, C. J.: 326, 1235 (2001) [van der Woerd et al.1984]woerd84 van der Woerd, H., de Kool, M., van Paradijs, J.: 131, 137 (1984)[Warner1986]warner86 Warner, B.: 219, 347 (1986)[Warner1996]warner96 Warner, B.: 241, 263 (1996)[Williams2003]williams03 Williams, G.: 115, 618 (2003)[Wynn and King1992]wynn92 Wynn, G. A., King, A. R.: 255, 83 (1992)Fig. 1.Longest differential light curves of V2069 Cyg.Fig. 2.Amplitude spectra of V2069 Cyg. The dotted lines mark the 743-s period and its first harmonic.Fig. 3.Average power spectra derived by the weighted averaging of 10 power spectra of long individual light curves of 2014 and of 10 power spectra of long individual light curves of 2015 from V2069 Cyg. The dotted lines mark the 743-s period and its first harmonic.Fig. 4.Power spectra derived for the data of 2014 and 2015 from V2069 Cyg. They reveal a coherent oscillation with periods of 743.4060±0.0034 and 743.4033±0.0036 s in 2014 and 2015, respectively. In the upper frame, on the left, one can also note a sign of the sideband oscillation. The principal peaks and one-day aliases of the two oscillations are labelled with 'F1', 'F2' and A1, 'A2' respectively.Fig. 5.Power spectra of the data of V2069 Cyg, from which the largest oscillation was subtracted. In the data of 2014, this subtraction allows us to detect one more coherent oscillation with a period of 764.5125±0.0049 s. In the data of 2015, however, this oscillation is completely absent. The dotted lines mark the location of the principal peak of the subtracted oscillation and its one-day aliases.Fig. 6.Power spectra of the data of V2069 Cyg in the vicinity of the first harmonic of the spin oscillation. In the data of 2014, the first harmonic is clearly present, whereas, in the data of 2015, this harmonic is nearly inconspicuous among noise peaks. The principal peak and one-day aliases of the first harmonic are labelled with 'F3' and 'A3', respectively.Fig. 7.Segment of the power spectrum calculated for all data from V2069 Cyg in the vicinity of the main oscillation. It reveals a period of 743.406 50±0.000 48 s. The upper frame shows the window function.Fig. 8.Pulse profiles of two oscillations obtained for the data of 2014 and 2015 from V2069 Cyg. The oscillation with a period of 743.406 50 s (on the left) reveal an unstable pulse profile, which varies from an asymmetric shape in 2014 to a quasi-sinusoidal shape in 2015. In addition, this profile has changeable amplitude and shows a small hump in phases 0.1–0.3. The oscillation with a period of 764.5125 s (on the right) has a quasi-sinusoidal pulse profile and is detected only in the data of 2014.Fig. 9.(O–C) diagram for all data from V2069 Cyg, which are subdivided into four groups and folded with a period of 743.406 50 s.Fig. 10.Low-frequency parts of the power spectra for the data of 2014 and 2015 from V2069 Cyg in the frequency range of the expected orbital variability of V2069 Cyg. The dotted lines mark the expected orbital period and its nearest one-day aliases. | http://arxiv.org/abs/1706.08930v1 | {
"authors": [
"V. P. Kozhevnikov"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170627162958",
"title": "Extensive photometry of the intermediate polar V2069 Cyg (RX J2123.7+4217)"
} |
[email protected] Department of Physics, Yale University, New Haven, Connecticut 06511, USAJILA and the Department of Physics, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309, USADepartment of Nuclear Engineering, University of California Berkeley, Berkeley, California 94720, USAWe describe in detail the analysis procedure used to derive the first limits from the Haloscope at Yale Sensitive to Axion CDM (HAYSTAC), a microwave cavity search for cold dark matter (CDM) axions with masses above 20 μeV. We have introduced several significant innovations to the axion search analysis pioneered by the Axion Dark Matter eXperiment (ADMX), including optimal filtering of the individual power spectra that constitute the axion search dataset and a consistent maximum likelihood procedure for combining and rebinning these spectra. These innovations enable us to obtain the axion-photon coupling |g_γ| excluded at any desired confidence level directly from the statistics of the combined data. The HAYSTAC Axion Search Analysis Procedure K. A. van Bibber December 30, 2023 ===========================================§ INTRODUCTIONThe axion <cit.> is a hypothetical pseudoscalar field originally postulated to explain the absence of CP violation in the theory of quantum chromodynamics (QCD); light axions (m_a ≲ 1 meV) have since been recognized as attractive candidates for a microscopic description of cold dark matter (CDM) <cit.>. Axions constituting our galactic halo with masses in the range 1 ≲ m_a ≲ 50 μeV may be detected via their resonant conversion into nearly monochromatic microwave photons in an “axion haloscope:” a high-Q cryogenic cavity immersed in a strong magnetic field and coupled to a low-noise receiver <cit.>. All haloscope detectors to date have used spectrally resolved coherent receivers, in which an axion signal would appear as an extremely weak but spectrally sharp persistent power excess over the noise floor at frequency ν_a = m_ac^2/h. In practice, the axion mass is unknown, so the cavity must be tunable. It is typical to assume that the halo axions are virialized, in which case the spectral distribution of the conversion power is inherited from the halo's kinetic energy distribution, with fractional width of order <v^2>/c^2 ∼ 10^-6. One example of a haloscope detector is the Haloscope At Yale Sensitive To Axion CDM (HAYSTAC), which recently demonstrated cosmological sensitivity to halo axions with m_a > 20 μeV for the first time <cit.>. The HAYSTAC detector is described in detail in Ref. <cit.>; the purpose of the present paper is to provide a detailed pedagogical account of the analysis procedure used to generate the exclusion limit reported in Ref. <cit.>. The basic framework of our analysis owes much to the procedure developed by the Axion Dark Matter eXperiment (ADMX) <cit.>; we have introduced a number of refinements that collectively enable us to obtain the relationship between search sensitivity and confidence directly from the statistics of the combined data without recourse to Monte Carlo. These innovations can easily be adapted to the analysis of data from other haloscope detectors such as ADMX and CULTASK <cit.> and perhaps also to “dielectric haloscopes” like MADMAX <cit.> and resonant hidden photon detectors like DM Radio <cit.>. The remainder of the paper is organized as follows. Section <ref> briefly reviews the aspects of the HAYSTAC detector most relevant to understanding the analysis and describes the axion search data set. Section <ref> presents a big-picture overview of the analysis procedure, whose distinct stages are discussed in greater detail in Sec. <ref> – <ref>. In Sec. <ref> we present our limit and conclude with a summary of our main innovations. Various tangential topics that are nonetheless important to a full understanding of the analysis procedure are discussed in appendices. § EXPERIMENT §.§ DetectorHAYSTAC is sited at the Wright Laboratory of Yale University, and housed within a cryogen-free dilution refrigerator integrated with a 9 T superconducting solenoid. The cavity hangs in the center of the magnet bore from a gold-plated copper gantry anchored to the dilution refrigerator's mixing chamber plate at temperature T_C =127 mK.Our current cavity is a 2 L copper-plated stainless cylinder whose axion-sensitive TM_010 mode may be tuned over the range 3.6< ν_c < 5.8 GHz via rotation of an off-axis copper rod occupying 25% of the cavity volume. We can also independently adjust the insertion into the cavity of a thin dielectric shaft and a coaxial antenna, used to fine-tune the mode's frequency and control its coupling to the receiver, respectively.The most notable feature of the HAYSTAC receiver is its use of a tunable Josephson parametric amplifier (JPA) as a preamplifier. The JPA is essentially a nonlinear LC circuit that exhibits parametric gain when driven with a sufficiently strong microwave pump tone near its resonant frequency. For a small signal detuned completely to one side of the pump, a JPA acts like a phase-insensitive linear amplifier whose added noise is close to the fundamental limits imposed by quantum mechanics <cit.>. Our current JPA may be tuned over the range 4.5–6.4 GHz via application of a small DC magnetic flux bias. The first element in the receiver signal path is a microwave switch that allows us to calibrate the cavity noise by comparison with a known blackbody source at T_H = 775 mK, the temperature of the dilution refrigerator's still plate. Signals at the JPA output are amplified further at 4 K and room temperature, and downconverted to an intermediate frequency (IF) band using an IQ mixer whose local oscillator (LO) is set 780 kHz above the cavity resonance. After further amplification and filtering the IF signals are digitized at 25 MS/s.For the first HAYSTAC data run we scanned over the range 5.7–5.8 GHz in two continuous passes followed by several shorter scans to compensate for nonuniform tuning. This nonuniformity was a consequence of fine tuning with the dielectric shaft and moving the copper rod less frequently to mitigate imperfection in the rotary tuning system. §.§ Axion search dataA haloscope axion search consists of a sequence of iterations separated by discrete tuning steps, with a cavity noise measurement of duration τ and various auxiliary measurements at each iteration.[τ is the axion-sensitive averaging time, not the total data collection time including inefficiency.] We construct and average power spectra in parallel with acquisition of the cavity noise timestream data from the HAYSTAC detector, so only a single heavily averaged power spectrum is written to disk at each iteration. The auxiliary data consists of vector network analyzer (VNA) measurements of the cavity mode and JPA gain profile at each step and periodic Y-factor measurements to calibrate the noise; temperatures and pressures at various points in the cryogenic system are also logged independently. The purpose of the auxiliary data is to characterize detector parameters that can vary during the run, both to define data quality cuts (Sec. <ref>) and optimally rescale spectra (Sec. <ref>).The principal data from the first run consisted of 6936 power spectra with bin width Δν_b = 100 Hz, each obtained from τ = 15 minutes of averaging. We acquired the first 2244 spectra in winter 2016, and the rest in summer 2016 following a power outage that damaged the system and disrupted operations. Filters limit the usable IF bandwidth of each spectrum to roughly 2.5 MHz, well below the 12.5 MHz Nyquist frequency. Fig. <ref> shows a schematic layout of the regions of interest in each spectrum. It may be useful at this point to summarize the relations between the various frequency scales that will play a role in our subsequent discussions. When appropriately biased, the JPA has about 21 dB peak gain in a bandwidth Δν_JPA≈ 2.3 MHz centered on the pump tone. Δν_JPA is larger than the typical cavity linewidth Δν_c ≈ 500 kHz, which ensures that the total noise referred to the JPA input remains low over all frequencies of interest in each spectrum. The cavity linewidth Δν_c, which sets the width of the axion-sensitive region in each spectrum, is in turn much larger than the typical axion linewidth Δν_a ∼ν_a<v^2>/c^2 ≈ 5 kHz for a virialized axion in the initial HAYSTAC scan range. Finally, Δν_a ≫Δν_b, which helps us reject spurious single-bin features (Sec. <ref>) and take the axion lineshape into account in our analysis (Sec. <ref>). In principle fine frequency resolution also enables us to search for non-virialized structure in the axion energy spectrum (see Sec. <ref>); for the present analysis, we restrict our focus to virialized axions.As illustrated in Fig. <ref>, the dependence of the haloscope signal power on the detuning δν_a = ν_c - ν_a is Lorentzian with FWHM Δν_c; at δν_a = Δν_c, the signal power is thus down from its peak value by a factor of 5. Ultimately the quantity we care about is the signal-to-noise ratio (SNR) throughout the tuning range, to which individual spectra will contribute in quadrature [See Eq. (<ref>)]. Including bins further than Δν_c from the cavity mode in each spectrum in our analysis would improve the SNR only very marginally. Thus we can restrict our focus to an analysis band of full width ≈2Δν_c centered on the cavity mode in each spectrum without appreciably affecting our sensitivity.During the data run we fit the TM_010 resonance in transmission after each tuning step, and set the LO frequency by adding 780 kHz to the measured mode frequency and rounding to the nearest 100 Hz.[Coercing the LO frequency to the nearest 100 Hz ensures that the bin boundaries in different spectra are always aligned. As a result the analysis band is not exactly centered on ν_c in each spectrum, but the maximum offset is always <Δν_b.] We then set the JPA pump frequency 1.6 MHz below the LO. Setting the JPA pump frequency at a fixed offset from the LO instead of the cavity resonance ensures that the 1/Δν_b = 10 ms integration time of each subspectrum is an integer number of periods at the pump frequency, and thus minimizes spreading of the pump power throughout the spectrum.[Sinusoidal signals of arbitrary frequency will generally not be confined to single bins in the spectrum because we do not apply a window function to the timestream data in the process of computing the power spectrum of each 10 ms record. The “rectangular window” (equivalent to not windowing at all) is the correct choice for a haloscope search as it has the smallest equivalent noise bandwidth. Given the constraint of the rectangular window, a small bin width Δν_b≪Δν_a also ensures that distortion of the axion signal lineshape by the FFT is negligible.] The analysis band is defined as the set of bins between 129 kHz and 1.431 MHz in each spectrum; this is a conservative choice that accounts for variation of Δν_c over the scan range. § ANALYSIS OVERVIEWThe goal of a haloscope analysis is to combine a set of overlapping axion-sensitive power spectra to produce a single spectrum that optimizes the SNR throughout the scan range. Put another way, if there exists an axion with ν_a within the scan range and photon coupling |g_γ| sufficiently large, the conversion power should almost always result in a large excess relative to noise in the bin corresponding to ν_a in the final spectrum. The minimum coupling |g^min_γ| for which this statement will hold is set primarily by the detector design, but we must still understand how much the analysis procedure degrades this intrinsic sensitivity. The analysis should ideally allow us to write down an explicit expression for |g^min_γ| as a function of the desired confidence level (which quantifies the “almost always” in the informal description above).When we consider how best to combine spectra, one issue that immediately arises is that the shape and normalization of each spectrum depend both on quantities that affect the SNR (e.g., the system noise temperature), and quantities that do not (e.g., the net gain of the receiver chain, including the frequency-dependent attenuation of all room-temperature components). Rather than try to tease apart the relevant and irrelevant contributions, we can remove the spectral baseline entirely using a fit or filter, then rescale the resulting spectra using parameters extracted from the auxiliary data. In this way we can properly account for variation in sensitivity among spectra and within each spectrum. After baseline removal the bins in each spectrum may be regarded as samples drawn from a single Gaussian distribution.[The spectra are approximately Gaussian because each spectrum saved to disk is the average of a large number of subspectra, so the bin variance is much smaller than the mean squared bin amplitude. This point is discussed further in Sec. <ref>.] This is a convenient reference point for understanding the effects of subsequent processing on the statistics of the spectra. Of course, we need to make sure that the baseline removal procedure does not fit out bumps in the spectra on frequency scales comparable to Δν_a, or we will significantly degrade the axion search sensitivity. This point suggests that baseline removal is more fruitfully regarded as a problem in filter design than a fitting problem, as it has been described in previous ADMX analyses. The filter perspective will turn out to be quite useful in understanding the statistics of the spectra. The task of removing the spectral baseline without appreciably attenuating any axion signal is made tractable by their different characteristic spectral scales, or in other words by Δν_c ≫Δν_a. It is worth noting that this inequality is ultimately a consequence of the difficulty of achieving high cavity Q factors with normal metals at GHz frequencies; a detector with higher cavity Q and thus Δν_c ≈Δν_a would in principle be more sensitive. Because such a detector has yet to be built, we can exploit the fact that Δν_c ≫Δν_a where it simplifies the analysis. Because our spectra have Δν_b ≪Δν_a, the analysis procedure will generally involve taking appropriately weighted sums both “vertically” (i.e., combining IF bins from different spectra corresponding to the same RF bin) and “horizontally” (i.e., combining adjacent bins in the same spectrum). One of the main innovations of our analysis procedure is that we use the same maximum likelihood principle to obtain the optimal weights in both cases. Various statistical subtleties arise in the latter case because nearby bins in the same spectrum can be correlated. We will demonstrate below that we understand the origin of these correlations sufficiently well to obtain the relationship between|g^min_γ| and the confidence level from the statistics of the combined data, rather than from Monte Carlo as in previous ADMX analyses.In the preceding paragraphs we have emphasized what we regard as the main themes of this paper, which may be helpful to keep in mind as we work through the details. For ease of reference, we have outlined the steps of our procedure below, and indicated the section of the paper in which each step is discussed more thoroughly. * Use the auxiliary data to identify spectra that appear to be compromised and cut them from further analysis (Sec. <ref>).* Average the remaining raw spectra together aligned according to IF frequency to identify compromised IF bins and cut them from further analysis (Sec. <ref>). This procedure also yields an estimate of the average shape of the spectral baseline in the analysis band.* Normalize the analysis band in each raw spectrum to the average baseline, then use a Savitzky-Golay (SG) filter to remove the remaining spectral structure in each normalized spectrum (Sec. <ref>). Then subtract 1 from each spectrum to obtain a set of dimensionless processed spectra described by a single Gaussian distribution (Sec. <ref>).* Multiply each processed spectrum by the average noise power per bin and divide by the Lorentzian axion conversion power profile to obtain a set of rescaled spectra (Sec. <ref>). Construct a single combined spectrum across the whole scan range by taking an optimally weighted sum of all the rescaled spectra (Sec. <ref>).* Rebin the combined spectrum via a straightforward extension of the optimal weighted sum from the previous step to non-overlapping sets of adjacent combined spectrum bins (Sec. <ref>). Then, taking into account the expected axion lineshape (Sec. <ref>), construct the grand spectrum by adding an optimally weighted sum of adjacent bins to each bin in the rebinned spectrum (Sec. <ref>).* After correcting for the effects of the SG filter on both the statistics of the grand spectrum (Sec. <ref>) and the SNR (Sec. <ref>), set a threshold Θ for which some desired fraction of axion signals with a given SNR would result in excess power >Θ. Then flag all bins with excess power larger than Θ as rescan candidates (Sec. <ref>).* Acquire sufficient data around each rescan candidate to reproduce the sensitivity at that frequency in the original grand spectrum (Sec. <ref>). Follow the procedure above, with a few minor differences, to construct a grand spectrum for the rescan data, and determine if any candidate exceeds the corresponding threshold (Sec. <ref>). If no candidate exceeds the second threshold, the corrected SNR obtained in step 6 sets the exclusion limit. Any persistent candidates can be interrogated manually.A great deal of notation is introduced in the sections to follow; we have attempted to simplify it wherever possible by adopting consistent notational conventions. The notation used throughout the paper is summarized in Appendix <ref> for ease of reference.§ DATA QUALITY CUTS §.§ Cuts on spectraOur first task is to flag and cut any spectra whose sensitivity to axion conversion we cannot reliably calculate, due to e.g., large changes in the TM_010 mode frequency ν_c or the JPA gain during a noise measurement. We had reason to anticipate both of these effects in the first HAYSTAC data run: imperfections in the rotary tuning system noted in Sec. <ref> resulted in a slow drift of ν_c following actuation of the tuning rod, and the JPA gain is very sensitive to changes in the local magnetic flux. We sought to mitigate both gain and mode frequency drifts in the design of the data acquisition procedure (for example, by controlling the JPA's flux bias with feedback as described in Ref. <cit.>). However, the mode frequency still occasionally drifts sufficiently far during a single noise measurement to systematically distort the subsequent weighting of the spectrum by the Lorentzian profile of the cavity mode (see Sec. <ref>). Likewise, the flux occasionally drifts sufficiently far that the feedback system is unable to correct for it; the average JPA gain in such iterations is reduced and thus the input-referred noise is systematically higher than what we infer from periodic in situ noise calibrations.Cutting measurements compromised by mode frequency drift is straightforward, because we make VNA measurements of the cavity mode in transmission both before and after the cavity noise measurement at each iteration during the data run. Our analysis routine fits both measurements to Lorentzians and cuts iterations exceeding the conservative threshold |ν_c1-ν_c2|>60 kHz≈Δν_c/10 from subsequent analysis. We flag iterations compromised by gain drifts using the spectra themselves. The average level of each spectrum in a 100 kHz window close to the JPA pump is a good proxy for the average JPA gain during the noise measurement, though of course it will also reflect other changes in the net receiver gain. Another measure of the average gain accessible in the spectrum is the weak CW tone used to provide a signal for our flux feedback system. We set thresholds for both measures of the average JPA gain empirically to separate obvious outliers from the normal variation among spectra. In both cases, the thresholds were approximately 1 dB below the typical power averaged across all spectra.[These thresholds are consistent with independent measurements indicating that flux feedback holds the JPA gain constant to within 10% on timescales comparable to τ. Gain fluctuations during a cavity noise measurement will cause the normalization of each 10 ms subspectrum averaged by the in situ processing code to differ, but this variation is correlated across all the bins in each subspectrum; it affects the precision with which we can measure the mean noise power, but not the variance of the noise power within each spectrum, which is the quantity that determines our sensitivity to excess power on small spectral scales Δν_a ≪Δν_c. Thus absolute gain stability is not a critical parameter for haloscope experiments. At our operating gain, the effect of such small fluctuations on the system noise temperature is small compared to the uncertainty.]We also scanned the rest of the auxiliary data for any other anomalies that might motivate a cut, and observed a narrow (≈60 kHz) notch around 5.7046 GHz superimposed on measurements of the cavity response in transmission and reflection near this frequency. The absence of any analogous feature in the corresponding JPA gain profiles indicates that the notch originates in the cavity, most likely due to the TM_010 mode crossing with an “intruder” TE or TEM mode practically uncoupled to our antenna. The observation that the precise notch frequency depends on the insertion depth of the dielectric tuning shaft supports this interpretation. Because we used the dielectric shaft for fine tuning in our first data run, the notch frequency appeared to wander back and forth over a range of a few hundred kHz. We noticed that the notch was also visible in the spectra around the same frequency, which suggests that the effective temperature of the intruder mode was lower than that of the TM_010 mode.[As discussed in Ref. <cit.>, the TM_010 mode temperature was actually higher than the fridge temperature during our first data run due to a poor thermal link to the copper tuning rod.] All of these measurements collectively indicate that our basic assumption of the axion interacting with a single cavity mode fails around the intruder mode, and neither the VNA measurements of the cavity nor noise calibrations are likely to be reliable here. To be conservative, we simply cut all spectra containing any sign of the intruder mode.Other auxiliary data (such as the JPA-off receiver gain measurement at each iteration and the fridge temperature records) did not not prompt us to define additional cuts. Overall, of the 6936 spectra obtained during our first data run, we cut 170 from the subsequent analysis, of which 128 were cut in connection with the intruder mode. Of the remaining 42 spectra, 33 were cut because of JPA gain drifts, and 9 because of mode frequency drifts. §.§ Cuts on IF binsNarrowband interference can contaminate individual bins in spectra that are otherwise sensitive to axion conversion. Insofar as the intrinsic linewidth of these interference features is ≪Δν_b, a smaller bin width Δν_b helps reduce the number of contaminated bins that we fail to flag, whose collective effect is to distort the statistics of the spectra.It is useful to distinguish IF interference (resulting in excess power in the same bins in each spectrum) from RF interference (which would appear to propagate through spectra from adjacent tuning steps). RF interference is more insidious in that it can mimic an axion signal, and small excesses will be hard to flag until we have already combined the contributing spectra. Empirically, all of the most prominent sharp features in HAYSTAC power spectra are due to IF interference.The various IF features we observe have no single common origin. Some prominent features we eliminated during detector commissioning were associated with ground loops, others with switching power supplies in stepper motor drivers and other room-temperature electronics. Other features only appear when the system is cold, suggesting that cryocooler motors may be responsible. Single-bin IF features can also arise from small RF signals at fixed detuning from the LO or pump tones.We flag the “bad bins” contaminated by IF interference using the following procedure. First, we divide the set of spectra (ordered chronologically) into three approximately equally sized groups. We truncate each spectrum to the analysis band plus W=500 bins (50 kHz) on either side. We then average all truncated spectra within each group aligned according to IF frequency; this averaging reveals many sharp features due to IF interference too small to be visible above the noise floor of individual spectra. We apply an SG filter with polynomial degree d=10 and half-width W to the averaged spectrum to obtain an estimate of the spectral baseline. The SG filter is described in more detail in Sec. <ref>; for our present purposes it is sufficient to regard it as a low-pass filter with a very flat passband (i.e., it perfectly preserves features on large spectral scales).Dividing the averaged spectrum by the SG filter output and subtracting 1 produces a spectrum whose statistics (in the absence of IF interference) are Gaussian, with mean 0 and standard deviation σ^IF=(M_IFΔν_bτ)^-1/2, where M_IF≈2200 is the number of spectra in the group.[The procedure used here to flag IF interference is similar to the baseline removal procedure described in Sec. <ref> with a few key differences. Here the SG filter is applied to a spectrum that is more heavily averaged by a factor of M_IF, and we do not divide out the average shape of the spectrum before applying the SG filter. Both effects imply that the polynomial degree d of the SG filter must be higher here than in the main analysis.] The most obvious effect of IF interference is to produce a surplus of bins with large positive power excess. We flag all bins that exceed a threshold value of 4.5σ^IF; in the 14020 bins of the truncated spectrum, we expect on average only 0.05 bins exceeding this threshold due to statistical fluctuations. As noted in Sec. <ref>, the fact that we do not apply any windowing in the construction of HAYSTAC power spectra implies that the excess power associated with narrowband IF interference will not be entirely confined to isolated bins. To be conservative, for every set of contiguous bins exceeding the threshold, we add the 3 adjacent bins on either side to the list of bad bins. Empirically, while many features due to IF interference are indeed quite sharp, others consist of ∼30 consecutive bins exceeding the threshold. Averaging smaller numbers of adjacent spectra reveals that these broader features are the result of narrow IF peaks that wander back and forth across a range of a few kHz over the course of the data run. A second, more subtle effect of IF interference is to distort the local estimate of the spectral baseline around any sufficiently large power excess. To mitigate this effect we repeat the process described above iteratively. We remove all flagged bins from the averaged spectrum and apply the SG filter again to obtain a refined baseline estimate; using this improved baseline we generally find some additional bins with values exceeding the 4.5σ threshold; again 3 bins on either side are also flagged. In practice this procedure takes only 2 or 3 iterations to converge. The output of this iterative process is a list of bad bins within the truncated spectrum for each group of spectra; we also obtain an estimate of the average spectral baseline that we will use in the next stage of the analysis procedure.The bad bin lists we obtain from our three distinct groups of spectra are quite similar: roughly 75% of the bins that appear on each list also appear on the other two, and most discrepancies amount to shifting the boundaries of contiguous sets of bad bins. Because the three lists appear to describe IF interference that does not change throughout the run, we combined them into a single final list of bad bins to be cut from every spectrum. Any minimal group of 7 consecutive bins is included in the final list if it appears in two of the three lists and excluded if it appears on only one list. For all other features the final list is the union of the three lists. 11% of the bins in the analysis band (1456 bins) appear on this final list.Finally, we also want to flag narrowband interference that would average out in the procedure described above, so we set an additional threshold in each processed spectrum in units of the standard deviation σ^p (Sec. <ref>). We cannot afford to be as aggressive in cutting such features because Gaussian statistics dictates that roughly 300 bins will exceed 4.5σ^p across all 6766 processed spectra. Thus we set the processed spectrum threshold at 6σ^p, resulting in an additional 0–30 bins cut from each processed spectrum. The distribution of these bins throughout the spectra implicates temporally intermittent IF interference rather than RF interference.§ REMOVING THE SPECTRAL BASELINEA typical raw power spectrum from the HAYSTAC detector, truncated to the analysis band, is shown in black in Fig. <ref>(a). As emphasized in Sec. <ref>, the spectral baseline is in principle the product of the total input-referred noise (which affects the sensitivity of the axion search) and the net gain of the receiver (which does not). On large spectral scales the shape of the baseline is mainly due to three effects. Rolloff at the low-RF (high-IF; see Fig. <ref>) end of the spectrum is due to room-temperature IF components, rolloff on the high-RF side comes from the JPA gain profile, and the intermediate region around the cavity resonance is enhanced by the heightened temperature of the tuning rod <cit.>. We see that there can be as much as ∼4 dB variation in the “gain” within a single spectrum.An average baseline obtained via the process described in Sec. <ref> is shown in red in Fig. <ref>(a). Systematic deviations of the raw spectrum from the average baseline indicate that the spectral baseline can change from one iteration to the next. Such variation is not surprising, as the JPA is a narrowband amplifier for which gain fluctuations imply bandwidth fluctuations. The excess noise on resonance also depends on frequency-dependent parameters of the cavity mode, and there may be many other effects that can cause the spectral baseline to vary.Nonetheless, normalizing each raw spectrum to the average baseline does reduce the typical variation across each spectrum from ∼4 dB to ∼0.5 dB; the normalized spectrum (which is now dimensionless) is shown in black in Fig. <ref>(b). At this point we also remove all the bins compromised by IF interference from each spectrum. The normalized spectrum with bad bins removed is shown in green in Fig. <ref>(b). Although only the analysis band is shown in Fig. <ref>, we actually apply the above steps to the analysis band plus 500 bins on either side. These extra bins essentially serve as buffer regions for the SG filter that we now employ to remove the residual baseline of each spectrum. §.§ The Savitzky-Golay filterThe simplest way to understand the SG filter is as a polynomial generalization of a moving average characterized by two parameters d and W. For each point x_0 in the input sequence (assumed to be much longer than W), we fit a polynomial of degree d in a 2W+1-point window centered on x_0. The value of the SG filter output at x_0 is defined to be the least-squares-optimal polynomial evaluated at the center of the window, and this process is repeated for each x_0; thus the filter output is a smoothed version of the input sequence, with edge effects within W points of either end.Savitzky and Golay <cit.> showed that this moving polynomial fit is equivalent to a discrete convolution of the input sequence with an impulse response that depends only on d and W. This correspondence implies that we can fruitfully think about least-squares-smoothing from the perspective of filtering rather than fitting. The even symmetry of the SG filter impulse response implies that only even values of d generate unique filters. We can gain further insight into the properties of SG filters by considering their performance in the frequency domain <cit.>. In the haloscope analysis considered here, we convolve the SG filter impulse response with an input sequence which is itself a power spectrum. Describing the Fourier transform of the SG impulse response as the filter's “frequency response” may thus be misleading; we will instead refer to this Fourier transform as a transfer function in the “inverse bin domain.”Two SG filter transfer functions used in the HAYSTAC analysis are plotted in Fig. <ref>. In general, SG filters are low-pass filters with extremely flat passbands and mediocre stopband attenuation. The 3 dB point that marks the transition between these two regions scales approximately linearly with d and approximately inversely with W. In particular, the 3 dB point for an SG filter with d=4 and W=500 (black solid line in Fig. <ref>) is ≈ 1/(517 bins). Thus when this filter is applied to one of the normalized spectra discussed above, features in the residual baseline on spectral scales sufficiently large compared to 51.7 kHz will be essentially perfectly preserved in the filter output, and features on smaller spectral scales are suppressed to varying degrees. The output of the SG filter applied to the normalized spectrum in Fig. <ref>(b) is shown in red on the same plot. After dividing each normalized spectrum by the corresponding SG filter output to remove the residual baseline, we can discard the 500 bins at either edge of each spectrum, whose only purpose has been to keep edge effects out of the analysis band; all subsequent processing is applied to the analysis band of each spectrum only.The design of any digital filter involves some tradeoff between passband and stopband performance, and we have seen that SG filters generally sacrifice some stopband attenuation to optimize passband flatness. It remains to be shown that this is the correct choice for a haloscope analysis. To see this, note that appreciable passband ripple implies the presence of systematic structure on large scales in the processed spectra. Such structure in turn implies that we cannot assume all processed spectrum bins are samples drawn from the same Gaussian distribution (see Sec. <ref>); thus we cannot construct a properly normalized measure of excess power in an arbitrary IF bin, which is a central assumption of the rest of the analysis. Imperfect stopband attenuation, on the other hand, implies that features and fluctuations on small spectral scales are slightly suppressed when we divide each normalized spectrum by the SG filter output; equivalently, the SG filter slightly attenuates axion signals and imprints small negative correlations between processed spectrum bins. We will show that we can quantify both the filter-induced signal attenuation (Sec. <ref>) and the effects of correlations on the statistics of the grand spectrum in which we ultimately conduct our axion search (Sec. <ref>). Computing the axion search sensitivity directly from the statistics of the spectra requires a thorough understanding of both effects.[The application of SG filters to spectral baseline removal in a haloscope search was first explored by Ref. <cit.>, which did not however adopt our frequency-domain approach or consider the effects of filter-induced correlations. See Ref. <cit.> for further discussion of this experiment.]The above discussion implies that passband flatness is a more important consideration than stopband attenuation for estimating spectral baselines in a haloscope analysis, and thus the SG filter is a good choice.[An optimal Chebyshev filter with coefficients obtained from the Parks-McClellan algorithm may be able to achieve better attenuation than the SG filter in the relevant part of the stopband while retaining the requisite passband flatness. We did not explore this approach for the present analysis.] Acceptable values of the filter parameters d and W are constrained by the integration time at each tuning step. Longer integrations make us sensitive to smaller-amplitude systematic structure in the baseline on smaller spectral scales, and we must push the 3 dB point of the SG filter up towards smaller scales to ensure that this structure remains confined to the passband (see Appendix <ref> for a more detailed discussion). We will see in Sec. <ref> that different values of d and W are appropriate for the analysis of rescan data. §.§ Statistics of the processed spectraAt each data run iteration, the total noise referred to the receiver input is statistically equivalent to thermal noise at some effective (possibly frequency-dependent) temperature; thus the noise voltage distribution is Gaussian, and the fluctuations in each Nyquist-resolution subspectrum will have a χ^2 distribution of degree 2. During data acquisition we average Δν_bτ = 9×10^4 such subspectra together, so the noise power fluctuations about the slowly varying baseline of each raw spectrum will be Gaussian by the central limit theorem. The baseline removal procedure described above should thus yield a set of flat dimensionless spectra, each with small Gaussian fluctuations about a mean value of 1. Ultimately, we are interested in excess power (which may be positive or negative) relative to the average noise power in each bin, so we subtract 1 from each spectrum after dividing out the SG filter output. We refer to the set of spectra obtained this way as the processed spectra; a representative processed spectrum is shown in Fig <ref>(c). In the absence of axion conversion, the bins in each processed spectrum should be samples drawn from a single Gaussian distribution with mean μ^p=0 and standard deviation σ^p=1/√(Δν_bτ)=3.3×10^-3. In Fig. <ref>(a) we have histogrammed all IF bins from all processed spectra together in units of σ^p. The excess power distribution is indeed Gaussian out to ≈5σ, and the excess above 5σ is likely due to intermittent IF interference slightly too small to exceed our 6σ^p threshold (Sec. <ref>). These large single-bin power excesses will be significantly diluted when we combine and rebin spectra.Fig. <ref>(a) indicates that each bin in each processed spectrum may be regarded as a random variable drawn from the same Gaussian distribution, and this is an important check on our baseline removal procedure. It does not follow that each spectrum is a sample of Gaussian white noise, because nearby bins in each spectrum will be correlated due to the imperfect stopband attenuation of the SG filter. We can observe effects of these correlations if we regard each spectrum (rather than each bin) as a sample of the same Gaussian process. Let δ^p_ij represent the value of the jth IF bin in the ith processed spectrum, for i=1,…,M and j=1,…,n^p; M=6766 and n^p=11564 for the first HAYSTAC run after the cuts discussed in Sec. <ref>. The ith processed spectrum has sample mean μ^p_i = 1/n^p∑_jδ^p_ijand sample variance(σ^p_i)^2 = 1/n^p-1∑_j(δ^p_ij-μ^p_i)^2.In the absence of correlations, the set of sample means should be Gaussian distributed about μ^p with standard deviation σ_μ = σ^p/√(n^p), and the set of sample variances should be Gaussian distributed about (σ^p)^2 with standard deviation σ_σ^2 = √(2/(n^p-1))(σ^p)^2, again by the central limit theorem. The presence of negative correlations on small spectral scales will reduce σ_μ substantially and also increase σ_σ^2 slightly, without appreciably changing the mean value of either distribution. Empirically, we find that σ_μ is smaller than the above estimate by an order of magnitude, and σ_σ^2 is larger by about 8%. The distortions of the sample mean and variance distributions noted above do not themselves affect the axion search sensitivity. But the correlations responsible for them are still important, since the remainder of our analysis procedure will involve taking both horizontal and vertical weighted sums of processed spectrum bins. A weighted sum of any number of independent Gaussian random variables is another Gaussian random variable, with mean given by the weighted sum of component means, and standard deviation given by the quadrature weighted sum of component standard deviations. If instead the random variables are jointly normal but correlated, the sum is still Gaussian and has the same mean, but computing the variance of the sum requires knowledge of the full covariance matrix. We will return to this point in Sec. <ref>.§ COMBINING SPECTRA VERTICALLYThe Mn^p processed spectrum bins δ^p_ij correspond to n^c < Mn^p unique RF bins (n^c≈1.07×10^6 for the first HAYSTAC data run). For notational convenience we define the symbol Γ_ijk=1 if the jth IF bin in the ith spectrum is one of the m_k bins corresponding to the kth RF frequency (Γ_ijk=0 otherwise). Our next task is to construct a single combined spectrum by taking an optimally weighted vertical sum of all m_k IF bins corresponding to each RF bin k. The m_k bins in each sum will be statistically independent, since each processed spectrum contains at most one IF bin to corresponding to any given RF bin k.To gain insight into the form of the optimally weighted sum, let us consider the simple case where all axion conversion power is confined to a single RF bin k'. Then each processed spectrum bin with Γ_ijk'=1 may be regarded as a sample from a Gaussian distribution whose mean is nonzero. We will initially assume that all of these bins have the same mean μ_k'=1 but possibly different standard deviations; of course, all bins with Γ_ijk'=0 also share a mean value, namely 0.This assumption allows us to formulate the requirement for an optimally weighted vertical sum more precisely: for each k we will choose weights that yield the maximum likelihood (ML) estimate of the true mean value μ_k shared by all the contributing bins. ML estimation is briefly summarized in Appendix <ref>. In Sec. <ref> we will see that ML weighting maximizes the SNR among all choices that yield unbiased estimates of the power excess.In practice, the sensitivity of any given processed spectrum bin to axion conversion will generally depend on both i and j, so each of the bins with Γ_ijk'=1 is actually a Gaussian random variable with a different nonzero mean. Moreover, we saw in Sec. <ref> that each bin in each processed spectrum has the same standard deviation σ^p – we did not consider axion signals when discussing the statistics of the processed spectra, but we should expect the fluctuations of the noise power to be independent of the presence or absence of axion conversion power. Evidently the assumption we used above to motivate the ML-weighted vertical sum was precisely backwards. We can cast the problem into a form amenable to ML weighting by rescaling the processed spectra so that axion conversion would yield the same mean power excess in any rescaled spectrum bin. Determining the appropriate rescaling factor is the subject of the next section. After rescaling the spectra, we can meaningfully define ML weights and thus construct the combined spectrum. §.§ The rescaling procedureWe rescale the processed spectra by multiplying each spectrum by the mean noise power per bin and dividing by the signal power. The jth bin in the ith rescaled spectrum is thenδ^s_ij = k_BT_ijΔν_bδ^p_ij/P_ij,where T_ij is the system noise temperature referred to the receiver input,[We follow the convention of haloscope papers in using “system noise temperature” to denote the total noise power per unit bandwidth, including whatever thermal noise enters the receiver along with the axion signal.] and P_ij is the total conversion power we expect from an axion signal confined to the jth bin of the ith spectrum.[Note that to set a definite normalization for the rescaling factor we need to assume specific values for the theory parameters we hope to constrain; the assumption of single-bin signals likewise amounts to a simple but physically implausible choice of normalization for P_ij. The exclusion limit which is the final product of our analysis will not depend on either arbitrary choice of normalization.]It may be helpful to discuss qualitatively why Eq. (<ref>) is the appropriate form for the rescaling factor. An axion signal with any given conversion power will be relatively suppressed by baseline removal if it happens to appear in a noisier spectrum or a noisier region of a given spectrum; multiplying by T_ij undoes this suppression. Dividing by the signal power undoes the relative suppression of conversion power in spectra that are less sensitive overall due to e.g. smaller cavity Q, and undoes the Lorentzian suppression of the conversion power for axions at nonzero detuning from the cavity resonance.The net result is that in the absence of noise, the hypothetical single-bin axion signal we have considered will yield δ^s_ij=1 in each bin with Γ_ijk'=1. In the presence of noise, each of these bins is a Gaussian random variable with mean μ^s_ij=1 and every other bin is a Gaussian random variable with μ^s_ij=0. The rescaled spectra are no longer flat: each bin has a standard deviationσ^s_ij = k_BT_ijΔν_bσ^p_i/P_ij.Note that σ^s_ij = (R^ s_ij)^-1, where R^ s_ij is the SNR for our hypothetical single-bin axion signal (c.f. Eq. (3) in Ref. <cit.>); this is completely equivalent to the statement that an axion signal in any bin of any rescaled spectrum produces a mean power excess of 1. A representative rescaled spectrum is shown in Fig. <ref>(d). Its overall shape is primarily due to the Lorentzian cavity mode profile.We have not yet addressed how we actually obtain values for P_ij and T_ij. The axion conversion power <cit.> may be expressed as P_ij = U_0(ν_ciβ_i/1+β_iC_iQ_Li/1+[2(ν_ij-ν_ci)/Δν_ci]^2),where U_0=g_γ^2α^2/π^2ħ^3c^3ρ_a/Λ^42π/μ_0η_LB_0^2Vis a constant with dimensions of energy. The factors we have absorbed into the definition of U_0 are independent of both i and j and thus only affect the overall normalization of the rescaled spectra. Here g_γ is a dimensionless number characterizing the strength of axion-photon coupling in a particular axion model, α is the fine-structure constant, ρ_a is the local energy density of dark matter axions, Λ = 77.6 MeV is a fixed parameter that encodes the dependence of the axion mass on hadronic physics,[The value of Λ used in our analysis comes from a calculation in chiral perturbation theory (see Ref. <cit.>). Note also that Λ^4 = χ(T = 0), where χ is the QCD topological susceptibility that may be calculated on the lattice. A recent lattice calculation reported in Ref. <cit.> obtained Λ = 75.6 MeV. With the latter value the haloscope signal power would be enhanced by 11%.] η_L is a signal attenuation factor (see below), B_0 = 9 T is the applied magnetic field, and V = 1.545 L is the cavity volume excluding the tuning rod.[Of course B_0 can change in principle, but we operate our magnet in persistent mode so in practice it is extremely stable over the course of the run.] The parameters that experiment can constrain are |g_γ| and ρ_a; it is conventional to fix ρ_a=0.45 GeV/cm^3 and cite the results of any given haloscope search as constraints on |g_γ|. To set a definite normalization for the rescaled spectrum, we need to temporarily fix both parameters, so we set |g_γ|=|g^KSVZ_γ| = 0.97, corresponding to the standard KSVZ model <cit.>.The remaining factors in Eq. (<ref>) are all properties of the TM_010 mode of the cavity that can vary as it is tuned. The mode has resonant frequency ν_ci, bandwidth Δν_ci, and quality factor Q_Li = ν_ci/Δν_ci. Its coupling to the receiver is parametrized the dimensionless number β_i, defined implicitly by Q_Li=Q_0i/(1+β_i), where Q_0i is the unloaded quality factor. The form factor C_i parametrizes the overlap between the spatial profile of the mode's electric field and the applied magnetic field. Finally, ν_ij is the RF frequency of the j^th bin in the i^th spectrum. We use the auxiliary data to obtain values for all these parameters except the form factor C_i, whose frequency dependence is obtained from simulations of the cavity mode. As discussed in Sec. <ref>, we made VNA measurements of the cavity mode in transmission both before and after each cavity noise measurement to cut iterations with excessive drift. For the remaining iterations, the “before” and “after” measurements are very similar, so we average them and fit the average to a Lorentzian to obtain ν_ci and Q_Li. We also used the VNA to measure the cavity mode in reflection: the magnitude of the reflection coefficient on resonance and the net resonant phase shift together determine β_i.The system noise temperature T_ij may be parametrized in units of quanta ask_BT_ij = hν_ci[N_T + (N_cav)_ij + (N_A)_ij],where N_T is thermal noise at the known mixing chamber temperature T_C, N_cav is the excess thermal noise associated with the elevated tuning rod temperature, and the receiver added noise N_A includes the added noise of the JPA preamplifier, small contributions from subsequent amplifiers, and effective noise associated with loss between the microwave switch and the JPA.[Technically, N_T is a function of frequency evaluated at ν_ci, but it changes negligibly over our tuning range, so we suppress its i-dependence. j-dependence due to the finite width of the analysis band is of course much smaller still.] We calibrate the noise using Y-factor measurements (discussed in detail in Ref. <cit.>); in our first data run, Y-factor measurements were repeated every 10 iterations (roughly 3.5 hours).Assuming a cavity in thermal equilibrium with the mixing chamber plate of the dilution refrigerator (i.e., N_cav=0), Y-factor measurements are ideal for the haloscope search noise calibration because they naturally measure N_A as defined above, in contrast with measurements of the SNR improvement from switching on the preamplifier, which are not sensitive to the loss contribution. Neither method is sensitive to losses between the cavity and the microwave switch ( ≈0.6 dB throughout the initial HAYSTAC tuning range), which nonetheless degrade the axion search SNR. Thus the factor η_L=10^-0.6/10=0.87 must be included explicitly in Eq. (<ref>).As already noted above, the cavity was not in thermal equilibrium with the mixing chamber in the first HAYSTAC data run, and this resulted in a contribution to the system noise temperature with a roughly Lorentzian profile centered on ν_ci in each spectrum. In the presence of this additional unknown noise N_cav, the Y-factor measurement associated with the ith spectrum measures not (N_A)_ij+(N_cav)_ij but rather (N_A)_ij + Y_ij/(Y_ij-A_ij)(N_cav)_ij, where Y is the measured ratio of hot/cold noise power spectra and A is the measured hot/cold gain ratio.[The additional factors multiplying the N_cav term account for the fact that it contributes only to the cold load noise measurement, whereas N_A contributes to the noise in both the hot load and the cold load; see also discussion in Ref. <cit.>.]N_A should be independent of the presence of the cavity mode in the spectrum, and empirically it also exhibits no systematic dependence on RF frequency. Thus we can break the degeneracy in Y-factor measurements around the TM_010 resonance by subtracting (N̅_A)_j, the average receiver added noise obtained from off-resonance Y-factor measurements. By doing so we obtain an estimate of N_cav in each Y-factor measurement throughout the data run, though this method implies that deviations from N̅_A across spectra are attributed instead to variation in N_cav.We do expect N_cav to vary across spectra due to variation in Q and β. Moreover, the effective temperature of the cavity mode is determined by a competition between the walls, which are well coupled to the mixing chamber, and the rod, which was at a higher temperature throughout the first HAYSTAC data run; the relative strength of these contributions will depend on the shape of the cavity mode and thus on the mode frequency. Empirically, there were clearly correlations among the N_cav profiles obtained from nearby Y-factor measurements, but no deterministic frequency dependence strong enough to justify any particular interpolation scheme. Thus, we simply set T_ij for each spectrum at which we did not make a Y-factor measurement using the nearest measured value of N_cav. In Appendix <ref> we estimate the uncertainty in our exclusion limit resulting from possible miscalibration of the noise temperature. §.§ Constructing the combined spectrumWe have shown that the rescaled spectrum IF bins corresponding to each RF bin are independent Gaussian random variables with the same mean (1 in the presence of a single-bin KSVZ axion and 0 in the absence of a signal) and different variances. To obtain the ML estimate of this mean value (see Appendix <ref>) we weight each bin by its inverse variance:w_ijk = Γ_ijk(σ^s_ij)^-2/∑_i'∑_j'Γ_i'j'k(σ^s_i'j')^-2,where the denominator ensures that the weights are normalized.[Many of the expressions to follow contain sums over i and j in both the numerator and denominator. We will avoid cumbersome primes through slight abuse of notation by using the same indices i and j in both sums. k, which is not summed over, is understood to have the same value in the numerator and denominator. Sums whose upper and lower limits are elided are to be interpreted as running over all possible values of the index.] Then the ML estimate of the mean in each combined spectrum bin k is given by the weighted sum of contributing bins: δ^c_k = ∑_i∑_jw_ijkδ^s_ij = ∑_i∑_jΓ_ijk(P_ijδ^p_ij/k_BT_ijΔν_b(σ^p_i)^2)/∑_i∑_jΓ_ijk(P_ij/k_BT_ijΔν_bσ^p_i)^2.The standard deviation of each bin in the combined spectrum is the quadrature weighted sum of contributing standard deviations:σ^c_k = √(∑_i∑_jw_ijk^2(σ^s_ij)^2) = √(∑_i∑_jΓ_ijk(σ^s_ij)^-4(σ^s_ij)^2/[∑_i∑_jΓ_ijk(σ^s_ij)^-2]^2) ⇒σ^c_k = [∑_i∑_jΓ_ijk(P_ij/k_BT_ijΔν_bσ^p_i)^2]^-1/2. For each k, there are m_k nonvanishing contributions to the sums in the expressions above. In the first HAYSTAC data run, typical values of m_k ranged from 50 to 120 across the combined spectrum due to nonuniform tuning.[There are two ∼ MHz-width peaks in the distribution of m_k with peak values of 150 and 200, due to scans in which the tuning rod was temporarily stuck at a single frequency. m_k also drops precipitously around the frequency of the intruder mode where we cut spectra (Sec. <ref>) and at the edges of the scan range. On spectral scales small compared to the analysis band width, m_k fluctuates by ±2 due to the presence of missing bins in the processed spectra.]Two numbers are required to characterize the combined spectrum at each frequency: δ^c_k and σ^c_k describe respectively the actual power excess in each combined spectrum bin and the power excess we expect from statistical fluctuations. Absent any axion signals, each δ^c_k should be a Gaussian random variable drawn from a distribution with mean μ^c_k = 0 and standard deviation σ^c_k. Thus the distribution of normalized binsδ^c_k/σ^c_k = ∑_i∑_jΓ_ijk(P_ijδ^p_ij/k_BT_ijΔν_b(σ^p_i)^2)/√(∑_i∑_jΓ_ijk(P_ij/k_BT_ijΔν_bσ^p_i)^2)should be Gaussian with zero mean and unit variance; we can see in Fig. <ref>(b) that this is indeed the case.[In practice δ^c_k/σ^c_k will still appear to have a standard normal distribution even in the presence of axion conversion, since μ^c_k≠0 in only a few bins.]We can equivalently describe the combined spectrum by specifying the values of δ^c_k/σ^c_k and R^ c_k = (σ^c_k)^-1 for each k. The normalization of the ML weights implies that, for a single-bin KSVZ axion at frequency k', μ^c_k'=1 and thus E[δ^c_k'/σ^c_k']=R^ c_k'. Physically, R^ c_k is the SNR that a single-bin KSVZ axion would have in the kth bin of the combined spectrum (whether or not such an axion exists). In terms of the SNR, Eq. (<ref>) becomesR^ c_k = √(∑_i∑_jΓ_ijk(R^ s_ij)^2),which tells us that the SNR in each bin of the combined spectrum is simply the (unweighted) quadrature sum of the SNR across contributing bins. As discussed in Appendix <ref>, the ML estimate of the mean of a Gaussian distribution also has the smallest variance among unbiased estimates. The variance of the mean of a Gaussian distribution is simply proportional to the variance of the distribution, so equivalently ML weights yield the smallest σ^c_k and thus the largest R^ c_k among all possible weights that do not systematically bias δ^c_k. Thus, ML weighting is optimal for the haloscope analysis in a real physically intuitive sense.§ COMBINING BINS HORIZONTALLYThe parameterization of the combined spectrum in terms of δ^c_k/σ^c_k and R^ c_k lends itself naturally to identifying axion candidates and setting exclusion limits, via the procedure outlined in Sec. <ref>. However, R^ c_k is the (unrealistically large) SNR for an axion signal confined to a single bin, whereas our goal here is to construct an analysis tailored to the detection of virialized axions with Δν_a ≫Δν_b. Thus, our next task is to determine an explicit expression for the grand spectrum δ^g_ℓ/σ^g_ℓ as a weighted sum of adjacent combined spectrum bins. As in Sec. <ref>, we take the optimal weights to be those that yield the ML estimate of the mean grand spectrum power excess, after rescaling to make the expected excess due to axion conversion uniform across all contributing bins. The discussion above indicates that ML weights in the horizontal sum will maximize R^ g_ℓ, the SNR for a virialized axion signal concentrated in the ℓth grand spectrum bin. In the choice of ML weights for the vertical sums that define the combined spectrum, we have followed the published ADMX analysis procedure <cit.>, albeit with a somewhat different approach for pedagogical purposes.[See also Refs. <cit.> for different presentations of ML weighting in the ADMX analysis; note that there are a number of errors in the expressions corresponding to Eqs. (<ref>) and (<ref>) in Refs. <cit.>, <cit.>, and <cit.>.] In extending ML weighting to horizontal sums of adjacent bins in the combined spectrum, we are deviating from the procedure used by ADMX. We discuss the key differences between our present approach and the ADMX procedure further in Sec. <ref>.Though the principles of ML estimation remain valid, horizontal sums differ from the vertical sums considered in Sec. <ref> in two important respects. First, we can no longer assume that the bins in each sum are independent random variables; indeed, as noted in Sec. <ref>, we have reason to expect correlations on small spectral scales in the processed spectra, and thus also in the combined spectrum. ML estimation of the mean of a multivariate Gaussian distribution with arbitrary covariance matrix is in principle straightforward (see Appendix <ref>). In practice, it requires additional information about off-diagonal elements of the covariance matrix that are not as easily estimated as the variances. In the present analysis, we take ML weights that neglect correlations as approximations to the true ML weights, and define the horizontal sum using expressions appropriate for the uncorrelated case. We will quantify the effects of correlations in Sec. <ref>.Second, independent of any subtleties involving correlations, we have some additional freedom in how we define the horizontal sum besides the choice of weights. The simplest approach is to define each grand spectrum bin as a ML-weighted sum of all bins within a segment of length K≈Δν_a/Δν_b in the combined spectrum, such that the segments corresponding to different grand spectrum bins do not overlap. The total number of grand spectrum bins is then n^g≈ n^c/K. The disadvantage of this approach is that the signal power will generally be split across multiple bins unless ν_a happens to line up with our binning. We need to introduce an attenuation factor η_m (Sec. <ref>) to account for the average effect of misalignment on the SNR.We can minimize misalignment effects by allowing the segments of the combined spectrum corresponding to different grand spectrum bins to overlap: if each such segment is K bins long, then the first grand spectrum bin will be a ML-weighted sum of the first through Kth combined spectrum bins, the second grand spectrum bin will be a ML-weighted sum of the second through (K+1)th bins, and so on. But with K≈Δν_a/Δν_b this procedure implies a total of n^g≈ n^c grand spectrum bins, and thus the number of statistical rescan candidates (Sec. <ref>) will be larger at any given sensitivity than in the non-overlapping case; equivalently the total integration time required to exclude axions of a given coupling will be longer.The two approaches considered above may be regarded as limiting cases of a more general procedure in which we split the construction of the grand spectrum into two steps. First we take ML-weighted sums of adjacent bins in non-overlapping segments of the combined spectrum to yield a rebinned spectrum with resolution Δν_r = K^rΔν_b. Then we construct the grand spectrum via ML-weighted sums of adjacent bins in overlapping segments of length K^g in the rebinned spectrum. K^r and K^g should be chosen so that the product K^rK^g≈Δν_a/Δν_b; it should be emphasized that we have thus far cited only a very rough estimate for Δν_a, and we are free to choose K^r and K^g independently within a reasonable range. In the two-step procedure described above, the rebinned spectrum weights and grand spectrum weights are each obtained from the ML principle, but of course we must specify a supposed distribution of signal power before we can define ML weights. The ℓth grand spectrum bin should be a sum over bins in the rebinned spectrum frequency range [ν_ℓ,ν_ℓ+K^g-1] weighted so that the SNR is maximized if ν_a≈ν_ℓ. We will articulate this condition more precisely in Sec. <ref>, but we can already see that the grand spectrum weights will depend on the axion lineshape. The weights used to construct the rebinned spectrum cannot themselves depend on the lineshape: the above example demonstrates that any given ν_ℓ will correspond to the axion mass in one grand spectrum bin and the tail of the axion power distribution in another. We thus define weights to yield the ML estimate of the mean power excess in each bin of the rebinned spectrum assuming the axion signal distribution is uniform across contributing combined spectrum bins. As we reduce K^r, the distribution of signal power on scales smaller than Δν_r becomes more uniform, and we can also use a finer approximation to the axion lineshape in the grand spectrum weights. For the analysis of the first HAYSTAC data run we used K^r=10 and K^g=5, informed by the tradeoffs noted above. In the next section, we will briefly digress on the expected axion lineshape and its implications for the analysis. Then we will construct the rebinned spectrum in Sec. <ref> and the grand spectrum in Sec. <ref>. §.§ The expected axion signal lineshapeExperiments aiming to directly detect non-gravitational interactions of dark matter must make assumptions about the local dark matter mass and velocity distributions. Virialization of the dark matter in the galactic halo relates these two distributions. Searches typically assume a virialized halo which is moreover spherically symmetric and approximately isothermal, such that the dark matter velocity distribution is very nearly Maxwellian in the galactic rest frame. Such a pseudo-isothermal distribution <cit.> is fully specified by the values of two parameters, which we can take to be the local density ρ_a=0.45 GeV/cm^3 <cit.> and the local circular velocity v_c=220 km/s; the latter is the mode of the Maxwell-Boltzmann distribution. It is also possible that some fraction of the dark matter has not virialized due to cold, high-density streams of axions that fell into the galaxy relatively recently; such streams would manifest as sharp features in the spectrum of a putative haloscope signal. Fixing the values of the experimental parameters, a haloscope search specifically targeting non-virialized axions will generally be sensitive to smaller couplings |g_γ| because the signal bandwidth is smaller, but the converse is not true: the sensitivity of a search that assumes virialization is not appreciably degraded if there is non-virialized structure in the true signal. In this sense virialization is a conservative assumption.[The orbital motion of the earth <cit.> can also shift the frequency of a non-virialized axion signal by an amount comparable to its linewidth between repeated scans around the same frequency. Roughly speaking, searches for non-virialized signals of fractional width Δν_a/ν_a≲10^-7 must make further assumptions about the direction of the axion stream unless candidates were rescanned more frequently than once per week during the acquisition of the search data set, with correspondingly more stringent requirements on the frequency of rescans for narrower signals.]For the present analysis we assume a fully virialized pseudo-isothermal halo, emphasizing that its chief virtues are simplicity and the absence of strong evidence for any particular alternative; see Ref. <cit.> for a recent discussion of alternative halo models in the haloscope search. The form in which we save the HAYSTAC axion search data (Sec. <ref>) enables future searches for nonvirialized features with fractional width as small as Δν_b/ν_a∼2×10^-8. The spectral shape of a haloscope signal is proportional to the axion kinetic energy distribution. For a pseudo-isothermal halo in the galactic rest frame, axion velocities obey a Maxwell-Boltzmann distribution, and the corresponding kinetic energies have a χ^2 distribution of degree 3. As a function of the measured signal frequency ν≥ν_a, this distribution isf(ν) = 2/√(π)√(ν-ν_a)(3/ν_a<β^2>)^3/2e^-3(ν-ν_a)/ν_a<β^2>,where <β^2>=<v^2>/c^2 and the second moment of the Maxwell-Boltzmann distribution is <v^2> = 3v_c^2/2 = (270 km/s)^2. In a frame moving relative to the galactic rest frame, the dark matter velocity distribution is not in general Maxwellian. The motion of a terrestrial laboratory relative to the galactic halo is dominated by the orbital velocity of the sun about the center of the galaxy v_s≈ v_c. In the lab frame the spectrum of the axion signal thus becomes <cit.>f'(ν) = 2/√(π)(√(3/2)1/r1/ν_a<β^2>)sinh(3r√(2(ν-ν_a)/ν_a<β^2>)) ×exp(-3(ν-ν_a)/ν_a<β^2>-3r^2/2),where r=v_s/√(<v^2>)≈√(2/3). Eq. (<ref>) is not a χ^2 distribution, but is reasonably well approximated by Eq. (<ref>) with <β^2>→1.7<β^2>; of course, it approaches Eq. (<ref>) in the limit r→0. We used Eq. (<ref>) where we should have used Eq. (<ref>) in our original analysis.[We thank B. R. Ko for drawing our attention to this point.] Specific parameter values cited throughout Sec. <ref> and <ref> assume Eq. (<ref>), as this was used to derive the exclusion limit published in Ref. <cit.>, but we emphasize that the formal procedure outlined in the present paper is independent of any specific assumptions about the signal shape. If the spectrum of the axion signal is actually given by Eq. (<ref>), our exclusion limit will be degraded by ≈20% (quantified more precisely in Appendix <ref>) due to the combination of an irreducible effect from the wider signal bandwidth and the fact that our analysis was not optimized for this wider signal, as future HAYSTAC analyses will be.A haloscope analysis can ultimately depend on the spectral shape of the axion signal only through the grand spectrum weights, which in turn can only depend on slices of f(ν) integrated over the resolution of the rebinned spectrum Δν_r=K^rΔν_b. Thus we define the integrated signal lineshape to beL_q(δν_r) = K^g∫_ν_a+δν_r+(q-1)Δν_r^ν_a+δν_r+qΔν_rf(ν) dν,where q=1,…,K^g, and the misalignment δν_r is defined in the range -zΔν_r < δν_r ≤ (1-z)Δν_r, with 0<z<1. The value of z should be chosen so that for any δν_r in this range, η_c(δν_r) = ∑_qL_q(δν_r)/K^g is larger than the value we would obtain by shifting the range over which the q index is defined up or down by 1.[We might naively imagine a symmetric interval (corresponding to z=0.5) would be optimal in this sense. In practice, given the asymmetry of the axion lineshape, there will be more power in the K^g-bin sum if the lower bound of the integral in the q=1 bin is detuned below ν_a than at an equal detuning above ν_a. This implies that we should consider z>0.5; the optimal value will depend on the choice of K^r and K^g.] Physically, η_c is the fraction of signal power contained within a grand spectrum bin; it approaches 1 independent of δν_r for K^g sufficiently large. At any fixed value of K^g, the sum also depends on δν_r and thus on K^r.We can gain some insight into the considerations that enter into the choice of K^r and K^g by imagining for the moment that we take the grand spectrum weights to be uniform, as in Ref. <cit.>. Then, with K^r=1, η_c→1 as K^g increases, but the RMS noise power grows as √(K^g), so the grand spectrum SNR (∝η_c/√(K^g)) is maximized at a finite value of K^g. The SNR is relatively insensitive to δν_r at K^r=1; as we increase K^r, keeping K^rK^g fixed, η_c remains unchanged in the best-case scenario δν_r=0, but larger misalignments are possible, so dependence of the SNR on δν_r grows more pronounced.In order to define ML weights for the grand spectrum (Sec. <ref>), we will need an expression for some “typical” lineshape L̅_q that is independent of misalignment. The best approach is to define L̅_q as the average of L_q(δν_r) over the range in which δν_r is defined.[L̅_q has no ℓ index because in practice we evaluated Eq. (<ref>) with ν_a = 5.75 GHz both in the limits of integration and within f(ν). It would be trivial to instead calculate the lineshape with ν_a=ν_ℓ in the ℓth grand spectrum bin, but the variation of the lineshape over the initial HAYSTAC scan range was negligible.] Then the misalignment attenuation can be defined as η_m= SNR({L̅_q})/SNR({L_q(δν_r=0)}).[With this definition, η_m is a useful figure of merit for comparing different values of K^r and K^g, but we will not have to explicitly account for it in our analysis procedure, as the average effect of misalignment on the SNR is included in the definition of L̅_q.] In the ML-weighted grand spectrum the SNR is no longer proportional to η_c (indeed, it asymptotes to a constant value rather than degrading as we continue to increase K^g). However, the above prescription for defining η_m still holds if we use the correct expression for the SNR [Eq. (<ref>) in Appendix <ref>]. With K^r=10 and K^g=5, the optimal range for δν_r is obtained for z=0.7, and the misalignment attenuation is η_m=0.97. §.§ Rebinning the combined spectrumAfter choosing the values of K^r and K^g to be used in the remainder of the analysis, we rescale the combined spectrum, taking δ^c_k → (K^rK^g)δ^c_k and σ^c_k → (K^rK^g)σ^c_k. This rescaling leaves δ^c_k/σ^c_k formally unchanged and takes R^ c_k → R^ c_k/(K^rK^g), just what we would have obtained had we normalized Eq. (<ref>) to a more physically plausible fraction 1/(K^rK^g) of the expected KSVZ signal power in the first place. After this rescaling we expect μ^c_k'=1 if a KSVZ axion signal deposits a fraction 1/(K^rK^g) of its power in the combined spectrum bin k'.In Sec. <ref> we wrote rather verbose expressions for Eqs. (<ref>) and (<ref>) to make the dependence on physically meaningful quantities such as P_ij explicit. The ML-weighted sum can be written more succinctly in terms ofD^c_k = δ^c_k/(σ^c_k)^2 = 1/K^rK^g∑_i∑_jΓ_ijkδ^s_ij/(σ^s_ij)^2,which is just the sum in the numerator of Eq. (<ref>) rescaled by 1/(K^rK^g) as discussed above. Each D^c_k is a Gaussian random variable with standard deviation R^ c_k. We obtain the ML-weighted rebinned spectrum fromD^r_ℓ = ∑_k=k_i(ℓ)^k_f(ℓ)D^c_kand(R^ r_ℓ)^2 = ∑_k=k_i(ℓ)^k_f(ℓ)(R^ c_k)^2,where k_i(ℓ)=(ℓ-1)K^r+1, k_f(ℓ)=ℓ K^r, ℓ=1,…,n^r, and n^r≈ n^c/K^r≈1.07×10^5 for the first HAYSTAC data run. In the absence of correlations between combined spectrum bins, D^r_ℓ is a Gaussian random variable with standard deviation R^ r_ℓ. Defining σ^r_ℓ = (R^ r_ℓ)^-1 and δ^r_ℓ = D^r_ℓ(σ^r_ℓ)^2 as in the combined spectrum, it follows that each rebinned spectrum bin δ^r_ℓ is a Gaussian random variable with standard deviation σ^r_ℓ (and mean μ^r_ℓ=0 in the absence of axion signals). Each δ^r_ℓ is the ML-weighted estimate of the mean power excess in K^r adjacent combined spectrum bins δ^c_k if the axion power distribution is uniform on scales smaller than Δν_r. More precisely, μ^r_ℓ'=1 if a KSVZ axion deposits a fraction 1/(K^rK^g) of its power in each of the K^r adjacent combined spectrum bins corresponding to the rebinned spectrum bin ℓ', and R^ r_ℓ' is the SNR for such a signal. Neglecting small-scale variation in R^ c_k, Eq. (<ref>) implies that the SNR in each bin of the rebinned spectrum has increased by √(K^r). This is exactly what we should expect given that the signal power grows roughly linearly with bandwidth Δν (for Δν sufficiently small compared to Δν_a) and the RMS noise power grows as √(Δν). Empirically, the RMS variation in σ^c_k is typically ≲1% on 10-bin scales (and ≈3% on 50-bin scales), so the rebinned spectrum would not change much if we used uniform weights instead of ML weights. We will see in Sec. <ref> that ML weighting of the grand spectrum leads to a larger improvement relative to an unweighted analysis.In the absence of correlations, each δ^r_ℓ has standard deviation σ^r_ℓ, so δ^r_ℓ/σ^r_ℓ should have a standard normal distribution, like the analogous quantity in the combined spectrum. Empirically, in the first HAYSTAC data run, δ^r_ℓ/σ^r_ℓ was Gaussian with standard deviation ξ^r=0.98. ξ^r≠1 is a consequence of the fact that the expression for the variance of a sum of Gaussian random variables used in Eq. (<ref>) does not hold in the presence of correlations, as noted at the end of Sec. <ref>.[A similar reduction in the standard deviation following a horizontal sum was observed in Ref. <cit.>, pg. 122, and attributed to the baseline removal procedure, but not discussed further.] An analogous effect will arise in the construction of the grand spectrum, so we will defer further discussion of this point to Sec. <ref>. §.§ Constructing the grand spectrumTo extend the ML-weighted horizontal sum further, we must account for the fact that, for any given value of ν_a, the distribution of axion signal power in the K^g rebinned spectrum bins containing most of the signal is nonuniform. Specifically, for a KSVZ axion with ν_a≈ν_ℓ', we expect μ^r_ℓ'+q-1=L̅_q for q=1,…,K^g. As in Sec. <ref>, we must rescale the contributing bins so that they all have the same mean power excess before defining ML weights. For the ℓth grand spectrum bin, the appropriate rescaling is obtained by dividing both δ^r_ℓ+q-1 and σ^r_ℓ+q-1 by L̅_q, or equivalently by multiplying both D^r_ℓ+q-1 and R^ r_ℓ+q-1 by L̅_q. The quantities of interest in the ML-weighted grand spectrum are then given byR^ g_ℓ = √(∑_q(R^ r_ℓ+q-1L̅_q)^2)andδ^g_ℓ/σ^g_ℓ = D^g_ℓ/R^ g_ℓ = ∑_q D^r_ℓ+q-1L̅_q/√(∑_q(R^ r_ℓ+q-1L̅_q)^2),for ℓ=1,…,n^g, and n^g≈ n^r.Neglecting effects of the SG filter stopband, each δ^g_ℓ should be a Gaussian random variable with standard deviation σ^g_ℓ=(R^ g_ℓ)^-1 and mean μ^g_ℓ. Our definition of L̅_q in Sec. <ref> implies that μ^g_ℓ'=1 (equivalently, E[δ^g_ℓ'/σ^g_ℓ']=R^ g_ℓ') for a KSVZ axion signal with average misalignment in bin ℓ'.[Here and elsewhere in this paper, “an axion signal in the grand spectrum bin ℓ'” should be taken as shorthand for the condition -0.7Δν_r < ν_ℓ'-ν_a < 0.3Δν_r, where ν_ℓ refers to the frequency at the lower edge of bin ℓ. For detunings outside this range, the SNR will be larger in a different grand spectrum bin, and we will speak of the signal “in” that bin instead.] The small uncertainty in μ^g_ℓ' associated with the range of possible misalignments will contribute to the uncertainty in our exclusion limit, discussed in Appendix <ref>. Within K^g bins of ℓ', 0<μ^g_ℓ<1, because the overlapping horizontal sum correlates nearby grand spectrum bins.[It should be emphasized that these correlations are independent of, and would occur even in the absence of, the correlations between combined spectrum bins responsible for ξ^r≠1. The implications of these grand spectrum correlations for the analysis will be discussed further in Sec. <ref>.] Of course, μ^g_ℓ=0 for |ℓ-ℓ'|≥ K^g. Empirically, δ^g_ℓ/σ^g_ℓ [histogrammed in Fig. <ref>(c)] has a Gaussian distribution with mean 0 and standard deviation ξ=0.93. We saw above that correlations within each bin of the rebinned spectrum already reduced the width of the histogram by a factor ξ^r=0.98, which implies that the reduction we can attribute specifically to correlations between different rebinned spectrum bins is ξ^g=ξ/ξ^r=0.95.Setting aside the issue of correlations, we can gain further insight into the properties of our ML-weighting horizontal sum by considering how it differs from the corresponding step in the ADMX haloscope analysis procedure. ADMX analyses tailored to the detection of virialized axions have consistently used Δν_b/Δν_a approximately a factor of 10 larger than in the present analysis and K^r=1 (i.e., no rebinning after data combining). The original ADMX analysis <cit.> took the grand spectrum to be an unweighted sum of K^g=6 combined spectrum bins. This is not quite the same as setting L̅_q=1 in Eqs. (<ref>) and (<ref>) because our sums are still ML-weighted by (σ^r_ℓ)^-2 in this limit. However, as noted in Sec. <ref>, the variation in σ^c_k on the relevant scales is small enough that in practice there is not much difference. Thus we will compare our ML analysis to the unweighted K^g-bin sum in the limit that σ^r_ℓ (equivalently R^ r_ℓ) is equal in all contributing bins. In this limit, the grand spectrum SNR may be written in the formR^ g_ℓ=F(K^g,Δν_r,{L̅_q})K^g√(Δν_r)R^ r_ℓ,where we have introduced a figure of merit F to encode the dependence of R^ g_ℓ on K^r, K^g, and L̅_q. It becomes apparent that R^ g_ℓ only depends on these quantities through F when we rewrite the rebinned spectrum SNR in the form R^ r_ℓ=[(P_ℓ/K^g)/(k_BT_ℓ)]√(τ/Δν_r),where P_ℓ (T_ℓ) is an appropriately weighted average of the total axion conversion power (noise temperature) in all contributing processed spectrum bins.For our ML-weighted analysis, we obtain an explicit expression for F by comparing Eq. (<ref>) to Eq. (<ref>):F_ML = √(1/Δν_r∑_q(L̅_q/K^g)^2).The figure of merit for an unweighted sum follows from Eq. (<ref>) and R^ g_ℓ=√(K^g)η_cR^ r_ℓ (see Sec. <ref>):F_uw = 1/√(K^gΔν_r)∑_q L̅_q/K^g. For a meaningful comparison between analyses, we must assume the same underlying signal spectrum f(ν) in both cases. If we also assume that both analyses are characterized by the same values of K^g and Δν_r, and thus the same L̅_q, then F_uw is just the mean of L̅_q multiplied by (K^gΔν_r)^-1/2, whereas F_ML is the RMS of L̅_q times the same factor. Thus F_ML≥ F_uw independent of any specific features of the lineshape; this is another way to understand the improvement in sensitivity from ML weighting.[Eq. (<ref>) only quantifies the true improvement in the SNR from a ML analysis if our analysis has assumed the correct signal lineshape, but insofar as the true signal distribution is closer to the nominal lineshape than to a “boxcar” of width K^gΔν_r, the ML analysis will still be more sensitive than an unweighted sum.]We can also use Eqs. (<ref>) and (<ref>) to compare the sensitivity of analyses based on the same model f(ν) but characterized by different Δν_r and/or K^g and thus different L̅_q; this is a convenient way to quantify the considerations discussed in Sec. <ref>. For f(ν) given by Eq. (<ref>), the improvement in the SNR from an optimal ML-weighted analysis relative to an optimal unweighted analysis is about 7.5%.[Here “optimal” means the SNR is maximized with respect to Δν_r and K^g (or, for ML weighting, it is sufficiently close to its asymptotic value). The values of Δν_r and K^g adopted for the present analysis are not optimal in this sense, and indeed the SNR for our present analysis is only about 2% better than the SNR in the optimal unweighted case. However, this optimization does not take into account the fact that the integration time required for rescans increases as we reduce Δν_r, as emphasized at the beginning of Sec. <ref>. A better comparison would consider the improvement in SNR for a ML analysis relative to an unweighted analysis that results in comparable total rescan time. Our present ML analysis has 11.5% better SNR than the unweighted analysis with the same Δν_r and K^g.]In more recent ADMX analyses <cit.>, the grand spectrum is defined as a weighted sum of combined spectrum bins, with weights corresponding to the coefficients of a Wiener Filter (WF). In our notation, the WF weight for the bin δ^r_ℓ+q-1 isu^WF_q = L̅_q^2/L̅_q^2+(σ^r_ℓ+q-1)^2,up to a normalization factor. These weights are obtained as solutions to the least-squares minimization of the difference between the noisy observations δ^r_ℓ+q-1 and the mean power L̅_q independently in each bin. In the high-SNR limit σ^r_ℓ+q-1≪L̅_q, u^WF_q→1, whereas in the low-SNR limit, u^WF_q→ (L̅_q/σ^r_ℓ+q-1)^2. In neither limit do they agree with the unnormalized ML weights,[Here we are comparing the coefficients of the bins δ^r_ℓ+q-1 in the ML and WF analyses. The ML weights are more properly defined as the coefficients of the rescaled bins δ^r_ℓ+q-1/L̅_q. With this definition the numerator is (L̅_q/σ^r_ℓ+q-1)^2, but there is no such rescaling step in the WF analysis.] u^ML_q = L̅_q/(σ^r_ℓ+q-1)^2.The origin of this discrepancy is the fact that, while the ML and WF schemes are both based on least-squares optimization, they are obtained by minimizing the mean squared error with respect to different quantities: the ML procedure yields the least-squares optimal estimate of the mean power excess in the (appropriately rescaled) contributing bins (and thus results in larger SNR than all other unbiased analyses, as noted in Sec. <ref>), whereas the WF procedure yields the least-squares optimal estimates of the weights that most robustly undo the smearing of the axion lineshape due to the presence of noise. In our view, the ML scheme relates more directly to the fundamental quantities of interest in the haloscope search.Finally, we briefly note one more practical difference between the WF and ML methods: the WF weights depend on the SNR, whereas the ML weights only depend on the shape of the axion signal independent of any overall normalization. In practice the WF should be evaluated at an estimate of the average threshold sensitivity |g^min_γ| to be obtained from the analysis. In the high-SNR limit, the WF sum becomes unweighted, and the SNR improvement from ML weighting may be estimated from F_ML/F_uw as noted above. §.§ Accounting for correlationsIn the discussion above we noted two distinct effects on the rebinned spectrum [Eqs. (<ref>) and (<ref>)] and grand spectrum [Eqs. (<ref>) and (<ref>)] due to correlations between nearby combined spectrum bins. First, we have not used the correct expression for the variance of a weighted sum of correlated Gaussian random variables in Eqs. (<ref>) and (<ref>). Second, in the presence of correlations, the weights we have used are not actually the optimal ML weights. The former effect is responsible for ξ^r,ξ^g≠1; note that it is completely independent of whether or not the weights are optimal. We will consider the effect on the variance first; doing so will allow us to estimate the sum of off-diagonal elements in the relevant covariance matrices, and thus quantify the deviation from the optimal weights.The most general expression for the variance of a weighted sum of K Gaussian random variables X_q is Var(∑_qw_qX_q) =∑_q w_q^2 Var(X_q) +⋯ ⋯+2∑_q∑_q'=1^q-1w_qw_q'Cov(X_q,X_q').We will apply this expression to obtain the correct variance (σ̂^g_ℓ)^2 of the ℓth grand spectrum bin. With X_q=δ^r_ℓ+q-1/L̅_q and the grand spectrum weights used in Sec. <ref>, we obtain(σ̂^g_ℓ)^2 = (σ^g_ℓ)^2+2(σ^g_ℓ)^4∑_q=1^K^g∑_q'=1^q-1L̅_qL̅_q'Σ^r_ℓ qq'/(σ^r_ℓ+q-1σ^r_ℓ+q'-1)^2,where Σ^r_ℓ qq' = Cov(δ^r_ℓ+q-1,δ^r_ℓ+q'-1), and the factor of (σ^g_ℓ)^4 multiplying the second term comes from the normalization of the ML weights. The analogous expression for the correct variance of the ℓth rebinned spectrum bin is(σ̂^r_ℓ)^2 = (σ^r_ℓ)^2+2(σ^r_ℓ)^4∑_k=k_i(ℓ)^k_f(ℓ)∑_k'=k'_i(ℓ)^k-1Σ^c_kk'/(σ^c_kσ^c_k')^2,with Σ^c_kk' = Cov(δ^c_k,δ^c_k'). Having established the requisite formalism, we can now ask whether taking correlations into account explains the observed reduction of the grand spectrum and rebinned spectrum standard deviations. We see immediately that σ̂^g_ℓ can be smaller than σ^g_ℓ if the sum over off-diagonal elements of the covariance matrix is on average slightly negative. Formally the ratio σ̂^g_ℓ/σ^g_ℓ is frequency-dependent, but if nonzero Σ^r_ℓ qq' is a consequence of the stopband properties of the SG filter, we should expect the correlation matrix ρ^r_ℓ qq'=Σ^r_ℓ qq'/(σ^r_ℓ+q-1σ^r_ℓ+q'-1) to depend only on the bin spacing Δ q= q-q'. Analogous arguments also apply to the ratio σ̂^r_ℓ/σ^r_ℓ. Thus we expectξ^g=σ̂^g_ℓ/σ^g_ℓandξ^r=σ̂^r_ℓ/σ^r_ℓin the case of filter-induced correlations. We used a simulation to show that the observed values of ξ^r and ξ^g are indeed fully explained by processed spectrum correlations imprinted by the SG filter. Each iteration in the ξ^g simulation generates a set of m 14020-bin Gaussian white noise spectra with mean 1 and standard deviation σ^p=1/√(Δν_bτ), multiplies each spectrum by a random sample baseline derived from data, then uses the baseline removal procedure described in Sec. <ref> to obtain a set of simulated processed spectra.[The sample baselines used here and in the simulation described in Sec. <ref> were each obtained by applying a high-order SG filter (as in Sec. <ref>) to the average of about 50 consecutive raw spectra after removing contaminated bins.] The m processed spectra are averaged without weighting or offsets to obtain a single simulated combined spectrum, in which we average non-overlapping 10-bin segments. We calculate the product of each pair of bins with 0 ≤Δ q ≤ 5 in the simulated rebinned spectrum. Averaging each such product over ≈500 iterations of the simulation, we obtain reasonably precise estimates of (σ^r_ℓ)^2 and Σ^r_ℓ qq' for each bin ℓ in the rebinned spectrum. Then we calculate σ^g_ℓ and σ̂^g_ℓ from Eqs. (<ref>) and (<ref>), and ξ^g from Eq. (<ref>). We find that ξ^g=0.95 is constant throughout the analysis band, independent of m for values ranging from m=1 out to at least m=400>max(m_k) and independent of τ out to at least τ=900 s.[Our simulation and Eq. (<ref>) measure ξ^g rather than ξ=ξ^gξ^r because we use σ^r_ℓ rather than σ̂^r_ℓ in Eqs. (<ref>) and (<ref>). Note also that m_k is itself an upper bound on the averaging in each bin, because contributing spectra are not uniformly weighted.] From an analogous simulation to quantify the effects of correlations on the rebinned spectrum we obtain a constant ξ^r=0.98. To verify that the implementation of the simulation was correct, we calculate the same quantities from the simulated Gaussian white noise spectra directly (bypassing the steps where we imprint and then remove the baseline); we obtain ξ^g=ξ^r=1 as expected for this null test.These results demonstrate conclusively that the observed values of ξ^r and ξ^g depend only on the stopband properties of the SG filter. Fig. <ref> indicates that the filter-induced negative correlations increase at larger bin separations, consistent with the empirical result 1-(ξ^g)^2>1-(ξ^r)^2. The explicit demonstration that ξ^g and ξ^r are independent of m is critical because in the real data m_k varies throughout the combined spectrum: m-independence implies that nonuniform weighting and frequency offsets between processed spectra will not affect our results. We conclude that ξ^r and ξ^g are frequency-independent, as indeed the numerical agreement between the simulated and observed values already indicates.[The values of ξ^g and ξ^r obtained from the real data were unchanged when we divided the axion search dataset in half in various ways (winter/summer, high/low RF frequency, upper/lower half of analysis band) and constructed the grand spectrum separately from each subset of the data.]It follows that each grand spectrum bin δ^g_ℓ is a Gaussian random variable with standard deviation σ̃^g_ℓ=ξσ^g_ℓ=ξ^gξ^rσ^g_ℓ.and mean μ^g_ℓ=0 in the absence of axion signals. Now let us suppose there exists a KSVZ axion in bin ℓ' of the grand spectrum. If the only effect of the imperfect SG filter stopband were to correlate the statistical fluctuations of the noise in nearby bins, we would still have μ^g_ℓ'=1, since the mean of a weighted sum of Gaussian random variables is independent of whether or not they are correlated. However, the imperfect SG filter stopband will also lead to slight attenuation of any locally correlated power excess (e.g., an axion signal) in the raw spectra, so we should expect μ^g_ℓ'=η'<1. It follows that δ^g_ℓ'/σ̃^g_ℓ' is a Gaussian random variable with standard deviation 1 and meanR̃^ g_ℓ' = η'/σ̃^g_ℓ' = η R^ g_ℓ',where η=η'/ξ. Thus we see that filter-induced signal attenuation η' actually only reduces the SNR by the smaller factor η, because the RMS fluctuations of the noise power within the axion bandwidth are also reduced. The procedure we use to quantify η is described in detail in Sec. <ref>; though formally Eq. (<ref>) allows η>1, we will find that η<1, indicating that the net effect is indeed reduction of the SNR.Finally, we can return to the second effect of correlations neglected in the construction of the grand spectrum: in the presence of correlations, neither the rebinned spectrum weights nor the grand spectrum weights are actually the true ML weights. We are now equipped to show that in practice deviations from the optimal weights are negligibly small in both cases. We noted in Appendix <ref> that the true ML weights in the presence of correlations are sums over rows of the inverse covariance matrix. Applying the approximation in Eq. (<ref>),[It can be shown using Eq. (<ref>) that the average of the off-diagonal elements of the correlation matrix is 1.5[(ξ^g)^2-1]/(K^g-1)≈-0.035, where the numerical factor is due to lineshape weighting. Thus a first-order approximation is appropriate.] we find that the (properly normalized) true ML weights for the grand spectrum arew̃_ℓ q= (σ^g_ℓ)^2/2-(ξ^g)^2[L̅_q^2/(σ^r_ℓ+q-1)^2 - ∑_q'≠ qL̅_qL̅_q'Σ^r_ℓ qq'/(σ^r_ℓ+q-1σ^r_ℓ+q'-1)^2]= w_ℓ q^0 + δ w_ℓ q.Up to an overall change in the normalization, w_ℓ q^0=w_ℓ q, the ML weights in the absence of correlations. The mean value of δ w_ℓ q just compensates for this rescaling such that w̃_ℓ q remain normalized. The typical change in the relative weighting is given by the standard deviation of δ w_ℓ q, which is easy to calculate given the covariances obtained in our simulation: we find that the RMS fractional change in the weights is about 5%. The resulting fractional change in δ^g_ℓ will be much smaller because it is the average of K^g 5% deviations that are mutually negatively correlated (because the weights remain normalized). Thus, the systematic effect from neglecting correlations in the grand spectrum ML weights is small compared to the sources of error we consider in Appendix <ref>; the analogous effect in the rebinned spectrum is smaller still due to the smaller value of 1-(ξ^r)^2.§ CANDIDATES AND EXCLUSIONVia the procedure described in the previous sections, we have condensed our axion search data into the 2n^g numbers δ^g_ℓ/σ̃^g_ℓ and R̃^ g_ℓ. The statistical fluctuations of the total noise power result in a standard normal distribution for the corrected grand spectrum δ^g_ℓ/σ̃^g_ℓ in the absence of axion signals, and a KSVZ axion signal in a particular bin ℓ' would displace the mean of δ^g_ℓ'/σ̃^g_ℓ' by R̃^ g_ℓ'. Now we will explain how we use these quantities to interrogate the presence of axion conversion power in our scan range and derive an exclusion limit if there are no persistent signals.It should be emphasized that we have no a priori knowledge of which bin ℓ' (if any) corresponds to the axion mass, and the only qualitative difference between an axion signal and a positive excess power fluctuation in any given bin is that a true signal should be persistent across different scans at the same frequency. Thus the best we can do is set a threshold Θ and define any bin with δ^g_ℓ/σ̃^g_ℓ≥Θ as a rescan candidate. In the absence of grand spectrum correlations, we would expectŜ=n^g[1-Φ(Θ)]such rescan candidates from statistics alone, where Φ(x) is the cumulative distribution function of the standard normal distribution. We can then collect sufficient data at each rescan frequency to reproduce the sensitivity in the initial scan (Sec. <ref>), and thereby distinguish any real axion signal from statistical fluctuations (Sec. <ref>). In light of the above discussion, our proximate task is to determine an appropriate value for Θ. To simplify matters, let us first assume R̃^ g_ℓ=R_T is constant throughout the scan range. Perhaps the simplest choice of threshold is Θ=R_T. Takingn^g≈1.07×10^5 for the first HAYSTAC data run and assuming for now that R_T=5, we obtain Ŝ=0.03; thus any bin exceeding the threshold is extremely unlikely to be a statistical fluctuation. The problem with this choice of threshold becomes clear when we suppose there is an axion signal with SNR R_T in some bin ℓ': then δ^g_ℓ'/σ̃^g_ℓ'is a Gaussian random variable with mean R_T and standard deviation 1. Θ=R_T is a poor choice of threshold because the probability that δ^g_ℓ'/σ̃^g_ℓ'≥Θ is only 50%. For arbitrary Θ (again assuming a signal with SNR R_T in bin ℓ'), the probability that δ^g_ℓ'/σ̃^g_ℓ'≥Θ in the presence of noise is called the axion search confidence level. If we require a confidence level ≥ c_1 for the initial scan, the appropriate threshold isΘ=R_T - Φ^-1(c_1),and the expected rescan yield Ŝ follows from Eq. (<ref>). The relationship between all of these quantities is illustrated in Fig. <ref>. In Sec. <ref> we will see that grand spectrum correlations modify the expected rescan yield slightly, so we should actually expect S̅<Ŝ candidates.In the above discussion we assumed constant SNR throughout the scan range, when in fact R̃^ g_ℓ varied significantly on scales ≳1 MHz in the first HAYSTAC data run, with typical values between 0.7 and 1.2, due to nonuniform tuning and frequency-dependence of the cavity Q, form factor, etc. Recall that R̃^ g_ℓ is the SNR for an axion signal with photon coupling |g_γ|=|g^KSVZ_γ|, and as we emphasized in Sec. <ref>, our decision to normalize Eq. (<ref>) to the KSVZ coupling specifically was completely arbitrary. To obtain a frequency-independent threshold, we can simply defineG_ℓ=(R_T/R̃^ g_ℓ)^1/2,from which it follows that R_T is the SNR for an axion with frequency-dependent coupling|g^min_γ|_ℓ=G_ℓ|g^KSVZ_γ|. Eqs. (<ref>) – (<ref>) completely determine the confidence level at which we can exclude axions as a function of the two-photon coupling |g_γ| in each bin of the grand spectrum.[In contrast, in most ADMX analyses <cit.> the confidence level is obtained from Monte Carlo. The Monte Carlo procedure involves constructing a mock grand spectrum containing a large number of simulated axion signals with known SNR R_T, setting a threshold Θ, and defining c_1 as the fraction of simulated axions flagged as rescan candidates; the simulation may be repeated many times to determine the behavior of c_1 as a function of R_T and/or Θ. This more involved approach was originally adopted to circumvent the effects of correlations on the horizontal sum, which we have shown we can quantify.] By varying Θ we can adjust the tradeoff between S̅ (which determines the total time we need to spend acquiring rescan data) and |g^min_γ|_ℓ, the minimum coupling to which our search is sensitive.[A coupling |g_γ|_ℓ>|g^min_γ|_ℓ corresponds to a signal with SNR > R_T. At any given threshold Θ, a result δ^g_ℓ'/σ̃^g_ℓ'<Θ implies that axions with mass ν_ℓ' and coupling |g^min_γ|_ℓ' are excluded with confidence c_1, and axions with the same mass but larger coupling are excluded at higher confidence.] The validity of these expressions hinges crucially on our ability to regard each grand spectrum bin as a sample drawn from a Gaussian distribution with known mean and standard deviation. We demonstrated in Sec. <ref> that we are justified in treating any bin that does not contain an axion signal in this way. In Sec. <ref>, we will show that we can also quantify the mean and standard deviation for any bin containing an axion signal, thus validating the above procedure. Then we will return to the choice of threshold in Sec. <ref>. §.§ SG filter-induced attenuationIn Sec. <ref> we claimed that with a KSVZ axion signal in the grand spectrum bin ℓ', δ^g_ℓ'/σ̃^g_ℓ' is a Gaussian random variable with mean given by Eq. (<ref>) and standard deviation 1. Let us consider each of the claims here more carefully. In writing Eq. (<ref>) we have implicitly assumed that η is frequency-independent. While we could of course write a similar expression with η→η_ℓ, the utility of Eq. (<ref>) lies in the fact that we only need to specify a single correction factor to know the SNR in each bin. It is reasonable to expect η' (and thus η) to be frequency-independent, as η'≠1 is ultimately a consequence of the same imperfect SG filter stopband attenuation that led to frequency-independent ξ≠1. We will see more directly that η is constant below.In claiming that the distribution of excess power about the mean value R̃^ g_ℓ' is Gaussian with standard deviation 1, we are only assuming that the statistical fluctuations of the total noise power in any given bin are independent of whether or not that bin also includes excess power due to axion conversion. This is certainly a valid assumption for the raw data. We quantify η using a simulation which will also demonstrate explicitly that this assumption still holds in the grand spectrum.The simulation we use to quantify η begins by defining a set of m uniformly spaced simulated mode frequencies ν_ci and a frequency axis for a 14020-bin spectrum with resolution Δν_b=100 Hz centered on each mode frequency. With a tuning step size of 1.402 MHz/m, the low-frequency end of the last spectrum lines up with the high-frequency end of the first, and m_k (the number of spectra contributing to the kth combined spectrum bin) will vary from 1 to m over the tuning range. Each spectrum is initialized to the expected signal power for an axion with coupling |g_γ| and mass ν_a near the middle of the simulated frequency range. The signal power in the jth bin of the ith spectrum is proportional to the integral of Eq. (<ref>) over an interval Δν_b around the RF frequency ν_k for which Γ_ijk=1, multiplied by the inverse of the rescaling factor defined in Sec. <ref>. For simplicity we take Q_Li, C_i, β_i, and T_ij to be the same for each spectrum i, so that variation in the rescaling factor only comes from the j-dependence of T_ij and the Lorentzian mode profile.After the initialization described above, each iteration of the simulation adds simulated Gaussian white noise with mean 1 and standard deviation σ^p to each spectrum, and sends the full set of spectra through two analysis pipelines in parallel. The “standard” analysis multiplies each spectrum by a random sample baseline (see Sec. <ref>), then applies the baseline removal procedure of Sec. <ref> to obtain simulated processed spectra, and finally combines the simulated spectra both vertically and horizontally, following the procedure of Sec. <ref>-<ref>, to obtain a simulated grand spectrum. The “ideal” analysis is identical except that it bypasses the steps that imprint and then remove the baseline; thus we expect no effects associated with the SG filter in the ideal grand spectrum.From each iteration, we record the values of the normalized power excess δ^g_ℓ/σ^g_ℓ (not δ^g_ℓ/σ̃^g_ℓ) and the uncorrected SNR R^ g_ℓ in ≈ 2K^g bins around ν_ℓ'≈ν_a in both the standard and ideal grand spectra. We also record the value of δ^g_ℓ/σ^g_ℓ in a few other bins far from ν_ℓ' in different parts of the standard grand spectrum. The coupling |g_γ| is chosen to yield R^ g_ℓ'≈5 for m≈200. We let the simulation run for ∼10^4 iterations, after which we can histogram the distribution of any of the recorded bins across iterations. We are primarily interested in comparing the excess power distribution in bin ℓ' of the standard grand spectrum to the excess power distribution in the same bin of the ideal grand spectrum. This comparison is shown in Fig. <ref>. We see that in the ideal grand spectrum, the fluctuations of the noise power in the bin ℓ' containing an axion signal are Gaussian with standard deviation σ^g_ℓ', as they would be in any other bin; we can also see that our standard analysis procedure correctly calculates the SNR R^ g_ℓ in the absence of SG filter effects.[That is, E[δ^g_ℓ'/σ^g_ℓ']_i = (R^ g_ℓ')_i = (R^ g_ℓ')_s, where the subscripts “s” and “i” refer to the standard and ideal analyses, respectively. The calculated values of (R^ g_ℓ)_i and (R^ g_ℓ)_s are nearly equal in each bin ℓ because they only depend on the measured data through the distribution of processed spectrum standard deviations (Sec. <ref>), which is changed only very marginally by the presence of the SG filter.]In the standard grand spectrum, we find that the fluctuations of the noise power in bin ℓ' are still Gaussian, with a reduced standard deviation σ̃^g_ℓ'=ξσ^g_ℓ' and ξ=0.93 as in real data. We also obtain Gaussian fluctuations with standard deviation σ̃^g_ℓ in other bins ℓ far from the axion mass. This provides strong evidence for the assertion that each δ^g_ℓ is a Gaussian random variable with standard deviation σ̃^g_ℓ, whether or not bin ℓ contains an axion signal. Since we histogrammed δ^g_ℓ'/σ^g_ℓ' rather than δ^g_ℓ'/σ̃^g_ℓ' to more directly see the effects of the SG filter on σ^g_ℓ', the ratio of the two bin ℓ' excess power distributions measures η' rather than η; formally, E[δ^g_ℓ'/σ^g_ℓ']_s/E[δ^g_ℓ'/σ^g_ℓ']_i=(μ^g_ℓ')_s(R^ g_ℓ')_s/(R^ g_ℓ')_i=η'. Dividing the value of η' obtained this way by ξ we find η=0.90.This result for η is independent of m out to at least m=400 (c.f. the analogous result for ξ from the simulation described in Sec. <ref>). It also does not change if we vary |g_γ|^2 by ±50%; this linearity implies that we do not have worry about the simulation reproducing the precise value of R_T to be used in the analysis. Finally, η is independent of the misalignment of ν_a relative to the grand spectrum binning: with arbitrary misalignment E[δ^g_ℓ'/σ^g_ℓ']_i≠ R^ g_ℓ', but E[δ^g_ℓ'/σ^g_ℓ']_s always changes by the same factor.[For the simulation plotted in Fig. <ref>, we set ν_a to coincide with a bin boundary in the rebinned spectrum, and used L_q(δν_r=0) rather than L̅_q in the grand spectrum weights. This choice made it simpler to confirm E[δ^g_ℓ'/σ^g_ℓ']_i = R^ g_ℓ' and thereby verify the correct implementation of the analysis procedure (recall that with the lineshape L̅_q, E[δ^g_ℓ'/σ^g_ℓ']_i = R^ g_ℓ' if we average over the range of possible misalignments, but is not necessarily true for any given misalignment). We confirmed that we obtain the same value of η using L̅_q in the grand spectrum weights.]Taken together, the results of the simulation are entirely consistent with the interpretation of η≠1 as a result of the imperfect stopband attenuation of the SG filter. Thus we conclude that Eq. (<ref>) correctly describes the SNR in each bin of the grand spectrum. Although we have seen that filter-induced attenuation is a small effect, we may still ask whether we can avoid this slight SNR degradation by using different SG filter parameters. This question is explored further in Appendix <ref>.§.§ Setting the thresholdWe now return to the question of how we set appropriate values for c_1 and Θ. In many subfields of particle physics it is conventional to cite parameter exclusion limits at 90% or 95% confidence. For the analysis of the first HAYSTAC data run we set c_1=0.95, for which Eq. (<ref>) becomes R_T - Θ = 1.645.Given a value for c_1, the considerations that enter into the choice of Θ are best illustrated with an explicit example. For the first HAYSTAC data run we chose Θ=3.455, corresponding to a threshold SNR of R_T=5.1. S=28 grand spectrum bins exceeded this threshold and were flagged as rescan candidates. The corrected grand spectrum δ^g_ℓ/σ̃^g_ℓ and threshold Θ are shown in Fig. <ref>. Visual inspection suffices to demonstrate qualitatively the important point that many of the candidates are quite marginal; more precisely 11 of the 28 candidates exceed the threshold by less than ΔΘ=0.1, implying that we could have eliminated all of these candidates at the cost of a Δ G_ℓ/G_ℓ = [(R_T+ΔΘ)/R_T]^1/2-1≈ 1% degradation of our exclusion limit. Conversely, reducing the threshold by 0.1 would have improved the exclusion limit by 1% at the cost of 10 additional rescan candidates.Of course, this strong dependence of the rescan yield on the threshold is just what we expect from Gaussian noise statistics.[One consequence of the sensitivity of S to small changes in Θ is that the rescan lists for even relatively similar analyses (characterized by e.g., slightly different choices of K^r and/or K^g, or WF instead of ML weights) typically only overlap by ∼60-80%.] It is common for haloscope searches to set R_T=5 in estimates of the sensitivity that can be achieved with a given set of design parameters, but there is nothing special about this choice. In principle, Θ (and thus R_T) should be chosen to optimize the coupling sensitivity at fixed total integration time (initial scan plus rescans). For any haloscope detector using a coherent receiver, rescans are intrinsically less efficient than the initial scan, so the time spent on rescans should be a small fraction of the time spent acquiring the initial scan data.[This is because each measurement improves the SNR in ∼Δν_c/Δν_a non-overlapping grand spectrum bins simultaneously. In the initial scan each of these bins is relevant whereas in rescans we only care about the SNR within K^g bins of each candidate. In practice the discrepancy is further exacerbated by the fact that rescans are more difficult to fully automate than the continuous initial scan, and thus have worse live-time efficiency.] By this criterion, the optimal threshold is higher still than the value Θ=3.455 we adopted for the first HAYSTAC data run.Thus far in this section we have discussed the real data rescan yield S(Θ) without reference to any theoretical model. To confirm that we have obtained a rescan yield consistent with statistics, we must take into account the fact that any two grand spectrum bins ℓ and ℓ' will be positively correlated if |ℓ-ℓ'|≤ K^g-1 because the segments of the rebinned spectrum contributing to the bins ℓ and ℓ' will overlap. These correlations imply that both real axion signals and statistical fluctuations in the excess power are likely to result in several adjacent bins exceeding the threshold. We should not define all such bins as rescan candidates because they are largely redundant. Thus we add bins to the list of rescan candidates in order of decreasing excess power, and remove K^g-1 bins on either side of each candidate from consideration before moving on to the next candidate. The values of S(Θ) cited above were obtained using this procedure, which was originally proposed by Ref. <cit.>. Recall that Ŝ(Θ) defined by Eq. (<ref>) describes the expected rescan yield for a grand spectrum whose bins are samples drawn from a standard normal distribution: it does not depend on whether or not nearby bins are correlated provided n^g is much larger than the correlation length. Thus, Eq. (<ref>) would correctly describe the expected rescan yield if we did not exclude the correlated bins around each candidate; given that we do exclude these bins, we should actually expect a rescan yield S̅(Θ)<Ŝ(Θ). Note also that though the presence of grand spectrum correlations affects the rescan yield, it does not affect the initial scan confidence level c_1.[For any value of ν_a within our scan range, there will be some grand spectrum bin ℓ' in which the SNR is maximized, and the best limits we can set will come from this bin. R_T is the SNR in bin ℓ' if the axion has mass ν_a and coupling |g_γ| = |g^min_γ|_ℓ' (up to an uncertainty quantified in Appendix <ref>); it follows from Eq. (<ref>) that if bin ℓ' does not exceed the threshold Θ, we can exclude such axions with confidence c_1. The non-observation of excess power above the threshold in adjacent correlated bins just gives us an additional, strictly less restrictive constraint on the coupling of the axion with mass ν_a.] Our procedure for cutting correlated bins from the rescan yield will affect the rescan analysis procedure, discussed in Sec. <ref>.We obtain the Θ-dependence of the expected rescan yield S̅(Θ) from a simple simulation. We generate a simulated rebinned spectrum containing Gaussian white noise, apply the ML-weighted sum of Sec. <ref> to obtain a simulated grand spectrum, and then flag rescan candidates with the same procedure used for real data, cutting K^g-1 bins on either side of each candidate. We repeat this simulation with different values of Θ between 2.3 and 4.3, and then repeat it ≈50 times at our chosen value of Θ=3.455 to obtain a range of probable values for S̅.From this simulation we obtain S̅(Θ) consistently smaller than Ŝ(Θ) as expected: at Θ=3.455, Ŝ=29.5, S̅ = 24±5 and S=28.[The fact that S is closer to Ŝ than S̅ at Θ=3.455 is just a fluke made possible by the small candidate statistics at such a large threshold. At Θ=2.5, for example, we would have S=396, S̅=372, and Ŝ=588. Note that for any value of Θ, Ŝ > S̅ > Ŝ/K^g, where the latter is the rescan yield we would obtain from n^g/K^g≈ n^c uncorrelated bins (this was also noted by Ref. <cit.>). The second inequality gets at the reason (anticipated in Sec. <ref>) that we did not take K^r=1 and K^g=50 in constructing the grand spectrum: the number of rescan candidates would be much larger at comparable sensitivity even after we ensure that no two candidates fall within K^g bins of each other.] We conclude that the observed rescan yield S is consistent with statistics – by itself this result does not disfavor the hypothesis that any of our candidates could be a real axion signal, since the expected variation in S̅ is larger than one, and we expect at most one axion in the data set. To settle the question one way or another, we now turn to the acquisition and analysis of rescan data around each candidate.§ RESCAN DATA AND ANALYSISThree numbers are required to fully characterize each of the S=28 candidates obtained from the initial scan data set: the signal frequency ν_ℓ(s), the threshold coupling G_ℓ(s) (relative to the KSVZ coupling), and the properly normalized power excess δ^g_ℓ(s)/σ̃^g_ℓ(s), where ℓ(s) is the index of the grand spectrum bin that exceeded the threshold and s=1,…,S. Only the first two quantities explicitly appear in our subsequent analysis (though of course the power excess determines whether any given bin is flagged as a rescan candidate in the first place).To establish whether any of our rescan candidates is persistent, we must first determine for each candidate the rescan time τ^*_s required to obtain SNR R^*_T for an axion signal at frequency ν_ℓ(s) with coupling G_ℓ(s). Then we can acquire rescan data at each candidate frequency. The considerations that enter into these steps are described in Sec. <ref>.We can imagine two alternative approaches to processing the rescan data. One possibility is to process the rescan and initial scan data sets together to produce a single combined spectrum, from which we obtain a modified grand spectrum by following the procedure in Sec. <ref>. The extra integration time at each candidate frequency implies that each R̃^ g_ℓ(s) will increase by roughly a factor of √(2). Since we are interested in probing the same value of G_ℓ(s), we can impose a higher threshold Θ^*_ℓ(s) around each candidate. We can thus ensure that a real axion signal exceeds the new threshold with some desired confidence c_2, while simultaneously greatly reducing the probability that a statistical fluctuation does so.Alternatively, we can process the rescan data separately, following the procedure of Sec. <ref> – <ref> to produce a rescan grand spectrum, and leaving the initial scan grand spectrum unchanged. The rescan data set should allow us to set a coincidence threshold Θ^*_ℓ(s) around each candidate frequency which a real axion signal should exceed with confidence c_2. If c_2≈ c_1, we do not expect Θ^*_ℓ(s) to be substantially greater than Θ in this case, so the probability that a statistical fluctuation exceeds the threshold in any given bin will not change, but it is much less likely that this should happen in any of the same bins as in the initial scan.If no changes to the analysis procedure are required for rescan data, these two approaches are completely equivalent. Here we take “separate processing” approach, which is conceptually cleaner in that we process spectra together whenever we want to improve the coupling sensitivity |g^min_γ| and separately when we want to reproduce the coupling sensitivity of a previous scan. As we will see in Sec. <ref>, the rescan analysis differs from the initial scan analysis in a few crucial respects, such that we must use separate processing to obtain correct expressions for the coincidence thresholds Θ^*_ℓ(s). §.§ Rescan data acquisitionThe most efficient way to acquire rescan data at the candidate frequency ν_ℓ(s) is to take one long measurement with the axion-sensitive cavity mode fixed at frequency ν_cs≈ν_ℓ(s). We can calculate the integration time τ^*_s required to obtain SNR R^*_T by starting with an expression analogous to (<ref>) and using Eqs. (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>). The result isτ^*_s= 1/1-ε[R^*_Tk_BT_sH(δν_as)/η^*F_MLG^2_ℓ(s)U_0ν_csC_sQ_Lsβ_s/1+β_s]^2,where η^*=0.76 is the filter-induced attenuation for the rescan analysis (see Sec. <ref>), the noise temperature T_s is evaluated in the middle of the analysis band, and we have lumped all dependence on the detuning δν_as=ν_cs-ν_ℓ(s) into the factor H(δν_as) normalized so that H(0)=1; we have also assumed that only a fraction 1-ε of the integration time at each cavity setting ν_cs will contribute to improving the SNR at the candidate frequency.Eq. (<ref>) indicates that in order to know how long to integrate at each candidate frequency, we must estimate the values of the parameters Q_Ls, β_s, C_s, and T_s (see Sec. <ref>) and the detuning δν_as between the mode and candidate frequencies. If the true value any of these parameters during the rescan measurement deviates from the value we assume in the calculation of τ^*_s, the true SNR R̂^*_ℓ(s) calculated from the rescan data (see Sec. <ref>) will deviate from the target value R^*_T. This observation motivates the question of what nominal value to assign to R^*_T in Eq. (<ref>) – there is no a priori reason we must set R^*_T= R_T. Note that R̂^*_ℓ(s)≠ R^*_T for any given candidate is not a problem provided that the probability p_s of a statistical fluctuation exceeding the corresponding coincidence threshold Θ^*_ℓ(s) remains ≪ 1. This probability may be roughly estimated asp_s=n_K[1-Φ(Θ^*_ℓ(s))],whereΘ^*_ℓ(s) =R̂^*_ℓ(s) - Φ^-1(c_2)and we have defined an effective number of independent bins 1<n_K<2K^g-1 to account for the fact that we will reject the hypothesis of an axion in bin ℓ(s) only if δ^g*_ℓ/σ̃^g*_ℓ exceeds the appropriate coincidence threshold in neither the original bin ℓ(s) nor any of the (K^g-1) correlated bins on either side (see discussion in Sec. <ref>). n_K = 1 (n_K = 2K^g-1) would correspond to treating the 2K^g-1 bins associated with each candidate as perfectly correlated (uncorrelated); the appropriate value is clearly somewhere in between these two extremes.We would like to demand that ∑_sp_s ≪ 1 in order to avoid a second round of rescans in the absence of axion signals. If we assume for now that Θ^*_ℓ(s) will not vary too much around the nominal value obtained by taking R̂^*_ℓ(s)→ R^*_T in Eq. (<ref>), we should set R^*_T = Φ^-1(1-[∑_sp_s/(S× n_K)])+Φ^-1(c_2).For the first HAYSTAC analysis, we estimated n_K≈ K^g and demanded that simultaneously ∑_sp_s≤0.05 and c_2=0.95; with these choices, Eq. (<ref>) yields R^*_T=5.03 (equivalently, Θ_ℓ(s)^*≈3.28).Next we need to specify how we evaluate the other parameters that enter into the calculation of τ^*_s. For each candidate we set T_s by averaging T_ij over all initial scan Y-factor measurements i, and evaluating the average in the IF bin j corresponding to the cavity resonance. The form factor C_s and the unloaded cavity quality factor Q_0s depend deterministically on the cavity frequency and thus are easy to accurately estimate; Q_Ls=Q_0s/(1+β_s) then follows from our ability to control the cavity-receiver coupling β by adjusting the antenna insertion. We set β_s=2 in Eq. (<ref>) for each candidate, to match the average value of β_i throughout the initial scan.[β≈2 is optimal for a continuous data run because it maximizes the scan rate for a given sensitivity |g_γ| <cit.>. For a rescan measurement in which we only care about the SNR in a few bins around ν_ℓ(s), critical coupling (β=1) is better if δν_as≈0 and N_cav=0. However, with N_cav≠0 the system noise temperature also depends on β: with β_s=1, T_s would systematically underestimate the true noise temperature in the rescan measurement.]The detuning δν_as is trivial to measure but hard to control precisely, due to the mode frequency drifts discussed in Sec. <ref> and the backlash inevitably present in any mechanical tuning system. In practice we acquired the rescan data starting with the highest-frequency candidate and tuning down: for each candidate, we tuned the TM_010 mode ≈ 100 - 200 kHz above ν_ℓ(s) and waited 20 minutes for the mode frequency to settle before starting the measurement. We proceeded with the measurement only if |δν_as|<150 kHz after this interval. We set δν_as=0 for each candidate in Eq. (<ref>) for simplicity; since the cavity will be overcoupled and the noise temperature also decreases for δν_as≠0, τ^*_s is not too sensitive to small detunings. Another potentially more serious consequence of mode frequency drift is that for any given s, some or all of the processed spectrum bins contributing to the grand spectrum bin ℓ(s) may happen to coincide with a region of the analysis band contaminated by IF interference. We saw in Sec. <ref> that 11% of analysis band bins were contaminated in this way – thus there is a non-negligible chance that R̂^*_ℓ(s) will be substantially smaller than the target value R^*_T due to missing bins. We mitigate this effect by splitting the total integration time τ^*_s required for each candidate across 10 cavity noise measurements of duration τ^*_s/10, and step the LO and JPA pump frequencies together by 1 kHz (without tuning the cavity mode) between measurements. On average, we expect the candidate to fall in a contaminated part of the analysis band in about 1 of 10 such measurements: thus we set ε=0.1 in Eq. (<ref>).Finally, unlike the experimental parameters discussed above, η^* and F_ML depend on fixed parameters of the rescan analysis procedure and cannot change from one rescan measurement to the next. We will see in Sec. <ref> that whileF_ML will not change in the rescan analysis, η^* will not in general be equal to η and thus should be estimated in advance to avoid systematically biasing τ^*_s. Applying Eq. (<ref>) to the 28 rescan candidates from the first HAYSTAC data run, we obtained rescan times τ^*_s ranging from 5.8 hours (corresponding to G_ℓ(s)=2.72) to 17.9 hours (G_ℓ(s)=2.03). We had |ν_ℓ(s)-ν_ℓ(s+1)|<200 kHz for 3 of the 27 pairs of adjacent candidates: thus there is a 100 kHz range for ν_cs in which the condition |δν_as|<150 kHz can be satisfied simultaneously for both candidates. In each of these cases, we acquired rescan data for both candidates together, taking the larger of the two calculated integration times (which were generally very similar). Thus we made 25 rescan measurements, for a total of 282 hours of rescan time (c.f. Mτ=1692 hours of initial scan time).[We will use s to index quantities a_s associated with each rescan measurement as well as quantities b_s associated with each candidate, with the implicit understanding that in three cases, we will have a_s=a_s+1 but b_s≠ b_s+1]At each iteration s, after tuning the cavity to the appropriate frequency ν_cs and setting β_s≈2, we used a LabVIEW program to make 10 cavity noise measurements and acquire auxiliary data. Each cavity noise measurement was saved as an averaged power spectrum with frequency resolution Δν_b as in the initial scan. The auxiliary data at each iteration comprised VNA measurements of the cavity mode in transmission and the JPA gain profile both before and after the set of cavity noise measurements, a VNA measurement of the cavity mode in reflection, and a Y-factor measurement. We use this auxiliary data to quantify R̂^*_ℓ(s) as described in Sec. <ref>, and also to flag and cut anomalous iterations as in Sec. <ref>. Unlike in the initial scan analysis we must repeat any iterations we cut at this stage, to ensure that we have meaningful data around each rescan candidate. In the first HAYSTAC data run we had to repeat 6 of our 25 rescan measurements, in each case because of excessive mode frequency drift |ν_c1-ν_c2|>130 kHz.[Mode frequency drifts were generally larger than in the initial scan due to a combination of much larger tuning steps between iterations and much longer integration times. Four rescan measurements had drifts below 130 kHz but above the more conservative 60 kHz threshold used in the initial scan. The range over which the mode drifted was roughly centered on the candidate frequency ν_ℓ(s) in these cases, so the systematic deviation from the correct ML weight for any processed spectrum bin contributing to the combined spectrum around ν_ℓ(s) will be quite small. To bound this error we can consider the more extreme case where the mode frequency initially coincides with the candidate frequency and then drifts away slowly over the 10 subsequent measurements: with the maximum allowed drift and the minimum cavity bandwidth Δν_cs, the RMS fractional deviation from the true combined spectrum ML weights is 13%. As noted in Sec. <ref>, the systematic effect on the combined spectrum bin values δ^c*_k and the SNR R^ c*_k will be much smaller.] §.§ The rescan analysisOnce we have acquired a complete rescan dataset, the next step is to process and combine all rescan power spectra to produce a single rescan grand spectrum. We begin by truncating each of our 250 rescan spectra as in Sec. <ref>, normalizing each spectrum to the average baseline from the initial scan analysis, and using the list defined in Sec. <ref> to cut bins contaminated by IF interference from each spectrum.Next we must use an SG filter to remove the residual baseline from each spectrum. At this stage it becomes important that τ^*_s/10>τ even for the smallest value of τ^*_s obtained from Eq. (<ref>); moreover the residual baselines for the 10 spectra from each iteration will be very similar, since we do not tune the cavity or rebias the JPA between power spectrum measurements. Thus, although the total averaging at each candidate frequency in the rescan data is comparable to the total averaging at that frequency in the initial scan, we should expect the amplitude (relative to σ^p) of any small-scale systematic structure in the rescan processed spectra to be enhanced by a factor ∼√(τ^*_s/τ) if we use the same SG filter parameters as in the initial scan (see also discussion in Appendix <ref>).We have seen in the preceding sections that the statistics of the initial scan spectra are Gaussian at each stage of the processing, and in particular that the narrowing of the histogram of normalized grand spectrum bins δ^g_ℓ/σ^g_ℓ is completely explained by the stopband properties of the SG filter with parameters d=4 and W=500. This good agreement between the observed and expected statistics indicates that the amplitude of any small-scale systematic structure in the initial scan processed spectra must be ≪σ^p.The observation that baseline systematics will grow coherently over at least the single-spectrum integration time (and likely over the full rescan integration time) indicates that we cannot necessarily assume systematic structure will remain negligibly small in the rescan processed spectra. Studies of the effects of SG filters on simulated Gaussian white noise indicate that the parameters d and W used in the initial scan would produce unacceptable deviations from Gaussianity if applied to the rescan analysis. Thus we used an SG filter with d^*=6 and W^*=300 for the rescan analysis instead; Fig. <ref> suggests that with these parameters we should expect ξ^*<ξ and η^*<η, and we will see below that this is indeed the case. After applying the SG filter with parameters d^* and W^* to each rescan spectrum, we verify that the bins in each of the 10 processed spectra at iteration s have the expected Gaussian distribution with mean 0 and standard deviation σ^p*_s = 1/√(Δν_bτ^*_s/10). We then rescale the spectra to obtain a mean power excess of 1 in any rescaled spectrum bin in which a KSVZ axion deposits a fraction 1/(K^rK^g) of its total conversion power. Formally, the required rescaling is given by Eqs. (<ref>) and (<ref>), with the additional factor of 1/(K^rK^g) discussed at the beginning of Sec. <ref> absorbed into the definition of the signal power. Values for the factors in Eq. (<ref>) and Eq. (<ref>) are obtained from the auxiliary data at each rescan measurement via the procedure described in Sec. <ref>; unlike in the initial scan analysis, no interpolation is required for T^*_sj because we made a Y-factor measurement at each rescan iteration.[The j-dependent quantities in the rescaling factor should more properly be written with an additional index a=1,…,10 to account for the fact that the LO frequency varies across the 10 spectra at each iteration s. Apart from this small frequency offset, the rescaling factor is the same for all the spectra at a given iteration s.]We then follow the procedure of Sec. <ref> to construct a single ML-weighted combined spectrum from the set of 250 rescaled spectra. The frequency axis for the rescan combined spectrum extends from the smallest candidate frequency minus 651 kHz (i.e., half the analysis band) to the largest candidate frequency plus 651 kHz: there are thus formally a total of 1.02×10^6 combined spectrum bins, though about 70% of these bins are empty because we only took data around candidate frequencies. The typical spacing between candidate frequencies is such that most (non-empty) combined spectrum bins k are obtained by averaging only the m_k=10 spectra from a single rescan measurement. But the formal procedure of Sec. <ref> also correctly treats the cases where adjacent candidates are sufficiently close that spectra from different iterations overlap, and thus m_k>10. As expected, the distribution of combined spectrum bins δ^c*_k/σ^c*_k is Gaussian with mean 0 and standard deviation 1.Finally, we follow the procedure of Sec. <ref> and Sec. <ref> to obtain the rescan grand spectrum. Since we want to reproduce the initial scan sensitivity without changing any assumptions about the axion signal, we should use the same values of K^r, K^g, and L̅_q in Eqs. (<ref>), (<ref>), (<ref>), and (<ref>). However, we should expect ξ^r*≠ξ^r and ξ^g*≠ξ^g because we have used a different SG filter. Empirically, the distribution of rebinned spectrum bins δ^r*_ℓ/σ^r*_ℓ is Gaussian with mean 0 and standard deviation ξ^r*=0.96, and the distribution of grand spectrum bins δ^g*_ℓ/σ^g*_ℓ is Gaussian with mean 0 and standard deviation ξ^*=0.83, implying ξ^g*=ξ^*/ξ^r*=0.86.As in the initial scan analysis, we are ultimately interested in the quantities δ^g*_ℓ/σ̃^g*_ℓ=δ^g*_ℓ/(ξ^*σ^g*_ℓ) and R̃^ g*_ℓ=η^*R^ g*_ℓ that have been corrected for filter effects. As before, we obtain the value of ξ^* directly from the data; the value of η^* can only be obtained from simulation, but the common origin of ξ^* and η^* and good agreement between the observed and simulated values of ξ^* gives us confidence that we have applied the appropriate correction factor.We validate the observed value of ξ^* and measure η^* using simulations very similar to the ones described in Sec. <ref> and Sec. <ref>, respectively. Formally, the rescan simulations only differ in two respects: we multiply each simulated white noise spectrum by the same sample baseline instead of a random sample baseline, and we assign the same mode frequency to each spectrum in the simulation to quantify η^*.[We did not assign frequency offsets to the spectra in the simulation used to measure ξ in the initial scan analysis (see Sec. <ref>). The fact that we nonetheless obtained the same value of ξ as in real data indicates that small changes in the baseline shape (associated with tuning the cavity or rebiasing the JPA) can suppress the growth of small-scale systematics substantially, even without frequency offsets between spectra.] We reproduced the observed values of ξ^r* and ξ^* and obtained η^*=0.76 from these simulations; we verified that the results in each case were independent of the number of averages m and integration time τ, at least for mτ≤20 hours, and thus independent of frequency (see discussion in Sec. <ref>).At this point, we have obtained an explicit expression for the SNR R̃^ g*_ℓ for a KSVZ axion signal in each bin ℓ of the rescan grand spectrum, whereas we care about the SNR for an axion signal with the threshold coupling |g^min_γ|_ℓ. Naively we only care about evaluating the SNR in the S bins ℓ(s) that passed the threshold in the initial scan, as the hypothesis that an axion signal with coupling |g^min_γ|_ℓ is present in any other bin has already been excluded with confidence c_1. The presence of grand spectrum correlations complicates this picture slightly. If several adjacent bins pass the threshold together, we associate the candidate with the bin whose power excess was largest, but in the presence of fluctuations the bin with larger power excess does not necessarily have the largest SNR. Thus it is possible in principle that the rescan candidate we have associated with bin ℓ(s) actually corresponds to an axion signal in any of the 2K^g-1 grand spectrum bins ℓ'(s) correlated with ℓ(s). To be conservative we require each such hypothesis be rejected with confidence c_2 before we can reject the candidate. The above discussion implies that we should defineR̂^*_ℓ'(s) = G^2_ℓ'(s)R̃^ g*_ℓ'(s)with ℓ'(s) defined in the range [ℓ(s)-(K^g-1),ℓ(s)+(K^g-1)]. Values of R̂^*_ℓ'(s) in the first HAYSTAC data run ranged from 4.26 to 7.19, with an average of 5.19.[The SNR was consistently above 6.4 for all the bins associated with two adjacent candidates that were separated in frequency by only 270 kHz: this was above our threshold for acquiring data for both candidates together, but still close enough that the integration at each candidate contributed significantly to the SNR for the other. The average SNR among all other candidates was 5.09, close to our target value R^*_T=5.03. The RMS variation in R̂^*_ℓ'(s) among the bins ℓ'(s) associated with each candidate s was typically less than 1%, but was ∼5% in a few cases where the candidate frequency was close to a region of the grand spectrum with reduced exposure due to missing bins.] The effects of uncertainty in the factors used to calculate the rescan SNR are discussed in Appendix <ref>.The appropriate coincidence threshold Θ^*_ℓ'(s) for each bin correlated with each candidate is then obtained by using Eq. (<ref>) in Eq. (<ref>) with the substitution ℓ(s)→ℓ'(s). In the first HAYSTAC data run, δ^g*_ℓ'(s)/σ̃^g*_ℓ'(s) did not exceed Θ^*_ℓ'(s) for any of the bins ℓ'(s) associated with any of our S=28 rescan candidates.[Had we observed a small number of persistent candidates, we could easily have subjected them to an unambiguous test by repeating the rescan measurement with different applied magnetic fields. It is difficult to imagine any instrumental systematic capable of mimicking the B_0^2 scaling of the axion signal power.] The final result of the first HAYSTAC data run is thus a limit on the axion-photon coupling |g_γ|.§ CONCLUSIONThe absence of any persistent candidates in the first HAYSTAC data run implies that |g^min_γ|_ℓ given by Eq. (<ref>) may be interpreted as an exclusion limit on the dimensionless coupling |g_γ| in each bin ℓ in our initial scan range. The corresponding limit on the physical coupling |g_aγγ|=|g_γ|α/(πΛ^2)m_a that appears in the Lagrangian is plotted in Fig. <ref>. Assuming an axion signal lineshape described by Eq. (<ref>), we excluded |g_γ|≥2.3×|g^KSVZ_γ| on average over the mass range 23.55 < m_a < 24.0 μeV. What confidence should we ascribe to the exclusion of axions with the threshold coupling |g^min_γ|_ℓ? Following Ref. <cit.>, we initially chose c_1=c_2=0.95 to ensure the product c_1c_2≥0.9, and interpreted this product as the net confidence level. But this interpretation is overly conservative, because we only acquired and analyzed rescan data at frequencies that exceeded the initial scan threshold. The hypothesis of an axion signal with the threshold coupling in any given bin is excluded with confidence c_1 if that bin did not exceed the initial scan threshold. In the bins correlated with each candidate, the appropriate confidence level is the conditional probability that a true axion signal would fail to exceed the coincidence threshold, having already exceeded the initial scan threshold; since the two scans are independent, this probability is just c_2. Thus, our result |g^min_γ|_ℓ is properly interpreted as an exclusion limit at 95% confidence.[We can equivalently interpret this result as a marginally more sensitive exclusion limit at lower confidence. Our threshold coupling at 90% confidence would be smaller by a factor of [(R_T-Φ^-1(0.95)+Φ^-1(0.9))/R_T]^1/2≈0.964. HAYSTAC collaborators are also working on developing a Bayesian approach to the haloscope search analysis, which should offer an alternative prescription for defining rescan candidates and establishing confidence levels.]In this paper, we have described in detail the analysis procedure used to derive the first limits on cosmic axions from the HAYSTAC experiment. We have cited specific examples from the analysis of our first data run, but our formal procedure may easily be adapted to the analysis of data from other haloscope detectors. Throughout the preceding sections we have specifically emphasized our use of Savitzky-Golay filters to remove individual spectral baselines, our quantitative understanding of how filtering affects the statistics of the spectra, and our consistent application of maximum-likelihood weights to both the “vertical” sum of overlapping spectra and the “horizontal” sum of adjacent bins in the combined spectrum. All of these were innovations of our approach to the haloscope search analysis; taken together, they enable us to calculate our search sensitivity with minimal input from simulation, and obtain the relationship between sensitivity and confidence level directly from statistics. With the results of the first HAYSTAC data run we demonstrated that a sufficiently low-noise experiment can reach the QCD axion model band for m_a > 20 μeV, despite the unfavorable scaling of the haloscope signal power with increasing frequency <cit.>. A second run to extend this coverage is presently underway, with an improved thermal link to the tuning rod (and thus significantly reduced N_cav) and a new piezoelectric actuator with more reliable mechanical performance; these upgrades will be described in a forthcoming publication along with new results from the experiment. Ongoing cavity and amplifier R&D by members of the HAYSTAC collaboration also indicates several promising avenues for further improving the scan rate and extending the haloscope technique to still higher frequencies <cit.>.This work was supported by the National Science Foundation, under grants PHY-1362305 and PHY-1607417, and by the Heising-Simons Foundation under grants 2014-181, 2014-182, and 2014-183. We thank Ana Malagon and Dan Palken for fruitful discussions, and Marguerite Epstein-Martin and Miguel Goncalves for contributions to the project. § NOTATIONTab. <ref> summarizes the notation used in the formal description of the HAYSTAC analysis procedure above, and indicates where each commonly used symbol is first introduced in the text. We have omitted quantities which are not referenced outside the subsection in which they are defined, and haloscope physics parameters for which we have followed the standard notation in the field. In Tab. <ref>, the roman superscript y represents one of the spectrum labels defined in Tab. <ref> and the italic subscripts x and z represent indices defined in Tab. <ref>. We adhere to these conventions throughout the text: for example, R^ s_ij denotes the SNR for an axion confined to the jth bin of the ith rescaled spectrum, and the index k always runs from 1 to n^c, the number of bins in the combined spectrum. Primed indices (e.g., k', ℓ') are used throughout the text to single out a particular bin containing a putative axion signal; in Sec. <ref> a primed index is also used to specify the second bin in an expression for the covariance of nearby bins in a given spectrum.Finally, in Sec. <ref>, we use the superscript ^* to denote previously defined quantities whose values differ in the rescan analysis.§ MAXIMUM LIKELIHOOD ESTIMATIONTaking the discussion at the beginning of Sec. <ref> as motivation, we will assume we have m independent Gaussian random variables y_k drawn from distributions with the same mean μ but different variances σ_k^2. We are interested in finding an estimate of μ that maximizes the likelihood function, which is just the joint probability distribution of the observations y_k considered as a function of μ:ℒ(μ)=exp(-1/2∑_k(y_k-μ/σ_k)^2).We can equivalently maximize logℒ, since the logarithm is monotonically increasing. So we taked/dμlogℒ=∑_k(y_k-μ/σ_k^2)=0.Solving for μ yieldsμ=∑_k y_k/σ_k^2/∑_k 1/σ_k^2,which may be compared to Eq. (<ref>).If our observations are not independent but rather correlated, Eq. (<ref>) should be replaced withℒ(μ)=exp(-1/2(𝐲-μ𝐢)^⊺Σ^-1(𝐲-μ𝐢)),where 𝐢 is the m-vector (1,1,…,1), and Σ is the covariance matrix whose diagonal elements are σ_k^2. Maximizing with respect to μ we obtainμ=𝐲^⊺Σ^-1𝐢/𝐢^⊺Σ^-1𝐢.We see that the (unnormalized) ML weight for each y_k is a sum over the kth row of Σ^-1. A useful approximation to this sum for sufficiently small correlations is∑_k(Σ^-1)_kk'≈1/σ^2_k'[1 - ∑_k≠ k'Σ_kk'/σ^2_k],where we have neglected all terms that are higher than first order in the ratio of any off-diagonal element to any diagonal element of Σ; to first order the normalization is then just the sum of Eq. (<ref>) over k'. In Sec. <ref> we consider ML weighting in the presence of small correlations. We continue to use Eq. (<ref>) rather than Eq. (<ref>), and argue in Sec. <ref> that deviations from the true optimal weights are acceptably small.The ML estimate of the mean of a multivariate Gaussian distribution with arbitrary covariance matrix Σ can also be obtained from a least-squares perspective. To see this, consider a linear regression model 𝐲 = μ𝐱 + ϵ, where we would like to estimate the slope μ in the presence of noise ϵ, assumed to be drawn from a Gaussian distribution with zero mean and covariance matrix Σ. The generalized least squares (GLS) estimate of μ is the value that minimizes the mean squared errorχ^2(μ) = 1/m(𝐲-μ𝐱)^⊺Σ^-1(𝐲-μ𝐱).For 𝐱=𝐢, χ^2(μ)∝logℒ, so the estimate that extremizes either criterion will also extremize the other. This equivalence between the ML and GLS methods requires only that the statistics of the underlying noise distribution be Gaussian, and this condition will always be satisfied in our haloscope analysis. It can be proved that the variance of the GLS estimator is smaller than the variance of any other unbiased linear estimator <cit.>.Finally, we note as an aside that if we allow the elements of 𝐱 to vary, and take Σ to be diagonal for simplicity, the least squares estimate of μ becomesμ=∑_k x_ky_k/σ_k^2/∑_k (x_k/σ_k)^2,The elements of x_k here play the role of the rescaling factor discussed in Sec. <ref>; thus from a least-squares perspective the rescaling of the spectra need not be regarded as a distinct step of the analysis procedure. We stick to the ML perspective in the text to emphasize the value of using units in which the expected axion conversion power is 1, and thus the R=σ^-1 correspondence has an intuitive interpretation.§ OPTIMIZING SG FILTER PARAMETERSWe discussed the optimization of the SG filter parameters d and W briefly at the end of Sec. <ref>, but it is instructive to revisit this question after having observed the filter-induced narrowing ξ of the distribution of grand spectrum bins (Sec. <ref>) and the filter-induced attenuation of the SNR (Sec. <ref>). Fig. <ref> indicates that reducing d/W moves the 3 dB point of the SG filter down towards larger spectral scales and increases the stopband attenuation on the small spectral scales of interest (≤ K^rK^g bins). Thus we should expect ξ,η→1 as we reduce d/W. However, as noted in Sec. <ref>, reducing the 3 dB point of the SG filter invariably moves progressively larger-amplitude components of the baseline from the filter's passband into its stopband. This claim implicitly assumes that the power spectrum of the residual baseline falls off monotonically towards smaller spectral scales, and we can confirm this empirically: on small spectral scales the residual baseline power spectrum follows a power law distribution with spectral index α≈-2.The largest-amplitude baseline component that is not removed by the SG filter (and thus remains in the processed spectra) will coincide with the first zero of the filter's transfer function; let us call the corresponding bin separation κ. As we reduce d/W at fixed integration time τ (or increase τ for a given filter), the baseline amplitude a(κ) will grow relative to the statistical fluctuations σ^p=1/√(Δν_bτ). For a(κ)/σ^p sufficiently large, the distribution of processed spectrum bins δ^p_ij will appear non-Gaussian. Of course, each bin in each processed spectrum is still a Gaussian random variable with standard deviation σ^p; the apparent breakdown of Gaussianity just indicates that μ^p_ij=0 for each bin j has become a poor approximation given our failure to completely remove the spectral baseline.Even if the distribution of δ^p_ij exhibits no signs of non-Gaussianity, a(κ)≠0 implies positive correlations in the processed spectra on scales ≤κ/2; since κ>K^rK^g, this effect tends to counteract the negative correlations due to the SG filter stopband alone (i.e., independent of the spectrum of the baseline). In other words, systematic effects due to the shape of the baseline grow coherently in the horizontal sum over adjacent bins. They can also grow coherently in the vertical sum if the m_k contributing spectra have small detunings and similar baselines, as in the rescan data set (Sec <ref>).Thus, we find that unless a(κ) ≪ 1/√(Δν_bτ), ξ and η will depend on the integration time τ (and possibly also on m_k). The simulations discussed in Secs. <ref>, <ref>, and <ref> demonstrate that we are safely in the a(κ) ≪ 1/√(Δν_bτ) regime with filter parameters d,W (d^*,W^*) for the initial scan (rescan) analysis.§ PARAMETER UNCERTAINTIESThe corrected grand spectrum SNR R̃^ g_ℓ depends on many measured parameters whose uncertainties we have thus far ignored. Here we will quantify the effects of these uncertainties on the analysis, but first we note that there is potential for terminological confusion because “confidence level” is a generic statistical term often used to quantify uncertainty. The axion search confidence level c_1 we have defined in this paper is the probability that an axion with SNR R_T in any given grand spectrum bin will exceed the threshold – since the value of R_T is not fixed by measurement, c_1 is completely independent of parameter uncertainties. Rather, uncertainty in R̃^ g_ℓ translates [via Eqs. (<ref>) and (<ref>)] into uncertainty in the threshold coupling |g^min_γ|_ℓ for which we obtain SNR R_T in each bin ℓ.We can estimate the size of the fractional uncertainty δ|g^min_γ|/|g^min_γ| in a typical grand spectrum bin by first noting that|g^min_γ| ∝(T_eff/ηϕ(δν)η_LC_010)^1/2,where we have elided factors without uncertainty and quantities like Q_L and B_0 that are easily measured with fractional uncertainty ≤1%, and introduced an effective noise temperature T_eff and a function ϕ(δν) discussed below. It is easy to estimate the error in the factor η_L introduced in Sec. <ref> to quantify loss between the cavity and JPA. We estimated this loss to be -0.60 ± 0.15 dB, which implies δη_L/η_L≈3.5%.The filter-induced attenuation η and cavity mode form factor C_010 are both obtained from simulation, and thus estimating the uncertainty in these parameters is not necessarily straightforward. Nonetheless, our result for η is very robust against changes in the parameters of the simulation (see discussion in Sec. <ref>), and this implies a fractional uncertainty of δη/η≲1% which we can safely neglect. We did not include uncertainty in C_010 in our error budget because we did not have a reliable way to quantify it. Preliminary field profiling measurements suggest that the simulated form factors are reliable to better than 10%, so a careful treatment of the form factor uncertainty would likely change our final result δ|g^min_γ|/|g^min_γ|≈4% by at most a factor of 2 and possibly much less.In the denominator of Eq. (<ref>) we have definedϕ(δν)=√(∑_qL̅_qL_q(δν)/(K^g)^2) to encode the dependence of the SNR on the misalignment δν of the axion mass relative to the lower edge of the grand spectrum bin in which the SNR is maximized (see discussion in Sec. <ref>). The misalignment attenuation η_m≈ϕ̅/ϕ(0), where ϕ̅ is the mean value of ϕ(δν) over the range of possible misalignments; note also the formal similarity of Eq. (<ref>) to Eq. (<ref>).[Formally, η_m as defined in Sec. <ref> is obtained by replacing each L_q(δν) by its average value L̅_q and then normalizing to ϕ(0), which is not quite the same because ϕ is not linear in L_q. In practice, the difference is negligible.] With the misalignment error δϕ defined as the standard deviation of ϕ(δν) over this same range, we obtain δϕ/ϕ̅≈2%.Finally, in any given grand spectrum bin, the effective noise temperature T_eff is formally given by a ML-weighted average of T_ij across all contributing processed spectrum bins. Since we are only interested in estimating the typical fractional uncertainty in the noise temperature, we make the same approximation we used to set T_s in the calculation of the rescan time in Sec. <ref>: we average T_ij over all spectra and evaluate it in the IF bin j corresponding to the middle of the analysis band, where the ML weight is largest.Taking a typical cavity frequency ν_c=5.75 GHz in the middle of the first HAYSTAC scan range, we can then write T_eff=hν_c[N_T + N_cav + N_A]; the reader is referred to Sec. <ref> for the definition of these additive contributions and to Ref. <cit.> for detailed discussion of the noise calibration procedure. Briefly, we obtain N_A=1.35±0.05 quanta from off-resonance Y-factor measurements and N_cav=1.00±0.17 quanta from the average of all Y-factor measurements during the data run. Even allowing for a ±20 mK uncertainty in the calibration of the mixing chamber thermometer, the uncertainty in N_T=0.63 remains negligibly small, in part because the nominal HAYSTAC operating temperature T_C=127 mK is sufficiently far into the Wien limit that N_T depends only weakly on the physical temperature, and in part because errors in different contributions to the total noise T_eff are somewhat anti-correlated. Negative correlations arise because increasing any of the additive terms in T_eff while holding the others constant would reduce the measured value of the hot/cold noise power ratio Y.Adding the uncertainties cited in the above paragraph in quadrature and using k_BT_eff≈3hν_c we obtain δ T_eff/T_eff≈ 6%. This estimate (dominated by the variation in measurements of N_cav) is conservative in that we have neglected the fact that δ N_A and δ N_cav are negatively correlated, and because we have included the RMS systematic variation of N_cav across the tuning range in the “uncertainty” δ N_cav. Miscalibration of the still thermometer would need to be larger than ±20 mK to affect our estimate of δ T_eff.Combining the results of the preceding paragraphs, we obtainδ |g^min_γ|/|g^min_γ|≈√((1/2δ T_eff/T_eff)^2 + (1/2δϕ/ϕ̅)^2 + (1/2δη_L/η_L)^2)≈4%.This result (represented by the light green shaded region in Fig. <ref>) should be interpreted as a rough estimate of the uncertainty in our exclusion limit, not a formal 1σ error bar on the threshold coupling |g^min_γ|_ℓ in each bin.We should also consider the effects of miscalibrating the SNR in the rescan analysis. We can distinguish between “global” effects (e.g., overall miscalibration of the system noise temperature or uncertainty in η_L) and effects confined to the rescan analysis (e.g., miscalibration of η^* or mode frequency drifts in particular rescan measurements). The former affect R̃^ g*_ℓ and R̃^ g_ℓ in the same way: thus they do not change the candidate SNR R̂^*_ℓ'(s) obtained from Eq. (<ref>), and cannot change the results of the rescan analysis.Conversely, miscalibration of R̃^ g*_ℓ relative to R̃^ g_ℓ around any given candidate s implies that we have either underestimated or overestimated R̂^*_ℓ'(s), which in turn implies that the coincidence thresholds Θ^*_ℓ'(s) we imposed on the bins correlated with ℓ(s) were either unnecessarily low or too high. Clearly, the latter possibility is the one that should concern us: it implies that relative miscalibration of the rescan SNR can cause the probability that we miss a real persistent signal to exceed 1-c_2.Empirically, in the first HAYSTAC data run, we could reduce each R̂^*_ℓ'(s) by 17% before any of the (2K^g-1)S bins we examined exceeded the corresponding threshold.[The first bin to do so had δ^g*_ℓ/σ̃^g*_ℓ=2.7. Among S× n_K independent bins, we expect 0.5 bins with power excess this large, so the observation of one should not be surprising.] All of the parameter uncertainties whose contributions to δ |g^min_γ|/|g^min_γ| we have considered in this section are global effects to which the coincidence thresholds are insensitive. We conclude that miscalibration of R̃^ g*_ℓ relative to R̃^ g_ℓ by more than 17% is extremely unlikely. A more formal way to account for the possibility of relative miscalibration is to require a rescan confidence level c_2>c_1; we will adopt this approach in future HAYSTAC analyses.§ EFFECTS OF A WIDER LINESHAPEAs noted in Sec. <ref>, the analysis presented in this paper has assumed the spectral distribution of axion conversion power is given by Eq. (<ref>) instead of Eq. (<ref>), but we should actually expect the latter distribution in a terrestrial experiment if the halo axions are fully virialized with a pseudo-isothermal density profile and RMS velocity √(<v^2>)=270 km/s.To quantify the degradation of our exclusion limit |g^min_γ|_ℓ for an axion signal with the lab frame spectral distribution f'(ν), we repeated the simulation of Sec. <ref>, using Eq. (<ref>) instead of Eq. (<ref>) for the simulated axion signal but leaving the lineshape L̅_q used in both the “standard” and “ideal” analysis pipelines unchanged. As in Sec. <ref>, the main results of the simulation are two histograms (corresponding to the two analysis pipelines) representing the excess power distribution in the grand spectrum bin ℓ' best aligned with the simulated axion signal. These histograms are plotted in Fig. <ref>.We see that the mean value of the ideal analysis histogram E[δ^g_ℓ'/σ^g_ℓ']_i is no longer equal to the calculated SNR R^ g_ℓ' represented by the dashed vertical line. This is unsurprising, as R^ g_ℓ' is still calculated using the lineshape L̅_q obtained by integrating Eq. (<ref>). Thus, neglecting SG filter effects, the ratio ζ_0=E[δ^g_ℓ'/σ^g_ℓ']_i/R^ g_ℓ' = 0.69quantifies the reduction in SNR we should expect when we use an analysis optimized for signals with spectral distribution f(ν) to search for signals governed by the wider lab frame distribution f'(ν).Next we can consider how ζ_0 is modified by the imperfect SG filter stopband. From the width of the histogram obtained from the standard analysis, we obtain ξ=0.93, as we should expect given that we have not changed the parameters of the horizontal sum. Comparing the two histograms in Fig. <ref>, we obtain η_lab=E[δ^g_ℓ'/σ^g_ℓ']_s/(ξ E[δ^g_ℓ'/σ^g_ℓ']_i) = 0.83 [c.f. η=0.90 obtained in Sec. <ref> assuming the narrower distribution f(ν)]. The result η_lab<η is also expected, as the SG filter stopband attenuation gets worse towards larger spectral scales (See Fig. <ref>). The net reduction of the corrected KSVZ SNR R̃^ g_ℓ' is thusζ=(η_lab/η)ζ_0=0.64.Equivalently, at fixed R_T, |g_γ| is increased by a factor 1/√(ζ)=1.25. Since we cannot change the threshold in a reanalysis of a completed run without acquiring more rescan data, we conclude that our published exclusion limit |g^min_γ|_ℓ is degraded by 25% for axion signals with spectrum given by Eq. (<ref>). The modified limits still cut into the allowed parameter space for viable KSVZ and DFSZ models <cit.>; thus the qualitative conclusions of Ref. <cit.> remain unchanged.It should be emphasized that the value of ζ_0 derived from simulation above arises from the combination of two conceptually distinct effects. First, f'(ν) is wider than f(ν), and thus any analysis assuming the former will be less sensitive for a given noise temperature. Second, our analysis used values of K^g and L̅_q appropriate for the distribution f(ν), so the horizontal sum is not optimally weighted if the true signal spectrum is f'(ν). With K^r=10, K^g=7, and f(ν)→ f'(ν) in Eq. (<ref>), we can obtain ζ_0=0.78 analytically using Eq. (<ref>); simulation confirms this value and indicates that η_lab is unchanged. Thus we should expect ζ=0.72 for an analysis optimized for the wider signal distribution, or equivalently |g_γ| larger than our present limit by 18%, up to changes in other factors affecting the SNR. § SYNTHETIC AXION INJECTIONSIn Fig. <ref> we can see seven small notches in which |g^min_γ|_ℓ increases sharply over a very narrow range. These notches arise because we injected synthetic axion signals into the cavity at ten random frequencies during the initial data acquisition period in winter 2016, and cut data around each such signal before combining data from the winter and summer runs. In the two lowest-frequency notches, |g^min_γ|_ℓ increases by about a factor of 2^1/4 because roughly half the data contributing to the SNR at these frequencies was acquired during the winter run. At higher frequencies, a larger fraction of the data came from the summer run, and thus the depth of the notches gets progressively smaller. In particular, the effects of cutting data from the winter run around two injected signals above 5.76 GHz are not visible at the resolution of Fig. <ref>. The last injected signal happened to fall in the range where we cut spectra around an intruder mode (see Sec. <ref>), so it is also not visible in Fig. <ref>.The procedure we used to generate axion-like signals in HAYSTAC is summarized in Refs. <cit.> and <cit.>. Our goal in injecting these signals into the experiment was not to demonstrate an alternative approach to calibrating the search sensitivity, as obtaining sufficiently good statistics would entail polluting our spectrum with a large number of synthetic axions. Instead, we used synthetic signal injections as a simple fail-safe check on our data acquisition and analysis procedures, to verify that faint narrowband signals injected into the cavity did indeed result in large excess power in the expected grand spectrum bins. We decided on a nominal signal power of 10^-22 W, roughly equal to the expected conversion power for an axion with |g_γ|=4|g^KSVZ_γ| and sufficiently far above our target sensitivity to allow us to immediately establish the presence or absence of excess power with only a single pass over the tuning range. Due to a miscalculation, we set the power lower than this by a factor of 2.5 for the three highest-frequency signals, and moreover the exposure was lowest at these frequencies in the winter run: thus the expected SNR for these three signals was ≈1.5. We observed excess power consistent with this estimate (though of course also consistent with the absence of a signal) at these three frequencies. After correcting the signal power, we observed δ^g_ℓ/σ̃^g_ℓ > 5 in all bins corresponding to the remaining injected signals.Having demonstrated to our satisfaction that our analysis procedure can detect real axion-like signals, we opted not to inject signals during the summer run. Before constructing the combined spectrum used in the final analysis, we cut RF bins around each injected signal in which we expect more than 1% of the peak power given the measured signal lineshape.§ SCALING WITH INTEGRATION TIMEOn paper the expected SNR in a haloscope search is ∝√(τ), due to the τ^-1/2 scaling of the RMS noise power in each bin expected from Gaussian statistics. The observed standard normal distribution of the combined spectrum power excess δ^c_k/σ^c_k in both the initial scan and rescan analyses implicitly indicates that the RMS noise continues to decrease in this way with increasing averaging. We also demonstrated more directly that this τ^-1/2 scaling holds for real data out to τ>max(τ^*_s) with a dedicated measurement described below.For this measurement, we acquired 24 hours of noise data at a single frequency with the JPA gain maintained by feedback as in the data run. The data was saved to disk as a set of 17280 raw spectra obtained from τ_0=5 s of averaging each. In offline analysis we removed bins contaminated by knownIF interference, divided by the average baseline as in Sec. <ref>, and averaged every 10 adjacent spectra. We used this set of m=1728 averaged spectra to probe the behavior of the RMS noise σ_y as a function of the integration time τ_k=10kτ_0, for k=1,…,m.To measure σ_y(τ), we apply a Savitzky-Golay filter with parameters d^* and W^* to each of the m averages. Then for each k=1,…,m we average k filtered spectra and take σ_y(τ_k) to be the sample standard deviation of all bins in this k-spectrum average. We expect σ_y(τ)=1/√(Δν_bτ) – formallyσ_y=σ^p considered as a function of the integration time τ; we call this quantity σ_y in analogy to the Allan deviation, a time-domain measure of the dependence of the RMS noise on τ. 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"authors": [
"B. M. Brubaker",
"L. Zhong",
"S. K. Lamoreaux",
"K. W. Lehnert",
"K. A. van Bibber"
],
"categories": [
"astro-ph.IM",
"hep-ex"
],
"primary_category": "astro-ph.IM",
"published": "20170626141258",
"title": "The HAYSTAC Axion Search Analysis Procedure"
} |
Ioffe Physico-Technical Institute, 194021St.Petersburg, RussiaSt.Petersburg Institute of Nuclear Physics, 188300 Gatchina, Russia The conventional ways to calculate the perturbative component ofthe DIS structure function F_1 singlet involve approaches based on BFKL which account for the single-logarithmic contributions accompanying the Born factor 1/x. In contrast, we account for the double-logarithmic (DL) contributions unrelated to 1/x and because of that they were disregarded as negligibly small. We calculate F_1 singlet in the Double-Logarithmic Approximation (DLA) and account at the same time for the running α_s effects. We start with total resummation of both quark and gluon DL contributions and obtainthe explicit expression for F_1 in DLA. Then, applying the saddle-point method, we calculate the small-x asymptotics of F_1, which proves to be of the Regge form with the leading singularity ω_0 = 1.066.Itslarge value compensates for the lack of the factor 1/x in the DLA contributions. Therefore, this Reggeon can be named a new Pomeron which can be quite important for description of all QCD processes involving the vacuum (Pomeron) exchanges at very high energies. We prove that the expression for the small-x asymptotics of F_1 scales: it depends on a single variable Q^2/x^2 only instead of x and Q^2 separately. Finally,we show that the small-x asymptotics reliably represent F_1 at x ≤ 10^-6. 12.38.CyStructure Function F_1 singlet in Double-Logarithmic Approximation S.I. Troyan December 30, 2023 ==================================================================§ INTRODUCTION Description of the structure function F_1 singlet in the framework of Collinear Factorization usually involves DGLAP<cit.> to calculate the perturbative contributions. In this case F_1is represented in the form of two convolutions: F_1 = C_q (x/y) ⊗Δ q (y,Q^2)+ C_g (x/y) ⊗Δ g (y,Q^2),where C_q and C_g are the coefficient functionsand Δ q and Δ g denote the evolved (with respect to Q^2) quark and gluon distributions respectively.These distributions are solutions to the DGLAP equations which govern the Q^2-evolution of the initial quark and gluon distributions δ q (x,μ^2) and δ g(x,μ^2), evolving them from the scale μ^2 to Q^2. Bothδ q and δ g are defined at x ∼ 1 and Q^2 = μ^2 ∼ 1GeV^2.The parameter μis also called the factorization scale. The x-dependence of F_1 is described bythe coefficient functions C_q,g as well as by the phenomenological factors in δ q, δ g. In the framework of DGLAP the evolution in the k_⊥-space is is separated from evolution with respect to x. Such a separation takes place at x ∼ 1 only and breaks at small x as was shown in Ref. <cit.>. It is the theoretical reason not to use DGLAP at small x. A practical reason is that DGLAP, by its design, accounts for the total resummation of ln^n Q^2 while contributions ∼ln^n x are present in the DGLAP expressions in few first orders in α_s only (through the coefficient functions in NLO,NNLO, etc.).On the other hand, such contributions are very important at small x, so it would be appropriate to substitute the DGLAP expressions for the DIS structure functions by new ones which include the total resummation of all double-logarithmic (DL) contributions. In the first place there are DL terms ∼ (α_s ln^2 (1/x))^n, then the terms ∼ (α_s ln (1/x)ln Q^2)^n, etc. Expressions accounting for resummation of DL contributions and for the running α_s effects were obtained for severalstructure functions with non-vacuum exchanges in the t-channel:the spin structure function g_1 (the singlet and non-singlet components) and the non-singlet component of F_1 (see the overview<cit.> and refs therein).Besides, there were obtained the expressions for g_1 and non-singlet F_1 combining the DGLAP results and resummation of the DL contributions, which made possible to apply these expressions at arbitrary x and Q^2.However, a similar generalization of DGLAP was not obtained for the singlet F_1. The point is that by that time F_1 in the small-x region has been intensively investigated in terms of approaches based on BFKL<cit.> and this looked as the only way to study F_1 at small x. Indeed, the leading x-dependent contributions to F_1proved to bethe single-logarithmic (SL) terms accompanying the "Born" factor 1/x: (1/x) [1 + c_1 α_s ln (1/x) + c_2 (α_s ln (1/x))^2 + ...]while the DL contributions proportional to 1/x , i.e. the terms (1/x) [1 + c^DL_1 α_s ln^2 (1/x) + c_2^DL (α_s ln (1/x))^4 + ...], cancel each other (i.e. c_k^DL = 0 for k= 1,2,..) as was found first in Ref. <cit.>. As a result, the common strategy for investigating the QCD processes with vacuum exchanges in the t-channel was based on the use of the BFKL results. In particular, SL contributions to the structure functions F_1,2 was presented in Refs. <cit.>; SL contributions to F_2 in Ref. <cit.> were calculated with inclusion of resummed anomalous dimensions in the renormalization group equation while F_2in Ref. <cit.> was calculated with direct unification of DGLAP and FFKL.Solution to the BFKL equation is expressed through the series of the high-energy asymptotics of the Regge form, with the leading asymptotics commonly addressed as the BFKL Pomeron, so at x → 0 F_1 ∼ x^-(1 + Δ_P),whereΔ_P is the Pomeron intercept.As Δ_P > 0 for the both LO and NLO BFKL Pomerons, they are called the supercritical ones. As we are not going to use BFKL or its modifications like <cit.> in the present paper, we just mention that the extensive literature on this issue can be found in Ref. <cit.>.Instead of using the BFKL results or trying to increase the accuracy of the method of Ref. <cit.>, in the present paper we account for total resummation of the double-logarithmic contributions to F_1. In the first place we account for the x-dependent contributions 1 + c'_1 α_s ln^2 (1/x) + c'_2 (α_s ln^2 (1/x))^2 + ...and then for DL terms combining logs of x and Q^2. These DL contributions do not involve the large factor 1/x and by this reason they have been neglected in the BFKL approach. We calculate the singlet structure function F_1 in DLA, summing DL contributions coming from virtual gluon and quark exchanges. As a result, our expressions for coefficient functions and anomalous dimensions contain total resummations of appropriate DL terms. To calculate F_1 we compose and solve Infra-Red Evolution Equations (IREE) in the same way as we did for calculating the DIS structure function g_1 singlet (see Ref. <cit.>), investigating the cases of fixed and running α_s. We remind that the IREE method was suggested by L.N. Lipatov in Ref. <cit.>. It is based on factorization of DL contributions of the partons with minimal transverse momenta first noticed by V.N. Gribov in Ref. <cit.> in the context of QED of hadrons. Technology of implementation of this method to DIS is described in detail in Ref. <cit.>. In contrast to DGLAP and BFKL equations, we compose the two-dimensional evolution equations: They control evolutions in both x and Q^2. We obtain the explicit expression for F_1 and then, applying the saddle-point method, we calculate the small-x asymptotics of F_1 automatically complemented by the asymptotic Q^2-dependence. The asymptotics proves to be of the Regge form. The large value ofthe intercept compensates for the lack of the factor 1/x in the DL contributions and thereby makes the DLA asymptotics be of the same order as the BFKL one. This proves that the DL contributions to F_1 at small x are, at least, no less important than the contributions coming from the BFKL Pomeron.Our paper is outlined as follows: in Sect. II we compose and solve IREE for the Compton amplitudes A_q,g related toF_1 by the Optical theorem. In this Sect. we express A_q,g through the amplitudes of the 2 → 2 scattering of partons. Those amplitudes are calculated in Sect. III. In Sect. IV we apply the saddle-point method to obtain explicit expression for the small-x asymptotics of F_1 and prove that this asymptotics depends on the single variable Q^2/x^2 instead of separate dependence on Q^2 and x. In Sect. Vwe consider in detail the intercept of the Pomeron in DLA, embracing the cases of fixed and running α_s. We also fix the region where the small-x asymptotics can reliably represent F_1. Finally, Sect. VI is for our concluding remarks.§ IREE FOR THE AMPLITUDES OF COMPTON SCATTERING OFF PARTONS Following the DGLAP pattern, we consider F_1 in the framework of Collinear Factorization and represent F_1 through the convolutions of the perturbative components T_q and T_g with non-perturbative initial quark and gluon distributions ϕ_q,g respectively: F_1 = F^q_1 ⊗ϕ_q + F_1^gg ⊗ϕ_g. Throughout the paper we will consider the perturbative objects F_1^q,g only.It is convenient to consider the Compton amplitudes A_q and A_g related to T_1^q,g by Optical theorem: F_1^q,g (x,Q^2/μ^2) = -1/2π A_q,g(x,Q^2/μ^2),where we have introduced the factorization scale μ and used the standard notation x = Q^2/w, with w = 2pq and Q^2 = - q^2. The next step is to represent A_q,g in terms of the Mellin transform:A_q,g(w/μ^2, Q^2/μ^2) = ∫_- ∞^∞d ω/2 π(w/μ^2)^ωξ^(+)(ω) F_q,g(ω, Q^2/μ^2) ≈∫_- ∞^∞d ω/2 π e^ωρ F_q,g(ω, y),where we have introduced the signature factor ξ^(+)(ω) = (1 + e^- ω)/2 ≈ 1 and the logarithmic variables ρ, y (using the standard notation w = 2pq):ρ = ln (w/μ^2), y = ln (Q^2/μ^2). In what follows we will address F_q, F_g as Mellin amplitudes and will use the same form of the Mellin transform for other amplitudes as well.For instance, the Mellin transform for the color singlet amplitude A_gg of the elastic gluon-gluon scattering in the forward kinematics isA_gg = ∫_- ∞^∞d ω/2 π(w/μ^2)^ωξ^(+)(ω) f_gg (ω) ≈∫_- ∞^∞d ω/2 π e^ωρ f_gg (ω).We have presumed in Eq. (<ref>) that virtualities of all external gluons are ∼μ^2. Let us notice that the only difference between the Mellin representation for the Compton amplitudes A_q,g and the similar amplitudes related to the singlet g_1 is in the signature factors only: the signature factor for g_1 is ξ^(-)(ω) = (-1 + e^- ω)/2. Otherwise, technology of composing and solving IREE for A_gg and g_1 singlet is the same. Because of that we present IREE for F_q, F_g (and for auxiliary amplitudes as well) with short comments only. The full-length derivation of all involved IREE can be found in Ref. <cit.>. Now all set to construct IREEs for F_q,g.In the kinematics where w ≫ Q^2 ≫μ^2,the amplitudes F_q, F_g obey the partial differential equations:[∂/∂ y + ω] F_q(ω,y)=F_q (ω, y) h_qq (ω) + F_g (ω, y) h_gq (ω),[∂/∂ y + ω] F_g(ω, y)=F_q (ω, y) h_qg (ω) + F_g (ω, y) h_gg (ω),where we have used the following convenient notations:h_rr' = 1/8 π^2 f_rr',with r,r' = q,g and f_rr' being the parton-parton amplitudes. We will calculate h_rr' in the next Sect. Actually, the equations in (<ref>) manifest strong resemblance with the DGLAP equations. Indeed, the first factor in brackets in the l.h.s. of (<ref>) exists in DGLAP too. The second term vanishes when the Mellin factor (s/μ^2)^ω is replaced by the factor x^-ω which is used in the DGLAP equations. When the parton amplitudes f_rr' are in the Born approximation, Eq. (<ref>) coincides with the DGLAP equations. A general solution to Eq. (<ref>) is F_q(ω,y)=e^- ω y[C_(+) e^Ω_(+) y + C_(-) e^Ω_(-) y], F_g(ω, y)=e^- ω y[ C_(+)h_gg - h_qq + √(R)/2h_qg e^Ω_(+) y + C_(-)h_gg - h_qq - √(R)/2h_qg e^Ω_(-) y],where C_(±) (ω) are arbitrary factors whereas Ω_(±) = 1/2[ h_gg + h_qq±√(R)]and R = (h_gg + h_qq)^2 - 4(h_qqh_gg - h_qgh_gq) = (h_gg - h_qq)^2+ 4 h_qgh_gq . We specify the factors C_(±) (ω)by the matching with the Compton amplitudes f_q, f_g calculated in the kinematics Q^2 ≈μ^2, i.e. at y = 0. The matching condition is F_q(ω,y)|_y = 0 = f_q(ω), F_g(ω,y)|_y = 0 = f_g(ω),which leads to the following expressions: C_(+) = h_qg f_g(ω) - (h_gg - h_qq -√(R)) f_q(ω)/2 √(R), C_(-) =- h_qg f_g(ω) + (h_gg - h_qq +√(R)) f_q(ω)/2 √(R).Now let us express f_q, f_g through the parton-parton amplitudes h_rr'. To this end, we construct IREE for them. As f_q, f_g do not depend onQ^2, the IREE for them are algebraic: ω f_q(ω)=a_γ q + f_q (ω) h_qq (ω) + f_g (ω) h_gq (ω),ω f_g(ω)=f_q (ω) h_qg (ω) + f_g (ω) h_gg (ω),where a_γ q = e^2, with e^2 being the total electric charge of the involved quacks, so that a_γ q/ω is the Born value of amplitude f_q(ω). There is no a similar term in the equation for f_g(ω). The only difference between the r.h.s. of (<ref>) and (<ref>) is the factor a_γ q in Eq. (<ref>).The solution to Eq. (<ref>) is f_q (ω)=a_γ q (ω - h_gg)/G(ω),f_g (ω)= a_γ q h_qg/G(ω),with G(ω) being the determinant of the system (<ref>): G = (ω - h_qq)(ω - h_gg)- h_ggh_qg. Combining Eqs. (<ref>) and (<ref>), we express C_± through the parton-parton amplitudes: C_(+) =a_γ qh_qgh_gq - (ω - h_gg)(h_gg - h_qq - √(R))/2 G √(R), C_(-) =a_γ q -h_qgh_gq + (ω - h_gg)(h_gg - h_qq + √(R))/2 G √(R). Combining Eqs. (<ref>,<ref>) and (<ref>), we can easily express F_q,g in terms of the parton-parton amplitudes h_rr'. § PARTON-PARTON AMPLITUDES In this Sect. we obtain explicit expressions for the parton amplitudes h_rr'. The IREE for h_rr' are quite similar to Eq. (<ref>): ω h_qq =b_qq + h_qqh_qq + h_qgh_gq, ω h_qg = b_qg + h_qqh_qg+ h_qgh_gg,ω h_gq =b_gq + h_gqh_qq+ h_ggh_gq, ω h_gg = b_gg + h_gqh_qg+ h_ggh_gg,where the terms b_rr' include the Born factors a_rr' and contributions of non-ladder graphs V_rr':b_rr' = a_rr' + V_rr'. The Born factors are (see Ref. <cit.> for detail): a_qq = A(ω)C_F/2π, a_qg = A'(ω)C_F/π, a_gq = -A'(ω)n_f/2 π. a_gg = 2N A(ω)/π,where A and A' stand for the running QCD couplings: A = 1/b[η/η^2 + π^2 - ∫_0^∞d z e^- ω z/(z + η)^2 + π^2],A' = 1/b[1/η - ∫_0^∞d z e^- ω z/(z + η)^2],with η = ln(μ^2/Λ^2_QCD) and b being the first coefficient of the Gell-Mann- Low function. When the running effects for the QCD coupling are neglected, A(ω) and A'(ω) are replaced by α_s. The terms V_rr' are represented in a similar albeit more involved way (see Ref. <cit.> for detail): V_rr' = m_rr'/π^2 D(ω) ,withm_qq = C_F/2 N , m_gg = - 2N^2 , m_gq = n_f N/2 , m_qg = - N C_F ,andD(ω) = 1/2 b^2∫_0^∞ d z e^- ω zln( (z + η)/η) [ z + η/(z + η)^2 + π^2 - 1/z + η] . Let us note that D = 0 when the running coupling effects are neglected. It corresponds the total compensation of DL contributions of non-ladder Feynman graphs to scattering amplitudes with the positive signature as was first noticed in Ref. <cit.>. When α_s is running, such compensation is only partial. Solution to Eq. (<ref>) ish_qq = 1/2[ ω - Z - b_gg - b_qq/Z], h_qg = b_qg/Z , h_gg = 1/2[ ω - Z + b_gg - b_qq/Z], h_gq =b_gq/Z ,whereZ = 1/√(2)√( Y + W ) ,withY = ω^2 - 2(b_qq + b_gg)andW = √((ω^2 - 2(b_qq + b_gg))^2 - 4 (b_qq - b_gg)^2 - 16b_gq b_qg) The algebraic equations (<ref>) are non-linear, so they yield four expressions for Z. We selected in Eq. (<ref>) the solution obeying the matching with the Born amplitudes h^Born_rr': at large ω h_rr'→ h^Born_rr' =a_rr'/ω. Substituting the expressions of Eq. (<ref>) in (<ref>), we obtain explicit expressions for amplitudes f_q, f_g. Combining them with Eqs. (<ref>,<ref>) and (<ref>), we obtain explicit expressions for F_q and F_g. Substituting them in Eq. (<ref>), we arrive at the explicit expressions for theCompton amplitudes A_q and A_g. Finally, applying the Optical theorem (<ref>) to A_q and A_g, wearrive at the structure function F_1 singlet. § SMALL-X ASYMPTOTICS OF THE STRUCTURE FUNCTION F_1The regular way to obtain the small-x asymptotics ofA_q and A_g is to write explicit expressions for F_q and F_g in Eq. (<ref>), then push x → 0 and apply the saddle-point method. However before doing this, let us consider in derail how to calculate the asymptotics of the gluon-gluon scattering amplitude A_gg, presuming virtualities of all external gluons ∼μ^2.§.§ Asymptotics of F_1 The small-x asymptotics of A_q and A_g can be obtained with applying the saddle-point method to Eq. (<ref>).As Ω_(+) > Ω_(-), we neglectthe terms C_(-) in (<ref>) and represent Eq. (<ref>) to the following form: A_q≈ ∫_- ∞^∞d ω/2 π e^ωξF_q (ω) e^Ω_(+) y = ∫_- ∞^∞d ω/2 π e^Ψ_q, A_g≈ ∫_- ∞^∞d ω/2 π e^ωξF_g (ω) e^Ω_(+) y = ∫_- ∞^∞d ω/2 π e^Ψ_g,with ξ = ln(1/x) and F_q = C_(+), F_g = C_(+)(h_gg - h_qq + √(R))/2 h_qgand Ψ_q = ωξ + lnF_q, Ψ_q = ωξ + lnF_q. The stationary point at x → 0 of Ψ_q is given by the rightmost root ω_0 of the following equation: d Ψ_q/dω = ξ + F'_q(ω_0)/F_q (ω_0) = 0. When ξ→∞, it must be equated by some negative singular contribution in the second term of Eq. (<ref>). Using the explicit formulae for F_q,g, one can conclude that such contribution comes from the factor 1/W. So, the stationary point ω_0 is the rightmost root of the equation (ω^2 - 2b_qq -2 b_gg)^2 - 4(b_qq- b_gg)^2 - 16 b_qgb_gq = 0. We consider in detail solutions to Eq. (<ref>) at fixed and running α_s in the next Sect. In vicinity of ω_0 we can represent Eq. (<ref>) as Ψ'_q = ξ + ∂F_q/F_q ∂ Wd W/d ω = ξ + ∂F_q/F_q ∂ Wλ/W = ξ - φλ/W = 0,with φ_q = - ∂lnF_q/ ∂ W and λ = 2 ω(ω^2 - 2 (b_+) + b_(-)),so in vicinity of the singularity ω_0 W ≈ W_0 =φλ/ξExpanding Ψ_q in the series, we obtain Ψ_q (ω) ≈Ψ_q (ω_0) + (1/2) Ψ''_q (ω_0) (ω - ω_0)^2. In order to calculate Ψ''_q we notice that the most singular contributions comes from differentiation of the numerator in Eq. (<ref>), so Ψ''_q ≈ - ∂F_q/F_q ∂ Wλ d W^-1/d ω =φλ^2/W^3 = ξ^3/λφ_q^2and therefore the asymptotics of A_q at x → 0 is A_q(x, Q^2/μ^2) ∼ A_q^as(x, Q^2/μ^2) = C_(+)(ω_0) φ_q (ω_0) √(λ/2πξ^3)(1/x)^ω_0(Q^2/μ^2)^Ω_(+)(ω_0). Repeating the reasoning above for A_g and applying to them the Optical theorem, we conclude that the small-x asymptotics of F_1 is F_1 ∼Π(ω_0, ξ) (1/x)^ω_0(Q^2/μ^2)^Ω_(+)(ω_0) ,where the factor Π(ω_0, ξ)isΠ (ω_0, ξ) = C_(+)√(λ/2 πξ^3)[φ_q δ q + φ_g (h_gg(ω_0)- h_qq(ω_0) + √(R(ω_0))/2 h_qg(ω_0)) δ g],with δ q and δ q being the initial quark and gluon densities. They do not include singular factors ∼ x^-a, with positive a. The Regge form of the asymptotics is brought entirely by the perturbative contributions. Let us notice that Π∼ln^-3/2 (1/x). Eq. (<ref>) exhibits that the total resummation of DL contributions leads to the Regge behavior of F_1 at small x. §.§ Asymptotic scaling Substituting the explicit expressions for h_rr' of Eq. (<ref>) inEq. (<ref>) and using Eq. (<ref>),we obtain that Ω_(+)(ω_0) = ω_0/2. This allows us to write the asymptotics of F_1 of Eq. (<ref>) in the following way: F_1 ∼Π(ω_0, ξ) (1/x)^ω_0(Q^2/μ^2)^ω_0/2 = Π (ω_0, ξ) (Q^2/x^2 μ^2)^ω_0/2 Eq. (<ref>) manifests that F_1(x, Q^2) at asymptotically high energies depends on the single variable Q^2/x^2only. We name such confluence of the x and Q^2 dependence the asymptotic scaling. The same form of the asymptotic scaling was obtained earlier for the structure function g_1 and the non-singlet component of F_! (see Ref. <cit.> for detail). We stress that the asymptotic scaling for F_1 can be checked with analysis of available experimental data. Moreover, F_2 = 2x F_1 at very small x, which proves the asymptotic scaling for F_2. Finally, let us notice that the leading singularity ω_0 in Eq. (<ref>) does not depend on Q^2.§ ANATOMY OF THE LEADING SINGULARITY Ω_0 In this Sect. we consider in detail the leading singularity ω_0 which is the rightmost root of Eq. (<ref>). In order to make the asymptotics of F_1 be looking similarly to Eq. (<ref>), we denote ω_0 = 1 + Δso that ω_0 could look similarly to the BFKL leading singularity, see Eq. (<ref>). Now let us discuss different scenarios for calculating ω_0. In what follows we will address Δ as the DL Pomeron intercept. We remind that in the straightforwardReggeology concept Δ = 0 and the Pomeron withΔ > 0 is called the supercritical Pomeron. §.§ Intercept under approximation of fixed QCD coupling In the first place let us estimate ω_0 for the case of fixed α_s. In this case DL contributions of non-ladder graphs totally cancel each other, so that D = 0 and b_rr' = a_rr', with a_rr' defined in Eq. (<ref>), where A(ω) and A'(ω) should be replaced by α_s. Then the solution to Eq. (<ref>) is ω_0^fix =(α_s^fix/π)^1/2[4 N + C_F + √((4N - C_F)^2 - 8 n_f C_F)]^1/2≈ 2.63 √(α_s),with the standard notations of the color factors N = 3, C_F = (N^2-1)/(2N) and n_f=4 is the flavour number. According to Ref. <cit.>, in this case α_s^fix≈ 0.24 which gives ω_0^fix = 1.29. Using the representation of ω_0 of Eq. (<ref>), we obtain Δ^fix =ω_0^fix - 1 = 0.29 which fairly coincides with the well-known LO BFKL intercept Δ_LO. However, Δ_LO corresponds to accounting for gluon contributions only while Δ^fix accommodates both gluon and quark contributions. When the quark contributions in Eq. (<ref>) are dropped, the purely gluonic intercept Δ_g^fix becomes somewhat greater:Δ_g^fix = 0.35which again bears a strong resemblance to the LO BFKL intercept. However, we are positive that the approximation of fixed α_s can used for rough estimating only, so we will not pursuit this approximation any longer. §.§ Intercept for the case of running coupling Now we account for the running coupling effects in Eq. (<ref>). Because of that, Eq. (<ref>) can be solved only numerically. As the couplings A and A' included in the factors b_rr' depend on μ through η = ln (μ^2/Λ^2), the solution, ω_0 is also μ-dependent. Numerical calculations yield the plot of the η-dependence of ω_0 presented in Fig. <ref>.The curve in Fig. <ref> has the maximum ω_0^DLA = 1.066 at μ/Λ = 13.8. We address μ_0 = 13.8 Λas the optimal mass scale and call Δ = ω_0^DLA - 1 = 0.066the intercept of the Pomeron in DLA. It is interesting to notice that Δ is close to the NLO BFKL intercept. In contrast, when the quark contributions are neglected, the purely gluonic intercept Δ_g^DLA is much greater: Δ_g = 0.254 . Confronting Eq. (<ref>) to (<ref>) andEq. (<ref>) to (<ref>) demonstrates that accounting for the quark contributions decreases the intercept. Similarly, confronting Eq. (<ref>) to (<ref>) exhibits that accounting for the running α_s effects essentially decreases the intercept value. We also would like to stress that despite that our values of Δ in Eqs. (<ref>) and (<ref>) are close to the values of the LO BFKL and NLO BFKL intercepts respectively, this similarity if just a coincidence: our intercepts are obtained from resummation of DL contributions while the BFKL sums the single-logarithmic terms.Moreover, Eq. (<ref>) corresponds to the case of α_s running in every vertex of all involved Feynman graph while BFKL operates with fixed α_s and includes setting of its scale a posteriori. §.§ Applicability region of the small-x asymptotics It is obvious that the small-x asymptotic expressions, like Eq. (<ref>) are always much simpler than non-asymptotic expressions. However, it is important to know at which values of x the asymptotics can reliably be used. To answer this question we numerically investigateR_as defined as follows: R_as (x, Q^2) = A_q^as(x,Q^2)/A_q (x,Q^2) The x-dependence of R_as at fixed Q^2 is shown in Fig. 2 for the case when Q^2 ≈μ^2:Fig. 2 demonstrates that R_as = 0.9 at x ≈ 8. 10^-5 while the curve in Fig. 3, where Q^2 = 100 μ^2, grows slower and achieves the value R_as = 0.9 much later, at x ≈ 3.10^-7:Therefore, the applicability region of the small-x asymptotics essentially depends on the Q^2 value. The plots in Figs. 2,3 lead us to conclude that the small-x asymptotics reliably represent F_1 in the wide range of Q^2 when x < x_max, with x_max≈ 10^-6.§ SUMMARY AND OUTLOOK In this paper we have calculated the perturbative contributions F_1^q and F_1^g to the structure function F_1 in the Double-Logarithmic Approximation, by collecting the DL contributions and at the same time accounting for the running α_s effects. We obtained the explicit expressions for F_1^q,gand then, applying the saddle-point method, calculated the small-x asymptotics of F_1, arriving at the new, DL contribution to the QCD Pomeron. We demonstrated that despite the lack of the factor 1/x in the DL contributions, the impact of their total resummation makes this Pomeron be supercritical, albeit the value of the intercept strongly depends on the accuracy of calculations. The maximal value of the intercept corresponds to the roughest approximation where quark contributions are neglected and α_s is fixed. Then, the value of the intercept decreases when accuracy of the calculations increases: first, when the quark contributions are accounted for and then, notably, when the running α_s effects are taken into account. Nevertheless, the Pomeron remains supercritical as ω_0 = 1.066. Such monotonic decrease allows us suggest that further accounting for sub-leading contributions can decrease the value of the intercept down to zero, so that eventually the intercept will satisfy the Froissart bound. We proved thatthe x and Q^2 -dependencies of F_1 converge at small x in dependence on the single variable Q^2/x^2.We call this convergence the asymptotic scaling. We stress that this prediction of the asymptotic scaling can beconfirmed by analysis of available experimental data.As asymptotically F_2 ∼ 2x F_1, the asymptotic scaling should also take place for F_2. Investigating the applicability region for the asymptotics, we found that F_1can reliably be represented by its asymptotics at x ≤ x_max, with x_max≈ 10^-6. Although we have discussed the structure function F_1, we would like to notice that theexperimental date available in the literature are mostly on the structure functions F_2 and F_L, soit would be interesting to apply our approach to calculate F_2 and F_L as well. Calculating F_2 in DLA can be done in the way quite similar to that we have used for F_1. As a result, we obtain that F_2 in DLA can be represented through F_1: F_2 = 2 x F_1,which coincides with the well-known Born relation between F_1 and F_2. Eq. (<ref>) entails that in DLA F_L = 0. In order to estimate deviation of F_L from zero, one should account for sub-leading contributions to both F_1 and F_2. In the first place, such contributions are the single-logarithmic (SL) ones. In this regard we remind that the SL contributions to F_2 following from emission of gluons with momenta widely separated in rapidity and not ordered in transverse momenta were accounted in Refs. <cit.>-<cit.>, which involved dealing with the BFKL characteristic function. However, there are the SL contributions unrelated to BFKL, i.e., for the case of F_1, the SL terms unaccompanied by the factor 1/x similarly to the DL terms in Eq. (<ref>). In contrast to the DL contributions (<ref>), there is not a general technology in the literature forresummations of such non-BFKL SL terms.On the other hand, we were able to modify the IREE method for the spin structure function g_1 (see Ref. <cit.> and refs therein) to account for the SL contributionswhich are complementary to theones calculated in Refs. <cit.>-<cit.> namely, the SL following from emission of the partons with momenta ordered in the k_⊥-space and disordered in the longitudinal space. We plan to adapt this approach to calculate the SL contributions to F_1,2.Finally, we stress that in contrast to DGLAP we do not need singular factors ∼ x^-a in fits for the initial parton distributions for F_1. Such factorscause a steep rise of the structure functions at small x and lead to the Regge asymptotics of F_1. However, we have shown in Sect. IV that the resummation of the DL contributions to F_1 automatically leads to the Regge asymptotics, which makes unnecessary inclusion of the singular terms into the fits. This result agrees with our earlier results (see Ref. <cit.>) for the structure functions g_1 and F_1 non-singlet and also agrees with theresults of Refs. <cit.> obtained for the small-x behavior of the structure function F_2. The latter agreement isespecially interesting because approaches used in Refs. <cit.> and in the present paper are totally different.§ ACKNOWLEDGEMENT We are grateful to Mario Greco for interesting discussions of the running α_s effects. Work of B.I. Ermolaev was supported in part by the RFBRGrant No. 16-02-00790-a. 99dglap G. Altarelli and G. Parisi, Nucl. Phys.B126 (1977) 297; V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; L.N.Lipatov, Sov. J. Nucl. Phys. 20 (1972) 95; Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.ggfl V.G. Gorshkov, V.N. Gribov, G.V. Frolov, L.N. Lipatov. Yad.Fiz.6(1967)129; Yad.Fiz.6(1967)361; V.G. Gorshkov. Uspekhi Fiz. Nauk 110(1973)45.egtg1sum B.I. Ermolaev, M. Greco, S.I. Troyan. Riv.Nuovo Cim. 33 (2010) 57.bfkl E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov. Phys. JETP 44, 443 (1976); E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov. Phys. JETP 45, 199 (1977); I.I. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978); V.S. Fadin and L.N. Lipatov. Phys. Lett. B429 (1998) 127; G. Camici and M. Ciafaloni. Phys. Lett. B430 (1998) 349.ttwu B.M. McCoy, T.T. Wu. Phys. Rev. D 12 (1975) 3257.catciaf S. Catani, M. Ciafaloni, F. Hautmann. Nucl. Phys. Proc. Suppl. 29 A (1992) 182.cathaut S. Catani and F. Hautmann. Phys. Lett. 315B (1993) 157; Nucl. Phys. B 427 (1994) 475. ehb R.K. Ellis, F. Hautmann, B.R. Webber. Phys. Lett. 348B (1995) 582.haut F. Hautmann. CAVENDISH-HEP-95-04,hep-ph/9506303.kms J. Kwiecinski, A.D. Martin, A.M. Stasto. Phys.Rev. D56 (1997) 3991.balkov I. Balitsky. Nucl. Phys. B 463 (1996) 99; Y.V. Kovchegov. Phys. Rev. D 60 (1999) 034008.iancu E. Iancu, J.D. Madrigal, A.H. Mueller, G. Soyez, D.N. Triantafillopoulos. Nucl. Phys. A 956 (2016) 560.kl L.N. Lipatov. Zh.Eksp.Teor.Fiz.82 (1982)991; Phys.Lett.B116 (1982)411. R. Kirschner and L.N. Lipatov. ZhETP 83(1982)488; Nucl. Phys. B 213(1983)122.grib V.N. Gribov. Sov.J.Nucl.Phys. 5 (1967) 280.nest V.G. Gorshkov, L.N. Lipatov, M.M. Nesterov.Yad.Fiz. 9 (1969) 1221-1231.egtfix B.I. Ermolaev, M. Greco, S.I. Troyan. Eur.Phys.J.Plus 128 (2013) 34. | http://arxiv.org/abs/1706.08371v2 | {
"authors": [
"B. I. Ermolaev",
"S. I. Troyan"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170626135040",
"title": "Structure Function F_1 singlet in Double-Logarithmic Approximation"
} |
December 30, 2023H. A. Folonier and S. Ferraz-Mello Instituto de Astronomia Geofísica e Ciências Atmosféricas, Universidade de São Paulo, São Paulo, Brasil [email protected] and [email protected] synchronization of an anelastic multi-layered body: Titan's synchronous rotation. Hugo Folonier ˙ Sylvio Ferraz-Mello December 30, 2023 ======================================================================================= Tidal torque drives the rotational and orbital evolution of planet-satellite and star-exoplanet systems. This paper presents one analytical tidal theory fora viscoelastic multi-layered body with an arbitrary number of homogeneous layers. Starting with the static equilibrium figure, modified to include tide anddifferential rotation, and using the Newtonian creep approach, we find the dynamical equilibrium figure of the deformed body, which allows us to calculatethe tidal potential and the forces acting on the tide generating body, as well as the rotation and orbital elements variations. In the particular case ofthe two-layer model, we study the tidal synchronization when the gravitational coupling and the friction in the interface between the layers is added. Forhigh relaxation factors (low viscosity), the stationary solution of each layer is synchronous with the orbital mean motion (n) when the orbit is circular,but the rotational frequencies increase if the orbital eccentricity increases. This behavior is characteristic in the classical Darwinian theories and inthe homogeneous case of the creep tide theory. For low relaxation factors (high viscosity), as in planetary satellites, if friction remains low, each layercan be trapped in different spin-orbit resonances with frequencies n/2,n,3n/2,2n,…. When the friction increases, attractors with differentialrotations are destroyed, surviving only commensurabilities in which core and shell have the same velocity of rotation. We apply the theory to Titan. Themain results are: i) the rotational constraint does not allow us confirm or reject the existence of a subsurface ocean in Titan; and ii) thecrust-atmosphere exchange of angular momentum can be neglected. Using the rotation estimate based on Cassini's observation (Meriggiola et al. in Icarus275:183-192, 2016), we limit the possible value of the shell relaxation factor, when a deep subsurface ocean is assumed, to γ_s≲10^-9 s^-1, which correspond to a shell's viscosity η_s≳10^18 Pa s, depending on the ocean's thickness and viscosity values. In the casein which a subsurface ocean does not exist, the maximum shell relaxation factor is one order of magnitude smaller and the corresponding minimum shell'sviscosity is one order higher. § INTRODUCTION Tidal torque is a key physical agent controlling the rotational and orbital evolution of systems with close-in bodies and may give important clues on thephysical conditions in which these systems originated and evolved. The viscoelastic nature of a real body causes a non-instantaneous deformation, and thebody continuously tries to recover the equilibrium figure corresponding to the varying gravitational potential due to the orbital companion. In standardDarwin's theory (e.g. Darwin, 1880; Kaula, 1964; Mignard, 1979; Efroimsky and Lainey, 2007; Ferraz-Mello et al., 2008), the gravitational potential of thedeformed body is expanded in Fourier series, and the viscosity is introduced by means of ad hoc phase lags in the periodic terms or, alternatively,an ad hoc constant time lag.All these theories predict the existence of a stationary rotation. If the lags are assumed to be proportional to the tidal frequencies, the stationaryrotation has the frequency Ω_stat≃ n(1+6e^2), where n is the mean motion and e is the orbital eccentricity. The synchronous rotation isonly possible when the orbit is circular, but the stationary rotation becomes super-synchronous in the non-zero eccentricity case. In these theories, theexcess of rotation 6ne^2 does not depend on the rheology of the body. However, this prediction is not confirmed for Titan, where the excess provided bythe theory is ∼ 38^∘ per year, and the Cassini mission, using radar measurement, has not shown discrepancy from synchronous motion larger than∼0.02^∘ per year (Meriggiola, 2012; Meriggiola et al., 2016). Standard theories circumvent this difficulty by assuming that the satellite has anad hoc triaxiality, which is permanent and not affected by the tidal forces acting on the body.Recently, a new tidal theory for viscous homogeneous bodies has been developed by Ferraz-Mello (2013, 2015a) (hereafter FM13 and FM15, respectively). ANewtonian creep model, which results from a spherical approximate solution of the Navier-Stokes equation for fluids with very low Reynolds number, is usedto calculate the surface deformation due to an anelastic tide. This deformation is assumed to be proportional to the stress, and the proportionalityconstant γ, called the relaxation factor, is inversely proportional to the viscosity of the body. In the creep tide theory, the excess of synchronousrotation is roughly proportional to 6nγ^2e^2/(n^2+γ^2). This result reproduces the result obtained with Darwin's theory in the limitγ >> n (gaseous bodies), but tends to zero when γ→ 0 reproducing the almost synchronous rotation of stiff satellites, without theneed of assuming an ad hoc permanent triaxiality. The asymmetry created by the tidal deformation of the satellite is enough to create the torquesresponsible for its almost synchronous rotation.Tidal theories founded onhydrodynamical equations were also developed by Zahn, (1966) andRemus et al. (2012).A planar theory using a Maxwell viscoelastic rheology and leading to similar results was developed by Correia et al. (2014) and generalized later to thespatial case by Boué et al. (2016). Despite the different methods used to introduce the elasticity of the body, this approach is virtually equivalent tothe creep tide theory (Ferraz-Mello, 2015b).Other general rheologies were studied by Henning et al. (2008) and Frouard et al. (2016).However, real celestial bodies are quite far from being homogeneous, and how the tide influences its dynamic evolution is not entirely clear yet.Differentiation is common in our Solar System, and several satellites present evidence of a subsurface liquid ocean. We may cite, for instance, Europa (Wahret al., 2006; Khurana et al., 1998) and Enceladus (Porco et al., 2006; Nimmo et al., 2007). One paradigmatic case is Titan, where, in addition, the exchangeof a certain amount of angular momentum between the surface and the atmosphere may be important (Tokano and Neubauer, 2005; Richard et al., 2014); in addition,the presence of an internal ocean (Tobie et al., 2005; Lorenz et al., 2008, Sohl et al., 2014) may decouple rotationally the crust from the interior(Karatekin et al., 2008). The rotation of the crust has been studied by Van Hoolst et al. (2008) using the static tide and internal effects as gravitationalcoupling and pressure torques. They found that the crust rotation is influenced mainly by the atmosphere and the Saturn torque and claim that the viscouscrust deformation and the non-hydrostatic effects could play an important role in the amplitude of the crust oscillation.Here, we extend the planar creep tide theory to the case of a viscoelastic body formed by N homogeneous layers and study the stationary rotation of theparticular case N=2. Adapting the multi-layered Roche static figure, given by Folonier et al. (2015), to include differential rotation, we solve the creeptide equation for each layers interface. Moreover, we add the gravitational coupling and the friction in the interface between the layers. The layout of the paper is as follows: in Sect. 2 we present the creep tide model for a multi-layered body, using the static equilibrium figure of Folonieret al. (2015), adapted to include the differential rotation. In Sect. 3, we compute the disturbing potential of the deformed body. The forces and toques arecalculated in Sects. 4 and 5. In Sect. 6 we calculate the work done by the tidal forces acting on the bodies. The variations in semi-major axis andeccentricity are shown in Sect. 7. In Sect. 8, we develop the two-layer model, adding the interaction torques between the core and the shell. In Sect. 9, wecompare the two-layer model with the homogeneous theory. In Sect. 10, we apply to Titan, and, finally, the conclusions are presented in Sect. 11. The paperis completed by several appendices where are given technical details of some of the topics presented in the forthcoming sections. In addition, an OnlineSupplement is provided with further details, not worthy of inclusion in the paper but useful for the reproduction of several developments. § NON-HOMOGENEOUS NEWTONIAN CREEP TIDE THEORY Let us consider one differentiated body m of mass m_T, disturbed by one mass point M of mass M orbiting at a distance r from thecenter of m. We assume that the body is composed of N homogeneous layers of densities d_i (i=1,…,N) and angular velocitiesΩ⃗_i, perpendicular to the orbital plane.The outer surface of the ith layer is ζ_i(φ_i,θ_i,t), where ζ_i is the distance of the surface points to thecenter of gravity of m and the angles φ_i, θ_i are their longitudes and co-latitudes in a fixed reference system.At each instant, we assume that the static equilibrium figure of each layer under the action of the tidal potential and the rotation may be approximated bya triaxial ellipsoidal equilibrium surface ρ_i(φ_i,θ_i,t), whose semi-major axis is oriented towards M.The adopted rheophysical approach is founded on the simple lawζ̇_i = γ_i(ρ_i-ζ_i),where γ_i is the relaxation factor at the outer surface of the ith layer. This is a radial deformation rate gradient related to the viscositythrough (see Appendix 1)γ_i = (d_i-d_i+1)g_iR_i/2η_i,where R_i and g_i are the equatorial mean radius and the gravity acceleration at the outer surface of the ith layer. η_i is the viscosity of theinner layer (assumed to be larger than that of the outermost layers).Although the creep equation is valid in a reference system co-rotating with the body, we can use the coordinates in a fixed reference system. This isbecause only relative positions appear in the right-hand side of the creep equation. If φ_F is the longitude of a point in one frame fixedin the body, then we haveφ_i = φ_F + Ω_i t.§.§ The static equilibrium figure The static equilibrium figure of one body composed by N homogeneous layers, under the action of the tidal potential and the non-synchronous rotation, whenall layers rotate with the same angular velocity, was calculated by Folonier et al. (2015).[Although considering first-order deformations (lineartheory for the flattenings) the results of Folonier et al. (2015) are in excellent agreement with the results founded on a high-order perturbative method ofWahl et al. (2017).]In this work, we need, beforehand, to extend these results to the case in which each layer has one different angular velocity. We assume that each layer has an ellipsoidal shape with outer semiaxes a_i, b_i and c_i, where the axis a_i is pointing towards M and c_iis the axis of rotation. Then, the equatorial prolateness ϵ_ρ^(i) and polar oblateness ϵ_z^(i) of the outer surface of the ithlayer can be written asϵ_ρ^(i) = a_i-b_i/R_i = ℋ_iϵ_ρ; ϵ_z^(i) = b_i-c_i/R_i = 𝒢_iϵ_z,where R_i=√(a_ib_i) is the outer equatorial mean radius of the ith layer, ϵ_ρ is the flattening of the equivalent Jeans homogeneousspheroid and ϵ_z is the flattening of the equivalent MacLaurin homogeneous spheroid in synchronous rotation:ϵ_ρ = 15MR_N^3/4m_Tr^3; ϵ_z = 5n^2R_N^3/4Gm_T.Here, G is the gravitation constant, R_N is the equatorial mean radius of m and n is the mean motion of M. The Clairaut's coefficientsℋ_i and 𝒢_i depend on the internal structure and are (see Appendix 2)ℋ_i = ∑_j=1^N (E^-1)_ijx_j^3; 𝒢_i = ∑_j=1^N (E^-1)_ijx_j^3(Ω_j/n)^2,where (E^-1)_ij are the elements of the inverse of the matrix E, whose elements are(E)_ij = {[- 3/2f_N(d_j-d_j+1)x^3_i,i<j; - 3/2f_N(d_i-d_i+1)x^3_i + 5/2- 5/2f_N∑_k=i+1^N (d_k-d_k+1) (x_k^3-x_i^3),i=j;- 3/2f_N(d_j-d_j+1)x^5_j/x^2_i,i>j ].where x_i=R_i/R_N and d_i=d_i/d_1 are the normalized mean equatorial radius and density, respectively, and f_N=3∫_0^1d(z)z^2 dz.Finally, the static ellipsoidal surface equation of the outer boundary of the ith layer, to first order in the flattenings, can be written asρ_i = R_i(1+1/2ϵ_ρ^(i)sin^2θcos(2φ_i-2φ_M)-(1/2ϵ_ρ^(i)+ϵ_z^(i)) cos^2θ),(see Section A in the Online Supplement), where φ_M is the longitude of M in the same fixed reference system used to defineφ_i.§.§ The creep equation Using the static equilibrium surface (<ref>), the creep equation (<ref>) becomesζ̇_i + γ_iζ_i = γ_i R_i(1+1/2ϵ_ρ^(i)sin^2θcos(2φ_i-2ϖ-2v)-(1/2ϵ_ρ^(i)+ϵ_z^(i)) cos^2θ),where ϖ and v are the longitude of the pericenter and the true anomaly, respectively.For resolving the creep differential equation, we proceed in a similar way as FM13 and FM15. We consider the two-body Keplerian motion. The equations of theKeplerian motion of M arer=a(1-e^2)/1+ecosv,andv = ℓ + (2e-e^3/4)sinℓ+5e^2/4sin2ℓ+13e^2/12sin3ℓ+𝒪(e^4),where a, e, ℓ are the semi-major axis, the eccentricity, and the mean anomaly, respectively. The resulting equation is then an ordinary differential equation of first order with forced terms that may be written asζ̇_i + γ_iζ_i = γ_i R_i(1+∑_k∈ℤ(𝒞_iksin^2θcosΘ_ik+𝒞”_ikcos^2θcosΘ”_ik)),where the arguments of the cosines Θ_ik, Θ”_ik are linear functions of the timeΘ_ik = 2φ_i-2ϖ+(k-2)ℓ; Θ”_ik = kℓ. The constants 𝒞_ik,𝒞”_ik are𝒞_ik = 1/2ℋ_iϵ_ρ E_2,k; 𝒞”_ik = - 1/2ℋ_iϵ_ρ E_0,k - δ_0,k𝒢_iϵ_z,where δ_0,k is the Kronecker delta (δ_0,k=1 when k=0 and δ_0,k=0 when k≠0), the constant ϵ_ρ isϵ_ρ = 15MR_N^3/4m_Ta^3,and E_q,p are the Cayley functions (Cayley, 1861), defined byE_q,p(e) = 1/2π∫_0^2π(a/r)^3cos(qv+(p-q)ℓ) dℓ. After integration, we obtain the forced termsδζ_i = R_i∑_k∈ℤ(𝒞_iksin^2θcosσ_ikcos(Θ_ik-σ_ik)+𝒞”_ikcos^2θcosσ”_ikcos(Θ”_ik-σ”_ik)).The phases σ_ik and σ”_ik aretanσ_ik= ν_i+kn/γ_i;cosσ_ik = γ_i/√(γ_i^2+(ν_i+kn)^2);sinσ_ik = ν_i+kn/√(γ_i^2+(ν_i+kn)^2) tanσ”_ik= kn/γ_i;cosσ”_ik = γ_i/√(γ_i^2+(kn)^2);sinσ”_ik = kn/√(γ_i^2+(kn)^2),where ν_i=2Ω_i-2n,is the semi-diurnal frequency. These phases are introduced during the exact integration of the creep equation (<ref>).If we define the anglesδ_ik = 2ϖ-(k-2)ℓ+σ_ik; δ”_ik = kℓ-σ”_ik,and the equatorial and polar flatteningsϵ_ρ^(ik) = 2𝒞_ikcosσ_ik;ϵ_z^(ik) = -𝒞”_ikcosσ”_ikcosδ”_ik-ϵ_ρ^(ik)/2,the solution (<ref>) can be written asδζ_i = R_i∑_k∈ℤ(1/2ϵ_ρ^(ik)sin^2θcos(2φ_i-δ_ik)-(1/2ϵ_ρ^(ik)+ϵ_z^(ik))cos^2θ),which has a simple geometric interpretation: using Eq. (<ref>), we can identify each term of the Fourier expansion of the heightδζ_i, with the boundary height of one ellipsoid, with equatorial and polar flattenings ϵ_ρ^(ik) and ϵ_z^(ik),respectively, rotated at an angle δ_ik/2, with respect to the axis x.§ THE DISTURBING POTENTIAL The potential of the ith layer of m at a generic point M^*(r^*,θ^*,φ^*) external to this layer, can be written as thepotential of one spherical shell of outer and inner radii R_i and R_i-1, respectively, plus the disturbing potential due to the mass excess or deficitcorresponding to the outer and the inner boundary heights δζ_i and δζ_i-1. It is important to note that since these excesses ordeficits are very small, we may calculate the contribution of each term of the Fourier expansion separately and then sum them to obtain the totalcontribution.In this way, we assume that the ith layer has outer and inner boundary heights given by the kth term of the Fourier expansion. The equatorialand polar flattenings of the outer boundary, ϵ_ρ^(ik) and ϵ_z^(ik), are given by Eq. (<ref>), and the bulge isrotated at an angle δ_ik/2 with respect to the axis x. Similarly, the inner boundary height δζ_i-1^(1), can be identified with theboundary height of one ellipsoidal surface, with equatorial and polar flatteningsϵ_ρ^(i-1,k) = 2𝒞_i-1,kcosσ_i-1,k;ϵ_z^(i-1,k) = -𝒞”_i-1,kcosσ”_i-1,kcosδ”_i-1,k-ϵ_ρ^(i-1,k)/2,rotated at an angle δ_i-1,k/2, with respect to the axis x.The disturbing potential at an external point M^*(r^*,θ^*,φ^*), due to the mass excess or deficit, corresponding to the kth term ofthe Fourier expansion of the outer and the inner boundary heights δζ_i and δζ_i-1, isδ U_ik(r⃗^*)=-3GC_i/2r^*3sin^2θ^*Δ(R_i^5𝒞_ikcosσ_ikcos(2φ^*-δ_ik))/R_i^5-R_i-1^5-G C_i/2r^*3(3cos^2θ^*-1)Δ(R_i^5𝒞”_ikcosσ”_ikcosδ”_ik)/R_i^5-R_i-1^5,where C_i is the axial moment of inertia of the ith layer (see Section A in the Online Supplement) and Δ(f_i) = f_i-f_i-1, denotes theincrement of one function f_i, between the inner and the outer boundaries of this layer.Taking into account that the total disturbing potential of the ith layer, can be approximated by the sum of the contribution of each term of the Fourierexpansion, we obtainδ U_i(r⃗^*)=∑_k∈ℤδ U_ik(r⃗^*). § FORCES AND TORQUES To calculate the force and torque due to the ith layer of m, acting on one mass M^* located in M^*(r^*,θ^*,φ^*),we take the negative gradient of the potential of the ith layer at the point M^* and multiply it by the mass placed in the point, that is,F⃗_i=-M^*∇_r⃗^*δ U_iF_1i =-9GM^*C_i/2r^*4sin^2θ^*∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_ikcos(2φ^*-δ_ik))/R_i^5-R_i-1^5-3GM^*C_i/2r^*4(3cos^2θ^*-1)∑_k∈ℤΔ(R_i^5𝒞”_ikcosσ”_ikcosδ”_ik)/R_i^5-R_i-1^5 F_2i = 3GM^*C_i/2r^*4sin2θ^*∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_ikcos(2φ^*-δ_ik))/R_i^5-R_i-1^5 -3GM^*C_i/2r^*4sin2θ^*∑_k∈ℤΔ(R_i^5𝒞”_ikcosσ”_ikcosδ”_ik)/R_i^5-R_i-1^5 F_3i =-3GM^*C_i/r^*4sinθ^*∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_iksin(2φ^*-δ_ik))/R_i^5-R_i-1^5. The corresponding torque is M⃗_i=r⃗^*×F⃗_i, or since r⃗^*=(r^*,0,0),M_1i=0; M_2i=-r^*F_3i; M_3i=r^*F_2i,that isM_2i = 3GM^*C_i/r^*3sinθ^*∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_iksin(2φ^*-δ_ik))/R_i^5-R_i-1^5 M_3i = 3GM^*C_i/2r^*3sin2θ^*∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_ikcos(2φ^*-δ_ik))/R_i^5-R_i-1^5 -3GM^*C_i/2r^*3sin2θ^*∑_k∈ℤΔ(R_i^5𝒞”_ikcosσ”_ikcosδ”_ik)/R_i^5-R_i-1^5. § FORCES AND TORQUES ACTING ON M Since we are interested in the force acting on M due to the tidal deformation of the ith layer of m, we must substitute(M^*,r^*,θ^*,φ^*) by (M,r,π/2,ϖ+v). Replacing the angles δ_ik and δ”_ik given their definitions(Eq. <ref>), the forces, then areF_1i =-9GMC_i/2r^4∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_ikcos(2v+(k-2)ℓ-σ_ik))/R_i^5-R_i-1^5+3GMC_i/2r^4∑_k∈ℤΔ(R_i^5𝒞”_ikcosσ”_ikcos(kℓ-σ”_ik))/R_i^5-R_i-1^5 F_2i =0F_3i =-3GMC_i/r^4∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_iksin(2v+(k-2)ℓ-σ_ik))/R_i^5-R_i-1^5.and the corresponding torques areM_2i = 3GMC_i/r^3∑_k∈ℤΔ(R_i^5𝒞_ikcosσ_iksin(2v+(k-2)ℓ-σ_ik))/R_i^5-R_i-1^5 M_3i =0,After Fourier expansion, the torque along to the axis z (M_zi=-M_2i), can be written asM_zi = 3GMC_i/a^3∑_k,j∈ℤE_2,k+jΔ(R_i^5𝒞_ikcosσ_iksin(jℓ+σ_ik))/R_i^5-R_i-1^5. Finally, replacing the coefficient 𝒞_ik given by Eq. (<ref>), the time average of the tidal torque over one period⟨ M_zi⟩ = 1/2π∫_0^2πM_zidℓ is⟨ M_zi⟩ = 45GM^2R_N^3C_i/16m_Ta^6∑_k∈ℤE_2,k^2Δ(ℋ_iR_i^5sin2σ_ik)/R_i^5-R_i-1^5. The above expression for the time average, which is equivalent to take into account only the terms with j=0, is only valid if ν_i is constant. Thiscondition is satisfied, for example, by homogeneous bodies with γ≫ n, as stars and giant gaseous planets, where the stationary rotation is∼6nγ e^2/(n^2+γ^2). However, the final rotation of the homogeneous rocky bodies, with γ≪ n, as satellites and Earth-like planets, isdominated by a forced libration ∼ B_1cos(ℓ+ϕ_1) with the same period asthe orbital motion of the system (see Chap. 3 of FM15). In this case,any time average that involves the rotation, should also take into account this oscillation. It is worth emphasizing that in this paper we calculate thetime average of some quantities, as the work done by the tidal forces and the variations in semi-major axis and eccentricity, assuming that ν_i isconstant, which is valid only for bodies with low viscosity. The applications to Titan in this paper were done using the complete equations, where thedistinction between these extreme cases is not necessary.§ WORK DONE BY THE TIDAL FORCES ACTING ON M The time rate of the work done by the tidal forces due to the ith layer is Ẇ_i=F⃗_i·v⃗, where v⃗ is the relativevelocity vector of the external body[The definition of power (the time derivative of work) used in this section is the most general definition ofthe power done by the force couple formed by the disturbing force 𝐅_i acting on the external body and its reaction -𝐅_i acting on theith layer of the deformed body. It may be written as 𝐅_i ·𝐕_M + (-𝐅_i) ·𝐕_i where𝐕_M and 𝐕_i are, respectively, the velocities of the body M and of the ith layer of the body m, w.r.t. afixed reference frame. It is equivalent to 𝐅_i (̇𝐕_M - 𝐕_i), that is, 𝐅_i ·𝐯 (see Scheeres,2002; Ferraz-Mello et al., 2003).] whose components in spherical coordinates arev_1 = naesinv/√(1-e^2);v_2 = 0;v_3 = na^2√(1-e^2)/r.Using the tidal force, given by the Eq. (<ref>), the rate of the work corresponding to the ith layer isdW_i/dt =-3GMC_in/2a^3∑_k∈ℤ1/R_i^5-R_i-1^5Δ(R_i^5𝒞_ikcosσ_ik ×[cosσ_ik(3e/√(1-e^2)a^4/r^4sinvcos(2v+(k-2)ℓ)+a^5/r^52√(1-e^2)sin(2v+(k-2)ℓ))+sinσ_ik(3e/√(1-e^2)a^4/r^4sinvsin(2v+(k-2)ℓ)-a^5/r^52√(1-e^2)cos(2v+(k-2)ℓ))])+GMC_in/2a^3a^4/r^4∑_k∈ℤ1/R_i^5-R_i-1^5Δ(R_i^5𝒞”_ikcosσ”_ik3e/√(1-e^2)sinvcos(kℓ-σ”_ik)),or after Fourier expansion[For the details of the calculation see the Online Supplement of FM15.]dW_i/dt =-3GMC_in/2a^3∑_k,j∈ℤ(k+j-2)E_2,k+jΔ(R_i^5𝒞_ikcosσ_iksin(jℓ+σ_ik))/R_i^5-R_i-1^5+GMC_in/2a^3∑_k,j∈ℤ(k+j)E_0,k+jΔ(R_i^5𝒞”_ikcosσ”_iksin(jℓ+σ”_ik))/R_i^5-R_i-1^5.The time-average over one period is⟨dW_i/dt⟩ = 45GM^2R_N^3C_in/32m_Ta^6∑_k∈ℤ((2-k)E_2,k^2Δ(ℋ_iR_i^5sin2σ_ik)/R_i^5-R_i-1^5-k/3E_0,k^2Δ(ℋ_iR_i^5sin2σ”_ik)/R_i^5-R_i-1^5). The average of the term involving δ_0,k𝒢_iϵ_z in the last term of Eq. (<ref>), for k=0, is1/2π∫_0^2πn^2ℒ'_i(a/r)^4sinv dℓ = ∑_j=1^N Δ(R_i^5(E^-1)_ijx_j^3)/R_i^5-R_i-1^51/2π∫_0^2πΩ_j^2(a/r)^4sinv dℓ=0,(see Section C in the Online Supplement).§ VARIATIONS IN SEMI-MAJOR AXIS AND ECCENTRICITY In this section, we calculate the variation in semi-major axis and eccentricity. As in FM13 and FM15, we use the energy and angular momentum definitions.[We use the conservation laws of the two-body problem because they are universally known. However, it is worth emphasizing that the resultsobtained are the same obtained if instead of them we use the Lagrange variational equations.] If we differentiate the equation W=-GMm_T/2a,where a is the semi-major axis of the relative orbit, we obtain the equation for the variation in semi-major axis:ȧ= 2a^2Ẇ/GMm_T. Replacing Ẇ by the Eq. (<ref>) and summing over all layers, we obtainȧ =-∑_i=1^N3C_in/m_Ta∑_k,j∈ℤ(k+j-2)E_2,k+jΔ(R_i^5𝒞_ikcosσ_iksin(jℓ+σ_ik))/R_i^5-R_i-1^5+∑_i=1^NC_in/m_Ta∑_k,j∈ℤ(k+j)E_0,k+jΔ(R_i^5𝒞”_ikcosσ”_iksin(jℓ+σ”_ik))/R_i^5-R_i-1^5. After the averaging over one period, we obtain⟨ȧ⟩ = ∑_i=1^N45MR_N^3C_in/16m_T^2a^4∑_k∈ℤ((2-k)E_2,k^2Δ(ℋ_iR_i^5sin2σ_ik)/R_i^5-R_i-1^5-kE_0,k^2Δ(ℋ_iR_i^5sin2σ”_ik)/R_i^5-R_i-1^5). In the same way, if we differentiate the total angular momentum equationL=Mm_T/M+m_Tna^2√(1-e^2)=GMm_T/na√(1-e^2),where e is the eccentricity of the relative orbit, and use ṅ/n=-3ȧ/2a, we obtain the equation for the variation in eccentricityeė/1-e^2= ȧ/2a-L̇/L,where L̇=ℳ_z is the total torque exerted by the tidal forces. The interaction torques between the layers do not affect the orbital motion,because they are action-reaction pairs (that is ℳ_ij=-ℳ_ji, ∀ i,j=1,…,N and i≠ j), then they mutually cancelthemselves.Replacing Ẇ and ℳ_z by the Eqs. (<ref>) and (<ref>), and summing over all layers, we obtainė =-∑_i=1^N3C_in/m_Ta^2(1-e^2)/2e∑_k,j∈ℤ(2/√(1-e^2)+(k+j-2))E_2,k+jΔ(R_i^5𝒞_ikcosσ_iksin(jℓ+σ_ik))/R_i^5-R_i-1^5+∑_i=1^NC_in/m_Ta^2(1-e^2)/2e∑_k,j∈ℤ(k+j)E_0,k+jΔ(R_i^5𝒞”_ikcosσ”_iksin(jℓ+σ”_ik))/R_i^5-R_i-1^5. After the time-average over one period, we obtain that the variation in eccentricity are⟨ė⟩ = ∑_i=1^N45MR_N^3C_in/16m_T^2a^4(1-e^2)/2ae ×∑_k∈ℤ(((2-k)-2/√(1-e^2))E_2,k^2Δ(ℋ_iR_i^5sin2σ_ik)/R_i^5-R_i-1^5-kE_0,k^2Δ(ℋ_iR_i^5sin2σ”_ik)/R_i^5-R_i-1^5). § THE TWO-LAYER MODEL In the previous sections we have studied the tidal effect on one body composed of N homogeneous layers. However, in contrast with a homogeneous body, inone differentiated body we must also take into account the interaction between the different layers. In this paper we consider the gravitational coupling ofthe layers and the friction that occurs at each interface of two layers in contact. An important point to keep of mind is that the number of freeparameters increases significantly as the number of layers increases.In this section, we study the simplest non-homogeneous problem: one body formed by two independent rotating parts. The inner layer, or core, isdenoted with the subscript c and the outer layer, or shell, is denoted with the subscript s. Despite its simplicity, thetwo-layer model allows us to study the main features of the stationary rotations, introducing a minimum number of free parameters.§.§ The tidal torques The tidal torques due to the core and the shell, along the axis z, are (see Eq. <ref>)M_zc = T_ccC_c𝒯_c M_zs = T_ssC_s𝒯_s-T_scC_s𝒯_c,where the function 𝒯_i (i=c,s) is𝒯_i = ∑_k,j∈ℤ E_2,kE_2,k+jγ_i(ν_j+kn)cosjℓ+γ_i^2sinjℓ/γ_i^2+(ν_i+kn)^2,the constants T_ij areT_cc = 𝒯ℋ_c;T_sc = 𝒯ℋ_cR_c^5/R_s^5-R_c^5;T_ss = 𝒯ℋ_sR_s^5/R_s^5-R_c^5,and the tidal parameter 𝒯, is defined as𝒯 = 45GM^2R_s^3/8m_Ta^6≈3n^2ϵ_ρ/2,R_c, C_c are the mean outer radius and moment of inertia of the core, and R_s, C_s are the mean outer radius and moment of inertia of the shell. Theparameters ℋ_c, γ_c are the Clairaut parameter and the relaxation factor at the core-shell interface and ℋ_s, γ_s arethe Clairaut parameter and the relaxation factor at the body's surface. §.§ The gravitational core-shell coupling When the principal axes of inertia of two layers are not aligned, a restoring torque appears which tends to align these axes again. This torque wascalculated by several authors (e.g. Buffett, 1996; Van Hoolst et al., 2008; Karatekin et al., 2008; Callegari et al., 2015) when the layers are rigid.Here, we use one similar expression for this torque adapted to a body formed by two layers whose boundaries are prolate ellipsoids, whose flattenings aredefined by the composition of the main elastic and anelastic tidal components. If we follow the same composition adopted in FM13, these flattenings areϵ'_c = ℋ_cϵ_ρ√(λ_c^2 + cos^2σ_c0(1 + 2λ_c)); ϵ'_s = ℋ_sϵ_ρ√(λ_s^2 + cos^2σ_s0(1 + 2λ_s)),where 0<λ_c<1 and 0<λ_s<1 are the relative measurements of the actual maximum heights of the elastic tides of the core and the shell,respectively. The geodetic lags of these two ellipsoidal surfaces, when one elastic component is added areϑ_c = 1/2tan^-1(sin2σ_c0/1+2λ_c+cos2σ_c0); ϑ_s = 1/2tan^-1(sin2σ_s0/1+2λ_s+cos2σ_s0). In this case, the torques, along the axis z, areΓ_c=K sin2ξ Γ_s= -K sin2ξ,where ξ=ϑ_s-ϑ_c is the offset of the geodetic lags of the two ellipsoidal boundaries and the constant of gravitational coupling K isK = 32π^2 G/75ϵ'_c ϵ'_s d_cd_sR_c^5,(see Appendix 3 for more details).We may pay attention to the sign of these torques. If ϑ_s > ϑ_c, the motion of the shell is braked, while the motion of the core isaccelerated. This is consistent with the signs of the above equations. §.§ Linear drag The model considered here also assumes that a linear friction occurs between the two contiguous layers. For the two-layer model, the torques actingon the core and the shell, along the axis z, areΦ_c= μ (Ω_s-Ω_c) Φ_s= -μ (Ω_s-Ω_c),where the friction coefficient μ is an undetermined ad-hoc constant that comes from assuming that a linear friction occurs between two contiguous layers.When we consider that the body m has solid layers, but not rigid, we can assume that between the core and the shell exists one thin fluid boundarywith viscosity η_o and thickness h. If this interface is a Newtonian fluid, the Eq. (<ref>) is the law corresponding to liquid-solidboundary for low speeds, and μ can be written asμ = 8π/3 η_o/hR_c^4,(see Appendix 4 for more details). §.§ Rotational equationsPutting together all contributions to the torque, we obtain the rotational equationsC_cΩ̇_c=M_z^core= -M_zc+Γ_c+Φ_cC_sΩ̇_s=M_z^shell = -M_zs+Γ_s+Φ_s,where M_z^core and M_z^shell are the z-components of the total torque acting on the core and on the shell. These torquesinclude the reaction of the tidal torque M_zi, the gravitational coupling Γ_i and the friction Φ_i.§ COMPARISON WITH THE HOMOGENEOUS CASE In this section, we compare some of the main features of the homogeneous creep tide theory, developed in FM15, with the non-homogeneous creep tidetheory for the two-layer model developed in this article. The main difficulty lies in the number of free parameters in these approaches. In the homogeneouscase, with a suitable choice of dimensionless variables, the final state of rotation depends only on the ratio n/γ and on the eccentricity e (Eq.42 of FM15). However, even in the most simple non-homogeneous case (the two-layer model), we need to set 12 free parameters. In order to proceed, we usethe typical values for Titan and also Titan's eccentricity e=0.028 (see Tables <ref>-<ref> in Sect. <ref>), and letas free parameters, only n/γ_i, e and μ.Following FM15, we introduce the adimensional variables y_i=ν_i/γ and the scaled time x=ℓ/γ, where γ=2γ_c γ_s(γ_c+γ_s)^-1. If we consider the case in which γ_c=γ_s, the behavior of the evolutions of y_c and y_s is similar tothat observed in the homogeneous case. Figure <ref> shows the time evolution of y_s, with initial conditions y_c=0.3, y_s=0.15 and differentsvalues of α=log_10(n/γ_c)=log_10(n/γ_s). When γ_i≪ n (i.e. rocky bodies), after a transient, the solution oscillatesaround zero, independently of the initial conditions (left panel), and the amplitude of oscillation decreases when α decreases. In the casewhere α=4, we also plot the solution with initial conditions y_c=0.3 and y_s=-0.15 (dashed black line). This solution increases quickly, becomingindistinguishable from the solution with initial value y_s=0.15.When γ_i∼ n, the stationary solution becomes a super-synchronous rotation withthe amplitude of oscillation tending to zero, and, finally, when γ_i≫ n, the stationary solution of y_s becomes closer zero (rightpanel). The evolution of y_c is very similar, and the friction does not have any relevant role. However, when we analyze the time evolution of ν_s instead of y_s, we observe that when γ_i≪ n, the solution oscillates around zeroand the amplitude of the oscillation of ν_s increases when α decreases (left panel of Fig. <ref>). When γ_i≫ n. thisamplitude decreases when α decreases and ν_s tends to 12ne^2, independently of the value of α (right panel of Fig. <ref>). When γ_c≠γ_s, we can have different core and shell rotation behavior. In Fig. <ref>, we show the core and shell rotation (leftand right, respectively) for log_10(n/γ_c)=2 and log_10(n/γ_s)=4. We also set two very different values for the friction:the frictionless case μ=0 (black) and a very high value of friction μ=10^28 kg km^2s^-1 (red lines), larger than theexpected value in the case of Titan (μ=10^11-10^13 kg km^2s^-1), which corresponds to a typical ocean viscosity η_o=η_H_2O≈10^-3 Pa s and a large range for the ocean thickness h (see Eq. <ref>). In the frictionless case, we can observe the differentialrotation between the core and the shell. After a transient, both solutions oscillate around zero with very different amplitudes, depending on the value ofγ of each surface. For very high friction parameter, both layers rotate with the same angular velocity. The core and the shell have the sameamplitude of oscillation and phase, keeping the relative velocity equal to zero. Finally, we study the dependence of the stationary solutions on the eccentricity. For that sake, we choose a grid of initial conditions ν_c/n andν_s/n, and integrate the system (<ref>) until the stationary solution is reached. When n/γ_c=n/γ_s≪ 1, all initialconditions lead to the same equilibrium point (a super-synchronous rotation), independently of the value of the friction parameter. The value of the excessof rotation depends only on the eccentricity. In the left panels of Fig. <ref>, we show the family of stationary solutions, where each pointcorresponds to a different eccentricity value in 0≤ e≤ 0.5. If the eccentricity is zero, the rotations are synchronous to the orbital motion. Whenthe eccentricity increases, the rotations become super-synchronous, and the excess of rotation ν_i/n is proportional to e^2 (right panels).When n/γ_c and n/γ_s increase, that is, when the viscosities increase, the excess in the super-synchronous rotation decreases. If theeccentricity is low, the only attractor is the super-synchronous solution. When the eccentricity increases, captures in other attractors ν_i≃ n,2n,3n,…appear gradually. This behavior is the same studied by in FM15 and also in Correia et al. (2014) in the case of homogeneous bodies.Figure <ref> shows the families of stationary rotation for n/γ_c=n/γ_s=1, 0≤ e≤ 0.5 and two values of the friction parameter: thefrictionless case, with μ=0 (top panels), and a very high friction case, with μ=10^20 kg km^2s^-1 (bottom panels).In the frictionless case, when the eccentricity is smaller than ∼0.48, only the super-synchronous solution is possible. If the eccentricity is largerthan 0.48, besides the super-synchronous solution, three new stationary configurations appear: The core and the shell in the 3/2 commensurability(ν_c≃ n and ν_s≃ n), the core in super-synchronous rotation and the shell in the 3/2 commensurability (ν_c≃ 0 and ν_s≃ n),and the core in the 3/2 commensurability and the shell in super-synchronous rotation (ν_c≃ n and ν_s≃ 0). Figure <ref> shows in moredetail these stationary solutions. The labels R_pq denote the stationary families indicating the resonances ν_c=pn and ν_s=qn. It is importantto note that the excesses in the rotations are large because the eccentricity is high. In the high friction case (bottom panels of Fig. <ref>),only the stationary solutions with the same commensurabilities survive because in these configurations, the relative velocity of rotation between the coreand the shell is zero. If n/γ_c and n/γ_s continue to increase and the friction parameter is low (not necessarily zero), the core and the shell may tend to differentresonances, depending on the eccentricity. If the friction increases, the attractors with higher differential rotation, begin to disappear, until eventually,as from a certain value limit of μ only survive the attractors with differential rotation zero Fig. <ref>. § APPLICATION TO TITAN'S ROTATION§.§ The modelTitan's interior was largely discussed in many papers (e.g. Tobie et al., 2005; Castillo-Rogez and Lunine, 2010; McKinnon and Bland, 2011; Fortes, 2012).The existing general data of the Titan-Saturn system is given in Table <ref>. In this section, we assume the interior model given by Sohl et al.(2014) (hereafter reference model), which is given in Table <ref>. In this model, Titan is formed by four homogeneous layers: i) aninner hydrated silicate core (inner core); ii) a layer of high-pressure ice (outer core); iii) a subsurface water-ammonia ocean and iv) a thin ice crust.For the sake of simplicity, we construct one two-layer equivalent model, where the core is a layer formed by the inner core and the high-pressureice layer, and the shell is a layer formed by the subsurface ocean and the ice crust, but keeping some features of the four-layer model (e.g. axialmoments of inertia and Clairaut numbers). In this way, we can use the rotational equations (<ref>), retaining the main features of therealistic reference model. This simplified model is given in Table <ref>, and some calculated parameters of each layer are listed in Table<ref>. The existence of relative translational motions due to the non-coincidence of the barycenters of the several layers, asdiscussed by Escapa and Fukushima (2011) in the case of an icy body with an internal ocean and solid constituents, has not been taken into account.In order to estimate the relative height of the elastic tide λ_s, we assume that the difference between the observed surface flattening ϵ'_swith the tidal flattening ϵ_s=ℋ_sϵ_ρ E_2,0cosσ_s0≈ℋ_sϵ_ρcosσ_s0(calculated) is due to the existence of an elastic component, with flattening ϵ_s^(el)=λ_sℋ_sϵ_ρ (see Appendix3 for more details). If we use Eq. (<ref>), and assume that near the synchronous rotation cos^2σ_s0≈ 1, we obtain λ_s≈ϵ'_s/ℋ_sϵ_ρ - 1.For the relative heights of the elastic tide λ_c, we assume λ_c ≈λ_sλ. §.§ Atmospheric influence on Titan's rotation The seasonal variation in the mean and zonal wind speed and direction in Titan's lower troposphere causes the exchange of a substantial amount of angularmomentum between the surface and the atmosphere. The variation calculated from the observed zonal wind speeds shows that the atmosphere angular momentumundergoes a periodic oscillation between 3 × 10^18 and 3 × 10^19kgkm^2 s^-1 (Tokano and Neubauer, 2005,hereafter TN05) with a period equal to half Saturn's orbital period and maxima at Titan's equinoxes (when the Sun is in the satellite's equatorial plane).The angular momentum of the atmosphere may be written as L_ atm=L_0+L_1cos 2α_⊙ where L_0=1.65×10^19 kg km^2 s^-1,L_1=1.35×10^19 kg km^2 s^-1 and α_⊙ is the Saturnian right ascension of the Sun. The variation of the angular momentum isL̇_ atm=-2L_1 n_⊙sin 2α_⊙. If we neglect external effects (as atmospheric tides), this variation may be compensated by an equalvariation in the shell's angular momentum: δL̇_s= -L̇_ atm, which corresponds to an additional shell acceleration δΩ̇_s=2L_1 n_⊙/C_ksin 2α_⊙=A_⊙sin 2α_⊙. We must emphasize that we have considered in these calculations the moment of inertia of the ice crust C_k, since the winds are acting on the crust and donot have direct action on the liquid part of the shell.In a more recent work, Richard et al. (2014) (hereafter R14) re-calculate the amplitude of the variation of the angular momentum with the Titan IPSL GCM(Institut Pierre-Simon Laplace General Circulation models) (Lebonnois et al., 2012). They obtain L_1=8.20×10^17 kg km^2 s^-1, which is∼16.5 times less than the TN05 value.§.§ The results We fix the outer radius of the inner core R_ic and the outer radius of the high-pressure ice layer R_oc, the densities of the inner and outercores d_ic and d_oc and the density of the crust d_k, to the reference model values in Table <ref>. The density of theinner core is calculated so as to verify the value of Titan's mass m_T=13.45×10^22 kg.Figure <ref> shows the weak dependence of the parameters on the thickness of the ocean h: the density of the inner core d_ic (solid orangeline) and densities of the reference model (left panel); the Clairaut numbers ℋ_c, ℋ_s (middle panel); and theaxial moments of inertia C_c and C_s (right panel).The main consequence of the weak dependence of these parameters with the thickness of the subsurface ocean, is that both the effect of the tide and thegravitational coupling parameter also depend weakly on h. The strength of the acceleration of the rotation, due to the tide, is given by the productT_ij𝒯_k (see Eqs. <ref> and <ref>). While the parameter T_ij only depends on the internal structure of Titan, thefunction 𝒯_k do not depend on h. The left panel of Fig. <ref> shows T_ij and the gravitational coupling amplitudeK_i=K/C_i, as function of h. We also observe that the thickness of the ocean does not have any relevant role. Then, for the tide and the gravitationalcoupling, the rotational evolution is driven by the ratios n/γ_c, n/γ_s and the orbital eccentricity e. The right panel of Fig. <ref> shows the quantity nμ_i=nμ/C_i as function of the thickness h, when we consider the realistic oceanviscosity η_o=η_H_2O≈ 10^-3 Pa s. The rotational acceleration of each layer, due to the friction, is μ_i(Ω_s-Ω_c). Insuper-synchronous rotation, the excess of rotation of each layer is of order ne^2, thenμ_i(Ω_s-Ω_c)≪ nμ_i≪ T_ij,K_i.Therefore, in Titan's case, the friction term is negligible compared with the tide and the gravitational coupling terms, independently of the h value.Equations (<ref>) and (<ref>), allow us to calculate the velocities of rotation of the shell and of the core of Titanfor a wide range of relaxation factors γ_c and γ_s, when different effects are considered. For that sake, we have to adopt the values of theinvolved parameters. We use four different values for the viscosity of the subsurface ocean: a realistic value η_o=η_H_2O=10^-3 Pa s, amoderate value η_o=10^0 Pa s and two very high values η_o=10^6 Pa s and η_o=10^9 Pa s. For the thickness of the ocean,we use the values h=15, 178 and 250 km, and for the variation of the atmospheric angular momentum, we use the values given by Tokano and Neubauer(2005) and Richard et al. (2014). When we integrate the rotational equations, assuming the values of relaxation factor typical for rock bodies (γ_i<n),the results show that the excess of rotation of the shell is damped quickly and the final state is an oscillation around the synchronous motion with a periodof ∼15 days (a periodic attractor), equal to the orbital period (Fig. <ref>). The amplitude of this oscillation depends on the relaxationfactors and the ocean thickness. The periodic attractor of the spin rate ν_i of each layer can be approximated by the trigonometric polynomialν_i≃ B_i0 + B_i1cos(ℓ+ϕ_i1) + B_i2cos(2ℓ+ϕ_i2),where the constants B_ij and the phases ϕ_ij, depend on the relaxation factors. The tidal drift B_i0 also depends on e^2, while theamplitude of oscillation B_ij, depends on e^j. The coefficients B_ij and ϕ_ij gives rise to intricate analytical expressions, but are easyto calculate numerically (an analytical construction of these constants is presented in the Section B of the Online Supplement). Figure <ref> showsone example for Titan's core and shell constants B_cj and B_sj, as a function of the shell relaxation factor, when the core relaxation factor isγ_c=10^-8 s^-1, and the ocean's viscosity and thickness are η_o=10^-3 Pa s and h=178 km, respectively. We can observethat if γ_s≳10^-7.5 s^-1, the shell oscillates around the super-synchronous rotation. When γ_s≲10^-7.5 s^-1,the tidal drift B_s0 tends to zero and the shell oscillates around the synchronous rotation, with a period of oscillation equal to the orbital period.Finally, if γ_s≲10^-8 s^-1, the amplitude of the shell rotation decreases, tending to zero when γ_s decreases. On the otherhand, the core oscillates around the synchronous rotation, with a period of oscillation equal to the orbital period, independently of the shell relaxationfactor. In Fig. <ref>, fixing η_o=10^-3 Pa s and L_1=1.35×10^19 kg km^2 s^-1 (TN05), we plot the resulting maximum andminimum of the final oscillation of the shell rotation Ω_s-n, or, equivalently, the length-of-day variation Δ LOD= 2π/n-2π/Ω_s,in function of γ_s, for two dynamical models: i) tidal forces, gravitational coupling and linear friction (solid black lines); and ii) tidal forces,gravitational coupling, linear friction and the atmospheric influence (dashed red lines). The horizontal lines show the intervals corresponding to 1σuncertainties of the observed values: the blue dashed lines, labelled , correspond to Meriggiola (2012) and Meriggiola et al. (2016) and greendashed lines, labelled , correspond to the Stiles et al. (2010). The core relaxation factor γ_c increases from γ_c = 10^-9 s^-1 (top panels) to 10^-6 s^-1 (bottom panels) and the ocean thickness h increases from 15 km (leftpanels) to 250 km (right panels). Figure <ref> shows that if γ_s<10^-7 s^-1, the shell's rotation oscillate around the synchronous motion and the amplitude ofoscillation depends on the relaxation factors and the ocean thickness. The average rotation (central orange line) is synchronous; it only becomessuper-synchronous for relaxation values larger than ∼10^-6.5 s^-1. We also observe that when γ_s<10^-8 s^-1,independently of the values of γ_c and h, the amplitude of oscillation of the shell tends to zero when the relaxation factor γ_s decreases.Particularly, if γ_s<10^-9 s^-1, the amplitude of the oscillation of the excess of rotation reproduces the dispersion of the Ω_svalue of ± 0.02 deg/yr around the synchronous value, observed as reported by Meriggiola (2012) and Meriggiolla et al. (2016). The results are notconsistent with the previous drift reported by Stiles et al. (2008, 2010). We note that for larger values of the relaxation, e.g. 10^-8 s^-1,the large short period oscillation due to the tide would be much larger than the reported values and would introduce big dispersion in the measurements,much larger than the reported dispersion due to the difficulties in the precise localization of Titan's features. On the other hand, the effect of theatmospheric torque is completely negligible in the range of possible γ_s that reproduces the observed values of the shell rotation, even for the highvalue of L_1 given Tokano and Neubauer (2005). When we consider the amplitude of the variation of the angular momentum given by Richard et al. (2014), thecontribution to the rotation variations tends to zero. The results shown in Fig. <ref> remain virtually unchanged when the ocean viscosity is increased up to a value of η_o= 10^6 Pa s. But ifthe ocean viscosity is increased to η_o= 10^9 Pa s, the transfer of angular momentum between the shell and the core induces in the shellaccelerations of the same order as the rotational acceleration due to the others forces. As a consequence, the shell rotation will follow the core rotationclosely (which is shown in Fig. <ref>). This high value of η_o can be interpreted as the ocean thickness tending to zero. In this case, toobtain the dispersion of Titan's observed rotation as determined by Meriggiola et al. (2016) we should have a value of γ_s smaller than the valuesobtained in the previous cases, where a low viscosity ocean was assumed between the shell and the core. It is worth noting yet that, in this case,the observed dispersion could also be obtained taking for γ_c an extremely low value (10^-9 s^-1) and for γ_s a much larger andunexpected value (10^-5 s^-1).It is important to note that, in any case, the rotational constraint does not allow us to estimate the value of the core relaxation factor γ_c. Forrealistic values of the ocean viscosity (η_o=10^-3-10^6 Pa s), the shell relaxation factor may be such that γ_s≲10^-9s^-1. The actual value will depend on the values of h and γ_c and on the interpretation of the dispersion determined by Meriggiola, whichmay include the forced short-period oscillation of Ω_s. Equivalently, using Eq. (<ref>), the shell viscosity may be such that η_s ≳10^18 Pa s. These values remain without significant changes if η_o<10^9 Pa s. For the case in which a subsurface ocean does notexist, the shell relaxation factor may be such that γ_s≲10^-10 s^-1, one order less than when an ocean is considered. Equivalently,the shell viscosity may be such that η_s≳10^19 Pa s. It is worth noting that in this case, when γ_s≲10^-7 s^-1,the rotation of the core remains stuck to the rotation of the shell even when γ_c is larger, notwithstanding the larger moment of inertia of the core(Fig. <ref>). § CONCLUSION In this article we extended the static equilibrium figure of a multi-layered body, presented in Folonier et al. (2015), to the viscous case, adapting it toallow the differential rotation of the layers. For this sake, we used the Newtonian creep tide theory, presented in Ferraz-Mello (2013) and Ferraz-Mello(2015a). Once solved the creep equations for the outer surface of each layer, we obtained the tidal equilibrium figure, and thereby we calculated thepotential and the forces that act on the external mass producing the tide.In order to apply the theory to satellites of our Solar System, we calculated the explicit expression in the particular case of one body formed by twolayers. We may remember that the number of free parameters and independent variables increases quickly when the number of layers increases. The simplestversion of the non-homogeneous creep tide theory (the two-layer model), allows us to obtain the main features due to the non-homogenity of the body, byintroducing a minimal quantity of free parameters. In the used model, we have also calculated the tidal torque, which acts on each layer and also thepossible interaction torques, such as the gravitational coupling and the friction at the interface between the contiguous layers (general development ofthese effects are given in Appendices 3 and 4). The friction was modeled assuming two homogeneous contiguous layers separated by one thin Newtonian fluidlayer. This model of friction is particularly appropriate for differentiated satellites with one subsurface ocean, as are various satellites of our SolarSystem (e.g. Titan, Enceladus, and Europa).The two-layer case was compared with the homogeneous case. For that sake, we fixed the free parameters of Titan and studied the main features ofthe stationary solution of this model in function of a few parameters, such as the relaxation factors γ_i, the friction parameter μ and theeccentricity e. When γ_c≈γ_s, the behavior of the stationary rotations turned out to be identical to the homogeneous case. Whenγ_c≈γ_s≪ n, the stationary solutions oscillate around the synchronous rotation. When γ_c and γ_s increase, theoscillation tends to zero. Finally, if γ_c≈γ_s≫ n, the stationary solution is damped to super-synchronous rotation. We have alsocalculated the possible attractors when the eccentricity and the friction parameter μ are varied. We recovered the resonances trapping incommensurabilities Ω_c≈Ω_s≈2+k/2n (where k=1,2,3,…∈ℕ) as shown in Ferraz-Mello (2015a) and Correia etal. (2014) for the homogeneous case, and we found that if friction remains low, the non-zero differential rotation commensurabilitiesΩ_c∼2+i/2n and Ω_s∼2+j/2n, with i,j=1,2,3,…∈ℕ and i≠ j, are possible. When the frictionincreases, the resonances with higher differential rotation are destroyed. If μ continues increasing, only the resonances in which core and shell havethe same rotation survive. This behavior is also observed in the non-homogeneous Darwin theory extension, when one particular ad hoc geodetic lagand one dynamical Love number for each layer are chosen (Folonier, 2016).The two-layer model was applied to Titan, but adding to it the torques due to the exchange of angular momentum between the surface and the atmosphere, asmodeled by Tokano and Neubauer (2005) and by Richard et al. (2014), and the results were compared to the determinations of Titan's rotational velocity asdetermined from Cassini observations by Stiles et al. (2010) and Meriggiola et al. (2016). These comparisons allowed us to constraint the relaxation factorof the shell to γ_s≲10^-9 s^-1. The integrations show that for γ_s≲10^-7.5 s^-1 the shell may oscillatearound the synchronous rotation, with a period of oscillation equal to the orbital period, and the amplitude of this oscillation depends on the relaxationfactors γ_c and γ_s and the ocean's thickness and viscosity. The tidal drift tends to zero and the rotation is dominated by the main periodicterm.The main result was that the rotational constraint does not allow us to confirm or reject the existence of a subsurface ocean on Titan. Only the maximumshell's relaxation factor γ_s can be determined, or equivalently, the minimum shell's viscosity η_s, because the icy crust is rotationallydecoupled from the Titan's interior. When a subsurface ocean is considered, the maximum shell's relaxation factor is such that γ_s≲10^-9s^-1, depending on the ocean's thickness and viscosity values considered. Equivalently, this maximum value of γ_s, corresponds with aminimum shell's viscosity η_s≳10^18 Pa s, some orders of magnitude higher than the modeled by Mitri et al. (2014). When the non-oceancase is considered, the maximum shell's relaxation factor is such that γ_s≲10^-10 s^-1 and the corresponding minimum shell'sviscosity is η_s≳10^19 Pa s. For these values of γ_s, the amplitude of the oscillation of the excess of rotation reproduces thedispersion of the Ω_s value of ± 0.02 deg/yr around the synchronous value, observed as reported by Meriggiola (2012) and Meriggiolla et al.(2016). It is important to note that in all the cases studied, the influence of the atmosphere can be neglected, since it does not affect the results in theranges of γ_c and γ_s where the excess of rotation calculated is compatible with the excess of rotation observed. We wish to thank Michael Efroimsky and one anonymous referee for their comments and suggestions that helped to improve the manuscript. we also thank toNelson Callegari for our fruitful discussions about the gravitational coupling. This investigation was supported by the National Council for Scientific andTechnological Development, CNPq 141684/2013-5 and 302742/2015-8, FAPESP 2016/20189-9, and by INCT Inespaço procs. FAPESP 2008/57866-1 and CNPq574004/2008-4. § APPENDIX 1: RELAXATION FACTOR Let us consider the equilibrium surface ρ_i(ϕ,θ) between two adjacent homogeneous layers of the body m whose densities are d_i (inner)and d_i+1 (outer). We consider that at a given instant, the actual surface between the two layers ζ_i(ϕ,θ) does not coincide with theequilibrium surface (Fig. <ref>). In some parts, the separation surface is above the equilibrium surface (as in region I) and in other parts it isbelow the equilibrium surface (as in region II). Let us now consider one small element of the equilibrium surface in region I. The pressure in the baseof this element is positive because the weight of the column above the element is larger than its weight in the equilibrium configuration. Note that thecolumn is now partly occupied by the fluid with density d_i and d_i > d_i+1. The pressure surplus is given byp_I=Δw h,where Δw = (d_i-d_i+1)g is the difference of the specific weight of the two columns in the neighborhood of the separation surface, and h is thedistance of the element of the equilibrium surface to the actual separation surface. g is the local acceleration of gravity. The radial flow in the considered element is ruled by the Navier-Stokes equation:0=F_ext - ∇ p_I + η_i Δu⃗where F_ext is the external force per unit volume, u⃗ is the radial velocityand η_i is the viscosity of the layer i (assuming η_i>η_i+1). We notice that Δ is operating on a vector, contrary to the usualΔ. Actually, in this pseudo-vectorial notation, the formula refers to the components of u⃗ and means thevector formed by the operation of theclassical Δ on the three components of the vector u. We assume that the flow is orthogonal to the equilibrium surface. We remind that, by the definition of the equilibrium surface, the tangential component ofthe resultant forces[The forces considered in the determination of the equilibrium surface are the self-gravitation, the tidal forces acting on thebody and the inertial forces due to the rotation of the body.] acting on the fluid vanishes at the equilibrium surface. Since the equilibrium surface is analmost spherical ellipsoid, we may consider in a first approximation that the motion of the fluid in that region is a radial flow.If we consider that F_ext=0 (no other external forces are acting on the fluid) and restricting u⃗ to its radial component u_r, there follows0 ≈Δ w + η_i ∇^2u_r.Hence, ∇^2u_r=∂^2 u_r/∂ r^2+2/r∂ u_r/∂ r-2u_r/r^2 = -Δ w/η_i. The general solution of this equation is u_r(r) =C_1r + C_2/r^2-Δ w/4η_ir^2,where C_1 and C_2 are integration constants. The task of interpreting and determining its integration constants becomes easier if the solution islinearized in the neighborhood of r=ρ_i (i.e. h=0):u_r(r)=u_r(ρ_i)+u_r'(ρ_i)(r-ρ_i)+1/2 u_r”(ρ_i)(r-ρ_i)^2 +… Hence, u_r(ρ_i)=0, that is, there is no pressure surplus (or deficit) when the actual separation surface coincides with the equilibrium and the linearapproximation of the solution is obtained when we assume u_r”(ρ_i)=0.Therefore,C_1 = ρΔ w/6 η_i C_2 = ρ^4 Δ w/ 12 η_i. Hence, u_r'(ρ_i)=ρ_i Δ w/2 η_i, and the linear approximation corresponding to the Newtonian creep of the fluid is u_r(r)=γ_i (r-ρ_i),whereγ_i = u_r'(ρ_i) =Δ wρ_i/2 η_i. In the region II, the calculation is similar; however, instead of a pressure surplus we have a pressure deficit because the equilibrium assumes one fluidwith density d_i below the equilibrium surface, which is now occupied byfluid of density d_i+1 < d_i. The equations are the same as above. We notethat in the new equations, the adopted viscosity continues being η_i since we assumed it larger than η_i+1. The relaxation of the surface to theequilibrium is governed by the larger of the viscosities of the two layers.In the homogeneous case we have one layer body (N=1). If we consider d_N+1=0 (neglected the density of the atmosphere), we recover the expression ofthe relaxation factor given by Ferraz-Mello (2013; 2015a)γ_N ≈w R_N/2η_N,where w= d_N g is the specific weight and ρ_N≈ R_N.§ APPENDIX 2: EQUILIBRIUM ELLIPSOIDAL FIGURES In this appendix we calculate the equatorial and the polar flattenings of the equilibrium ellipsoidal figures for differentiated non-homogeneousbodies innon-synchronous rotation when each layer has a different angular velociy. For this, we extend the results obtained by Folonier et al. (2015), where therigid rotation hypothesis has been assumed.Let us consider one body m of mass m_T and one mass point M of mass M orbiting at a distance r from the center of m. Weassume that the body is composed of N homogeneous layers of density d_i (i=1,…,N) and angular velocity Ω⃗_i=Ω_ik̂⃗̂,perpendicular to the orbital plane (Fig. <ref>). We also assume that each layer has an ellipsoidal shape with outer semiaxes a_i, b_i and c_i;the axis a_i is pointing towards M while c_i is along the axis of rotation.The equatorial prolateness and polar oblateness of the ith ellipsoidal surface, respectively, areϵ_ρ^(i) = a_i-b_i/R_i;ϵ_z^(i) = b_i-c_i/R_i,where R_i=√(a_ib_i) is the outer equatorial mean radius of the ith layer.Following Folonier et al. (2015), and carrying out modifications to account for the different velocities of rotations, the equilibrium equations can be written as0 =-3GM/r^3 + ∑_j=1^i-1Gm'_j/R_i^3(2ϵ_ρ^(i) - 6ϵ_ρ^(j)/5(R_j/R_i)^2) + Gm'_i/R_i^34ϵ_ρ^(i)/5 + ∑_j=i+1^N Gm'_j/R_j^3(2ϵ_ρ^(i)-6ϵ_ρ^(j)/5) Ω_i^2= ∑_j=1^i-1Gm'_j/R_i^3(2ϵ_z^(i)- 6ϵ_z^(j)/5(R_j/R_i)^2) + Gm'_i/R_i^34ϵ_z^(i)/5+ ∑_j=i+1^N Gm'_j/R_j^3(2ϵ_z^(i) -6ϵ_z^(j)/5),where G is the gravitation constant andm'_k = 4π/3(d_k-d_k+1) R^3_k. If we assume thatϵ_ρ^(i) = ℋ_iϵ_J;ϵ_z^(i) = 𝒢_iϵ_M,where ϵ_M is the flattening of the equivalent MacLaurin homogeneous spheroid in synchronous rotation and ϵ_J is the flatteningof the equivalent Jeans homogeneous spheroids:ϵ_M = 5R_N^3n^2/4m_TGϵ_J = 15MR_N^3/4m_Tr^3,the Eq. (<ref>) can be written asγ_i ℋ_i=x_i^3 + ∑_j=1^i-1α_i jℋ_j + ∑_j=i+1^N β_i jℋ_j γ_i 𝒢_i= (Ω_i/n)^2x_i^3 + ∑_j=1^i-1α_i j𝒢_j + ∑_j=i+1^N β_i j𝒢_j,where x_i=R_i/R_N is the normalized mean equatorial radius and the coefficients α_i j, β_i j and γ_i areα_i j = 3m'_j/2m_T(R_j/R_i)^2β_i j = 3m'_j/2m_T(R_i/R_j)^3 γ_i=1 + 3(m_T-m'_i)/2m_T - ∑_k=i+1^N 5m'_k/2m_T(R_k^3-R_i^3)/R_k^3. Then, the Clairaut's coefficients ℋ_i and 𝒢_i areℋ_i = ∑_j=1^N (E^-1)_ijx_j^3 𝒢_i = ∑_j=1^N (E^-1)_ijx_j^3(Ω_i/n)^2,where (E^-1)_ij are the elements of the inverse of the matrix E, whose elements are(E)_ij = {[ α_ij=3/2f_N(d_j-d_j+1)x^5_j/x^2_i,i>j; γ_i=3/2f_N(d_i-d_i+1)x^3_i + 5/2- 5/2f_N∑_k=i+1^N (d_k-d_k+1) (x_k^3-x_i^3),i=j; β_ij=3/2f_N(d_j-d_j+1)x^3_i,i<j ].where d_i=d_i/d_1 is the normalized density of the ith layer and f_N=3∫_0^1d(z)z^2 dz.§ APPENDIX 3: GRAVITATIONAL COUPLING When the principal axes of inertia of two layers (of one body composed by N homogeneous layers) are not aligned, a restoring gravitational torque, whichtends to align these axes appears. The torque acting on the inner jth layer due to the outer ith layer (not necessarily contiguous) isΓ_ji = -∫_m_j (r⃗×∇δU_i) dm_j =-∫_0^2π∫_0^π∫_ζ'_j-1^ζ'_jd_j ( r⃗×∇δU_i) r^2sinθ dr dθ dφ,where d_j, m_j are the density and the mass in the jth layer and δ U_i is the disturbing potential of the ith layer at an external point (Fig.<ref>). The limits of the integral in Eq. (<ref>), ζ'_j and ζ'_j-1, are the real outer and inner boundaries of the jth layer,respectively. In our model we have to consider the actual flattening of the surfaces, which is the composition of the main elastic and anelastic tidalcomponents (see Sec. 10 of Ferraz-Mello, 2013). The addition of the two components is virtually equivalent to the use the Maxwell viscoelastic model abinitio as done by Correia et al. (2014) (Ferraz-Mello, 2015b).Assuming that the elastic and the anelastic components have ellipsoidal surfaces (not aligned), the resulting surface can be approximated by a prolate ellipsoidwith equatorial flattening ϵ' and rotated by an angle ϑ with respect to M. For the sake of simplicity, we also assume that therelative motion of the outer body M is circular. Then, neglecting the axial term does not contribute to the calculation of the gravitational coupling,the height of the outer surface of the jth layer with respect to the one sphere of radius R_j, in polar coordinates, rotated by an angle ϑ_jwith respect to M and to first order in the flattenings (see Fig. <ref>), isδζ'_j = 1/2R_jϵ'_jsin^2θcos(2φ-2ϑ_j)=1/2R_jℋ_jϵ_ρλ_jsin^2θcos2φ+1/2R_jℋ_jϵ_ρcosσ_j0sin^2θcos(2φ-σ_j0),where 0<λ_j<1 is a relative measurement of the maximum height of the elastic tides of the outer boundary of the jth layer. The angle ϑ_jis often called the geodetic lag of the surface. If we open the trigonometric functions, by identification of the terms with same trigonometric arguments, the resulting equatorial flattening of the outerboundary of the jth layer isϵ'_j = ℋ_jϵ_ρ√(λ_j^2 + cos^2σ_j0(1 + 2λ_j)),and the geodetic lag isϑ_j = 1/2tan^-1(sin2σ_j0/1+2λ_j+cos2σ_j0). The height of the inner boundary of the jth layer, taking into account the composition of the main elastic and anelastic tides has an identicalexpression:δζ'_j-1 = 1/2R_j-1ϵ'_j-1sin^2θcos(2φ-2ϑ_j-1)= 1/2R_j-1ℋ_j-1ϵ_ρλ_j-1sin^2θcos2φ+1/2R_j-1ℋ_j-1ϵ_ρcosσ_j-1,0sin^2θcos(2φ-σ_j-1,0),where0<λ_j-1<1 is the relative measurement of the maximum height of the elastic tides of the inner boundary of the jth layer. Then, theresulting equatorial flattening isϵ'_j-1 = ℋ_j-1ϵ_ρ√(λ_j-1^2 + cos^2σ_j-1,0(1 + 2λ_j-1)),and the geodetic lag isϑ_j-1 = 1/2tan^-1(sin2σ_j-1 0/1+2λ_j-1+cos2σ_j-1,0). In the same way, we assume that the ellipsoidal shape of this layer is also given by the composition of the main elastic and anelastic tidal components.Then, the inner and outer equatorial flattenings, respectively, areϵ'_i-1 = ℋ_i-1ϵ_ρ√(λ_i-1^2 + cos^2σ_i-1,0(1 + 2λ_i-1)); ϵ'_i = ℋ_iϵ_ρ√(λ_i^2 + cos^2σ_i0(1 + 2λ_i)),and the corresponding geodetic lags areϑ_i = 1/2tan^-1(sin2σ_i0/1+2λ_i+cos2σ_i0); ϑ_i-1 = 1/2tan^-1(sin2σ_i-1,0/1+2λ_i-1+cos2σ_i-1,0),where0<λ_i,λ_i-1<1 are the relative measurements of the maximum heights of the elastic tides of the outer and inner boundaries of theith layer.Using the expression of the disturbing potential, given in Section A in the Online Supplement, and neglecting the axial term, we obtainδ U_i = -3GC_i/4r^3sin^2θΔ(R_i^5ϵ'_icos(2φ-2ϑ_i))/R_i^5-R_i-1^5,where Δ(f_i) = f_i-f_i-1, denotes the increment of one function f_i between the inner and the outer boundaries of this layer. Then, thevectorial product in Eq. (<ref>) isr⃗×∇δU_i = -2π Gd_i/5r^3(2sinθΔ(R_i^5ϵ'_isin(2φ-2ϑ_i)) θ⃗+sin2θΔ(R_i^5ϵ'_icos(2φ-2ϑ_i)) φ⃗). Using the polar unitary vectors in Cartesian coordinatesθ⃗ = cosθcosφ x⃗ + cosθsinφ y⃗ - sinθ z⃗ φ⃗ =-sinφ x⃗ + cosφ y⃗,and the approximation of lnζ'_j/ζ'_j-1 to first order in the flatteningslnζ'_j/ζ'_j-1≈lnR_j/R_j-1 + 1/2ϵ'_jsin^2θcos(2φ-2ϑ_j) - 1/2ϵ'_j-1sin^2θcos(2φ-2ϑ_j-1),then, we may perform the integrals of Eq. (<ref>) and obtain the torque acting on the inner jth layer due to the outer ith layerΓ_ji =-32π^2 G/75 d_id_j Δ_ij(R_i^5ϵ'_iϵ'_jsin(2ϑ_j-2ϑ_i))z⃗,where Δ_ij(f_ij)Δ(f_ij)-Δ(f_i,j-1)=f_ij-f_i-1,j-f_i,j-1+f_i-1,j-1.As the torque acting on the outer ith layer, due to the inner jth layer, is the reactionΓ_ij =-Γ_ji,then, the total gravitational coupling, acting on the jth layer can be written asΓ_j = ∑_p=1; p≠ j^N Γ_jp = ∑_p=1^j-1Γ_jp - ∑_p=j+1^N Γ_pj. If we consider the two-layer model, the torque acting on the core and the shell, areΓ_c=Ksin(2ϑ_s-2ϑ_c) Γ_s=-Ksin(2ϑ_s-2ϑ_c),where the gravitational coupling parameter K isK=32π^2 G/75 d_cd_sϵ'_cϵ'_sR_c^5. The equatorial flattenings areϵ'_c = ℋ_cϵ_ρ√(λ_c^2 + cos^2σ_c0(1 + 2λ_c)); ϵ'_s = ℋ_sϵ_ρ√(λ_s^2 + cos^2σ_s0(1 + 2λ_s)),and the geodetic lags areϑ_c = 1/2tan^-1(sin2σ_c0/1+2λ_c+cos2σ_c0); ϑ_s = 1/2tan^-1(sin2σ_s0/1+2λ_s+cos2σ_s0).The parameters 0<λ_c,λ_s<1 are relative measurements of the heights of the elastic tides of the outer surfaces of the core and the shell,respectively. The trigonometric functions in (<ref>)-(<ref>) are frequency functions. Using Eq. (<ref>), anelementary calculation shows thatcos^2σ_i0 = γ_i^2/γ_i ^2+ν_i^2; sin2σ_i0 = 2γ_iν_i/γ_i ^2+ν_i^2; cos2σ_i0 = γ_i^2-ν_i^2/γ_i ^2+ν_i^2. It is important to note that some works, as Karatekin et al. (2008), use a different gravitational coupling parameter K. When applied to a two-layer model,their K differs from Eq. (<ref>) by a multiplicative factor (1-d_s/d_c). The reason for this difference is simple: while here we calculate thetorque due to the mutual gravitational attraction of two layers, through Eq. (<ref>), they calculate the gravitational coupling between regionsthat involve various layers simultaneously (see Fig. 2 and Eq. 16 of Van Hoolst et al., 2008).§ APPENDIX 4: LINEAR DRAGThe model considered here also assumes that a linear friction occurs between two contiguous layers. We assume that between two contiguous layers (forinstance, the inner boundary of the ith layer and the outer boundary of the (i+1)th layer) exists a thin liquid boundary with viscosityη_i and thickness h_i.We assume that the torque, along the axis z, acting on the inner ith layer due to the outer (i+1)th layer is Φ_i,i+1 = μ_i (Ω_i+1-Ω_i),and vice-versa. The friction coefficient μ_i of the ith boundary is an undetermined constant. Let d𝐅_i,i+1 be the force acting tangentially on the area element of an sphere of radius R_i. If the fluid in contact with the surface of thesphere is a Newtonian fluid, and the thickness of the liquid boundary is thin enough to allow us to consider a plane-parallel geometry (a plane Couetteflow), the modulus of the force is (Papanastasious et al., 2000, Chap. 6, Eq. 6.15)dF_i,i+1=η_i/h_i V_i R_i^2sinθ dϕ dθwhere V_i=R_isinθ(Ω_i+1-Ω_i) is the relative velocity of the (i+1)th layer with respect to the ith layer at the latitude θand R_i,ϕ,θ are the spherical coordinates of the center of the area element. The modulus of the torque of the force d𝐅_i,i+1, alongthe axis z, is dΦ_i,i+1=R_isinθ dF_i,i+1.The element of area is R_idθ× R_isinθ dϕ. 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Let us consider the equation of surface of this homogeneous triaxial ellipsoid, in a reference system where the semi axes a and c are aligned to thecoordinates axes x and z, respectively:x^2/a^2+y^2/b^2+z^2/c^2=1.If we use the semi axes (<ref>), the spherical coordinatesx=ρsinθcosφ;y=ρsinθsinφ;z=ρcosθ,and expand to first order in the flattenings, we obtainρ= R(1+ϵ_ρ/2sin^2θcos2φ-(ϵ_ρ/2+ϵ_z)cos^2θ). The mass of this ellipsoids ism = 4π/3d abc ≈4π/3d R^3(1-ϵ_ρ/2-ϵ_z). The principal moments of inertia areA= 1/5m(b^2+c^2) ≈2/5mR^2 (1-ϵ_ρ-ϵ_z) B= 1/5m(a^2+c^2) ≈2/5mR^2 (1-ϵ_z) C= 1/5m(a^2+b^2) ≈2/5mR^2,and its differences areC-A≈ 2/5mR^2 (ϵ_ρ+ϵ_z) ≈ C (ϵ_ρ+ϵ_z)C-B≈ 2/5mR^2 ϵ_z ≈ Cϵ_z B-A≈ 2/5mR^2 ϵ_ρ≈ C ϵ_ρ. The corresponding gravitational potential of this homogeneous triaxial ellipsoid, at an external point r⃗^⃗*⃗, isU(r⃗^⃗*⃗)=-Gm/r^* - G(B-A)/2r^*5(3x^*2-r^*2) + G(C-B)/2r^5(3z^*2-r^*2) ≈-Gm/r^* - GC/2r^*3ϵ_ρ(3cos^2φsin^2θ-1) + G C/2r^*3ϵ_z(3cos^2θ-1),orU(r⃗^⃗*⃗)=-Gm/r^* - 3GC/4r^*3ϵ_ρsin^2θcos2φ + G C/2r^*3(ϵ_ρ/2+ϵ_z)(3cos^2θ-1).§.§ Ellipsoidal layer Let us consider a homogeneous triaxial ellipsoidal shell with density d_i, outer semi axes a_i>b_i>c_i, outer equatorial mean radius R_i= √(a_ib_i)and outer equatorial and polar flatteningsϵ_ρ^(i) = a_i-b_i/R_i;ϵ_z^(i) = b_i-c_i/R_i. At the inner ellipsoidal boundary, the semi axes are a_i-1>b_i-1>c_i-1 (not necessarily aligned with the axes of the outer surface). The innerequatorial mean radius is R_i-1= √(a_i-1b_i-1) and the inner equatorial and polar flattenings areϵ_ρ^(i-1) = a_i-1-b_i-1/R_i-1;ϵ_z^(i-1) = b_i-1-c_i-1/R_i-1. The semi axes of the outer boundary, to first order in flattenings, area_i = R_i (1+ϵ_ρ^(i)/2); b_i = R_i (1-ϵ_ρ^(i)/2); c_i = R_i (1-ϵ_ρ^(i)/2-ϵ_z^(i)),and the semi axes of the inner boundary area_i-1 = R_i-1(1+ϵ_ρ^(i-1)/2); b_i-1 = R_i-1(1-ϵ_ρ^(i-1)/2); c_i-1 = R_i-1(1-ϵ_ρ^(i-1)/2-ϵ_z^(i-1)). Let us consider the equation of the surface of the outer triaxial ellipsoidal layer, in a reference system where the semi axes a_i and c_i are alignedto the coordinates axes x and z, respectively:x^2/a_i^2+y^2/b_i^2+z^2/c_i^2=1.If we use the semi axes (<ref>), the spherical coordinates (<ref>) and expand to first order in the flattenings, weobtainρ_i = R_i(1+ϵ_ρ^(i)/2sin^2θcos2φ-(ϵ_ρ^(i)/2+ϵ_z^(i))cos^2θ). For the inner boundary we have the same expression, when the reference system is again such that the semi axes a_i-1 and c_i-1 are aligned to thecoordinate axes x and z, respectively. If we use the semi axes (<ref>), the spherical coordinates (<ref>) andexpand to first order in the flattenings, we obtainρ_i-1 = R_i-1(1+ϵ_ρ^(i-1)/2sin^2θcos2φ-(ϵ_ρ^(i-1)/2+ϵ_z^(i-1))cos^2θ). The mass m_i of this layer, can be written as the subtraction of the masses of the two homogeneous ellipsoids of same density d_i: the homogeneousellipsoid of mass m'_i and same surface as the outer boundary of the layer, less the homogeneous ellipsoid of mass m”_i and same surface as the innerboundary of the layer Fig. <ref>. The total mass of the layer then ism_i=m'_i-m”_i≈4π/3d_i(R_i^3(1-ϵ_ρ^(i)/2-ϵ_z^(i))-R_i-1^3(1-ϵ_ρ^(i-1)/2-ϵ_z^(i-1))).Note that this result is independent of the orientation of the ellipsoidal boundaries semi axes. The masses m'_i and m”_i arem'_i =d_i4π/3a_ib_ic_i ≈m_iR_i^3 (1-ϵ_ρ^(i)/2-ϵ_z^(i))/R_i^3(1-ϵ_ρ^(i)/2-ϵ_z^(i))-R_i-1^3(1-ϵ_ρ^(i-1)/2-ϵ_z^(i-1))m”_i=d_i4π/3a_i-1b_i-1c_i-1≈m_iR_i-1^3(1-ϵ_ρ^(i-1)/2-ϵ_z^(i-1))/R_i^3(1-ϵ_ρ^(i)/2-ϵ_z^(i))-R_i-1^3(1-ϵ_ρ^(i-1)/2-ϵ_z^(i-1)). To calculate the principal moments of inertia A_i,B_i,C_i of a homogeneous triaxial ellipsoidal layer when the inner and the outer boundaries are notaligned is particularly complicated because the orientation of the principal axes of inertia do not coincide with the axes of symmetry of both boundaries.In the sequence we focus in the particular case in which the inner and the outer boundaries are aligned.In this case, we can use the same scheme used to calculate the mass of the layer. The principal moments of inertia of the layer, can be written as thesubtraction of the principal moments of inertia of two homogeneous ellipsoids of same density d_i: the principal moments of inertia of one homogeneousellipsoid of mass m'_i and the same surface as the outer boundary of the layer, less the principal moments of inertia of the homogeneous ellipsoid of massm”_iand the same surface as the inner boundary of the layer. Using the semi axes (<ref>) and the masses (<ref>), the principalmoments of inertia can be approximated to first order in the flattenings asA_i= 1/5m'_i(b_i^2+c_i^2)-1/5m”_i(b_i-1^2+c_i-1^2)≈ 2/5m_iR_i^5-R_i-1^5/R_i^3-R_i-1^3(1+Δ(R_i^3ϵ_ρ^(i))/R_i^3-R_i-1^3+Δ(R_i^3ϵ_z^(i))/R_i^3-R_i-1^3-3/2Δ(R_i^5ϵ_ρ^(i))/R_i^5-R_i-1^5-2Δ(R_i^5ϵ_z^(i))/R_i^5-R_i-1^5) B_i= 1/5m'_i(a_i^2+c_i^2)-1/5m”_i(a_i-1^2+c_i-1^2)≈ 2/5m_iR_i^5-R_i-1^5/R_i^3-R_i-1^3(1+Δ(R_i^3ϵ_ρ^(i))/R_i^3-R_i-1^3+Δ(R_i^3ϵ_z^(i))/R_i^3-R_i-1^3-1/2Δ(R_i^5ϵ_ρ^(i))/R_i^5-R_i-1^5-2Δ(R_i^5ϵ_z^(i))/R_i^5-R_i-1^5)C_i= 1/5m'_i(a_i^2+b_i^2)-1/5m”_i(a_i-1^2+b_i-1^2)≈ 2/5m_iR_i^5-R_i-1^5/R_i^3-R_i-1^3(1+Δ(R_i^3ϵ_ρ^(i))/R_i^3-R_i-1^3+Δ(R_i^3ϵ_z^(i))/R_i^3-R_i-1^3-1/2Δ(R_i^5ϵ_ρ^(i))/R_i^5-R_i-1^5-Δ(R_i^5ϵ_z^(i))/R_i^5-R_i-1^5),and its differences areC_i-A_i≈C_i (Δ(R_i^5ϵ_ρ^(i))/R_i^5-R_i-1^5+Δ(R_i^5ϵ_z^(i))/R_i^5-R_i-1^5) C_i-B_i≈C_i Δ(R_i^5ϵ_z^(i))/R_i^5-R_i-1^5 B_i-A_i≈C_i Δ(R_i^5ϵ_ρ^(i))/R_i^5-R_i-1^5,where Δ(f_i) = f_i-f_i-1, denotes the increment of one function f_i, between the inner and the outer boundaries of this layer.Using the same scheme used to calculate the mass and the principal moments of inertia, the corresponding gravitational potential of this homogeneoustriaxial layer at an external point r⃗^* isU_i(r⃗^*)=-Gm_i/r^* - G(B_i-A_i)/2r^*5(3x^*2-r^*2) + G(C_i-B_i)/2r^*5(3z^*2-r^*2) ≈-Gm_i/r^* - GC_i/2r^*3Δ(R_i^5ϵ_ρ^(i))/R_i^5-R_i-1^5(3cos^2φ^*sin^2θ^*-1) + G C_i/2r^*3Δ(R_i^5ϵ_z^(i))/R_i^5-R_i-1^5(3cos^2θ^*-1),orU_i(r⃗^*)=-Gm_i/r^* -3GC_i/4r^*3Δ(R_i^5ϵ_ρ^(i))/R_i^5-R_i-1^5sin^2θ^*cos2φ^* +G C_i/2r^*3Δ(R_i^5(ϵ_ρ^(i)/2+ϵ_z^(i)))/R_i^5-R_i-1^5(3cos^2θ^*-1). If we consider the static equilibrium figure, the flattenings can be written asϵ_ρ^(k) = ℋ_kϵ_ρ;ϵ_z^(k) = 𝒢_kϵ_z,where ℋ_k and 𝒢_k are the Clairaut numbers. Then, the difference of the principal moments of inertia can be approximated to firstorder in the flattenings asC_i-A_i≈C_i (ℒ_iϵ_ρ+ℒ'_iϵ_z) C_i-B_i≈C_i ℒ'_iϵ_z B_i-A_i≈C_i ℒ_iϵ_ρ,where the parameters ℒ_i and ℒ'_i areℒ_i= ℋ_iR_i^5-ℋ_i-1R_i-1^5/R_i^5-R_i-1^5; ℒ'_i = 𝒢_iR_i^5-𝒢_i-1R_i-1^5/R_i^5-R_i-1^5.The coefficients ℒ_i and ℒ'_i play a role equivalent to the coefficients ℋ_i and 𝒢_i for the quantitiesC_i-A_i, C_i-B_i and B_i-A_i. In this case, the moments of inertia B_i-A_i (resp. C_i-B_i) of the ith layer can be written as the homogeneousmoments multiplied by the coefficients ℒ_i (resp. ℒ'_i), characteristics of this layer. The difference between ℒ_i andℒ'_i comes from the fact that the body has a differential rotation. If we assume a rigid rotation, then ℒ'_i=ℒ_i(Ω/n)^2.The corresponding gravitational potential of this homogeneous triaxial layer at an external point r⃗^* isU_i(r⃗^*)=-Gm_i/r^* - G(B_i-A_i)/2r^*5(3x^*2-r^*2) + G(C_i-B_i)/2r^*5(3z^*2-r^*2) ≈-Gm_i/r^* - GC_iℒ_i/2r^*3ϵ_ρ(3cos^2φ^*sin^2θ^*-1) + G C_iℒ'_i/2r^*3ϵ_z(3cos^2θ^*-1),orU_i(r⃗^*)=-Gm_i/r^* - 3GC_iℒ_i/4r^*3ϵ_ρsin^2θ^*cos2φ^*+GC_i/2r^*3(ℒ_iϵ_ρ/2+ℒ'_iϵ_z)(3cos^2θ^*-1). Although we do not calculate the principal moments of inertia when the inner and the outer boundaries are not aligned, it is possible to calculate easilythe gravitational potential with the same scheme used to calculate the mass of the layer and the principal moments of inertia. The potential of the layer,can be written as the subtraction of the potential of two homogeneous ellipsoids of same density d_i: the potential of one homogeneous ellipsoid of massm'_i and the same surface as the outer boundary of the layer, given by the Eq. (<ref>), less the potential of the homogeneous ellipsoid of massm”_i and the same surface as the inner boundary of the layer, given by the Eq. (<ref>).The corresponding gravitational potential isU_i(r⃗^*)=-Gm_i/r^* -3GC_i/4r^*3Δ(R_i^5ϵ_ρ^(i)cos(2φ^*-2ϕ_i))/R_i^5-R_i-1^5sin^2θ^*+G C_i/2r^*3Δ(R_i^5(ϵ_ρ^(i)/2+ϵ_z^(i)))/R_i^5-R_i-1^5(3cos^2θ^*-1). § NEAR-SYNCHRONOUS SOLUTION OF THE ROTATIONAL EQUATIONS Using the convention 1=core and 2=shell, the rotational system of the two-layer model, given by Eq. (<ref>), can be written asẏ_1=- T_11^*𝒯_1+𝒦_1sin2ξ+ℱ_1(γ_2y_2-γ_1y_1) ẏ_2=T_21^*𝒯_1-T_22^*𝒯_2-𝒦_2sin2ξ-ℱ_2(γ_2y_2-γ_1y_1),where, the rotational variables arey_1=ν_1/γ_1=2Ω_1/γ_1-2n/γ_1; y_2=ν_2/γ_2=2Ω_2/γ_2-2n/γ_2,the tidal function 𝒯_i is𝒯_i = ∑_k,j∈ℤ E_2,kE_2,k+j(y_i+P_ik)cos(jnt)+sin(jnt)/1+(y_i+P_ik)^2.The constants areT_ij^* = 2𝒯/γ_iℋ_jR_j^5/R_i^5-R_i-1^5; 𝒦_i=2K/γ_iC_i; ℱ_i=μ/γ_iC_i; P_ik = kn/γ_i = kp_i,and the tidal parameter 𝒯, is defined as𝒯 = 45GM^2R_s^3/8m_Ta^6≈3n^2ϵ_ρ/2. We assume that the solution, to second order in eccentricity, can be written asy_1=b_10e^2 + c_11ecosℓ + s_11esinℓ + c_12e^2cos2ℓ + s_12e^2sin2ℓ y_2=b_20e^2 + c_21ecosℓ + s_21esinℓ + c_22e^2cos2ℓ + s_22e^2sin2ℓ,where b_i0, c_ij and s_ij are undetermined coefficients. Introducing the solution (<ref>) into the rotational system (<ref>)and expanding to second order in eccentricity, by identification of the terms with same trigonometric argument, we can calculate these coefficients.The derivatives of (<ref>) areẏ_1=n s_11ecosℓ - n c_11esinℓ + 2n s_12e^2cos2ℓ - 2n c_12e^2sin2ℓ ẏ_2=n s_21ecosℓ - n c_21esinℓ + 2n s_22e^2cos2ℓ - 2n c_22e^2sin2ℓ, The tidal function can be approximated by𝒯_i≃ (b_i0-12p_i/1+p_i^2+q_ic_i1+r_is_i1)e^2+(c_i1-4p_i/1+p_i^2)ecosℓ+(s_i1-4p_i^2/1+p_i^2)esinℓ+(c_i2-17p_i/1+4p_i^2+q_ic_i1-r_is_i1)e^2cos2ℓ+(s_i2-34p_i^2/1+4p_i^2+r_ic_i1+q_is_i1)e^2sin2ℓ,where the coefficients q_i and r_i areq_i = 3(2+p_i^2+p_i^4)/2(1+p_i^2)^2; r_i = 3p_i/(1+p_i^2)^2. In the same way, the trigonometric function of the gravitational coupling can be approximated bysin2ξ = sin[tan^-1(y_2/1+λ_2(1+y_2^2))-tan^-1(y_1/1+λ_1(1+y_1^2))]≃ (b_20/1+λ_2 -b_10/1+λ_1)e^2+(c_21/1+λ_2-c_11/1+λ_1)ecosℓ+(s_21/1+λ_2-s_11/1+λ_1)esinℓ+(c_22/1+λ_2-c_12/1+λ_1)e^2cos2ℓ+(s_22/1+λ_2-s_12/1+λ_1)e^2sin2ℓ,and the amplitude of oscillation isK = 32π^2 G/75ℋ_1ℋ_2ϵ_ρ^2 d_1 d_2R_1^5E_2,0^2√(λ_1^2 +1 + 2λ_1/1+y_1^2)√(λ_2^2 +1 + 2λ_2/1+y_2^2)≃ 32π^2 G/75ℋ_1ℋ_2ϵ_ρ^2 d_1 d_2R_1^5 (1+λ_1)(1+λ_2)+𝒪(e^2). The friction term isγ_2y_2-γ_1y_1≃ (γ_2b_20-γ_1b_10)e^2+(γ_2c_21-γ_1c_11)ecosℓ+(γ_2s_21-γ_1s_11)esinℓ+(γ_2c_22-γ_1c_12)e^2cos2ℓ+(γ_2s_22-γ_1s_12)e^2sin2ℓ. Replacing (<ref>)-(<ref>) into (<ref>) and colecting the terms with same trigonometric argument, we can find threelinear sub-systems for the undetermined b_i0, c_ij and s_ij, which can be written in vectorial notation asD_1 Λ_1= T_1P_1 D_2 Λ_2= T_1P_2-T_1R_2 DΛ_0= TP-TR,whereΛ_0=[ [ b_10; b_20 ]]; Λ_1=[ [ c_11; c_21; s_11; s_21 ]]; Λ_2=[ [ c_12; c_22; s_12; s_22 ]],are the undetermined coefficients vectors. The constants matrices are defined asT=[ [T_11^* 0; -T_21^*T_22^* ]]; T_1=[ [ T 0; 0 T ]]; D=[ [ d_11 d_12; d_21 d_22 ]]; D_1=[ [ DnI; -nI D ]]; D_2=[ [D2nI; -2nID ]],where I is the identity matrix and the coefficients d_ij are d_11 = T_11^*+𝒦_1/1+λ_1+ℱ_1γ_1;d_12 =-𝒦_1/1+λ_2-ℱ_1γ_2;d_21 =-T_21^*+𝒦_2/1+λ_1+ℱ_2γ_1;d_22 = T_22^*-𝒦_2/1+λ_2-ℱ_2γ_2,and the vectors P_i and R_i areP=12[ [ p_1/(1+p_1^2); p_2/(1+p_2^2) ]]; R=[ [ q_1c_11+r_1s_11; q_2c_21+r_2s_21 ]]; P_1=4[ [ p_1/(1+p_1^2); p_2/(1+p_2^2); p_1^2/(1+p_1^2); p_2^2/(1+p_2^2) ]]; P_2=17[ [p_1/(1+4p_1^2);p_2/(1+4p_2^2); 2p_1^2/(1+4p_1^2); 2p_2^2/(1+4p_2^2) ]]; R_2=[ [ q_1c_11-r_1s_11; q_2c_21-r_2s_21; r_1c_11+q_1s_11; r_2c_21+q_2s_21 ]]. The solution of these sub-systems areΛ_1= D_1^-1T_1P_1 Λ_2= D_2^-1T_1P_2-D_2^-1T_1R_2 Λ_0= D^-1TP-D^-1TR.Finally, the rotational solutions can be written asν_1=B_10 + B_11cos(ℓ+ϕ_11) + B_12cos(2ℓ+ϕ_12) ν_2=B_20 + B_21cos(ℓ+ϕ_21) + B_22cos(2ℓ+ϕ_22),where the constants B_ij and the phases ϕ_ij areB_i0 = γ_ib_i0e^2; B_ij = γ_i√(c_ij^2+s_ij^2) e^j; ϕ_ij = -tan^-1(s_ij/c_ij).§.§ Tidal drift and the periodic terms The tidal drift is the term B_i0 of the solution (<ref>). It isν_1^(stat) = B_10 = γ_1b_10e^2+𝒪(e^3)ν_2^(stat) = B_20 = γ_2b_20e^2+𝒪(e^3). This result can be rewritte asν_1^(stat) = 12nκ_11γ_1^2e^2/γ_1^2+n^2 +12nκ_12γ_2^2e^2/γ_2^2+n^2-κ_11γ_1(q_1c_11+r_1s_11)e^2 -κ_12γ_2(q_2c_21+r_2s_21)e^2+𝒪(e^3) ν_2^(stat) = 12nκ_21γ_1^2e^2/γ_1^2+n^2 +12nκ_22γ_2^2e^2/γ_2^2+n^2-κ_21γ_1(q_1c_11+r_1s_11)e^2 -κ_22γ_2(q_2c_21+r_2s_21)e^2+𝒪(e^3).The coefficient κ_ij can be written as κ_ij=f_ij/g, where f_ij isf_ij = δ_i,j𝒯C_1C_2/Cℋ_1ℋ_2R_2^5/R_2^5-R_1^5+𝒟_jγ_i/γ_j(1+λ_i)K/(1+λ_1)(1+λ_2)+𝒟_jγ_iμ/2,δ_i,j is the Kronecker delta (δ_1,1=δ_2,2=1 and δ_1,2=δ_2,1=0), the parameter 𝒟_1 and 𝒟_2are defined as𝒟_1 = (C_1-C_2R_1^5/R_2^5-R_2^5)ℋ_1/C; 𝒟_2 = C_2R_2^5/R_2^5-R_1^5ℋ_2/Cand the constant g isg = f_11+f_22-𝒯C_1C_2/Cℋ_1ℋ_2R_2^5/R_2^5-R_2^5. The two first terms of each Eq. (<ref>)N_i= 12nκ_i1γ_1^2e^2/γ_1^2+n^2 +12nκ_i2γ_2^2e^2/γ_2^2+n^2come from the non-periodic terms with |j|=0, while the terms that involve c_i1 and s_i1.P_i= -κ_i1γ_1(q_1c_11+r_1s_11)e^2-κ_i2γ_2(q_2c_21+r_2s_21)e^2,come from the periodic terms with |j|=1. The harmonic terms with |j|=2, do not contribute to the stationary rotation at order e^2.It is worth emphasizing that in the absence of friction and gravitational coupling, that is, K=μ=0, the coefficient κ_ij=δ_i,j. Then,the non-periodic tidal drift of the ith layer has the same expression that the excess rotation in the case of a homogeneous body, with γ_i insteadof γν_i^(stat)=12nγ_i^2e^2/γ_i^2+n^2+𝒪(e^3). In the case n/γ_1≫1, n/γ_2≫1, an elementary calculation shows that each coefficient κ_ij becomes independent of the frictionparameter μ, depending only on the internal structure and on the relaxation factors γ_1 and γ_2, with f_ij tending tof_ij = δ_i,j𝒯C_1C_2/Cℋ_1ℋ_2R_2^5/R_2^5-R_1^5+𝒟_jγ_i/γ_j(1+λ_i)K/(1+λ_1)(1+λ_2). In the case n/γ_1≪1, n/γ_2≪1, each coefficient κ_ij becomes independent of 𝒯, K and μ, depending only on theinternal structure and on the relaxation factors γ_1 and γ_2, tending toκ_ij= 𝒟_jγ_i/𝒟_1γ_1+𝒟_2γ_2,and the stationary solution tends to synchronous rotation.The periodic terms have amplitudes B_i1 and B_i2, given by the Eq. (<ref>). The coefficients c_ij and s_ij gives rise to intricateanalytical expressions, but are easy to calculate numerically. Fig. <ref> shows one example for the Titan's core and the shell constants B_1j andB_2j, respectively, in function of the shell relaxation factor γ_2 (see Table <ref>-<ref>). We use that the corerelaxation factor is γ_1=10^-8 s^-1, and fix the ocean's viscosity and thickness values to η_o=10^-3 Pa s and h=178 km, respectively. We also plot the non-periodic N_i and periodic P_i terms, separately, and the total tidal drift B_i0=N_i+P_i. We can observethat if γ_2≳10^-7.5 s^-1, the shell oscillates around the super-synchronous rotation. When γ_2≲10^-7.5 s^-1,the tidal drift B_20 becomes negative and tends to zero, that is, the shell oscillates around the synchronous rotation, with a period of oscillationequal to the orbital period. The negative sign of the tidal drift B_20, is due to the contribution of the periodic terms P_2, which becomes negativeand |P_2|≫ N_2. Finally, if γ_2≲10^-8 s^-1, the amplitude of the shell rotation decreases, tending to zero when γ_2decreases. On the other hand, the core oscillates around the synchronous rotation, with a period of oscillation equal to the orbital period, independentlyof the shell relaxation factor. This behavior is confirmed by the numerical simulations of non-approximate system (see Sec. <ref>).In Fig. <ref>, we show the comparison of the Titan's shell rotation in the complete non-linear system given by Eq. (<ref>) andin the approximate analytical solution given by Eq. (<ref>), for some values of the core's relaxation factor γ_1 and ocean thickness h.The dashed red lines show the maximum and minimum values of Ω_2-n given by the approximate solution, taking into account only the first harmonic(|j|≤1), while the solid black lines show the maximum and minimum values of Ω_2-n when the complete non-linear system is integrated (using |j|≤7).The approximate solution is in excellent agreement with numerical integration of the equations.§.§ Atmospheric influence When we consider the effect of the atmosphere, the rotational system becomes ẏ_1=- T_11^*𝒯_1+𝒦_1sin2ξ+ℱ_1(γ_2y_2-γ_1y_1) ẏ_2=T_21^*𝒯_1-T_22^*𝒯_2-𝒦_2sin2ξ-ℱ_2(γ_2y_2-γ_1y_1)+𝒜_⊙sin2α_⊙.where𝒜_⊙=2A_⊙/γ_2. We assume that the particular solutiony_1⊙ = C_1⊙cos2α_⊙ + S_1⊙sin2α_⊙y_2⊙ = C_2⊙cos2α_⊙ + S_2⊙sin2α_⊙,can be added to (<ref>) to obtain the general solutions of the complete equation. C_j⊙ and S_j⊙ are undetermined coefficients tobe obtained by substitution of the parts of the solution into Eq. (<ref>) and identification.The derivative of (<ref>) isẏ_1⊙ =-2n_⊙ C_1⊙sin2α_⊙ + 2n_⊙ S_1⊙cos2α_⊙ ẏ_2⊙ =-2n_⊙ C_2⊙sin2α_⊙ + 2n_⊙ S_2⊙cos2α_⊙. The tidal function can be approximated by𝒯_i≃C_i⊙cos2α_⊙+S_i⊙sin2α_⊙,the trigonometric function of the gravitational coupling can be approximated bysin2ξ ≃ (C_2⊙/1+λ_2-C_1⊙/1+λ_1)cos2α_⊙+(S_2⊙/1+λ_2-S_1⊙/1+λ_1)e^2sin2α_⊙,and the friction term isγ_2y_2-γ_1y_1≃ (γ_2C_2⊙-γ_1C_1⊙)cos2α_⊙+(γ_2S_2⊙-γ_1S_1⊙)sin2α_⊙. Defining the constant matrix D_⊙ and the constant vectors Λ_⊙, P_⊙, asD_⊙=[ [D2n_⊙I; -2n_⊙ID ]]; Λ_⊙=[ [ C_1⊙; C_2⊙; S_1⊙; S_2⊙ ]]; P_⊙=𝒜_⊙[ [ 0; 0; 0; 1 ]],the undetermined coefficient vector isΛ_⊙ = D^-1_⊙P_⊙. In Fig. <ref>, we show the same comparison of the Titan's shell rotation in the complete non-linear system and in the approximateanalytical solution of the above section. The approximate solution, also is in excellent agreement with numerical integration. It is important to note thatthe fact that the approximate solution of the non-linear system (<ref>) can be expressed as the sum of solutions (<ref>) and(<ref>), it means that this system has a behavior quasi-linear, at least for the Titan's problem.§ THE INTEGRAL OF SECTION <REF> §.§.§ Proposition: 1/2π∫_0^2πΩ_j^2(a/r)^4sinv dℓ = 0.To prove (<ref>), we consider only the tidal force. Introducing the adimensinals variables and timey_i=ν_i/γ_i;x=nt=ℓ,the rotational system can be written asẏ_1=- T_11^*∑_k,j∈ℤ E_2,kE_2,k+j(y_1+P_1k)cos(jx)+sin(jx)/1+(y_1+P_1k)^2 ẏ_2=- T_22^*∑_k,j∈ℤ E_2,kE_2,k+j(y_2+P_2k)cos(jx)+sin(jx)/1+(y_2+P_2k)^2+ T_21^*∑_k,j∈ℤ E_2,kE_2,k+j(y_1+P_1k)cos(jx)+sin(jx)/1+(y_1+P_1k)^2 ⋮ ẏ_N=- T_NN^*∑_k,j∈ℤ E_2,kE_2,k+j(y_N+P_Nk)cos(jx)+sin(jx)/1+(y_N+P_Nk)^2+ T_NN-1^*∑_k,j∈ℤ E_2,kE_2,k+j(y_N-1+P_N-1k)cos(jx)+sin(jx)/1+(y_N-1+P_N-1k)^2.where the constantsT_ij^* =2𝒯/γ_inℋ_jR_j^5/R_i^5-R_i-1^5; P_ik = kn/γ_i. In low-γ approximation (γ_i≪ n), we can neglet the terms k≠ 0. If we consider only the terms j=0, the system becomesẏ_1=- T_11^* E_2,0^2y_1/1+y_1^2 ẏ_2=- T_22^* E_2,0^2y_2/1+y_2^2+T_21^* E_2,0^2y_1/1+y_1^2 ⋮ ẏ_N=- T_NN^* E_2,0^2y_N/1+y_N^2+T_NN-1^* E_2,0^2y_N-1/1+y_N-1^2. In the same way in Ferraz-Mello (2015), each solution of this system tends to zero. The role of the terms j≠ 0 are periodic fluctuations which are theharmonics of the orbital period are that added to the solution. If we consider the terms j≠ 0, we have that y_i≪ 1, and the rotational system isẏ_i=- ∑_j∈ℤ j≠ 0 K_ijsin(jx),where K_ij=(T_ii^*-T_ii-1^*)E_2,0E_2,j. The solution of this differential equation isy_i(x) = y_i0 -K_ij + ∑_j∈ℤ j≠ 0K_ij/jcos(jx),or, in term of the angular velocity, we obtainΩ_i = Ω_i0 -γ_iK_ij/2+ ∑_j∈ℤ j≠ 0γ_iK_ij/2jcos(jnt). Therefore, the square of the angular velocity of the jth layer can be written asΩ_j^2 = ∑_k=0^∞ A_jkcoskℓ. Finally, the integral (<ref>) is1/2π∫_0^2πΩ_j^2(a/r)^4sinv dℓ = ∑_k=0^∞A_jk/2(1/2π∫_0^2π(a/r)^4sin(v+kℓ) dℓ-1/2π∫_0^2π(a/r)^4sin(v-kℓ) dℓ)=0. In high-γ approximation (γ_i≫ n),we can neglet P_ik, then, the system can be written asẏ_1=- T_11^*∑_k,j∈ℤ E_2,kE_2,k+jy_1cos(jx)+sin(jx)/1+y_1^2 ẏ_2=- T_22^*∑_k,j∈ℤ E_2,kE_2,k+jy_2cos(jx)+sin(jx)/1+y_2^2 + T_21^*∑_k,j∈ℤ E_2,kE_2,k+jy_1cos(jx)+sin(jx)/1+y_1^2 ⋮ ẏ_N=- T_NN^*∑_k,j∈ℤ E_2,kE_2,k+jy_Ncos(jx)+sin(jx)/1+y_N^2 + T_NN-1^*∑_k,j∈ℤ E_2,kE_2,k+jy_N-1cos(jx)+sin(jx)/1+y_N-1^2. If we consider only the terms j=0, the system becomesẏ_1=- T_11^*∑_k∈ℤ E_2,k^2y_1/1+y_1^2 ẏ_2=- T_22^*∑_k∈ℤ E_2,k^2y_2/1+y_2^2+ T_21^*∑_k∈ℤ E_2,k^2y_1/1+y_1^2 ⋮ ẏ_N=- T_NN^*∑_k∈ℤ E_2,k^2y_N/1+y_N^2 + T_NN-1^*∑_k∈ℤ E_2,k^2y_N-1/1+y_N-1^2,which is identical to the system <ref>, with ∑_k∈ℤ E_2,k^2 instead of E_2,0^2. Therefore, each solution of this systemtends to zero. As in low-γ approximation, the role of the terms j≠ 0 are periodic fluctuations which are the harmonics of the orbital period areadded to the solution. If we consider the terms j≠ 0, we have that y_i≪ 1, and the rotational system isẏ_i=- ∑_j∈ℤ j≠ 0 K'_ijsin(jx),where K'_ij=(T_ii^*-T_ii-1^*)∑_k∈ℤ E_2,kE_2,k+j. Using the solution of the low-γ approximation, then, the angular velocity isΩ_i = Ω_i0 -γ_iK'_ij/2+ ∑_j∈ℤ j≠ 0γ_iK'_ij/2jcos(jnt),and the integral (<ref>) is zero.99 ferraz.2015 Ferraz-Mello, S.: “Tidal synchronization of close-in satellites and exoplanets. II. Spin dynamics and extension to Mercury and exoplanetshost stars.”Celest. Mech. Dyn. Astron. 122, 359-389 (2015).meriggiola.2016 Meriggiola, R., Iess, L., Stiles, B.W., Lunine, J., Mitri, G.: “The rotational dynamics of Titan from Cassini RADARimages.”Icarus 275, 183-192 (2016).sties.2008 Stiles, B.W., Kirk, R.L., Lorenz, R.D., Hensley, S., Lee, E., et al.: “Determining Titan's spin state from Cassini RADARimages.”Astron. J. 135, 1669-1680 (2008) and Erratum: Astron. J., 139, 311 (2010). | http://arxiv.org/abs/1706.08603v2 | {
"authors": [
"Hugo A. Folonier",
"Sylvio Ferraz-Mello"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170626212640",
"title": "Tidal synchronization of an anelastic multi-layered body: Titan's synchronous rotation"
} |
International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China Chinese Academy of Sciences (CAS) Center for Excellence and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China [email protected] Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China Chinese Academy of Sciences (CAS) Center for Excellence and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China Chinese Academy of Sciences (CAS) Center for Excellence and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, [email protected] International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China We propose a hierarchy set of minimal optical Raman lattice schemes toward the experimental realization of spin-orbit (SO) couplings of various types for ultracold atoms, including two-dimensional (2D) Dirac type of C_4 symmetry, 2D Rashba and 3D Weyl types. These schemes are well accessible with current cold-atom technology, and in particular, a long-lived Bose-Einstein condensation of the 2D Dirac SO coupling has been experimentally proved. The generation of 2D Rashba and 3D Weyl types has an exquisite request that two sources of laser beams have distinct frequencies of factor-two difference. Surprisingly, we find that ^133Cs atoms provide an ideal candidate for the realization. A common and essential feature is the absence of any fine-tuning and phase-locking in the realization, and the resulted SO coupled ultracold atoms have a long lifetime. These schemes essentially improve over the current experimental accessibility and controllability, and open new experimental platforms to study high-dimensional SO physics and novel topological phases with ultracold atoms. Dirac, Rashba and Weyl type spin-orbit couplings: toward experimental realization in ultracold atoms Xiong-Jun Liu December 30, 2023 ====================================================================================================Topological phase of matter has become a mainstream of research in condensed matter physics. Recent outstanding examples include topological insulators, which have been predicted and experimentally discovered in two-dimensional (2D) and 3D materials <cit.>, topological semimetals, which exhibit linear dispersion around nodes termed the Dirac or Weyl points, were found recently in a number of materials like Cd_3As_2 <cit.> and TaAs <cit.>, and topological superconductors <cit.>, which host exotic zero-energy states called Majorana modes and have attracted considerable experimental efforts <cit.>. In these matters, the spin-orbit (SO) interactions, manifesting Dirac, Rashba andWeyl types, play essential roles in driving the phases to be topologically nontrivial.Very recently, the experimental realizations of 2D SO couplings have been respectively reported in ^87Rb Bose-Einstein condensate (BEC) <cit.> and ^40K degenerate Fermi gases <cit.>, where nontrivial band topology and Dirac point are observed. The reports signify an important achievement in the field of quantum simulator with ultracold atoms and optical lattice, opening a great deal of possibilities to explore novel quantum states beyond natural conditions <cit.>. Nevertheless, the current experiments suffer challenges in controllability, since the realizations rely on the blue-detuned optical lattices <cit.> or delicate manipulation of resonant Raman couplings between multiple ground states <cit.>. To be broadly applicable to studying exotic topological phases mentioned above, the SO coupled systems should exhibit the following criteria, the high controllability and high stability with long lifetime, which however cannot be satisfied in the current experiments. In this Letter, we propose a hierarchy set of minimal optical Raman lattice schemes to realize high-dimensional SO couplings of various types in cold atoms, including 2D Dirac,Rashba and 3D Weyl types. The schemes make use of spin-independent or spin-dependent lattice and periodic Raman potentials, the latter of which are generated via a double -Λ internal level configuration. For the Dirac SO coupling, the current new scheme has essential advantages in two primary aspects. First, the new realization is of high controllability, well suited for both red- and blue-detuned lattices. Further, it exhibits a precise C_4 symmetry, leading to much broader topological phase region and high stability with a lifetime one-order over that in our previous realization <cit.>. These remarkable features are confirmed by the successful generation of a long-lived 2D Dirac SO coupling for ^87Rb BEC <cit.>. Realization of the Rashba and Weyl type SO couplings applies a spin-dependent lattice, and it further requests that the Raman-coupling beams have a frequency which is double of that for the lattice beam. This seemingly stringent requirement can be surprisingly well satisfied by ^133Cs atoms which provide an ideal candidate for the present scheme. A long lifetime over several seconds is also predicted for the realization with ^133Cs atoms, which shows high feasibility of the scheme for Rashba and Weyl SO couplings. Dirac type 2D SO coupling with C_4 symmetry. The realization is sketched in Fig. <ref>(a). The basic elements include a pair of laser beams which generate both the optical lattice and the Raman couplings, with two sets of “dotted boxes" (denoted as DB) containing λ/4 wave plates used to manipulate the symmetry of the realized Hamiltonian. The spin states |g_↑,↓⟩ refer to as two hyperfine levels, which are split by a bias magnetic field along the ẑ direction. The laser beam E_x running in the x direction has frequency ω_x and polarization in the y - z plane, and E_y in the y direction has frequency ω_y and polarization in the x - z plane, with their wave vector k_0 = ω_x / c≈ω_y/c. The two beams generate both the lattice and Raman potentials when the frequency difference δω = ω_y - ω_x compensates the Zeeman splitting between |g_↑⟩ and |g_↓⟩. We show below that this minimized simple setting generates the following Hamiltonian <cit.>H = [ p^2/2m + V_ latt (x,y) ] ⊗1+ m_z σ_z+ ℳ_x (x,y) σ_x + ℳ_y (x,y) σ_y ,where V_ latt denotes the square lattice potential, ℳ_x/y are the Raman coupling potentials, and m_z = δ/2 measures the two-photon detuning δ of Raman coupling.The “DB" box containing a λ/4 wave plate induces an additional π/2-phase shift for ẑ-component field. This gives the light fields as E_x =ŷ E_xycos k_0x + i ẑ E_xzsin k_0x, and E_y = x̂ E_yxcos k_0y + i ẑ E_yzsin k_0y, where E_μν (μ, ν = x,y,z) is amplitude of the field in the μ direction and with the ν polarization. All the irrelevant phase factors in E_x,y have been ignored. For alkali atoms, the scalar optical potential generated by linearly polarized lights is spin-independent for typical detuning Δ which is less than fine structure splitting but much larger than hyperfine structure splitting. It follows that V_ latt (x,y) = V_0xcos^2 k_0x + V_0ycos^2 k_0y, with the amplitudes V_0x∝ ( E_xy^2 - E_xz^2)/Δ and V_0y∝(E_yx^2 - E_yz^2)/Δ contributed from D_1 and D_2 lines in alkali atoms <cit.>.The Raman couplings are also induced by the two lights E_x and E_y, as illustrated in Fig. <ref>(b), through the double -Λ type configuration. One Raman potential ℳ_x (x,y) = M_10cos k_0y sin k_0x, with M_10∝ E_xz E_yx, is generated by the E_xy and E_yz components, and the other by E_xz and E_yx components reads ℳ_y (x,y) = M_20cos k_0x sin k_0y, with M_20∝ E_xy E_yz. The Raman ℳ_x (ℳ_y) and lattice potentials satisfy a relative antisymmetric configuration along x (y) direction, a crucial property to realize the minimal SO coupled quantum anomalous Hall model <cit.>, which exhibits novel bulk and edge physics <cit.>. Moreover, the present realization has an exact C_4 symmetry: (x,y;σ_x,σ_y) → (y,-x;σ_y,-σ_x), giving C_4HC_4^-1=H, which is broken by a term cos k_0xcos k_0y(σ_x+iσ_y) in the previous scheme <cit.>. The C_4 symmetry has a profound effect on the topological phase boundary, as shown in Fig. <ref>. In particular, Fig. <ref>(a1,a2) and (b1,b2) show the phase diagrams in the V_0-m_z and M_0-m_z planes (M_0=M_10=M_20) for the C_4-symmetric and C_4-symmetry-breaking systems, respectively. A marked distinction is that the topological region of the C_4 symmetric system can expand over the whole V_0 and M_0 axes, while the C_4-symmetry breaking term induces couplings to high orbital bands and reduces the topological region to a finite range versus V_0 and M_0 <cit.>.The key merits of the new scheme are below. First, due to the C_4 symmetry, the realization is valid for any type of detuning Δ, blue or red. Moreover, while the phases of two Raman potentials can be random due to the independent light beams along x and y directions, their relative phase is automatically fixed, since M_x,y are generated from the same two light beams. In contrast, the previous scheme necessitates a long spatial loop to correlate the light beams in x and y directions <cit.>. These features ensures much greater controllability and stability. Generalization to Rashba and Weyl SO couplings. The above scheme of Dirac type SO coupling can be generalized to Rashba and Weyl types by replacing the spin-independent lattice with spin-dependent one. As showed in Fig. <ref>(a), a laser beam E_V (k_0,ω_0) is incident along x-direction with y-polarization, and then reflected by three mirrors. The λ/4-wave plate before the mirror M_3 is applied to rotate the polarization vector of E_V field by 90 degrees. As a result, the total electric field can be written as E_V = ŷ E_0 e^ik_0x + ẑ E_0 e^-ik_0x + x̂ E_0 e^-ik_0y + ẑ E_0 e^ik_0y. We again neglected the irrelevant phases of the laser beams, which have no effect on the realization <cit.>. The atom-light coupling induces an effective magnetic field B_ eff∝E^* ×E, for which different hyperfine states experience different lattice potentials V_ eff=B_ eff·F <cit.>. Thus the lattice potentials of the internal states |g_↑⟩ and |g_↓⟩ read generically V_σ(x,y) = V_σ[ sin(2k_0x) +sin(2k_0y) ], with σ = (↑, ↓). In experiment, |g_↑⟩ and |g_↓⟩ are chosen so that sgn(V_↑) = -sgn(V_↓), implying that the lattices for |g_↑,↓⟩ are staggered [Fig. <ref>(b) <cit.>].We next show how to generate Raman coupling potentials. As illustrated in Fig. <ref>(a), the first Raman beam E_R1(2k_0,2ω_0) with double frequency 2ω_0 is incident from x-direction with y-polarization, and running along the same path ofthe lattice beam E_V. Being of twice the frequency of E_V, the polarization direction of E_R1 is not affected by λ/4-wave plate, and therefore it forms a standing wave in x - y plane: E_R1 = ŷ E_R1cos(2k_0x ) + x̂ E_R1cos(2k_0y). Further, the second Raman beam is incident along z-direction E_R2(2k_0,2ω_0+δω) = (i x̂ + ŷ) E_R2 e^i2k_0z. As sketched in Fig. <ref>.(c), E_R1 and E_R2 form the double -Λ type configuration to induce the Raman couplings between |g_↑⟩ and |g_↓⟩. Similar to the generation of Dirac SO coupling, the Raman potentials take the form M(x,y,z) = M_1 + M_2 = M_0 e^i2k_0z[ cos(2k_0x) + i cos(2k_0y) ], where M_0 is the amplitude <cit.>. The relative π/2-phase between M_1 and M_2 is a consequence of the phase difference between x̂ and ŷ components in E_R2, and is crucial to induce the SO coupling in x - y plane. Moreover, a momentum transfer of 2k_0 along z direction is induced by M(x,y,z), and leads to additional SO coupling along z direction. Together with a Zemman term m_z σ_z, the total Hamiltonian isH_ 3D =p^2/2m + m_z σ_z + V_0 ( sin 2k_0x +sin 2k_0y ) σ_z +M_0 e^i2k_0zσ_z( cos 2k_0x σ_x+ cos 2k_0y σ_y).We show below that H_ 3D gives rise to 3D Weyl SO coupling. Further, one can reduce the above Hamiltonian to the 2D regime by considering a confinement along z direction. In this case the phase factor e^i2k_0z is replaced with a constant, giving a 2D Rashba SO coupling.The Rashba and Weyl type SO couplings can be best understood with the tight-binding model. We note that the system is in a free space in the z direction, so the tight-binding model shall be derived only in the x - y plane. Performing a unitary transformation defined by U_z=e^-ik_0zσ_z yields H_ 3D→H̃_ 3D=U_zH_ 3DU^-1_z thatH̃_ 3D =p^2/2m+ (m_z +λ_zp_z)σ_z + V_0 ( sin 2k_0x +sin 2k_0y ) σ_z +M_0 ( cos 2k_0xσ_x + cos 2k_0yσ_y ),where λ_z=ħ k_0/m. We consider the physics of s orbital ϕ_s σ(σ=↑,↓), and take into account the nearest neighbor and diagonal hopping in the x-y plane. The tight-binding Hamiltonian can be straightforwardly written as H=p_z^2/2m+λ_zp_zσ_z-t_σ∑_i,j,σĉ^†_σ(r_i) ĉ_σ(r_i + S_j) + ∑_i m_z ( n_i ↑ - n_i ↓ ) + ∑_i,j t_ so^i jĉ^†_↑(r_i) ĉ_↓(r_i + N_j) + h.c., where t_σ denotes spin-conserved hopping, N_j and S_j are vectors connecting a lattice site to its four nearest and four second-nearest neighbor sites, respectively, and t_ so^i j represent spin-flip hopping [Fig. <ref>(b)]. With the basis ( ĉ_q, ↑, ĉ_q, ↓) the Bloch Hamiltonian readsℋ_ 3D(q) =p_z^2/2m-4 t_0cos(q_x a)cos(q_y a)+ (m_z+λ_zp_z)σ_z + 2t^(0)_ soσ_xsin (q_xa) + 2t^(0)_ soσ_ysin (q_ya),where a=π /(√(2)k_0). For convenience we consider that t_↑ = t_↓ = t_0 and a=1. Expanding the Hamiltonian around the Γ point, we reach the generic Weyl type SO coupling ℋ(q)≃ 4 t_0(q^2_x + q^2_y+α p_z^2) + 2t^(0)_ so ( q_x σ_x + q_y σ_y+β p_z σ_z)+m_zσ_z. Here α=1/(8mt_0) and β=λ_z/(2t^(0)_ so). Accordingly, by restricting the system to a 2D plane, Rashba SO coupling can be readily obtained withℋ (q)≃ 2 t_0 (q_x^2 + q_y^2) + t^(0)_ so ( q_x σ_x + q_y σ_y)+m_zσ_z. For m_z=0 the Rashba Hamiltonian respects time-reversal symmetry.Realization for ^133Cs Atoms. In realizing Rashba and Weyl SO couplings the spin-dependent lattice and Raman couplings are generated by two sources of laser beams with one's frequencies being twice of the other. Note that both spin-dependent lattice and Raman couplings have to be induced by optical transitions with detunings less than the fine structure splitting of excited levels. This is clearly not achievable with a single D_1/D_2 transitions. Remarkably, we find surprisingly that ^133Cs atoms provide an ideal candidate for the realization.Our key idea is to consider the transitions from ground states to both 6^2P_J and 7^2P_J levels for generating lattice and Raman potentials [Fig. <ref>(a)]. The ground manifold of ^133Cs atoms is 6^2S_1/2, of which we take |F=4,m_F=-4 ⟩ and |3,-3 ⟩ as spin up |g_↑⟩ and spin down |g_↓⟩, respectively. The Landé g-factor g_F = 1/4 (-1/4) for |4,-4 ⟩ (|3,-3 ⟩), leading to the opposite lattice potentials by B_ eff for |g_↑⟩ and |g_↓⟩. In particular, the spin-dependent lattice potentials are contributed from transitions to 6^2P_1/2,3/2 states, giving V(x,y) = (V_0 σ_z + δ V_0 1) ( sin 2k_0x + sin 2k_0y), where V_0 =7/12 E_V1^2 α^2_D_1 /Δ_1, with α_D_1 being the dipole moment of D_1 line, and δ V_0 ∼ 0.1 V_0 denotes a small distinction of strength of lattice potentials for spin-up and spin-down, with 1/Δ_1 ≡ 1/Δ_1/2 - 1/Δ_3/2 <cit.>. The δ V_0-term corrects m_z in 2D Rashba SO coupling, and shifts the Weyl points in 3D case. The Raman potentials are generated from transitions via 7^2 P_3/2,1/2 states and read M = M_0 e^2ik_0z( cos 2k_0x + i cos 2k_0y ), where M_0 =√(7)α^2_D_1/12 √(2) E_R1 E_R2/Δ_2, with Δ_2 defined similar as Δ_1 <cit.>. We choose the wavelength for Raman (λ_R) and lattice (λ_V) beams λ_V = 2 λ_R = 916.6 nm. One can verify that the typical parameter condition (V_0, M_0)= (4E_r, 1E_r), with the recoil energy E_r = ħ k_R^2/2m, can be achieved for ^133Cs atoms by taking the lattice beams with a waist ϖ=200μ m and power P=15.4mW, and the Raman beams with the same waist and the power P=21mW. This condition can be easily satisfied in experiment.Fig. <ref>(b) shows the numerical results of the band structure around Γ point with the parameters (4E_r,1E_r) and m_z=0.68E_r. Interestingly, the band crossing at Γ point is obtained, due to the cancellation of m_z by nonzero δ V_0 term at Γ point, while the crossings at other symmetric points like (k_x,k_y)=(±π,±π) are avoided, implying that the time-reversal symmetry is broken. As detailed in section II.E of supplementary material <cit.>, the δ V_0 term can bring about novel spin texture in momentum space beyond a pure Rashba SO coupling and can have interesting applications. Furthermore, for the 3D Weyl type SO coupling the spin texture around Weyl point is characterized by a skyrmion and protected by Chern numbers, as shown in Fig. <ref>(c)Discussion on lifetime and conclusion. The high controllability and stability are particularly reflected by long lifetime in the present minimized schemes. For the Dirac type SO coupling, we first consider ^87Rb BEC. The heating rate of the dipole trap is about 10nK/s <cit.>. The lattice and Raman beams are applied with wavelength λ = 786nm (between D_1 and D_2 lines ) and the heating rate caused by photon scattering is about 63.7nK/s for parameters (4E_r,1E_r) <cit.>. Thus the lifetime of ^87Rb BEC (the time taken for heating to critical temperature) is about τ∼100nK/73.7nK/s≈ 1.36s. We note that this lifetime has been confirmed in our latest experiment <cit.>. For ^40K degenerate fermion gas, we take the lattice and Raman lights with wavelength λ = 768nm, and can estimate the heating rate to be about 665nK/s for parameters (4E_r,1E_r), giving lifetime τ∼ 150ms. The lifetimes are much longer than those in the recent experiments <cit.>.The lifetime for Rashba and Weyl SO couplings is even much longer for ^133Cs atoms. The condensate lifetime in a typical trap is about 15 s <cit.>. Under the parameters (4E_r,1E_r), the heating rate of the lattice beam is only 3nK/s due to the large detuning Δ_1, and of Raman beams is also small ≃10nK/s because of the long lifetime of 7^2P_J manifolds. Thus the lifetime can be τ>5s <cit.>.In conclusion we have proposed a hierarchy set of minimized experimental schemes to realize Dirac, Rashba, and Weyl SO couplings. These schemes are well accessible with current cold-atom technology, and the resulted SO coupled ultracold atoms can have a long lifetime. For the Dirac type SO coupling, a long-lived Bose-Einstein condensation has been experimentally proved <cit.>. For Rashba and 3D Weyl SO couplings, we uncover an ideal candidate provided by ^133Cs atoms for the realization. This work shall essentially push forward cold-atom experimental progresses in realizing high dimensional SO couplings and topological phases in the near future.The work is supported by the Ministry of Science and Technology of China (under grants 2016YFA0301604 and 2016YFA0301601), National Natural Science Foundation of China (grant 11574008), Thousand-Young-Talent Program of China, the CAS, the National Fundamental Research Program (under grant 2013CB922001), and Fundamental Research Funds for the Central Universities (under grants 2030020028 and 2340000034).§ SUPPLEMENTAL MATERIALIn this supplementary material we provide the details of deriving the effective Hamiltonians in the main text based on real cold atom candidates, discuss the effects of the different types of spin-orbit (SO) couplings, and calculate the heating and lifetime for the schemes.§ I. SCHEME FOR DIRAC TYPE 2D SO COUPLING In this section we discuss the details of deriving the effective Hamiltonian. The experimental setup for ^40K atoms is shown in Fig. 1 in the main text. Two laser beams form the standing wavesE_x = ŷ E_xy e^i(α + α_L/2)cos(k_0x - α_L/2)+ẑ E_xz e^i(α' + α'_L/2)cos(k_0x - α'_L/2)E_y = x̂ E_yxe^i(β + β_L/2)cos(k_0y - β_L/2) + ẑ E_yze^i(β' + β'_L/2)cos(k_0y - β'_L/2).Here α, α', β, β' are the initial phases forE_xy,E_xz,E_yx,E_yz, and α_L is the phase acquired by E_xy through the optical path from intersecting point to mirror M_1, then back to the intersecting point. The phases α'_L,β_L,β'_L have the similar meanings for E_xz,E_yx,E_yz. E_xy, E_xz, E_yx, and E_yz are real amplitudes. We shall find that the initial phases are irrelevant for the realization. §.§ A. Raman Fields For convenience, we take ^40K atoms for our consideration, while all the results are straightforwardly applicable to ^87Rb atoms. For ^40K atoms we define the spin-1/2 by the two ground states |g_↑⟩ = |9/2,+9/2⟩ and |g_↓⟩ = |9/2,+7/2⟩.Raman coupling scheme for ^40K atom is shown in Fig. <ref>, where two independent Raman transitions are driven by the light components E_xz,E_yx and E_xy,E_yz. Taking into account the contributions from both D_1(2^2P_1/2) and D_2(2^2P_3/2) lines, we haveM_1= ∑_F Ω^(3/2)*_↑ F,xz·Ω^(3/2)_↓ F,yx/Δ_3/2 + ∑_F Ω^(1/2)*_↑ F,xz·Ω^(1/2)_↓ F,yx/Δ_1/2= √(2)/9(α^2_D_1/Δ_1/2- α^2_D_2/2 Δ_3/2)E^*_xzE_yx^(+), M_2= ∑_F Ω^(3/2)*_↑ F,xy·Ω^(3/2)_↓ F,yz/Δ_3/2 + ∑_F Ω^(1/2)*_↑ F,xy·Ω^(1/2)_↓ F,yz/Δ_1/2= √(2)/9(α^2_D_1/Δ_1/2- α^2_D_2/2 Δ_3/2) E^(-)*_xyE_yz,where Ω^J_↑ F,γ z = ⟨↑| ez|F,0,J ⟩ê_z ·E_γ z and Ω^J_↑ F,γ x = ⟨↑| ex|F,+1,J ⟩ê_+ ·E_γ x + ⟨↑| ex|F,-1,J ⟩ê_- ·E_γ x, and E_yx^(+)=ê_+ ·E_yx = E_yx/√(2) represents the right-handed light of E_yx, which couples the states |9/2,+7/2⟩ and |F,+9/2⟩ as shown in Fig. <ref>. Similarly, E^(-)_xy=-i E_xy/√(2) represents the left-handed light of E_xy, which couples the states |9/2,+9/2⟩ and |F,+7/2⟩. Since α_D_2≈√(2)α_D_1≈ 5.799 ea_0 <cit.>, with a_0 being the Bohr radius, then we haveM_1= M_10cos(k_0x - α'_L/2)cos(k_0y - β_L/2) e^-i(α' + α'_L/2) e^i(β + β_L/2),M_2= i M_20cos(k_0x - α_L/2) cos(k_0y - β'_L/2) e^-i(α + α_L/2)e^i(β' + β'_L/2), where M_10/20 = α^2_D_1/9(1/Δ_1/2 - 1/Δ_3/2) E_xz/xy E_yx/yz. To realize a Dirac type 2D SO coupling with nontrivial topology, we require the following two conditions to be satisfied. (1) The phase difference δθ between M_1 and M_2 must be non-zero to have a 2D SO coupling. Then the Raman potential can be written as (|M_1| + |M_2| cosδθ) σ_x + |M_2| sinδθσ_y. Here we shall consider the optimal regime with δθ = ±π/2, which gives the Raman coupling as |M_1| σ_x ± |M_2|σ_y accordingly. Hence the phases should meet the following conditionδθ = (α - α') + (β - β') + 1/2[ (α_L - α'_L) + (β_L - β'_L)] + π/2 = π/2 + n π ,where n is an integer number. Note that α and α' (similar for β and β') are phases of two components of the same laser beam. The relative value between them is automatically fixed and can be easily manipulated by wave plates. Also, α_L and α'_L (β_L and β'_L) are the phases acquired through the same optical path, so their relative value is also automatically fixed and can be controlled by wave plates. The system is stable against any phase fluctuations. (2) To let the SO coupled system to be topological non-trivial, we consider a λ/4 wave plate before each mirror (M_1 and M_2). Then the phases will meet the following conditions1/2 (α_L - α'_L) = π/2 + p π and 1/2 (β_L - β'_L) = π/2 + q π,with p and q are integer numbers. Accordingly, the Raman potentials turn intoM_1 = M_10sin(k_0x - α_L/2)cos(k_0y - β_L/2),and M_2 = ± i M_20cos(k_0x - α_L/2) sin(k_0y - β_L/2).We shall see below that the Raman coupling potentials M_1 and M_2 are antisymmetric with respect to the lattice in x and y directions, respectively, a key point to obtain the nontrivial topology.In the real experiment the conditions (<ref>) and (<ref>) can be satisfied by taking (α - α') + (β - β') + π/2 = π/2 + n π, (α_L - α'_L)/2 = π/2 + p π, and (β_L - β'_L)/2 = π/2 + q π. Since E_xy and E_xz come from the same laser beam, one can naturally take that α = α'. Similarly, we have β = β', and then the phase difference δθ=π/2. On the other hand, we can control that each λ/4 wave plate induces an additional π/2-phase shift to the ẑ-component light when the light pass through the wave plate one time. This leads to (α_L - α'_L)/2=(β_L - β'_L)/2=π/2, satisfying the condition.It is trivial to know that the Raman coupling potentials combine to induce a 1D SO coupling when δθ=0,π. If controlling the phase difference continuously, the crossover between 1D and 2D SO couplings can be induced. §.§ B. Spin-independent lattice potentials As shown in Fig. <ref>, the optical lattice is contributed from both D_2 (2^2 S_1/2→ 2^2 P_3/2) and D_1 (2^2 S_1/2→ 2^2 P_1/2) lines, and it is a spin-independent latticeV_↑ = V_↓ = ∑_F 1/Δ_3/2( |Ω^(3/2)_↑ F,xz|^2 +|Ω^(3/2)_↑ F,xy|^2 +|Ω^(3/2)_↑ F,yz|^2 +|Ω^(3/2)_↑ F,yx|^2) + ∑_F 1/Δ_1/2( |Ω^(1/2)_↑ F,xz|^2 +|Ω^(1/2)_↑ F,xy|^2 +|Ω^(1/2)_↑ F,yz|^2 +|Ω^(1/2)_↑ F,yx|^2)= V_0xcos^2(k_0x - α_L/2) + V_0ycos^2(k_0y - β_L/2) ).Here the constant is neglected in the last line, and we have V_0x = α_D_1^2/3 (2/Δ_3/2 + 1/Δ_1/2)( E_xy^2 - E_xz^2 ) and V_0y= α_D_1^2/3 (2/Δ_3/2 + 1/Δ_1/2) (E_yx^2 - E_yz^2). Note that when E_xy > E_xz and E_yx > E_yz, the lattice potential takes the form cos^2(k_0x) + cos^2(k_0y). In comparison, when E_xy<E_xz and E_yx<E_yz, the lattice potential becomes sin^2(k_0x) + sin^2(k_0y), which corresponds to a translation along the diagonal direction compared to cos^2(k_0x) + cos^2(k_0y). Effectively, the sign of the optical potentials is reversed for the two cases, equivalent to changing the sign of the detuning (e.g. from blue- to red-detuned or vise versa). §.§ C. Effective Hamiltonian As clarified previously, the phase fluctuations for lattice and Raman fields are the same: ϕ^ fluc_x = - α_L/2, and ϕ^ fluc_y = - β_L/2. Hence the relative spatial configuration of M_x/y and V are always automatically fixed, and the fluctuationsonly lead to a global shift of the lattice and Raman fields. Therefore, we can set α_L =β_L = 0 safely. The global phase of Raman potentials are irrelevant and can also be removed. Then the Hamiltonian can be written asH=p^2/2m + V_0xcos^2(k_0x) + V_0ycos^2(k_0y) + ℳ_x σ_x + ℳ_y σ_y + m_z σ_z,the strength of Raman coupling ℳ_x = |M_1| + |M_2| cosδθ and ℳ_y = |M_2| sinδθ, which reduce to an 1D SO coupling for δθ = n π and an optimal 2D Dirac type SO coupling for δθ = π/2 + n π. This enables a fully controllable study of the crossover between 2D and 1D SO couplings by tuning δθ.§ II. SCHEME FOR RASHBA AND WEYL TYPE SO COUPLINGS§.§ A. Spin-dependent lattice Potential We generate the spin-dependent square lattice potential in x - y plane from the traveling-wave beams, described by the electric field E_VE_Vx = ŷ E_0 e^ik_0x+iϕ_0 + ẑ E_0 e^-ik_0x + iϕ_0 + 2iϕ_L + iϕ'_L E_Vy = x̂ E_0 e^-ik_0y + iϕ_0 + iϕ_L + ẑ E_0 e^ik_0y + iϕ_0 + iϕ_L + iϕ'_L,where E_V,x/y represents the laser propagating along x/y-direction, ϕ_0 is the initial phase of the laser beam, and ϕ_L/ϕ'_L is the phase acquired on path L/L'. The path L denotes the loop from lattice center to mirror M_1, then to M_2, and finally back to the lattice center, while the path L' denotes the one from lattice center to the mirror M_3 and back to lattice center [Fig. <ref>(a)].In general, the optical potential generated for atoms in the ground state is related to the electromagnetic field byV(r) = u_s |E|^2 + i u_v ( E^* ×E) ·S,where the first term is the scalar potential with u_s=-1/12 Δ_s | ⟨ l=0 | d | l=1 ⟩ |^2 and u_s |E|^2, and the second term denotes a vector light shift, with u_v = A_FS/ħΔ_eu_s = 2 Δ_FS/3 Δ_e u_s and Δ_FS being the fine-structure splitting. The scalar potential is spin-independent for linearly polarized lights. In comparison, the vector light shift is spin-dependent. The effective Hamiltonian H_ eff = u_s |E|^2 + μ_B g_J/ħ( B + B_ eff) ·J, where we have replaced S with J = S + L sinceL=0 for the ground state. The symbol B denotes the external magnet field, andB_ eff = i ħ u_v ( E^* ×E) /μ_B g_J is the effective magnetic field. Considering the hyperfine structure, we further replace g_J J with g_F F and then haveH_ eff = u_s |E|^2 + μ_B g_F/ħ( B + B_ eff) ·F.In our proposal, the effective magenetic field produced by the lasers has the equal components in x and y directions. Therefore, we must apply external magnetic field along the diagonal direction (B_x^ eff,B_y^ eff≠0) of x-y plane. This ensures that B_ eff has nonzero components along the direction of B.With this setup, the effective magnetic field should be B_x^ effê_x + B_y^ effê_y. The external magnetic field B along the diagonal direction satisfies B ≫ B_x^ eff, B_y^ eff. Thus, the total field reads B_t =√((B + B_x^ eff/√(2) + B_y^ eff/√(2) )^2 + (B_x^ eff/√(2) - B_y^ eff/√(2))^2). We expand the total field B_t around the point B_x^ eff = B_y^ eff=0, keeping the first-order term, and finally have B_t ≈ B +B_x^ eff/√(2) +B_y^ eff/√(2). In other words, if we decompose the effective magnetic field in the traverse and longitudinal direction of the external magnetic field, only the longitudinal component matters.Next we focus on the expression of the effective magnetic fieldB_ eff ∝E^* ×E = ( E^*_Vx×E_Vx + E^*_Vy×E_Vy ) +( E^*_Vx×E_Vy + E^*_Vy×E_Vx )=( B_x^ eff +B_y^ eff )+ B_xy^ eff.The cross term B_xy^ eff can be obtainedB_xy^ eff∝ŷ E_0^2 sin(k_0x-k_0y-ϕ_L-ϕ'_L) -x̂ E_0^2 sin(k_0x-k_0y-ϕ_L-ϕ'_L) + ẑE_0^2 sin(k_0x+k_0y-ϕ_L).However, this term is perpendicular to external field B, so the projection of this term in the direction of B is zero, hence it can be neglected. It is ready to verify that the scalar potential is a constant, and can be neglected. Finally the lattice potential takes the form qualitativelyV_ latt = ( B_x^ eff +B_y^ eff ) ·F∝E^2_0 sin(2k_0x - 2ϕ_L - ϕ'_L) + E^2_0 sin(2k_0y + ϕ'_L).Note that we can use another method to obtain the result more accurately. We first consider the laser propagating along x-direction, write the electric fields in the basis of the circular polarized lightE_Vx = E_0/√(2) e^i ϕ_0 + iϕ_L + iϕ'_L/2 +iπ/4 2[ i ê_+sin(k_0x - ϕ_L - ϕ'_L/2 - π/4) + ê_-cos(k_0x - ϕ_L - ϕ'_L/2 - π/4) ]= ê_+ E_+ +ê_- E_-,and|E_+|^2= 2 E^2_0 sin^2(k_0x - ϕ_L -ϕ'_L/2 - π/4) = E^2_0 - E^2_0 sin(2k_0x - 2ϕ_L -ϕ'_L) |E_-|^2= 2 E^2_0 cos^2(k_0x - ϕ_L -ϕ'_L/2 - π/4) = E^2_0 + E^2_0 sin(2k_0x - 2ϕ_L -ϕ'_L) . As shown in Fig. <ref>, the spin-dependent lattice potential is contributed from both D_2 (6^2 S_1/2→ 6^2 P_3/2) and D_1 (6^2 S_1/2→ 6^2 P_1/2) lines <cit.>. The light E_V drive the σ transitions from ground states |F=4,m_F=-4 ⟩ and |F=3,m_F=-3 ⟩ to all possible excited levels which satisfy the selection rule. The state |4,-4 ⟩ (|g_↑⟩) is coupled to excited states |F,-5 ⟩ and |F,-3 ⟩, while the state |3,-3 ⟩ (|g_↓⟩) is coupled to other states |F,-4 ⟩ and |F,-2 ⟩. The detunings Δ_1/2 and Δ_3/2 are much larger than the hyperfine-structure splitting, and have the same order of magnititude of the fine-structure splitting, which is theenergy difference between D_1 and D_2 lines. Then the lattice potential can be obtained by summing over the contributions of all the allowed transitionsV_↑(x)= ∑_F 1/Δ_3/2( |Ω^(3/2)_↑ F,+1|^2 + |Ω^(3/2)_↑ F,-1|^2) + ∑_F 1/Δ_1/2( |Ω^(1/2)_↑ F,+1|^2 + |Ω^(1/2)_↑ F,-1|^2) =2/3 E_ V^2α^2_D_1( 1/Δ_3/2 - 1/Δ_1/2) sin(2k_0x - 2ϕ_L -ϕ'_L).We have neglected all the constants in the final result. Taking V_0 = E_0^2α^2_D_1/Δ_1, with 1/Δ_1= ( 1/Δ_3/2 - 1/Δ_1/2), we have V_↓(x) = - 1/2 V_0sin(2k_0x - 2ϕ_L -ϕ'_L) for the atoms staying in the state |3,-3 ⟩. Similarly,the spin-dependent lattice potential in y-direction is V_↑(y) = 2/3 V_0sin(2k_0y + ϕ'_L) and V_↓(y) = - 1/2 V_0 sin(2k_0y + ϕ'_L). The total lattice can be finally written asV_ latt =( 1/121 + 7/12σ_z) V_0[sin(2k_0x - 2ϕ_L -ϕ'_L) +sin(2k_0y + ϕ'_L)].§.§ B. Raman FieldsNow we study the generation of Raman fields by adding another two frequency-doubled lights E_R1(2k_0,2ω_0) and E_R2(2k_0,2ω_0+δω), where δω is of the same order of the energy difference between |g_↑⟩ and |g_↓⟩. The polarization of E_R1 can't be affected by λ/4 wave plate, hence E_R1 form a standing wave on x-y plane, andE_R2 is just a travelling wave, i.e.E_R1 = E_R1 e^iϕ_1 + 2iϕ_L + iϕ'_L[ ŷcos(2k_0x - 2ϕ_L - ϕ'_L) + x̂cos(2k_0y + ϕ'_L) ],E_R2 = ( i x̂ + ŷ ) E_R2 e^2ik_0z + iϕ_2.The Raman fields are contributed from 6^2 S_1/2→ 7^2 P_3/2 and 6^2 S_1/2→ 7^2 P_1/2 transitions. It's a little tedious to calculate Raman potentials because the magnetic field points to diagonal direction. We first need to decompose the vectors E_R1, E_R2 along the external magnetic field directionE_R1 = 1/√(2) E_R1 e^i θ[ cos(2k_0x - 2ϕ_L - ϕ'_L) + cos(2k_0y + ϕ'_L)] B̂+ 1/√(2) E_R1 e^i θ[ cos(2k_0x - 2ϕ_L - ϕ'_L) - cos(2k_0y + ϕ'_L)]B̂_⊥,E_R2 = E_R2/√(2)( 1 + i ) e^2ik_0z + iϕ_0B̂ + E_R2/√(2)( 1-i ) e^2ik_0z + iϕ_0B̂_⊥,where θ = ϕ_0 + 2 ϕ_L + ϕ'_L. Then the potentials can be obtained as followsM_1= ∑_FΩ^(3/2)*_↑ F,1·Ω^(3/2)_↓ F,2⊥/Δ̃_3/2 + ∑_F Ω^(1/2)*_↑ F,1·Ω^(1/2)_↓ F,2⊥/Δ̃_1/2= √(7)α^2_D̃_1/12 √(2)( 1/Δ̃_1/2 - 1/Δ̃_3/2) E_R1 E_R2 (1-i) e^2ik_0z [ cos(2k_0x - 2ϕ_L - ϕ'_L) + cos(2k_0y + ϕ'_L)],andM_2= √(7)α^2_D_1/12 √(2)( 1/Δ̃_1/2 - 1/Δ̃_3/2) E_R1 E_R2 (1+i) e^2ik_0z[ cos(2k_0x - 2ϕ_L - ϕ'_L) - cos(2k_0y + ϕ'_L) ].Note that in general the detunings Δ̃_1/2,3/2≠Δ_1/2,3/2. As discussed in the main text, lattice potential V and Raman fields M have the same phase fluctuations, so the fluctuations can be neglected and for simplicity we take ϕ^fluc = 0. We then findM = M_1 + M_2= M_0 e^2ik_0z( cos 2k_0x + i cos 2k_0y ),where M_0 =√(7)α^2_D_1/6 √(2) E_R1 E_R2/Δ_2, with 1/Δ_2=( 1/Δ_1/2 - 1/Δ_3/2). §.§ C. Effective HamiltonianWith the lattice and Raman potentials above, the total Hamiltonian can be written as H_ 3D =p^2/2m+ m_z σ_z + V_0 ( sin 2k_0x +sin 2k_0y ) σ_z + M_0 [ e^i2k_0z ( cos 2k_0x + i cos 2k_0y )|g_↑⟩⟨ g_↓| +h.c. ].To remove the phase term e^i2k_0z in the Raman potentials in order to make the Hamiltonian periodic, we define a rotation operator U=e^-ik_0z σ_z and perform a transformationH_ 3D→H̃_ 3D = U H_ 3D U^†.Then the Hamiltonian would change intoH̃_ 3D =p^2/2m+ (m_z + λ_z p_z) σ_z + V_0 ( sin 2k_0x +sin 2k_0y ) σ_z + M_0 ( cos 2k_0x σ_x + cos 2k_0y σ_y ) ,where λ_z = ħ k_0/m and we have ignored the constant above. §.§ D. The Tight-Binding ModelWe next derive the tight-binding model from the effective Hamiltonian, while we emphasize that the realization of SO couplings is not restricted by tight-binding model. For simplicity, we first discuss the tight-binding model in 2D system, and then extend it to 3D. We only consider the nearest and next-nearest neighbor hopping, and assume that atoms stay in the lowest s orbits. As shown in Fig. <ref>, the tight-binding model of this system in real space can be written asH=-t_σ∑_i,j,σĉ^†_σ(r_i) ĉ_σ(r_i + S_j) + ∑_i m_z ( n̂_i ↑ - n̂_i ↓ ) + ∑_i,j t_ so^ijĉ^†_↑(r_i) ĉ_↓(r_i + N_j) + h.c.where N_i and S_i represents the distances between the nearest neighbor sites and second-nearest neighbor sites, respectively. The particle number operators are defined as n̂_i σ = ĉ^†_i,σĉ_i,σ, and t_σ (σ=↑,↓) denotes the spin-conserved hopping, given byt_σ = ∫^2 rϕ_sσ^(i) (r) [ p^2/2m + V(r) ] ϕ_sσ^(j)(r). t_ so^ij is the spin-flipped hopping coefficient given by t_ so^i j =∫^2 rϕ_s ↑^(i) (r) M_1/2(r) ϕ_s ↓^(j)(r), representing the induced 2D SO interaction. It can be readily verified that the spin-flip hopping terms due to Raman fields satisfy t_ so^j_x,j_x ± 1 = ± t_ so^(0) and t_ so^j_y,j_y ± 1 = ± i t_ so^(0). Then we transform the above equation into momentum space and write theHamiltonian in matrix formH= ∑_q( [ ĉ^†_q, ↑ ĉ^†_q, ↓ ]) ℋ(q) ( [ ĉ_q, ↑; ĉ_q, ↓ ]).By setting t_↑ = t_↓ = t_0, the matrix ℋ in momentum space can be written asℋ(q)= m_z σ_z-4 t_0 cos(q_x a)cos(q_y a) ⊗1 + 2t_ so^(0)sin (q_xa) σ_x + 2t_ so^(0)sin (q_ya)σ_y ,where a=|N_i| is the lattice constant representing the distance between the nearest neighbor sites. Expanding the Hamiltonian around the Γ point yields ℋ(q) = 4 t_0 q^2 + 2 t_ so^(0) (q_x σ_x + q_y σ_y ) + m_zσ_z,which renders the expected 2D Rashba type SO coupling. To generate 3D weyl type SO coupling, we simply remove the confining potential along z direction and the tight binding model can be written asℋ_ 3D(q)= p^2_z/2m - 4 t_0 cos(q_x a)cos(q_y a)+ (m_z + λ_z p_z) σ_z+ 2t_ so^(0)sin (q_xa) σ_x + 2t_ so^(0)sin (q_ya)σ_y. §.§ E. Realization for ^133Cs Atoms As discussed above, we choose the state |4,-4 ⟩ from the ground state as spin up |g_↑⟩, and |3,-3 ⟩ as spin down |g_↓⟩. The lattice potential includes a small correction to the purely spin-dependent term and can be written asV_ latt =(V_0 σ_z + δ V_0 1) (sin 2k_0x +sin 2k_0y),where V_0 = 7/12 E_ V^2α^2_D_1( 1/Δ_3/2 - 1/Δ_1/2) and the correction δ V_0 ∼ 0.1 V_0. In the following we show that δ V_0 can induce novel effects. To simplify the proceeding analysis, we denote the tight binding model and the hopping terms as-4 (t_01 + δ t_0 ⊗σ_z) cos(q_x a)cos(q_y a),where t_0 = (t_↑ + t_↓)/2 and δ t_0 = (t_↑ - t_↓)/2. Obviously, there is δ t_0∼ 0.1 t_0 in the tight binding model. Then we redefine the effectZeeman term m^ eff_z asm^ eff_z = m_z - 4δ t_0 cos(q_x a)cos(q_y a).A few novel effects are followed. First, at the Γ point m^ eff_z=m_z - 4δ t_0, which vanishes when m_z = 4δ t_0, and the band crossing is obtained [Fig. <ref>(a)]. Nevertheless, the Zeeman splitting at other symmetric points k=(0,π),(π,0),(π,π) keeps, showing that the time-reversal symmetry is indeed broken with nonzero m_z and δ t_0. Secondly, in the case of a nonzero Zeeman splitting at Γ point, one can tune m_z properly so that the spins of states at a finite energy E exactly point to x-y plane, namly, at such energy m^ eff_z (q)=0 so that the polarizations for states at k and -k are opposite [Fig. <ref>(b)]. In such region the spin texture resembles the case without Zeeman splitting. This effect can have novel consequences. For bosons, one can tune that the purely in-plane spin texture results in the band bottom, leading to a strong spin uncertainty in the ground states. For fermions, one can tune that the purely in-plane spin texture is obtained around Fermi surface. In this case under an attractive interaction the superfluid pairing Δ = U_0 ⟨ĉ_k,↑ĉ_-k,↓⟩ can be largely enhanced even the Zeeman splitting is present, being different from a purely Rashba system where the Zeeman splitting always polarizes the Fermi surface and reduces the superfluid pairing. The topological gap of such a superfluid can then be greatly enhanced. Finally, when -4 δ_0 < m_z < 4 δ_0, each single energy band is topological nontrivial with the Chern number Ch_1 = ± 1. According to the former discussions, the Raman potential takes the formM = 2M_0 e^2ik_0z( cos 2k_0x + i cos 2k_0y ).Note that the Raman light E_R1 also forms a standing wave in x-y plane and generate a spin-independent lattice V_ in∝cos^2 2k_0x +cos^2 2k_0y.This potential has the same contribution to both spin-up and spin-down atoms. Hence, it does not change any result that we obtain above. §.§ F. Spin Texture Around The Weyl Point To simplify the proceeding analysis, we assume the Hamiltonian around the Weyl point is isotropic, which would not affect the topological property of the system. The Hamiltonian can be simplified asℋ(q) = q^2/2m_ eff + v_F q·σ.The eigenvalues are E_± = q^2/2m_ eff± v_F |q|, and the degeneracy is the Weyl point at q=0. The normalized eigenstates can be written asu_±(q) = 1/√(2q(q ± q_z))(q_x - i q_y, - q_z ∓ q ). We next consider the spin texture on the fixed-energy surfaces around the Weyl point. There are two closed spherical surfaces F_+ and F_- with fixed energy in the 3D Brillouin zone. And the spin polarization on these surfaces can be defined as S_± (q) = ⟨u_±(q) | σ̂ |u_±(q) ⟩. By substituting the equation (S20) into the defination, we obtain a simple resultS_± (q) = ±q/|q|.The spin textures on thetwo spherical surfaces F_+ and F_- are drawn in the Fig. <ref>. And we can define the first Chern numberon the 2D spherical surface to characterize the spin textureCh_1 = 1/4π∫_F_±S·( ∂_1 S×∂_2 S)^2 k.By employing spherical coordinates and parametrize the vector S_± (q) = ±( sinθcosϕ, sinθsinϕ, cosθ), the Berry curvature Ω_± can be easily caculatedΩ_± = S_±·( ∂_θ/qS_±×∂_ϕ/q sinθS_±)= ±1/q^2.We immediately find in the following for the Berry field strengthV = ±q/q^3.This field looks like a magnetic field which is generated by a monopole at the Weyl point q=0 with the strength ± 1. Hence the Weyl node is a monopole or antimonopole for Berry curvature. Two fixed-energy surfaces F_+ and F_- always contain the Weyl node with the strength +1 and -1. Hence the Chern number defined on the sphere F_+ and F_- would be Ch_1 = 1 and Ch_1 = -1.§ III. ESTIMATION OF THE LIFETIME Heating by the lattice and Raman lights is mainly induced by the spontaneous scattering of photons, which is random and causes fluctuations of the radiation force. For alkali atoms, a general expression of the photon scattering rate Γ_ sc for the D line doublet ^2S_1/2→ ^2P_1/2, ^2P_3/2 <cit.> can be written asΓ_ sc (r) = π c^2 Γ^2/2 ħω_0^3( 2+𝒫 g_F m_F/Δ^2_D_2 + 1 - 𝒫 g_F m_F/Δ^2_D_1) I(r),where g_F is the Landé factor, 𝒫 denotes the laser polarization (𝒫=0, ± 1 for linearly and circularly σ^± polarized light), ω_0 is the averaged resonant frequency of the D line doublet, and I(r) is the laser beam intensity. In experiment, we assume that the atoms stay at the center of a laser beam E, and then the laser beam intensity can be written as I(r) = 1/2 ϵ_0 c |E|^2. With the scattering rate, we can estimate the heating rate Ṫ byk_ BṪ = 2/3 E_r Γ_ sc.And then the lifetime can be estimated as τ∼ 100 /Ṫ, where 100 is the typical difference between the initial experimental temperature and the critical temperature.For the scheme of Dirac type SO couping, we consider the realization for ^87Rb Bose-Einstein condensate (BEC) and ^40K degenerate Fermi gases. For the ^87Rb atoms, the dipole trap is formed by two far-detuned laser beams and the estimated heating rate is about 10 / <cit.>. The lifetime with the dipole trap alone is about 10 seconds. Then we consider the heating from lattice and Raman beams, which are applied with the wavelength λ = 786and the beam waist ϖ = 200. Under typical parameter conditions V_0 = 4E_r and M_0 = 1E_r, we findthe scattering rate Γ_ sc≈ 0.54. Since the recoil energy E_r/ħ = 2π× 3707, the total heating rate of the lattice and Raman beams is about Ṫ=63.7 /. Therefore, the lifetime of ^87Rb BEC is about τ∼100/73.7/≈ 1.36. For the ^40K degenerate fermion gas, to realize typical parameter conditions V_0 = 4E_r and M_0 = 1E_r, we find the heating rate of lattice and Raman lights with wavelength λ = 768 is about 665 /, which gives the lifetime about τ∼ 150. If considering a smaller lattice, one can even enhance the lifetime up to several hundreds of milliseconds.For the scheme of Rashba and Weyl type SO coupling, the application for ^133Cs atoms has been considered, and the lifetime of ^133Cs atoms can be even much longer. The lifetime with only the dipole trap is approximately 15 <cit.>. The lattice potential is generated by the laser beam with the wavelength λ = 916.6, which couples the gound state and 6^2P_1/2(6^2P_3/2) levels. To produce the lattice potential with a typically depthV_0 = 4E_r, we apply the laser beam with the waist ϖ = 200 and the power P = 15.4. And then the heating rate of the lattice light can be estimated as 3/, which is very small because the detuning Δ is very large. Then we consider the heating rate of the Raman lights. The Raman potental is generated by two light beams with the wavelength λ = 458.3, which couples the ground state and 7^2P_1/2(7^2P_3/2) levels. To produce the Raman potentials with a typically strength M_0 = 1E_r, we apply the laser beams with the waistϖ = 200 and the power P = 21. The heating rate of the Raman lights is also considerably small about 10 /. With these results we can estimate the lifetime of the ^133Cs atoms to be τ >5. | http://arxiv.org/abs/1706.08961v2 | {
"authors": [
"Bao-Zong Wang",
"Yue-Hui Lu",
"Wei Sun",
"Shuai Chen",
"Youjin Deng",
"Xiong-Jun Liu"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci",
"quant-ph"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170627174708",
"title": "Dirac, Rashba and Weyl type spin-orbit couplings: toward experimental realization in ultracold atoms"
} |
Domain reduction techniques for global NLP and MINLP optimization [================================================================= This paper describes our submission to the 2017 BioASQ challenge. We participated in Task B, Phase B which is concerned with biomedical question answering (QA). We focus on factoid and list question, using an extractive QA model, that is, we restrict our system to output substrings of the provided text snippets. At the core of our system, we use FastQA, a state-of-the-art neural QA system. We extended it with biomedical word embeddings and changed its answer layer to be able to answer list questions in addition to factoid questions. We pre-trained the model on a large-scale open-domain QA dataset, SQuAD, and then fine-tuned the parameters on the BioASQ training set. With our approach, we achieve state-of-the-art results on factoid questions and competitive results on list questions. § INTRODUCTION BioASQ is a semantic indexing, question answering (QA) and information extraction challenge <cit.>. We participated in Task B of the challenge which is concerned with biomedical QA. More specifically, our system participated in Task B, Phase B: Given a question and gold-standard snippets (i.e., pieces of text that contain the answer(s) to the question), the system is asked to return a list of answer candidates.The fifth BioASQ challenge is taking place at the time of writing. Five batches of 100 questions each were released every two weeks. Participating systems have 24 hours to submit their results. At the time of writing, all batches had been released.The questions are categorized into different question types: factoid, list, summary and yes/no. Our work concentrates on answering factoid and list questions. For factoid questions, the system's responses are interpreted as a ranked list of answer candidates. They are evaluated using mean-reciprocal rank (MRR). For list questions, the system's responses are interpreted as a set of answers to the list question. Precision and recall are computed by comparing the given answers to the gold-standard answers. F1 score, i.e., the harmonic mean of precision and recall, is used as the official evaluation measure [The details of the evaluation can be found at <http://participants-area.bioasq.org/Tasks/b/eval_meas/>].Most existing biomedical QA systems employ a traditional QA pipeline, similar in structure to the baseline system by weissenborn2013answering. They consist of several discrete steps, e.g., named-entity recognition, question classification, and candidate answer scoring. These systems require a large amount of resources and feature engineering that is specific to the biomedical domain. For example, OAQA <cit.>, which has been very successful in last year's challenge, uses a biomedical parser, entity tagger and a thesaurus to retrieve synonyms.Our system, on the other hand, is based on a neural network QA architecture that is trained end-to-end on the target task. We build upon FastQA <cit.>, an extractive factoid QA system which achieves state-of-the-art results on QA benchmarks that provide large amounts of training data. For example, SQuAD <cit.> provides a dataset of ≈100,000 questions on Wikipedia articles. Our approach is to train FastQA (with some extensions) on the SQuAD dataset and then fine-tune the model parameters on the BioASQ training set.Note that by using an extractive QA network as our central component, we restrict our system's responses to substrings in the provided snippets. This also implies that the network will not be able to answer yes/no questions. We do, however, generalize the FastQA output layer in order to be able to answer list questions in addition to factoid questions.§ MODEL Our system is a neural network which takes as input a question and a context (i.e., the snippets) and outputs start and end pointers to tokens in the context. At its core, we use FastQA <cit.>, a state-of-the-art neural QA system. In the following, we describe our changes to the architecture and how the network is trained. §.§ Network architecture In the input layer, the context and question tokens are mapped to high-dimensional word vectors. Our word vectors consists of three components, which are concatenated to form a single vector:* GloVe embedding: We use 300-dimensional GloVe embeddings [We use theembeddings available here: <https://nlp.stanford.edu/projects/glove/>] <cit.> which have been trained on a large collection of web documents.* Character embedding: This embedding is computed by a 1-dimensional convolutional neural network from the characters of the words, as introduced by seo2016bidirectional.* Biomedical Word2Vec embeddings: We use the biomedical word embeddings provided by biomedical_word2vec. These are 200-dimensional Word2Vec embeddings <cit.> which were trained on ≈10 million PubMed abstracts.To the embedding vectors, we concatenate a one-hot encoding of the question type (list or factoid). Note that these features are identical for all tokens.Following our embedding layer, we invoke FastQA in order to compute start and end scores for all context tokens. Because end scores are conditioned on the chosen start, there are O(n^2) end scores where n is the number of context tokens. We denote the start index by i ∈ [1, n], the end index by j ∈ [i, n], the start scores by y_start^i, and end scores by y_end^i, j.In our output layer, the start, end, and span probabilities are computed as: p_start^i = σ(y_start^i)p_end^i, · = softmax(y_end^i, ·)p_span^i, j = p_start^i · p_end^i, j where σ denotes the sigmoid function. By computing the start probability via the sigmoid rather than softmax function (as used in FastQA), we enable the model to output multiple spans as likely answer spans. This generalizes the factoid QA network to list questions.§.§ Training & decoding Loss We define our loss as the cross-entropy of the correct start and end indices. In the case of multiple occurrences of the same answer, we only minimize the span of the lowest loss. Optimization We train the network in two steps: First, the network is trained on SQuAD, following the procedure by weissenborn2017fastqa (pre-training phase). Second, we fine-tune the network parameters on BioASQ (fine-tuning phase). For both phases, we use the Adam optimizer <cit.> with an exponentially decaying learning rate. We start with learning rates of 10^-3 and 10^-4 for the pre-training and fine-tuning phases, respectively.BioASQ dataset preparation During fine-tuning, we extract answer spans from the BioASQ training data by looking for occurrences of the gold standard answer in the provided snippets. Note that this approach is not perfect as it can produce false positives (e.g., the answer is mentioned in a sentence which does not answer the question) and false negatives (e.g., a sentence answers the question, but the exact string used is not in the synonym list).Because BioASQ usually contains multiple snippets for a given question, we process all snippets independently and then aggregate the answer spans, sorting globally according to their probability p_span^i, j. Decoding During the inference phase, we retrieve the top 20 answers span via beam search with beam size 20. From this sorted list of answer strings, weremove all duplicate strings. For factoid questions, we output the top five answer strings as our ranked list of answer candidates. For list questions, we use a probability cutoff threshold t, such that {(i, j)|p_span^i, j≥ t} is the set of answers. We set t to be the threshold for which the list F1 score on the development set is optimized. Ensemble In order to further tweak the performance of our systems, we built a model ensemble. For this, we trained five single models using 5-fold cross-validation on the entire training set. These models are combined by averaging their start and end scores before computing the span probabilities (Equations <ref>-<ref>). As a result, we submit two systems to the challenge: The best single model (according to its development set) and the model ensemble. Implementation We implemented our system using TensorFlow <cit.>. It was trained on an NVidia GForce Titan X GPU.§ RESULTS & DISCUSSION We report the results for all five test batches of BioASQ 5 (Task 5b, Phase B) in Table <ref>. Note that the performance numbers are not final, as the provided synonyms in the gold-standard answers will be updated as a manual step, in order to reflect valid responses by the participating systems. This has not been done by the time of writing[The final results will be published at <http://participants-area.bioasq.org/results/5b/phaseB/>]. Note also that – in contrast to previous BioASQ challenges – systems are no longer allowed to provide an own list of synonyms in this year's challenge.In general, the single and ensemble system are performing very similar relative to the rest of field: Their ranks are almost always right next to each other. Between the two, the ensemble model performed slightly better on average.On factoid questions, our system has been very successful, winning three out of five batches. On list questions, however, the relative performance varies significantly. We expect our system to perform better on factoid questions than list questions, because our pre-training dataset (SQuAD) does not contain any list questions.Starting with batch 3, we also submitted responses to yes/no questions by always answering yes. Because of a very skewed class distribution in the BioASQ dataset, this is a strong baseline. Because this is done merely to have baseline performance for this question type and because of the naivety of the method, we do not list or discuss the results here.§ CONCLUSION In this paper, we summarized the system design of our BioASQ 5B submission for factoid and list questions. We use a neural architecture which is trained end-to-end on the QA task. This approach has not been applied to BioASQ questions in previous challenges. Our results show that our approach achieves state-of-the art results on factoid questions and competitive results on list questions. acl_natbib | http://arxiv.org/abs/1706.08568v1 | {
"authors": [
"Georg Wiese",
"Dirk Weissenborn",
"Mariana Neves"
],
"categories": [
"cs.CL",
"cs.AI",
"cs.NE"
],
"primary_category": "cs.CL",
"published": "20170626191410",
"title": "Neural Question Answering at BioASQ 5B"
} |
The Second Leaper TheoremNikolai Beluhov336ptAbstract. A (p, q)-leaper is a fairy chess piece that, from a square a, can move to any of the squares a + (± p, ± q) or a + (± q, ± p). Let L be a (p, q)-leaper with p + q odd and C a cycle of L within a (p + q) × (p + q) chessboard. We show that there exists a second leaper M, distinct from L, such that a Hamiltonian cycle D of M exists over the squares of C. We give descriptions of C and M in terms of continued fractions. We introduce the notion of a direction graph, roughly a leaper graph from which all information has been abstracted away save for the directions of the moves, and we study C and D in terms of direction graphs. We introduce the notion of a dual generalized chessboard, a generalized chessboard B of more than one square such that the leaper graph of a leaper L over B is connected and isomorphic to the leaper graph of a second leaper M, distinct from L, over B, and we give constructions for dual generalized chessboards.§ PRELIMINARIES Fairy chess is the study of chess problems featuring unusual boards, pieces, or stipulations. A regular chessboard is a rectangular grid of unit squares. A generalized chessboard is a set of unit squares in the plane, with sides parallel to the coordinate axes, obtained from each other by means of integer translations. For instance, every polyomino is a generalized chessboard.The infinite chessboard is the one obtained by dissecting all of the plane into unit squares by means of two pencils of parallel lines.We reserve the term square for a unit square regarded as a part of a chessboard. We refer to both regular and generalized chessboards as boards for short.Consider any object O that consists of squares, possibly together with some structure imposed on them. For instance, O may be a square, a board, or a graph whose vertices are squares. We write O + v for the copy of O under a translation v. Let p and q be nonnegative integers, at least one of them positive. A (p, q)-leaper L is a fairy chess piece that, from a square a, can move to any of the squares a + (± p, ± q) or a + (± q, ± p). For instance, in orthodox chess the knight is a (1, 2)-leaper and the king is a combination of a (0, 1)-leaper and a (1, 1)-leaper.We refer to p and q as the proportions of L, and to translations of the form (± p, ± q) or (± q, ± p) as L-translations. The leaper graph of a leaper L over a board B is the graph whose vertices are the squares of B and whose edges join the pairs of squares of B that are joined by a move of L.A leaper L is free over a board B if the leaper graph of L over B is connected.An open tour of a leaper L over a board B is a Hamiltonian path in the leaper graph of L over B. A closed tour of L over B is a Hamiltonian cycle in the leaper graph of L over B. The problem of constructing knight tours over regular chessboards dates back to at least the ninth century. For a detailed historical overview, see the History sections of <cit.>.The study of leaper tours beyond the knight appears to have commenced sometime in the late nineteenth and early twentieth century. The concept of a free leaper was introduced in <cit.> by George Jelliss and Theophilus Willcocks. General questions about leaper graphs were raised in <cit.> and George Jelliss' <cit.>, the latter also establishing a number of general properties of leaper graphs. In <cit.>, Donald Knuth completely solved the question of whether a given leaper is free over a given regular chessboard, and studied the question of whether there exists a closed tour of a given leaper over a given regular chessboard. For a detailed historical overview, see the Leapers at Large section of <cit.>. A (p, q)-leaper L is orthogonal if p = 0 or q = 0, diagonal if p = q, and skew if p ≠ 0, q ≠ 0, and p ≠ q. There exist four possible directions for the moves of an orthogonal leaper (east, north, west, and south), four possible directions for the moves of a diagonal leaper (northeast, northwest, southwest, and southeast), and eight possible directions for the moves of a skew leaper (east-northeast, north-northeast, …, east-southeast). A (p, q)-leaper L is basic if p + q is odd and p and q are relatively prime. Equivalently, a (p, q)-leaper L is basic if p - q and p + q are relatively prime.All basic leapers apart from the (0, 1)-leaper are skew leapers.There are a number of reasons to single out the class of basic leapers.Firstly, let L be an arbitrary leaper. Then there exists a unique basic leaper L' such that every connected leaper graph of L is a scaled and rotated copy of a leaper graph of L'. For instance, when L is a (1, 3)-leaper, L' is a (1, 2)-leaper, as in Figure <ref>.More precisely, let L be a (p, q)-leaper, d the greatest common divisor of p and q, and p = dp' and q = dq'. If p' + q' is odd, then L' is a (p', q')-leaper, the scaling factor equals d, and without loss of generality the angle of rotation equals 0^∘. If p' + q' is even, then L' is a (1/2|p' - q'|, 1/2(p' + q')) leaper, the scaling factor equals √(2)d, and without loss of generality the angle of rotation equals 45^∘, as in Figure <ref>.Therefore, if a problem is only concerned with the intrinsic properties of leaper graphs, it suffices to study basic leapers.Secondly, a leaper L is free over the infinite chessboard (or, equivalently, over all sufficiently large regular chessboards) if and only if it is basic.§ THE SECOND LEAPER THEOREM Consider a (p, q)-leaper L, p < q, over a (p + q) × (p + q) chessboard B. The central (q - p) × (q - p) squares of B are of degree zero in the corresponding leaper graph G, and all remaining squares are of degree two. Therefore, G consists of (q - p)^2 isolated vertices and a number of disjoint cycles.The following theorem was discovered by the author in January 2006. Let L be a (p, q)-leaper with p + q odd and C a cycle of L within a (p + q) × (p + q) chessboard. Then there exists a second leaper M, distinct from L, such that a Hamiltonian cycle of M exists over the squares of C.A number of examples are in order.When p = 1 and q = 2, G contains a single eight-square cycle C which is also toured by a (0, 1)-leaper (Figure <ref>). In general, when p = 1 and q = 2k, G contains a single 4k-square cycle which is also toured by a (0, 1)-leaper (Figure <ref>).When p = 2 and q = 3, G contains one 16-square cycle which is also toured by a (0, 1)-leaper and one eight-square cycle which is also toured by a (1, 2)-leaper (Figure <ref>). In general, when q = p + 1, G contains p cycles, one toured by each (r, r + 1)-leaper, r = 0, 1, …, p - 1 (Figure <ref>).When p = 2 and q = 5, G contains one 32-square cycle which is also toured by a (0, 1)-leaper and one eight-square cycle which is also toured by a (1, 2)-leaper (Figure <ref>). We begin by introducing several useful notions. An (m, n)-frame F, m < n, is an (m + n) × (m + n) chessboard with an (n - m) × (n - m) hole in the center. The sections of F, F_, F_, …, F_ as in Figure <ref>, are the eight rectangular subboards that F is dissected into when the sides of the central hole are extended. Given an (m, n)-frame F, we write L_F for the corresponding (m, n)-leaper and G_F for the leaper graph of L_F over F.The , , …,stand for the eight directions east, northeast, …, southeast. Given a direction i out of , , …,and an integer k, we write -i for the direction opposite i and i + k for the direction k steps counterclockwise from i. For instance, - =, + 1 =, for all directions i, -i = i + 4, and, for all directions i and integers k, -(i + k) = -i + k.Given a path w through a number of squares, we write -w for the path obtained from w by traversing it in the opposite direction.We proceed to shed some light onto the structure of a second leaper's Hamiltonian cycle. A cycle D of a leaper M within a frame F is proper if D is the concatenation of eight disjoint nonempty paths,D = a^a^… a^,such that, for all directions i, a^i lies within F_i and the L_F-translation that maps F_i ∪ F_i + 1 onto F_-(i + 1)∪ F_-i also maps a^ia^i + 1 onto -a^-(i + 1)-a^-i. Let us write this out in more detail. In a proper cycle, we havea^a^ + (-n, -m) = -a^-a^, a^a^ + (-m, -n) = -a^-a^, a^a^ + (m, -n) = -a^-a^,anda^a^ + (n, -m) = -a^-a^. Eventually we will see that, in the statement of the theorem, “a Hamiltonian cycle of M” can be replaced with “a proper Hamiltonian cycle of M”.From here on, the plan of the proof is as follows. First we define three transformations, f, g, and h, that lift smaller frames to larger ones. Given a subset S of the squares of a smaller frame F, each lifting transformation constructs from it a subset T of the squares of a larger frame H.This is done in such a way that if S is the vertex set of an L_F-cycle within F then T is the vertex set of an L_H-cycle within H and if S is the vertex set of a proper cycle of a leaper M within F then T is the vertex set of a proper cycle of M within H.The proof is then completed by induction on p + q.We begin with f. Let F be an (m, n)-frame, m < n, and H an (m, 2m + n)-frame. Place F and H so that their centers coincide. (Or, equivalently, so that the outer contour of F coincides with the inner contour of H.) We define the transformation f, lifting F to H, as follows.An f-translation is any of the eight translations corresponding to a move of a (0, m + n)-leaper or an (m, m)-leaper. Given a direction i, we write v^f_i for the f-translation pointing in direction i. For instance, v^f_ = (m + n, 0) and v^f_ = (m, m).Let S be a subset of the squares of F. Then f(S) is a subset of the squares of H, the disjoint union of(S + v^f_i) ∩ H_iwhen i ranges over , , …, . Equivalently, f(S) is the disjoint union of a number of translation copies of the intersections of S with the sections of F as in Figure <ref>. Each subboard I of H labeled F_i in Figure <ref> contains a copy of S ∩ F_i under the f-translation that maps F_i onto I.We proceed to establish the two key properties of f. Let S be the vertex set of an L_F-cycle within F. Then f(S) is the vertex set of an L_H-cycle within H. Let S be the vertex set of the L_F-cycle C.Replace each square a in C with a path within H through the images of a under f as follows.Case 1. a belongs to F_. Replace a with the single-square patha + v^f_. Case 2. a belongs to F_. Then the two squares adjacent to a in C are a + (-n, -m) and a + (-m, -n). Replace a with the three-square path(a + v^f_)(a + v^f_)(a + v^f_),traversed in such a direction that the endpoint a + v^f_ is on the side of a + (-n, -m) and the endpoint a + v^f_ is on the side of a + (-m, -n).All other cases are obtained from Cases 1 and 2 by symmetry.The concatenation of the resulting sequence of paths is an L_H-cycle of vertex set f(S).Let S be the vertex set of a proper cycle of a leaper M within F. Then f(S) is the vertex set of a proper cycle of M within H. Let S be the vertex set of the proper cycleD = a^a^… a^of M within F.Letb^ = -a^-a^-a^ + v^f_and define b^, b^, and b^ analogously. Also letb^ = a^ + v^f_and define b^, b^, and b^ analogously. We proceed to show thatb^b^… b^is a proper cycle of M of vertex set f(S).We need to show that, for all directions i, the final square of b^i is joined by an M-move to the opening square of b^i + 1. We establish the case i =, and all other cases are obtained from it by symmetry.Since D is a proper cycle,-a^-a^ + (n, -m) = a^a^. It follows that the suffix -a^-a^ + v^f_ of b^ coincides with a^a^ + v^f_. Therefore, the final square of b^ and the opening square of b^ are in the same relative position as the final square of a^ and the opening square of a^.We are left to show that, for all directions i, the L_H-translation that maps H_i ∪ H_i + 1 onto H_-(i + 1)∪ H_-i also maps b^ib^i + 1 onto -b^-(i + 1)-b^-i. We establish the case i =, and all other cases are obtained from it by symmetry.We haveb^b^ = (-a^ + v^f_)(-a^-a^ + v^f_)(a^ + v^f_)and-b^-b^ = (-a^ + v^f_)(a^a^ + v^f_)(a^ + v^f_). Since D is a proper cycle,-a^-a^ + (n, -m) = a^a^. Therefore, a (-2m - n, -m) translation maps b^b^ onto -b^-b^. Before we continue to g and h, we need to introduce one more species of subdivision of a frame. Let m < n, 3m ≥ n, and F be an (m, n)-frame. The shell of F, F^-, is the union of eight equal, symmetrically placed square subboards of F defined as follows.When 2m ≥ n, F^- is the union of eight subboards of F of size (n - m) × (n - m), one in each corner and four adjacent by side to the central hole, as in Figure <ref>.When 2m ≤ n ≤ 3m, F^- is the union of eight subboards of F of size (3m - n) × (3m - n), one in the middle of each outer side and four adjacent by corner to the central hole, as in Figure <ref>.In both cases, the core of F, F^+, is F ∖ F^-. When 2m = n, both parts of the definition give the same shell which coincides with the complete frame.Let, for all directions i, F^-_i be the intersection of the shell of F and the section F_i of F, as in Figures <ref> and <ref>. Then, for all i, an L_F-translation maps F^-_i onto F^-_i + 3. Thus the shell of F is the union of the vertex sets of a number of disjoint eight-square L_F-cycles and the core of F is the union of the vertex sets of all other L_F-cycles within F.We go on to g and h. Let F be an (m, n)-frame, m < n, and H an (n, 2n - m)-frame. Place F and H so that their centers coincide. (Or, equivalently, so that their inner contours coincide.) We define the transformation g, lifting F to H, as follows.A g-translation is any of the eight translations corresponding to a move of a (0, n - m)-leaper or an (n, n)-leaper. Given a direction i, we write v^g_i for the g-translation pointing in direction i. For instance, v^g_ = (n - m, 0) and v^f_ = (n, n).Let S be a subset of the squares of F. Then g(S) is a subset of the core of H, the disjoint union of(S + v^g_i) ∩ H_iwhen i ranges over , , …, . Equivalently, g(S) is the disjoint union of a number of translation copies of the intersections of S with the sections of F as in Figure <ref>. Let S be the vertex set of an L_F-cycle within F. Then g(S) is the vertex set of an L_H-cycle within H. Analogous to the proof of Lemma <ref>.Let S be the vertex set of the L_F-cycle C and a a square in C. The transformation rules for a are as follows.Case 1. a belongs to F_. Then the two squares adjacent to a in C are a + (-n, m) and a + (-n, -m). Replace a with the three-square path(a + v^f_)(a + v^f_)(a + v^f_),traversed in such a direction that the endpoint a + v^f_ is on the side of a + (-n, m) and the endpoint a + v^f_ is on the side of a + (-n, -m).Case 2. a belongs to F_. Replace a with the single-square patha + v^f_. All other cases are obtained from Cases 1 and 2 by symmetry.Let S be the vertex set of a proper cycle of a leaper M within F. Then g(S) is the vertex set of a proper cycle of M within H.The proof is analogous to the proof of Lemma <ref>. Let F be an (m, n)-frame, m < n, and H an (n, m + 2n)-frame. Place F and H so that their centers coincide. (Or, equivalently, so that the outer contour of F coincides with the inner contour of H.) We define the transformation h, lifting F to H, as follows.An h-translation is any of the eight translations corresponding to a move of a (0, m + n)-leaper or an (n, n)-leaper. Given a direction i, we write v^h_i for the h-translation pointing in direction i. For instance, v^h_ = (m + n, 0) and v^h_ = (n, n).Let S be a subset of the squares of F. Then h(S) is a subset of the core of H, the disjoint union of(S + v^h_i) ∩ H_iwhen i ranges over , , …, . Equivalently, h(S) is the disjoint union of a number of translation copies of the intersections of S with the sections of F as in Figure <ref>. Let S be the vertex set of an L_F-cycle within F. Then h(S) is the vertex set of an L_H-cycle within H. Analogous to the proof of Lemma <ref>.Let S be the vertex set of the L_F-cycle C and a a square in C. The transformation rules for a are as follows.Case 1. a belongs to F_. Then the two squares adjacent to a in C are a + (-n, m) and a + (-n, -m). Replace a with the five-square path(a + v^f_)(a + v^f_)(a + v^f_)(a + v^f_)(a + v^f_),traversed in such a direction that the endpoint a + v^f_ is on the side of a + (-n, m) and the endpoint a + v^f_ is on the side of a + (-n, -m).Case 2. a belongs to F_. Then the two squares adjacent to a in C are a + (-n, -m) and a + (-m, -n). Replace a with the three-square path(a + v^f_)(a + v^f_)(a + v^f_),traversed in such a direction that the endpoint a + v^f_ is on the side of a + (-n, -m) and the endpoint a + v^f_ is on the side of a + (-m, -n).All other cases are obtained from Cases 1 and 2 by symmetry.Let S be the vertex set of a proper cycle of a leaper M within F. Then h(S) is the vertex set of a proper cycle of M within H.The proof is analogous to the proof of Lemma <ref>.We pause for a moment to point out a deep connection between the three lifting transformations.Let m and n be arbitrary integers.Introduce a Cartesian coordinate system Oxy over the infinite chessboard such that the integer points are the centers of the squares if m + n is odd, and the vertices of the squares if it is even, and write (x, y) for the square centered at (x, y).Define the standard (m, n)-frame to be the set of all squares (x, y) such that at least one of |x| and |y| exceeds the smaller of 1/2|m - n| and 1/2(m + n), and both of |x| and |y| are less than the larger. The -section of a standard (m, n)-frame would be the set of all squares (x, y) such that x lies between 1/2(n - m) and 1/2(m + n) and y lies between 1/2(m - n) and 1/2(n - m), the -section the set of all squares (x, y) such that both x and y lie between 1/2(n - m) and 1/2(m + n), and so on. In general, a standard frame is not the disjoint union of its sections.Let, then, 0 ≤ m < n. The standard (n, -m)-frame consists of the same squares as the standard (m, n)-frame. Therefore, we can view an (m, n)-frame F as an overlapping inside-out (n, -m)-frame F'. When we plug the values (n, -m) into the definition of f and lift F' by means of f, the result is precisely the same as when we lift F by means of g.Furthermore, the standard (n, m)-frame consists of the same squares as the standard (m, n)-frame. Therefore, we can view an (m, n)-frame F as an overlapping (n, m)-frame F”. When we plug the values (n, m) into the definition of f and lift F” by means of f, the result is precisely the same as when we lift F by means of h.So, all three lifting transformations are in a sense forms of the same fundamental lifting transformation.We have built all the tools we need and are ready to tackle the theorem.Let d be the greatest common divisor of p and q, p = dp', and q = dq'. Then every cycle of L within a (p + q) × (p + q) chessboard is a scaled, by a factor of d, copy of a cycle of the basic (p', q')-leaper L' within a (p' + q') × (p' + q') chessboard. Therefore, it suffices to consider the case when L is a basic leaper.Let, from here on, p < q and L be a basic leaper.We will show, by induction on p + q, that there exists a second (r, s)-leaper M, r + s < p + q, such that a proper Hamiltonian cycle of M exists over the squares of C.When p = 1 and q = 2, the theorem holds with r = 0 and s = 1 as in Figure <ref>.Let, from here on, p + q > 3 and H be a (p, q)-frame. We distinguish three cases for the proportions of H.Case f. 3p < q. Let m = p and n = q - 2p. Then m < n, m + n < p + q, p = m, and q = 2m + n.Let F be an (m, n)-frame. Then f lifts F to H.By Lemma <ref>, f lifts each L_F-cycle within F to an L_H-cycle within H. Since f maps the set of all squares of F onto the set of all squares of H, there exists an L_F-cycle A within F such that f lifts A to C.By the induction hypothesis, there exists a proper Hamiltonian cycle of an (r, s)-leaper M, r + s < m + n, over the squares of A. By Lemma <ref>, we are done.Case g. 2p > q. Let m = 2p - q and n = p. Then m < n, m + n < p + q, p = n, and q = 2n - m.Suppose first that C lies within the core of H. Let F be an (m, n)-frame. Then g lifts F to H and the proof continues as in Case f.Suppose, then, that C lies within the shell of H. Then C consists of eight squares, one in each H^-_i, and a proper Hamiltonian cycle of an (m, n)-leaper M exists over the squares of C.Case h. 2p < q < 3p. Let m = q - 2p and n = p. Then m < n, m + n < p + q, p = n, and q = m + 2n.Suppose first that C lies within the core of H. Let F be an (m, n)-frame. Then h lifts F to H and the proof continues as in Case f.Suppose, then, that C lies within the shell of H. The proof continues as in the second part of Case g.This completes the proof of the theorem. § DESCENTS AND EVEN CONTINUED FRACTIONS Let us look more closely into the concluding part of the proof of Theorem <ref>. Let L be a basic (p, q)-leaper distinct from the (0, 1)-leaper and the (1, 2)-leaper, and H a (p, q)-frame. Then there exist a unique (m, n)-frame F and a unique lifting transformation that lifts F to H. When we continue this process backwards, eventually it bottoms out at a (1, 2)-frame. Therefore, there exists a unique sequence of lifting transformations that lifts a (1, 2)-frame to H. The descent of a skew basic (p, q)-leaper L is the unique string e_1e_2… e_l, composed of the characters , , and , such that successively applying lifting transformations of types e_l, e_l - 1, …, e_1 to a (1, 2)-frame lifts it to a (p, q)-frame. This induces a one-to-one mapping between strings composed of the characters , , andand skew basic leapers.Let us look into several examples. The descent of a (1, 2)-leaper is the empty string. The descent of a (1, 2r)-leaper is …, where the characteroccurs r - 1 times. The descent of an (r, r + 1)-leaper is …, where the characteroccurs r - 1 times. And the descent of an (18, 41)-leaper is .The proof of Theorem <ref> is essentially a proof by induction on descent.Let G be the leaper graph of a basic (p, q)-leaper L over a (p + q) × (p + q) chessboard. Let us look at the cycles of G from the point of view of the descent of L.Consider a (p, q)-frame F. Lifting F by means of f extends all existing cycles without altering the second leapers that tour them, and does not create any new cycles. Lifting F by means of either g or h extends all existing cycles without altering the second leapers that tour them, and creates (q - p)^2 new eight-square cycles within the shell of the larger frame, all of which are translation copies of each other and each of which is also toured by a (p, q)-leaper. Hence the following theorem. Let e = e_1e_2… e_l be the descent of the skew basic (p, q)-leaper L.Consider all suffixes e_i + 1e_i + 2… e_l of e such that e_i is eitheror . Let k be one larger than the number of such suffixes, the i-th such suffix, i = 1, 2, …, k - 1, be the descent of a (p_i, q_i)-leaper, and p_k = 0 and q_k = 1.Then the leaper graph G of L over a (p + q) × (p + q) chessboard consists of (q - p)^2 isolated vertices and a number of disjoint cycles of k distinct types. For all i, G contains (q_i - p_i)^2 cycles of type i, all of which are translation copies of each other and each of which is also toured by a (p_i, q_i)-leaper.It is possible to word Theorem <ref> in terms of continued fractions without referencing descents. In order to do so, let us track what happens to the ratio r = q/p of the proportions of a (p, q)-frame F when we lift F.The transformation f lifts F to a (p, 2p + q)-frame H, and2p + q/p = 2 + q/p = 2 + r. The transformation g lifts F to a (q, 2q - p)-frame H, and2q - p/q = 2 - p/q = 2 - 1/r. The transformation h lifts F to a (q, p + 2q)-frame H, andp + 2q/q = 2 + p/q = 2 + 1/r. Chaining the right-hand sides of the above equations gives us an expression of the formq/p = c_1 ±1c_2 ±1⋱±1/c_k,where the terms c_1, c_2, …, c_k and the ± signs are determined as follows.Let e = e_1e_2… e_l be the descent of the skew basic (p, q)-leaper L ande = e'_1e”_1e'_2e”_2… e”_k - 1e'_ka partitioning of e into (possibly empty) substrings such that, for all i, e'_i is a (possibly empty) run of the characterof length c'_i and e”_i consists of a single character, eitheror . Then c_i = 2c'_i + 2 and the sign following c_i is - if e”_i = and + if e”_i =. An even continued fraction is an expression of the form[c_1±, c_2±, …, c_k] = c_1 ±1c_2 ±1⋱±1/c_k,where c_i is an even integer for all i, c_1 is nonnegative, and c_2, c_3, …, c_k are all positive. A nonnegative rational number q/p with p and q relatively prime possesses a finite even continued fraction representation if and only if p + q is odd. Moreover, that representation is unique.We are ready to give the alternative form of Theorem <ref>. Let L be a skew basic (p, q)-leaper. Letq/p = [c_1±, c_2±, …, c_k]be the representation of q/p as an even continued fraction,q_i/p_i = [c_i + 1±, c_i + 2±, …, c_k]for i = 1, 2, …, k - 1, where q_i/p_i is irreducible, and p_k = 0 and q_k = 1.Then the leaper graph G of L over a (p + q) × (p + q) chessboard consists of (q - p)^2 isolated vertices and a number of disjoint cycles of k distinct types. For all i, G contains (q_i - p_i)^2 cycles of type i, all of which are translation copies of each other and each of which is also toured by a (p_i, q_i)-leaper.Here follow a couple of noteworthy corollaries of Theorems <ref> and <ref>. The leaper graph of a (p, q)-leaper L, p < q, over a (p + q) × (p + q) chessboard contains a single cycle (or, equivalently, the leaper graph of L over a (p, q)-frame is nonempty and connected) if and only if p = 1 and q is even.Given a skew basic (p, q)-leaper L, there exists a unique cycle C of L within a (p + q) × (p + q) chessboard such that there exists a Hamiltonian cycle of a (0, 1)-leaper over the squares of C. We go on to give a concise expression for the length of each cycle of type i in terms of even continued fractions. First, however, we need to lay some groundwork. Let L be a (p, q)-leaper with p + q odd, C a cycle of L within a (p + q) × (p + q) chessboard B, and F the (p, q)-frame obtained from B by removing the central (q - p)^2 squares.Then C visits an odd number of squares (and, in particular, at least one square) in each section of F. Furthermore, C visits the same number of squares in each of the four side sections F_, F_, F_, and F_ of F, and the same number of squares in each of the four corner sections F_, F_, F_, and F_ of F. By induction on descent.Let, in the setting of Theorem <ref>, for all il_i/d_i = [c_1±, 2-, c_2±, 2-, …, c_i],where l_i/d_i is irreducible. Then the length of each cycle of type i is 4l_i. Let F be the (p, q)-frame obtained from the (p + q) × (p + q) chessboard by removing all squares isolated in G, and C a cycle in G.Let d be the number of squares that C visits within F_ and l the number of squares that C visits within F_∪ F_. By Theorem <ref>, the length of C is 4l.When C is contained within the shell of F (provided that F does indeed possess a shell), it is of type 1, c_1 = 2, d = d_1 = 1, and l = l_1 = 2.Let us track what happens to the ratio r = l/d when we lift F to a larger frame H. Let E be the image of C within H, and define d' and l' analogously to d and l, but based on E and H.When we lift F by means of f, d' = d, l' = 2d + l, andl'/d' = 2d + l/d = 2 + l/d = 2 + r. When we lift F by means of g, d' = 2l - d, l' = 3l - 2d, andl'/d' = 3l - 2d/2l - d = 2 - 12 - d/l = 2 - 12 - 1/r. When we lift F by means of h, d' = 2l - d, l' = 5l - 2d, andl'/d' = 5l - 2d/2l - d = 2 + 12 - d/l = 2 + 12 - 1/r. Chaining the right-hand sides of the above equations yields the theorem. Occasionally, there exists a third leaper besides the second one. Let, in the setting of Theorems <ref> and <ref>, c_k = 2 (or, equivalently, the final character e_l in the descent e_1e_2… e_l of L be eitheror ) and l_k be indivisible by three. Then there exists a Hamiltonian cycle of a (1, 2)-leaper over the squares of the unique cycle of type k.Let us look into a couple of examples.When p = 2 and q = 3, k = 2, c_2 = 2, and l_2 = 16. Figure <ref> shows that the unique cycle of type 2, previously depicted in Figure <ref>, top, is also toured by a (1, 2)-leaper.When p = 2 and q = 5, k = 2, c_2 = 2, and l_2 = 32. Figure <ref> shows that the unique cycle of type 2, previously depicted in Figure <ref>, top, is also toured by a (1, 2)-leaper. When k = 1, the theorem holds as in Figure <ref>. Let, from here on, k ≥ 2.Let a_1a_2… a_l_k be a proper Hamiltonian cycle of a (0, 1)-leaper over the squares of the unique cycle C in G of type k, as constructed in the proof of Theorem <ref>, a_i + l_k≡ a_i for all integers i, and F the (p, q)-frame obtained from the (p + q) × (p + q) chessboard by removing all squares isolated in G. The cycle C visits at least three squares in each section of F and the squares a_i and a_i + 3 are linked by a (1, 2)-move for all integers i. When p = 2 and either q = 3 or q = 5, the claim holds as in Figures <ref> and <ref>. The proof continues by induction on descent.Suppose that the lemma holds for C. Lift F to a larger frame H by means of any of the three lifting transformations. Let E be the image of C within H and b_1b_2… b_s a proper Hamiltonian cycle of a (0, 1)-leaper over the squares of E, as constructed in the proof of Theorem <ref>, with b_j + s≡ b_j for all integers j.By the definitions of f, g, and h and the induction hypothesis, E visits at least three squares in each section of H.It follows that, for all j, b_j and b_j + 3 are either within the same subboard of H in Figure <ref>, <ref>, or <ref>, or within the union of two subboards of H visited in direct succession by b_1b_2… b_s.Therefore, there always exist two squares a_i and a_i + 3 within F such that an f, g, or h-translation maps a_i and a_i + 3 onto b_j and b_j + 3.By the induction hypothesis, a_i and a_i + 3 are linked by a (1, 2)-move. Therefore, so are b_j and b_j + 3. Consider the sequence of squaresa_3, a_6, …, a_l_k,the i-th term of which equals a_3i for i = 1, 2, …, l_k.By Lemma <ref>, this sequence is a cycle of a (1, 2)-leaper. Furthermore, as l_k is indivisible by three, it contains each square of C precisely once. Therefore, it is a Hamiltonian cycle of a (1, 2)-leaper over the squares of C. The proof of Theorem <ref> does not appear to generalize beyond the (1, 2)-leaper. For more on this topic, see Questions <ref> and <ref>.§ CONNECTEDNESS We take a brief detour to expand on an application of the technique of induction on descent.In <cit.>, Donald Knuth shows that a (p, q)-leaper L, p ≤ q, is free over a regular chessboard B (of size greater than 1 × 1) if and only if L is basic and B contains a (p + q) × 2q subboard. We proceed to give a proof of the more involved “if” part by induction on descent. (Donald Knuth, <cit.>) Let L be a basic (p, q)-leaper, p < q. Then L is free over a (p + q) × 2q chessboard. Let B be a (p + q) × 2q chessboard. We view B as the union of q - p + 1 translation copies F + (t, 0) of a (p, q)-frame F, t = 0, 1, …, q - p.Let C_1, C_2, …, C_c be all L_F-cycles within F. Then the leaper graph of L over B is the union of the translation copies C_k + (t, 0) for k = 1, 2, …, c and t = 0, 1, …, q - p.Construct the graph K as follows. The vertices of K are the ordered pairs (k, t) where k = 1, 2, …, c and t = 0, 1, …, q - p. Two vertices (k', t') and (k”, t”) in K are joined by an edge if and only if there exists a square in B that belongs to both C_k' + (t', 0) and C_k” + (t”, 0). Then the leaper graph of L over B is connected if and only if K is.When p = 1 and q = 2, K consists of two vertices joined by an edge.Suppose, then, that K is connected. Lift F to an (m, n)-frame H by means of any of the three lifting transformations.For simplicity, we extend the definitions of shell and core as follows. If 3m ≤ n, the shell of H is empty and the core of H is H.Let d be the number of shell cycles in H (zero if the lifting transformation is f and H only possesses a shell in the extended sense) and D_1, D_2, …, D_c + d all L_H-cycles within H, so that lifting C_i gives D_i + d for all i.Define E and N analogously to B and K, but based on H. It suffices to show that N is connected.Let E^+ be the union of the translation copies H^+ + (t, 0), t = 0, 1, …, n - m, and N^+ the spanning subgraph of N over all vertices (k, t) such that k > d, corresponding to cycles in the core of H.Let (k', t') and (k”, t”), t' < t”, be joined by an edge in K. Then there exist two squares a' and a” in F that are joined by a (t” - t', 0) move such that a' belongs to C_k' and a” belongs to C_k”.Since t” - t' ≤ q - p does not exceed the side of the central hole of F, both of a' and a” belong to one of the following subboards of F:F_, F_∪ F_, F_∪ F_, F_, F_∪ F_, F_∪ F_. By the definitions of f, g, and h, there exist in H^+ two images b' and b” of a' and a” such that b' and b” are joined by a (t” - t', 0) move, b' belongs to D_k' + d and b” belongs to D_k” + d.It follows that, whenever an edge joins (k', t') and (k”, t”) in K and t” - t' = u” - u', t' < t”, 0 ≤ u' < u”≤ n - m, an edge joins (k' + d, u') and (k” + d, u”) in N^+. Therefore, N^+ is connected.We are left to take care of the shell cycles of H. However, since all of the translation copies H^-_ + (t, 0), t = 0, 1, …, n - m, are subboards of E^+, every vertex (k, t) of N such that k ≤ d, corresponding to a cycle in the shell of H, is joined by an edge to a vertex in N^+. Therefore, N is connected. § DIRECTION GRAPHS Let L be a skew leaper. Then there exist eight possible skew directions for the moves of L, east-northeast, north-northeast, …, east-southeast, which we labelthroughstarting from east-northeast and proceeding counterclockwise, as in Figure <ref>.Given a skew direction i and an integer k, we write -i for the direction opposite i and i + k for the direction k steps counterclockwise from i. For instance, - =, + 1 =, for all skew directions i, -i = i + 4, and, for all skew directions i and integers k, -(i + k) = -i + k.We associate a 2 × 2 matrix with each skew direction as follows. The direction matrix of a move of a skew (p, q)-leaper L, p < q, of direction i from a square a to a square b = a + v is the unique matrix A_i out of the following eight,A_ = ([ 0 1; 1 0 ]), A_ = ([ 1 0; 0 1 ]),A_ = ([ -10;01 ]), A_ = ([0 -1;10 ]),A_ = ([0 -1; -10 ]), A_ = ([ -10;0 -1 ]),A_ = ([10;0 -1 ]), A_ = ([01; -10 ]),such thatv^T = A_i([ p; q ]).For all skew directions i, A_-i = -A_i. A direction graph is a symmetric directed graph Φ whose arcs are labeled with skew directions in such a way that the sum of the associated direction matrices over every simple cycle in Φ is the zero matrix. In particular, in a direction graph Φ, for every arc pointing from x to y and labeled i, the arc pointing from y to x is labeled -i. Hence the following definition. A labeled oriented graph Ψ represents a direction graph Φ if Ψ is obtained from Φ by removing exactly one arc out of each symmetric pair. A direction graph is completely determined by any labeled oriented graph that represents it. A direction graph Φ is extracted from a leaper graph G of a skew leaper L if there exists a one-to-one mapping σ between the vertices of G and the vertices of Φ such that an L-move of direction i leads from a vertex a to a vertex b in G if and only if an arc labeled i points from σ(a) to σ(b) in Φ. In other words, a direction graph is extracted from a leaper graph by abstracting away all information (such as the precise positions of the squares and the proportions of the leaper) save for the directions of the moves.For instance, the direction graph extracted from the (1, 2)-cycle in Figure <ref> is depicted in Figure <ref>, and is represented by the oriented cycle labeled .It may happen that a direction graph cannot be extracted from a given leaper graph. A simple oriented cycle C of a skew leaper is trivial if the sum of the associated direction matrices over C is the zero matrix, and nontrivial otherwise. A simple cycle C of a skew leaper is trivial if any (or, equivalently, both) of its two orientations are, and nontrivial otherwise. Given a leaper graph G, it is possible to extract a direction graph Φ from G if and only if G does not contain a nontrivial cycle.For instance, the (1, 2)-cycle and the (1, 3)-cycle in Figure <ref> are both nontrivial, and it is not possible to extract a direction graph from either. Let Φ be a direction graph and L a leaper. A graph G is an L-instantiation of Φ if the vertices of G are squares and there exists a mapping τ from the vertices of Φ to the vertices of G such that, for every arc in Φ pointing from x to y and labeled i, an L-move of direction i leads from τ(x) to τ(y). Let Φ be the direction graph extracted from a leaper graph G of a leaper L. Then G is an L-instantiation of Φ.Given a direction graph Φ and a leaper L, there always exists an L-instantiation of Φ, unique up to translation provided that Φ is connected. It is well-defined for arbitrary leapers L, as when L is an orthogonal or diagonal (p, q)-leaper, p ≤ q, we can define an L-move of direction i as an L-move of translation v such that v^T = A_i ( [ p; q ]). However, it may happen that the mapping τ is not one-to-one or that the leaper graph of L over the vertex set of G is distinct from G.We go on to delineate the class of direction graphs for which instantiation is well-behaved.Let x and y be vertices in a direction graph Φ. Then the sum of the associated direction matrices over every path from x to y in Φ is the same. Let Φ be a direction graph and x and y vertices in the same connected component of Φ. The distance from x to y in Φ is the sum of the associated direction matrices over any path from x to y in Φ.A direction graph Φ is coherent if, for every pair of vertices x and y in the same connected component of Φ, the distance from x to y in Φ equals the zero matrix if and only if x and y coincide, and a direction matrix if and only if x and y are joined by an arc. Every direction graph extracted from a leaper graph is coherent.Given a coherent direction graph Φ, let L be a skew (p, q)-leaper, p < q, and G an L-instantiation of Φ such that the images under τ of vertices in different connected components of Φ do not coincide and are not joined by an L-move. Furthermore, let x and y be vertices in the same connected component of Φ, A the distance from x to y in Φ, and v the translation defined by v^T = A ( [ p; q ]). Then τ(x) + v = τ(y).When A is distinct from the zero matrix, there exists at most one skew basic (p, q)-leaper L such that v is the zero translation, and when A is distinct from the direction matrix A_i, there exists at most one skew basic (p, q)-leaper L such that v is an L-translation of direction i.It follows that the mapping τ is one-to-one and the leaper graph of L over the vertex set of G is G for all but finitely many skew basic leapers L. In other words, G is a leaper graph of L and Φ is extracted from G for all but finitely many skew basic leapers L.Every leaper graph of an orthogonal or diagonal leaper is an instantiation of a direction graph. A leaper graph of a skew leaper is an instantiation of a direction graph if and only if it does not contain a nontrivial cycle. A permutation π of the eight skew directions is an equivalence permutation if there exist two real-coefficient 2 × 2 matrices P and Q, inducing π, such thatA_π(i) = PA_iQfor all skew directions i. Given an equivalence permutation π induced by P and Q and a directed graph Φ whose arcs are labeled with skew directions, we write π(Φ) or PΦ Q for the labeled directed graph obtained from Φ by applying π to every arc label. If Φ is a direction graph, then so is π(Φ). Furthermore, if Φ is a coherent direction graph, then so is π(Φ). Two direction graphs Φ_1 and Φ_2 are equivalent if there exists an equivalence permutation π such that π(Φ_1) = Φ_2. We proceed to look at Theorems <ref>, <ref>, and <ref> from the point of view of direction graphs.Let L be a skew basic (p, q)-leaper, p < q, C a cycle of L within a (p + q) × (p + q) chessboard, and F the associated (p, q)-frame.Impose an orientation on C and consider a square a of C in the section F_ of F. The two squares adjacent to a in C are a + (-q, p) and a + (-q, -p). Therefore, the directions of the moves to and from a in C are eitherand , orand .Analogous reasoning applies to all sections of F. It follows that the directions of the moves to and from every square a in C are of the form i and i ± 3. Let L be a skew basic (p, q)-leaper and C a cycle of L within a (p + q) × (p + q) chessboard.Impose an orientation on C and label every square a in C +_, +_, -_, or -_ as follows. Let i be the direction of the move to a in C. Then the + or - signifies whether the direction of the move from a in C is i + 3 or i - 3, and theorsignifies whether a belongs to a side or a corner section of the associated frame F.Equivalently, label a according to the following table.Label of aDirections of moves to and from a +_ , , ,+_ , , ,-_ , , ,-_ , , ,Then the cyclic string formed by the vertex labels of C is a signature of C.For instance, a signature of the (1, 2)-cycle in Figure <ref> is +_+_+_+_+_+_+_+_. This is also a signature of every shell cycle, such as the (2, 3)-cycle in Figure <ref>, bottom, and the (2, 5)-cycle in Figure <ref>, bottom.Let us track how a signature evolves when we lift a cycle.Lift C by means of f to f(C). By the proof of Lemma <ref>, a signature of f(C) is obtained from a signature of C by replacing every character with a string composed of the characters +_, +_, -_, and -_, subject to the following system of rewriting rules.+_ → -_+_ → +_+_+_-_ → +_-_ → -_-_-_ We refer to this transformation as an f-rewrite. For instance, the f-rewrite of +_+_+_+_+_+_+_+_ is -_+_+_+_-_+_+_+_-_+_+_+_-_+_+_+_, and this is a signature of the (1, 4)-cycle in Figure <ref>.Next lift C by means of g to g(C). By the proof of Lemma <ref>, a signature of g(C) is obtained from a signature of C by replacing every character with a string composed of the characters +_, +_, -_, and -_, subject to the following system of rewriting rules.+_ → +_+_+_+_ → -_-_ → -_-_-_-_ → +_ We refer to this transformation as a g-rewrite. For instance, the g-rewrite of +_+_+_+_+_+_+_+_ is +_+_+_-_+_+_+_-_+_+_+_-_+_+_+_-_, and this is a signature of the (2, 3)-cycle in Figure <ref>, top.Lastly, lift C by means of h to h(C). By the proof of Lemma <ref>, a signature of h(C) is obtained from a signature of C by replacing every character with a string composed of the characters +_, +_, -_, and -_, subject to the following system of rewriting rules.+_ → +_+_+_+_+_+_ → -_-_-_-_ → -_-_-_-_-_-_ → +_+_+_ We refer to this transformation as an h-rewrite. For instance, the h-rewrite of +_+_+_+_+_+_+_+_ is +_+_+_+_+_-_-_-_+_+_+_+_+_-_-_-_+_+_+_+_+_-_-_-_+_+_+_+_+_-_-_-_, and this is a signature of the (2, 5)-cycle in Figure <ref>, top.Given a string e = e_1e_2… e_l composed of the characters , , andand a string s composed of the characters +_, +_, -_, and -_, we write R_e(s) for the string obtained by successively applying rewrites of types e_l, e_l - 1, …, e_1 to s. Let L be a skew basic (p, q)-leaper, C a cycle of L within a (p + q) × (p + q) chessboard, and e = e_1e_2… e_l a prefixof the descent of L such that C is the product of successively applying lifting transformations of types e_l, e_l - 1, …, e_1 to a shell cycle. Then e is the descent of C. Let e be the descent of C. Then a signature of C isR_e(+_+_+_+_+_+_+_+_). We proceed to show that C is trivial and that a signature of C completely determines the direction graph extracted from C. Let L be a (p, q)-leaper with p + q odd and C a cycle of L within a (p + q) × (p + q) chessboard. Then C possesses orientation-preserving fourfold rotational symmetry together with orientation-reversing axial symmetry along a vertical, a horizontal, and two diagonal axes. By induction on descent. By Theorem <ref>, the moves of C occur in pairs symmetric with respect to the center of symmetry of C. The sum of the associated direction matrices over each such pair is the zero matrix. Therefore, C is trivial.Let C = a_1a_2… a_4n and s_i be the label of square a_i in a signature s_1s_2… s_4n of C for i = 1, 2, …, 4n.Suppose that s_1 = +_. All other cases are analogous.Since s_1 = +_, the direction of the move to a_1 in C is , , , or . Since, for i = 1, 2, …, 4n - 1, the direction of the move to a_i in C and the label s_i of a_i completely determine the direction of the move from a_i to a_i + 1 in C, each of those four possibilities yields a unique possibility for the directions of all moves of C.The labeling rules in the definition of a signature are invariant under a 90^∘ rotation. By Theorem <ref>, it follows that s_i = s_i + n = s_i + 2n = s_i + 3n for i = 1, 2, …, n. Again by Theorem <ref>, the moves to a_1, a_n + 1, a_2n + 1, and a_3n + 1 in C are copies of each other under multiple-of-quarter-turn rotations. Therefore, their directions are , , , and .It follows that all four possibilities for the directions of all moves of C are cyclic shifts of each other and yield the same direction graph. Hence the following theorem. Let L be a skew basic (p, q)-leaper, C a cycle of L within a (p + q) × (p + q) chessboard, and e the descent of C. Then C is trivial and the direction graph extracted from C depends only on e. Equivalently, let C be a cycle of type k. Then the direction graph extracted from C depends only on the k-th convergent [c_1±, c_2±, …, c_k] of the even continued fraction representation of q/p.Theorem <ref> provides the basis for the following definition. The fundamental direction cycle Φ(e) of descent e is the direction graph extracted from a cycle of descent e of a skew basic (p, q)-leaper within a (p + q) × (p + q) chessboard. Equivalently, the fundamental direction cycle Φ[c_1±, c_2±, …, c_k] of type [c_1±, c_2±,…, c_k] is the direction graph extracted from a cycle of type k of a skew basic (p, q)-leaper within a (p + q) × (p + q) chessboard, where [c_1±, c_2±, …, c_k] is the k-th convergent of the even continued fraction representation of q/p. The two notations for a fundamental direction cycle are related as follows. The fundamental direction cycles of descent e and type [c_1±, c_2±, …, c_k] coincide if e admits a partitioning into (possibly empty) substrings of the form e = e'_1e”_1e'_2e”_2… e”_k - 1e'_k, where e'_i is a run of the characterof length c_i/2 - 1 for i = 1, 2, …, k and e”_i consists of a single character,if the sign following c_i is - andif it is +, for i = 1, 2, …, k - 1.We go on to second leaper cycles. Let L be a (p, q)-leaper with p + q odd and C a cycle of L within a (p + q) × (p + q) chessboard. Then the canonical second leaper associated with C and the canonical second leaper cycle over the squares of C are the ones constructed in the proof of Theorem <ref>. Let D be a canonical second leaper cycle of a canonical second leaper M over the squares of C.Then D is the product of successively applying a series of lifting transformations to a canonical second leaper cycle D' of M within the shell of a frame F'. If F' is a (1, 2)-frame, then M is a (0, 1)-leaper and no direction graph can be extracted from D. Otherwise, F' is the product of lifting a smaller frame by means of either g or h. Let L be a skew basic (p, q)-leaper, D a canonical second leaper cycle of a skew basic canonical second leaper over the squares of a cycle of L within a (p + q) × (p + q) chessboard, and e = e_1e_2… e_l a prefix of the descent e_1e_2… e_m, l < m, of L such that D is the product of successively applying lifting transformations of types e_l, e_l - 1, …, e_1 to a shell canonical second leaper cycle. Then e is the descent of D and e_l + 1 is the origin of D. The origin of a canonical second leaper cycle of a skew basic canonical second leaper is alwaysor .When D is of origin , the direction graph extracted from D' is represented by the oriented cycle labeled . For instance, such is the case with the (1, 2)-cycle in Figure <ref>.When D is of origin , the direction graph extracted from D' is represented by the oriented cycle labeled . For instance, such is the case with the (1, 2)-cycle in Figure <ref>.Impose an orientation on D. By induction on descent following the proofs of Lemmas <ref>, <ref>, and <ref>, two skew directions occur as the directions of the moves to and from a square in D if and only if they occur as the directions of the moves to and from a square in one of the two orientations of D'. Let L be a skew basic (p, q)-leaper and D a canonical second leaper cycle of a skew basic canonical second leaper over the squares of a cycle of L within a (p + q) × (p + q) chessboard.Impose an orientation on D and label every square a of D +_, +_, -_, or -_ as follows. Refer to the tableLabel of aDirections of moves to and from a +_ , , ,+_ , , ,-_ , , ,-_ , , ,if D is of origin , and to the tableLabel of aDirections of moves to and from a +_ , , ,+_ , , ,-_ , , ,-_ , , ,if D is of origin .Then the cyclic string formed by the vertex labels of D is a signature of D. For instance, +_+_+_+_+_+_+_+_ is a signature of every shell canonical second leaper cycle, such as the (1, 2)-cycle in Figure <ref> and the (1, 2)-cycle in Figure <ref>. Let L be a (p, q)-leaper with p + q odd and D a canonical second leaper cycle over the squares of a cycle of L within a (p + q) × (p + q) chessboard. Then D possesses orientation-preserving fourfold rotational symmetry together with orientation-reversing axial symmetry along a vertical, a horizontal, and two diagonal axes. Furthermore, all of the aforementioned symmetries preserve the partitioning of D into eight disjoint nonempty paths, one within each section of the associated frame, as in the definition of a proper cycle. By induction on descent. Analogously to the case of an L-cycle, it follows from Theorem <ref> that D is trivial and that the signature of D completely determines the direction graph extracted from D.Let us track how a signature evolves when we lift a cycle.Given a signature s of D, partition s into eight substrings s^, s^, …, s^ so that, for each direction i out of , , …, , s^i is the string formed by the vertex labels of the portion of D within the section F_i of the associated frame F.By Theorem <ref>, s^ = s^ = s^ = s^, s^ = s^ = s^ = s^, and each of the eight strings s^, s^, …, s^ is a palindrome.Therefore, a signature s of D is completely determined by its section pair, the ordered pair (s^, s^).Let+_ = -_ +_ = -_ -_ = +_ -_ = +_and+_ = -_ +_ = -_ -_ = +_ -_ = +_.Given a string w = w_1w_2… w_n composed of the characters +_, +_, -_, and -_, we write w for the string w_1 w_2…w_n, [w for the string w_1w_2… w_n, and w] for the string w_1w_2… w_n - 1w_n.Lift D by means of f to f(D). By the proof of Lemma <ref>,(s^,[s^ s^ s^])is the section pair of a signature of f(D).This transformation and its analogues for g and h suffice to establish Theorem <ref>, but they are not well-suited to a proof of Theorem <ref>. For this reason, we introduce a change of variables.If |s^| ≤ |s^|, then partition s^ into three substrings s^ = s^Lefts^Corners^Right such that |s^Left| = |s^Right| and |s^Corner| = |s^|, and set s^Side = s^Rights^s^Left.If |s^| ≥ |s^|, then partition s^ into three substrings s^ = s^Lefts^Sides^Right such that |s^Left| = |s^Right| and |s^Side| = |s^|, and set s^Corner = s^Rights^s^Left.In both cases, s equals, up to a cyclic shift, the concatenation s^Corners^Side s^Corners^Side s^Corners^Side s^Corners^Side. We refer to the ordered pair (s^Corner, s^Side) as the corner-side pair of s.We proceed to show that(s^Sides^Corners^Side, s^Side)is the corner-side pair of a signature of f(D). We consider the case |s^| ≤ |s^| in detail, and the opposite case is analogous.Sinces^ = s^Lefts^Corners^Rightand[s^ s^ s^]= [s^Left s^Corner s^Right s^ s^Left s^Corner s^Right] == [s^Left s^Corner s^Side s^Corner s^Right],a corner-side pair of a signature of f(D) is(s^Corner s^Right]s^Lefts^Corners^Right[s^Left s^Corner, s^Side). However, since D is a proper cycle,s^Corner s^Right] = s^Rights^and[s^Left s^Corner = s^s^Left. Therefore, we can rewrite the above as(s^Rights^s^Lefts^Corners^Rights^s^Left, s^Side),as needed.We refer to the transformation that maps (s^Corner, s^Side) to (s^Sides^Corners^Side,s^Side) as an f-rearrangement. For instance, the f-rearrangement of (+_, +_) is (+_+_+_, -_), and +_+_+_-_+_+_+_-_+_+_+_-_+_+_+_-_ is a signature of the (1, 2)-cycle in Figure <ref>.Next lift D by means of g to g(D). Analogously, by the proof of Lemma <ref>,(s^Corner,s^Corners^Sides^Corner)is the corner-side pair of a signature of g(D).We refer to this transformation as a g-rearrangement. For instance, the g-rearrangement of (+_, +_) is (-_, +_+_+_), and -_+_+_+_-_+_+_+_-_+_+_+_-_+_+_+_ is a signature of the (1, 2)-cycle in Figure <ref>.Lastly, lift D by means of h to h(D). Analogously, by the proof of Lemma <ref>,(s^Corners^Sides^Corners^Sides^Corner, s^Corner s^Side s^Corner)is the corner-side pair of a signature of h(D).We refer to this transformation as an h-rearrangement. For instance, the h-rearrangement of (+_, +_) is (+_+_+_+_+_, -_-_-_), and +_+_+_+_+_-_-_-_+_+_+_+_+_-_-_-_+_+_+_+_+_-_-_-_+_+_+_+_+_-_-_-_ is a signature of the (1, 2)-cycle in Figure <ref>.Given a string e = e_1e_2… e_l composed of the characters , , andand an ordered pair of strings (s', s”) composed of the characters +_, +_, -_, and -_, we write W_e(s', s”) for the ordered pair of strings obtained by successively applying rearrangements of types e_l, e_l - 1, …, e_1 to (s', s”).Let e be the descent of D. ThenW_e(+_, +_)is the corner-side pair of a signature of D. Hence the following theorem. Let L be a skew basic (p, q)-leaper, D a canonical second leaper cycle of a skew basic canonical second leaper over the squares of a cycle C of L within a (p + q) × (p + q) chessboard, e the descent of D, and o the origin of D. Then D is trivial and the direction graph extracted from D depends only on e and o. Equivalently, let C be a cycle of type k. Then the direction graph extracted from D depends only on the k-th convergent [c_1±, c_2±, …, c_k] of the even continued fraction representation of q/p and the sign following c_k in that representation.Theorem <ref> provides the basis for the following definition. The second fundamental direction cycle Φ^II_o(e) of descent e and origin o is the direction graph extracted from a canonical second leaper cycle of descent e and origin o. Equivalently, the second fundamental direction cycle Φ^II_ϵ[c_1±, c_2±, …, c_k] of type [c_1±, c_2±,…, c_k] and sign ϵ is the direction graph extracted from a canonical second leaper cycle over the squares of a cycle of type k of a skew basic (p, q)-leaper within a (p + q) × (p + q) chessboard, where [c_1±, c_2±, …, c_k] is the k-th convergent of the even continued fraction representation of q/p and the sign following c_k in that representation is ϵ. The two notations for a second fundamental direction cycle are related as follows. The direction graphs Φ^II_o(e) and Φ^II_ϵ[c_1±, c_2±, …, c_k] coincide if e and [c_1±, c_2±, …, c_k] are related as in the case of an L-cycle, and ϵ = - if o = and ϵ = + if o =.When D is a cycle of a (0, 1)-leaper, Theorem <ref> continues to apply in the sense that D is an instantiation of both Φ^II_(e) and Φ^II_(e).The equivalence permutationπ^II = ([; ]),induced by the unit 2 × 2 matrix and ( [10;0 -1 ]), maps the signature labeling rules for a canonical second leaper cycle of originto the signature labeling rules for a canonical second leaper cycle of origin . Since Φ^II_(e) and Φ^II_(e) are determined by a common signature, it follows that they are equivalent under π^II.We conclude this section by showing that the family of all fundamental direction cycles and the family of all second fundamental direction cycles are essentially the same family. Let =, =, and =. The flip of a string e = e_1e_2… e_l composed of the characters , , andis the string e_l e_l - 1…e_1. A string e' is the flip of e” if and only if e” is the flip of e'. Let the strings e' and e” composed of the characters , , andbe flips of each other. Then the fundamental direction cycle of descent e' and the two second fundamental direction cycles of descent e” are equivalent.In other words, there exists a one-to-two equivalence mapping between fundamental direction cycles and second fundamental direction cycles. The equivalence permutationπ_ = ([; ]),induced by the matricesP_ = ([ 1/√(2) 1/√(2); 1/√(2) - 1/√(2) ])andQ_ = ([ 1/√(2) - 1/√(2); 1/√(2) 1/√(2) ]),maps the signature labeling rules for an L-cycle to the signature labeling rules for a canonical second leaper cycle of origin .The equivalence permutationπ_ = ([; ]),induced by the matricesP_ = Q_ = ([ 1/√(2) 1/√(2); 1/√(2) - 1/√(2) ]),maps the signature labeling rules for an L-cycle to the signature labeling rules for a canonical second leaper cycle of origin .(Furthermore, π^II∘π_ = π_.)We are left to show that Φ(e'), Φ^II_(e”), and Φ^II_(e”) are determined by a common signature.A signature of Φ(e') isR_e'(+_+_+_+_+_+_+_+_) = R_e'(+_)R_e'(+_)… R_e'(+_). Therefore, it suffices to show that(R_e'(+_), R_e'(+_)) = W_e”(+_, +_). We proceed by induction on descent.When e' and e” are both the empty string, equality holds.Suppose, then, that equality holds for e' and e”. We need to show that equality holds for e' and e”, e' and e”, and e' and e”. We consider the case of e' and e” in detail, and all other cases are analogous.For all strings e composed of the characters , , andand all strings s composed of the characters +_, +_, -_, and -_,R_e(s) = R_e(s). Therefore,(R_e'(+_),R_e'(+_))= (R_e'(R_(+_)),R_e'(R_(+_)))= (R_e'(-_),R_e'(+_+_+_))= (R_e'(+_),R_e'(+_)R_e'(+_)R_e'(+_))= W_(R_e'(+_),R_e'(+_))= W_(W_e”(+_, +_))= W_e”(+_, +_). This completes the proof of the theorem. § DUAL BOARDS AND DUAL DIRECTION GRAPHS Theorem <ref> can be strengthened as follows. Let L be a (p, q)-leaper with p + q odd, C a cycle of L within a (p + q) × (p + q) chessboard, M the canonical second leaper associated with C, and D the canonical second leaper cycle of M over the squares of C. Then D is the leaper graph of M over the squares of C. It suffices to consider the case when L is a skew basic leaper and p < q. Let F be the associated (p, q)-frame and r and s, r < s, the proportions of M. The proof proceeds by induction on descent.When D is a shell cycle, the theorem holds.Suppose, then, that the theorem holds for D. The proportions of L and M satisfy s ≤ p. Furthermore, s ≤ q - p unless the origin of D isand the descent of D is a run of the character . By induction on descent. Case f. Lift F and D to H and E by means of f. For each direction i out of , , …, , let S_i be the subboard (F + v^f_i) ∩ H of H.Let a and b be two squares in E joined by a move of M. By Lemma <ref>, s ≤ p. Therefore, there exists a direction i out of , , …,such that both a and b belong to S_i. By the fact that D is a proper cycle and the definition of E, a and b are the images under v^f_i of two squares a' and b' of D.Since a' and b' are joined by a move of M, by the induction hypothesis they are adjacent in D. Therefore, by the fact that D is a proper cycle and the definition of E, a and b are adjacent in E.Case g. Lift F and D to H and E by means of g. For each direction i out of , , , and , let S_i be the subboard (F + v^g_i) ∩ (H_i - 1∪ H_i ∪ H_i + 1) of H, and, for each direction i out of , , , and , let S_i be the subboard (F + v^g_i) ∩ H of H.If s ≤ q - p, the proof continues as in Case f.Otherwise, by Lemma <ref> the origin of D isand the descent of D is a run of the character .Let O be the center of symmetry of D. Introduce a Cartesian coordinate system Oxy over the infinite chessboard and write (x, y) for the square centered at (x, y).Let l the length of the descent of D. Then the set S' of all squares ((s - r)x', (s - r)y') such that x' and y' are integers and |x'| + |y'| = l + 2, and the set S” of all squares (±[(s - r)x” + r], ±[(s - r)y” + r]) such that x” and y” are nonnegative integers and x” + y” = l + 1, form a partitioning of the vertex set of E.Suppose first that both squares a and b = a + v belong to S'. Then both coordinates of the translation v are even and, since M is a basic leaper, v cannot be an M-translation. Analogous reasoning applies to the case when both a and b belong to S”.Suppose, then, that a belongs to S'. Then all but two of the squares of S” lie outside of the (2s + 1) × (2s + 1) chessboard centered at a. Therefore, every square in S' is joined by a move of M to at most two squares in S”.It follows that two squares of E are joined by a move of M if and only if they are adjacent in E.Case h. Lift F and D to H and E by means of h. For each direction i out of , , …, , let S_i be the subboard (F + v^h_i) ∩ H of H. The proof continues as in Case f. In other words, in the setting of Theorem <ref>, the leaper graphs of L and M over the board formed by the squares of C are isomorphic. A board B is dual with respect to two distinct leapers L and M if B contains more than one square and the leaper graphs of L and M over B are connected and isomorphic. The notion of a dual board raises a number of questions. Given two distinct leapers L and M, does there exist a board dual with respect to L and M?A necessary condition is that L and M are obtained from two basic leapers by means of the same scaling and rotation. Therefore, it suffices to study the case of both of L and M being basic leapers.By Theorems <ref> and <ref> and Corollary <ref>, a sufficient condition for basic L and M is that either one of them is a (0, 1)-leaper or both of them are skew leapers and the descent of one of them is a suffix of the descent of the other. Given two distinct leapers L and M, does there exist a polyomino board dual with respect to L and M?When one of L and M is a (0, 1)-leaper, this question is equivalent to the previous one. Given two distinct leapers L and M such that a board dual with respect to L and M does exist, what is the least number of squares that it may contain? Is the number of squares that it may contain unbounded from above?A dual board given by Theorem <ref> is never minimal, as removing any square from it yields a board dual with respect to the same pair of leapers.Corollary <ref> answers one special case of the second part of the question. For what positive integers n ≥ 2 does there exist a board dual with respect to n pairwise distinct leapers?A strengthening of Theorem <ref> analogous to Theorem <ref> shows that, in the setting of Theorem <ref>, the leaper graph of a (1, 2)-leaper over the squares of the unique cycle of type k is a cycle if and only if e_l = and l_k is indivisible by three. This yields an infinite family of boards dual with respect to three pairwise distinct leapers.Let L be a skew basic (p, q)-leaper, C a cycle of L within a (p, q)-frame F, M the canonical second leaper associated with C, and D the canonical second leaper cycle of M over the squares of C.We go on to describe all isomorphisms between C and D.Let D visit m squares in each side section of F and n squares in each corner section of F, D = a^a^… a^ be a partitioning of D into eight disjoint paths such that, for all directions i out of , , …, , a^i lies within F_i, a^ = a_1a_2… a_m, a^ = a_m + 1a_m + 2… a_m + n, …, a^ = a_4m + 3n + 1a_4m + 3n + 2… a_4(m + n), and, for all integers j, a_j + 4(m + n)≡ a_j.Given an integer x and a positive integer y, we write x % y for the remainder of x upon division by y, with 0 ≤ x % y ≤ y - 1.For ϵ = 0, 1 and all integers j, letυ(ϵ, j) = m + n + 2[(ϵ m - j) % (m + n)] + 1. Since D is a proper cycle, for every integer j the square a_j is linked by a move of L to both of the squares a_j + υ(0, j) and a_j + υ(1, j).By Theorem <ref>, m + n is even. Therefore, the parity of j differs from the parity of j + υ(ϵ, j) for ϵ = 0, 1 and all integers j.Furthermore, a_j' = a_j” + υ(ϵ, j”) if and only if a_j” = a_j' + υ(ϵ, j'), for ϵ = 0, 1 and all integers j' and j”.It follows that drawing an arc from a_j to a_j + υ(0, j) for all even j and from a_j to a_j + υ(1, j) for all odd j yields an orientation of C.Let ψ(1) = 1,ψ(j + 1) = ψ(j) + υ(ψ(j) % 2, ψ(j))for j = 1, 2, …, 4(m + n) - 1, and ψ(j + 4(m + n)) = ψ(j) for all integers j.Then the mappinga_j → a_ψ(j)is an isomorphism from D onto C, and all other such isomorphisms are obtained from it by means of rotation and reflection.When j takes on the values 1, 2, …, 4(m + n), the difference ψ(j + 1) - ψ(j) = υ(ψ(j) % 2, ψ(j)) takes on the same values as υ(j % 2, j). Those are all integers between m + n and 3(m + n) congruent to n - m - 1 modulo four, each occurring precisely eight times. Hence the following theorem. Let L be a (p, q)-leaper with p + q odd, C a cycle of L within a (p + q) × (p + q) chessboard, M the canonical second leaper associated with C, D the canonical second leaper cycle of M over the squares of C, and 4μ the length of both C and D.Impose orientations on C and D. Given a square a of D, let the arc from a in C point to b and the D-displacement of a be the number of steps along D that lead from a to b.Then the multiset of all D-displacements consists of all integers between μ and 3μ congruent to α modulo four, each occurring precisely eight times, where α = 1 or α = 3 depending on the orientations of C and D.We proceed to look at the isomorphisms between C and D from the point of view of direction graphs.Let r and s, r < s, be the proportions of M. Suppose that M is a skew basic leaper, and let o be the origin of D and e the descent of C and D.By the proof of Theorem <ref>, the skew direction of the move of M from a_j to a_j + 1 depends only on o, e, and j.Therefore, there exists an enumerationa^II_1, a^II_2, …, a^II_4(m + n)(with a^II_j + 4(m + n)≡ a^II_j for all integers j) of the vertices of the second fundamental direction cycle Φ^II_o(e), depending only on o and e and not on the remainder of the descent of L, such that D is an instantiation of Φ^II_o(e) under the mapping a^II_j → a_j.The move of L from a_j to a_j + υ(0, j) is of skew directionwhen a_j belongs to F_∪ F_,when a_j belongs to F_∪ F_, and similarly for all other possible positions of a_j. Analogously, the move of L from a_j to a_j + υ(1, j) is of skew directionwhen a_j belongs to F_∪ F_,when a_j belongs to F_∪ F_, and similarly for all other possible positions of a_j.Since m and n are completely determined by e, the section of F that a_j belongs to is completely determined by e and j. Therefore, the skew direction of the move of L from a_j to a_j + υ(ϵ, j) is completely determined by e, j, and ϵ.It follows that there exists an enumerationa^I_1, a^I_2, …, a^I_4(m + n)(with a^I_j + 4(m + n)≡ a^I_j for all integers j) of the vertices of the fundamental direction cycle Φ(e), depending only on e and not on the remainder of the descent of L, such that C is an instantiation of Φ(e) under the mapping a^I_j → a_ψ(j).LetA_ = ([ 1 0; 2 1 ]),A_ = ([01; -12 ]),A_ = ([ 0 1; 1 2 ]),and, for all strings e_1e_2… e_l composed of the characters , , and ,A_e_1e_2… e_l = A_e_1A_e_2⋯ A_e_l. Then the proportions of L and M satisfy([ p; q ]) = A_eA_o([ r; s ]). Let A^I(j) be the direction matrix of the arc from a^I_j to a^I_j + 1 in Φ(e), Dist^I(j', j”) the distance from a^I_j' to a^I_j” in Φ(e), A^II(j) the direction matrix of the arc from a^II_j to a^II_j + 1 in Φ^II_o(e), and Dist^II(j', j”) the distance from a^II_j' to a^II_j” in Φ^II_o(e), for all integers j, j', and j”.ThenDist^II(ψ(j), ψ(j + 1))([ r; s ]) = A^I(j)([ p; q ]),or, equivalently,Dist^II(ψ(j), ψ(j + 1))([ r; s ]) = A^I(j)A_eA_o([ r; s ]),for all integers j.Since all objects in this identity apart from r and s depend only on o and e, it continues to hold when r and s are replaced with the proportions r' and s', r' < s', of any canonical second leaper. However, every basic leaper occurs as a canonical second leaper.Therefore,Dist^II(ψ(j), ψ(j + 1)) = A^I(j)A_eA_ofor all integers j.We refer to the above identity as the duality identity for Φ(e) and Φ^II_o(e). Its full significance will become evident once we arrive at Theorem <ref>.Let e' be the flip of e, C' of descent e' a cycle of a skew basic (p', q')-leaper L', p' < q', over a (p' + q') × (p' + q') chessboard, and D' of origin o and descent e' the canonical second leaper cycle of the canonical second leaper M' of proportions r' and s', r' < s', associated with C'.We go on to apply Theorem <ref> to the duality identity for Φ(e) and Φ^II_o(e) in order to obtain the isomorphisms from C' onto D' from the isomorphisms from D onto C.By Theorem <ref> and its proof, there exists an enumerationb^I_1, b^I_2, …, b^I_4(m + n)(with b^I_j + 4(m + n)≡ b^I_j for all integers j) of the vertices of Φ(e') such that, for all integers j,A^II(j) = P_oA^I_Flip(j)Q_o,where A^I_Flip(j) is the direction matrix associated with the arc from b^I_j to b^I_j + 1 in Φ(e').Analogously, by Theorem <ref> and its proof, there exists an enumerationb^II_1, b^II_2, …, b^II_4(m + n)(with b^II_j + 4(m + n)≡ b^II_j for all integers j) of the vertices of Φ^II_o(e') such that, for all integers j,A^II_Flip(j) = P_oA^I(j)Q_o,where A^II_Flip(j) is the direction matrix associated with the arc from b^II_j to b^II_j + 1 in Φ^II_o(e').Let, for all integers j' and j”, Dist^I_Flip(j', j”) be the distance from b^I_j' to b^I_j” in Φ(e'). Then we can rewrite the duality identity for Φ(e) and Φ^II_o(e) asP_oDist^I_Flip(ψ(j), ψ(j + 1))Q_o = P^-1_oA^II_Flip(j)Q^-1_oA_eA_ofor all integers j.Since P_o = P^-1_o in both cases o = and o =, this is equivalent toDist^I_Flip(ψ(j), ψ(j + 1)) = A^II_Flip(j)Q^-1_oA_eA_oQ^-1_o. Let A^Unit be the unit 2 × 2 matrix, e = e_1e_2… e_l, and k the number of occurrences of the characterin e.In both cases o = and o =,A_A_oQ^-1_oA_ = A_A_oQ^-1_oA_ = A_oQ^-1_o, A_A_oQ^-1_oA_ = -A_oQ^-1_o,andQ^-1_oA_oQ^-1_oA_o = -A^Unit. ThusQ^-1_oA_eA_oQ^-1_oA_e'A_o= Q^-1_oA_e_1A_e_2⋯ A_e_lA_oQ^-1_oA_e_lA_e_l - 1⋯ A_e_1A_o= ± Q^-1_oA_e_1A_e_2⋯ A_e_l - 1A_oQ^-1_oA_e_l - 1A_e_l - 2⋯ A_e_1A_o= ⋯ = (-1)^kQ^-1_oA_oQ^-1_oA_o= (-1)^k + 1 A^Unit. Therefore,Dist^I_Flip(ψ(j), ψ(j + 1))A_e'A_o = (-1)^k + 1 A^II_Flip(j). Since the proportions of L' and M' satisfy([ p'; q' ]) = A_e'A_o([ r'; s' ]),it follows thatDist^I_Flip(ψ(j), ψ(j + 1))([ p'; q' ]) = (-1)^k + 1 A^II_Flip(j)([ r'; s' ]). Let C' = b_1b_2… b_4(m + n) (with b_j + 4(m + n)≡ b_j for all integers j) so that C' is an instantiation of Φ(e') under the mapping b^I_j → b_j if k is odd, and b^I_j → b_j + 2(m + n) if it is even. Then D' is an instantiation of Φ^II_o(e') under the mapping b^II_j → b_ψ(j).Therefore, the mappingb_j → b_ψ(j)is an isomorphism from C' onto D'. All other such isomorphisms are obtained from it by means of rotation and reflection. Hence the following theorem. Let the strings e' and e” composed of the characters , , andbe flips of each other. Let e' be the common descent of the cycle C' of a skew basic (p', q')-leaper within a (p' + q') × (p' + q') chessboard and the canonical second leaper cycle D' over the squares of C', e” the common descent of the cycle C” of a skew basic (p”, q”)-leaper within a (p” + q”) × (p” + q”) chessboard and the canonical second leaper cycle D” over the squares of C”, and 4μ the common length of C', D', C”, and D”.Let ψ be a permutation of 1, 2, …, 4μ. Then there exists an enumeration C' = a'_1a'_2… a'_4μ of the vertices of C' such that the mapping a'_j → a'_ψ(j) is an isomorphism from C' onto D' if and only if there exists an enumeration D” = a”_1a”_2… a”_4μ of the vertices of D” such that the mapping a”_j → a”_ψ(j) is an isomorphism from D” onto C”.Theorem <ref> allows us to extend Theorem <ref> as follows. In the setting of Theorem <ref>, given a square c of C, let the arc from c in D point to d and the C-displacement of c be the number of steps along C that lead from c to d.Then the multiset of all C-displacements coincides with the multiset of all D-displacements. By Theorems <ref> and <ref>, the multiset of all C-displacements consists of all integers between μ and 3μ congruent to β modulo four, each occurring precisely eight times, where β = 1 or β = 3 depending on the orientations of C and D.Let the arc from a square a in C point to a square b. Then the number of steps along D that lead from a to b is congruent to α modulo four.Replace each of those steps with a path along C advancing from the same starting vertex to the same ending vertex as that step. Since each step along D is replaced with a number of steps along C congruent to β modulo four, the total number of steps along C in the resulting path, leading from a to b, is congruent to αβ modulo four.Thus αβ is congruent to one modulo four and α = β, as needed. The duality identity for Φ(e) and Φ^II_o(e) shows that the underlying source of the duality of a dual board given by Theorem <ref> is the duality of the associated direction graphs.We proceed to outline the key conditions that a direction graph needs to satisfy so that its instantiations give rise to a family of dual boards in a similar manner. A direction graph Φ is dual if there exist a direction graph Φ', a complement of Φ, a one-to-one mapping η between the vertex sets of Φ and Φ', and a real-coefficient 2 × 2 duality matrix A_Dual, distinct from all direction matrices, such that the following conditions hold.Both of Φ and Φ' are coherent and connected.The unlabeled symmetric directed graphs obtained from Φ and Φ' by discarding all arc labels are isomorphic.For every arc from η(a) to η(b) labeled i in Φ', the distance from a to b in Φ equals A_iA_Dual. The basic properties of dual direction graphs are summarized in the following theorem. Let Φ be a dual direction graph of complement Φ' and duality matrix A_Dual. Then:(a) π(Φ) is a dual direction graph for all equivalence permutations π.(b) A_Dual is a unit-determinant integer-coefficient matrix.(c) Φ' is a dual direction graph of complement Φ and duality matrix A^-1_Dual.(d) The vertex sets of all but finitely many basic leaper instantiations of Φ are dual boards. Let η be a one-to-one mapping between the vertex sets of Φ and Φ' as in the definition of a dual direction graph.Let the matrices P and Q induce an equivalence permutation. Then Q is invertible and PΦ Q is a dual direction graph of complement PΦ' Q and duality matrix Q^-1A_DualQ. This settles (a).Let L be a basic (p, q)-leaper, p < q, and G an L-instantiation of Φ under the mapping τ. Since Φ is coherent, G is the leaper graph of L over the vertex set of G for all but finitely many basic leapers L. Suppose that L is indeed such a basic leaper.Let i be the label of the arc from η(a) to η(b) in Φ' and A the distance from a to b in Φ.Since A_Dual = A^-1_iA, it is an integer-coefficient matrix.Let([ r'; s' ]) = A_Dual([ p; q ]),r = min{|r'|, |s'|}, and s = max{|r'|, |s'|}. Then r and s are nonnegative integers and r ≤ s. Let M be an (r, s)-leaper.The translation v from τ(a) to τ(b) satisfiesv^T = A_iA_Dual([ p; q ]) = A_i ([ r'; s' ]).Therefore, v is an M-translation and τ(a) and τ(b) are joined by a move of M.Since Φ' is connected, the leaper graph of M over the vertex set of G is connected. Therefore, M can imitate a move of L.Since L is a basic leaper, the leaper graph of L over the infinite chessboard is connected. Hence the leaper graph of M over the infinite chessboard is also connected and M is also a basic leaper.It follows that the coordinates of the column vector A_Dual( [ p; q ]) are relatively prime for all relatively prime p and q such that p + q is odd and p < q. Therefore, A_Dual is invertible.Let Dist(x, y) be the distance from x to y in Φ and Dist'(x', y') the distance from x' to y' in Φ'. It follows from the definition of a dual direction graph that, for all vertices x and y of Φ,Dist(x, y) = Dist'(η(x), η(y))A_Dual. Therefore,Dist'(x', y') = Dist(η^-1(x'), η^-1(y'))A^-1_Dualfor all vertices x' and y' of Φ', and in particular for all x' and y' such that η^-1(x') and η^-1(y') are adjacent in Φ.Thus Φ' is a dual direction graph of complement Φ and duality matrix A^-1_Dual, settling (c).Consequently, A^-1_Dual is also a duality matrix and thus integer-coefficient. It follows that A_Dual is unit-determinant, settling (b).Since A_Dual is distinct from all direction matrices, L and M are distinct leapers for all but finitely many basic leapers L. Suppose that L is indeed such a basic leaper.There exists a direction matrix A_j, unique in the case when M is a skew leaper, such that([ r; s ]) = A_j([ r'; s' ]). Consider the direction graph Φ'A^-1_j. The graph G' over the vertex set of G such that an arc points from τ(x) to τ(y) in G' if and only if an arc points from η(x) to η(y) in Φ' is an M-instantiation of Φ'A^-1_j under the mapping η(x) →τ(x).Since Φ' is coherent, so is Φ'A^-1_j for every direction matrix A_j. Therefore, G' is the leaper graph of M over the squares of G' for all but finitely many basic leapers M and, since A_Dual is invertible, for all but finitely many basic leapers L. Suppose that L is indeed such a basic leaper.Then the vertex set of the L-instantiation of Φ is a board dual with respect to L and M, settling (d) and completing the proof of the theorem.Let e be a string composed of the characters , , and , and o one of the charactersand . Then both the fundamental direction cycle Φ(e) and the second fundamental direction cycle Φ^II_o(e) are dual direction graphs. Each of them is a complement of the other, the duality matrix of Φ^II_o(e) being A_eA_o and that of Φ(e) its inverse. By the duality identity for Φ(e) and Φ^II_o(e) and Theorem <ref>. In general, dual boards and dual direction graphs are not related in as straightforward a manner as Theorems <ref> and <ref> might suggest. For instance, there exist boards dual with respect to two distinct skew leapers such that the extracted direction graphs are not complements of each other. One example of this is the board formed by the squares of the cycles depicted in Figures <ref>, top and <ref>, viewed as dual with respect to a (1, 2)-leaper and a (2, 3)-leaper. Do there exist boards dual with respect to two distinct skew leapers such that at least one of the extracted direction graphs is not a dual direction graph, or such that at least one of the associated leaper graphs contains a nontrivial cycle and does not allow the extraction of a direction graph at all?Similarly to the notion of a dual board, the notion of a dual direction graph raises a number of questions. Given a unit-determinant integer-coefficient 2 × 2 matrix A distinct from all direction matrices, does there exist a dual direction graph of duality matrix A?By Theorem <ref>, a sufficient condition for A is that it is a nonempty product of the matrices A_, A_, and A_. Given a matrix A such that a dual direction graph of duality matrix A does exist, what is the least number of vertices that it may contain? Is the number of vertices that it may contain unbounded from above?Analogously to the case of a dual board, a dual direction graph given by Theorem <ref> is never minimal, as removing any vertex from it yields a dual direction graph of the same duality matrix.Corollary <ref> answers one special case of the second part of the question. For what positive integers n ≥ 2 do there exist n dual direction graphs which are pairwise complements?In particular (as, unlike in the case of a dual board, Theorem <ref> does not yield an example of n = 3), do there exist three dual direction graphs which are pairwise complements?§ CONSTRUCTIONS OF DUAL BOARDSAND DUAL DIRECTION GRAPHS I We set out to add one more lifting transformation, ħ, to f, g, and h. To this end, first we extend the definitions of f, g, and h so that they act upon a more general family of cycles.Introduce a Cartesian coordinate system Oxy over the infinite chessboard such that the centers of the squares are the integer points. For each direction i out of , , …, , let α^i be the ray emanating from O and pointing in direction i.We say that a square lies inside, or strictly inside, a region if its center does.Given a path w through a number of squares, we write w_Start and w_End for the opening and final squares of w, and, if w is of odd length, w_Mid for the middle square of w.In the context of a cycle C and an orientation imposed on C, given two squares a and b of C, we write [a; b] for the portion of C that runs from a to b and (a; b) for the portion of C that runs from the square directly following a to the square directly preceding b.Given a path or a cycle w through a number of squares, we write κ(w) for the broken line formed by joining the centers of the squares of w in the order in which they are visited by w. A cycle D of a leaper M is (p, q)-perfect, p < q, if it admits a perfect partitioning into eight disjoint nonempty paths,D = a^a^… a^,satisfying the following conditions.Translation.a^a^ + (-q, -p)= -a^-a^a^a^ + (-p, -q)= -a^-a^a^a^ + (p, -q)= -a^-a^a^a^ + (q, -p)= -a^-a^ Symmetry. For all directions i out of , , …, , a^i is of odd length, a^i_Mid lies on the ray α^i, and a^i is symmetric with respect to α^i. Furthermore, a 90^∘ rotation about O maps a^i onto a^i + 2.Separation. For all directions i out of , , …, , the counterclockwise portion (a^i_Mid; a^i + 1_Mid) of D lies strictly inside the 45^∘ angle between the rays α^i and α^i + 1.Simplicity. The broken line κ(D) is the contour of a simple polygon. In particular, D is a simple cycle.Coherence. D is the leaper graph of M over the squares of D. Protocoherence. No square of D is joined by a move of M to a square of either D + (2q, 0) or D + (p + q, p + q).Figure <ref> illustrates the definition of a perfect cycle.We begin with a couple of lemmas. Let D be a (p, q)-perfect cycle. Then D lies strictly inside the square bounded by the lines y = ± x ± (p + q), and strictly inside the square bounded by the lines x = ± q and y = ± q. Let D = a^a^… a^ be a perfect partitioning of D and L a (p, q)-leaper.By the translation property of D, every square in a^a^ is joined by an L-move to a square of a^a^. By the separation property of D, a^a^ lies strictly below and to the left of the line y = -x. Therefore, a^a^ lies strictly below and to the left of the line y = -x + p + q. Analogously, so does a^a^.Since the counterclockwise portion [a^_Mid; a^_Mid] of D is a subpath of a^a^a^, it also lies strictly below and to the left of the line y = -x + p + q. By this and the separation property of D, [a^_Mid; a^_Mid] lies inside the triangle Δ formed by the lines x = 0, y = 0, and y = -x + p + q, with only a^_Mid and a^_Mid on the contour of Δ, in the interiors of the two legs.Analogous reasoning applies to [a^i_Mid; a^i + 2_Mid] for i =, , and . Therefore, D lies strictly inside the square bounded by the lines y = ± x ± (p + q).Analogously, D lies strictly inside the square bounded by the lines x = ± q and y = ± q.Let L be a (p, q)-leaper, p < q, and D a (p, q)-perfect cycle. Then two squares of D are joined by an L-move if and only if they are joined by an L-move by the translation property of D. Let D = a^a^… a^ be a perfect partitioning of D.Consider any L-translation v, say, (p, q). It suffices to show that D and D + v do not have any squares in common other than the squares of a^a^.By the separation property of D and Lemma <ref>, the line y = q strictly separates a^a^ + v and D.Consider a^a^ + v.Let R be the point (p - q/2, p + q/2). Consider the cyclic list of broken linesκ(a^a^), κ(a^a^), κ(a^a^ + v), κ(a^a^ + (-q, p)). By the translation and symmetry properties of D, 90^∘ rotation about R maps each item in this list onto the following one, cyclically. By the simplicity property of D, no two items in direct succession have a common point. It follows that no two items have a common point at all. Consequently, a^a^ + v has no squares in common with a^a^ and a^a^.By the separation property of D and Lemma <ref>, the line y = 0 strictly separates a^a^ + v and a^a^.Lastly, by the separation property of D, the line y = x strictly separates a^a^ + v and a^a^.It follows that a^a^ + v does not have any squares in common with D.Analogously, neither does a^a^ + v. By Lemma <ref>, a (p, q)-perfect cycle admits a unique perfect partitioning. Indeed, a^ is uniquely determined as the intersection of D, D + (q, p), and D + (q, -p), and analogously a^, a^, …, a^ are uniquely determined as well.Here, then, follow the extended definitions of f, g, and h. Let D be a (p, q)-perfect cycle of perfect partitioning D = a^a^… a^. Letb^ = -a^-a^-a^ + (p + q, 0)andb^ = a^ + (p, p),and define b^, b^, …, b^ symmetrically. Then the f-lift of D is the cycle f(D) = b^b^… b^ together with its partitioning into b^, b^, …, b^.Let D be a (p, q)-perfect cycle of perfect partitioning D = a^a^… a^. Letb^ = a^ + (q - p, 0)andb^ = -a^-a^-a^ + (q, q),and define b^, b^, …, b^ symmetrically. Then the g-lift of D is the cycle g(D) = b^b^… b^ together with its partitioning into b^, b^, …, b^.Let D be a (p, q)-perfect cycle of perfect partitioning D = a^a^… a^. Letb^ = -a^-a^-a^-a^-a^ + (p + q, 0)andb^ = a^a^a^ + (q, q),and define b^, b^, …, b^ symmetrically. Then the h-lift of D is the cycle h(D) = b^b^… b^ together with its partitioning into b^, b^, …, b^. We are ready to introduce ħ. Let D be a (p, q)-perfect cycle of perfect partitioning D = a^a^… a^. Letb^ = a^a^a^ + (p + q, 0)andb^ = -a^-a^-a^-a^-a^ + (q, q),and define b^, b^, …, b^ symmetrically. Then the ħ-lift of D is the cycle ħ(D) = b^b^… b^ together with its partitioning into b^, b^, …, b^. The lifting transformations h and ħ are companions in a way analogous to how f and g are. In a sense, the two lifting transformations in each companion pair are 45^∘ rotations of each other.The following lemma establishes the key properties of the extended f, g, and h and ħ as they relate to perfect cycles. Let D be a (p, q)-perfect cycle of an (r, s)-leaper M. Then:(a) Each of the four lifts of D is a cycle of M.(b) f(D) possesses the translation property with parameters (p, 2p + q), g(D) possesses the translation property with parameters (q, 2q - p), and both of h(D) and ħ(D) possess the translation property with parameters (q, p + 2q).(c) Each of the four lifts of D possesses the symmetry property.(d) Each of the four lifts of D possesses the separation property.(e) Each of the four lifts of D possesses the simplicity property.(f) Provided that s ≤ p and r + s ≤ q - p, each of the four lifts of D possesses the coherence property.(g) Provided that s ≤ p and r + s ≤ q - p, each of the four lifts of D possesses the protocoherence property with the same parameters as in (b). The proofs of parts (a) and (b) of the lemma in all cases f, g, h, and ħ are analogous to the proofs of Lemmas <ref>, <ref>, and <ref>.Part (c) follows by the symmetry property of D.The main motif of the proofs of parts (d), (e), (f), and (g) is that, locally, a lift of D looks precisely like D. In other words, for small regions R, the subgraph of a lift of D within R is a translation copy of a subgraph of D under one of the associated lifting translations.Let D = a^a^… a^ be the perfect partitioning of D.Case f. Let b^, b^, …, b^ be as in the definition of f.Consider the counterclockwise portion (b^_Mid; b^_Mid) of f(D).By the separation property of D and Lemma <ref>, -[a^_Start; a^_Mid) + (p + q, 0) lies in the interior of the 45^∘ angle between α^ and α^. By the separation property of D, so does [a^_Start; a^_Mid) + (p, p). Since it is the concatenation of those two paths, so does (b^_Mid; b^_Mid) as well. This and the symmetry property of f(D) settle (d).For (e), by the symmetry and separation properties of f(D) it suffices to establish that the broken line κ([b^_Mid; b^_Mid]) does not intersect itself.By the translation property of D, [b^_Mid; b^_Mid] is a subpath of a^a^a^ + (p, p). By the simplicity property of D, the broken line tracing the latter does not intersect itself, and this settles (e).For (f), say that two squares of f(D) form an illicit pair if they are joined by a move of M, but this move is not an edge of f(D).Let i be any direction out of , , …, .By the coherence property of D, no illicit pair can occur within b^i.Suppose that an illicit pair has one square a in b^b^ and another square b in b^b^.By the coherence property of D, no illicit pair can occur within Π = a^a^a^a^a^ + (p, p). Thus either a belongs to Π_1 = b^b^∖Π or b belongs to Π_2 = b^b^∖Π.By the translation property of D, Π_1 = -a^ + (p + q, 0). Thus, by the separation property of D and Lemma <ref>, the strip bounded by the lines y = 0 and y = p strictly separates Π_1 and b^b^. However, since s ≤ p, no illicit pair can have its two squares on different sides of this strip.By the translation property of D, Π_2 = (-a^ + (0, p + q))b^. Thus, by the separation property of D and Lemma <ref>, the strip bounded by the lines x = 0 and x = p strictly separates b^b^ and Π_2. However, since s ≤ p, no illicit pair can have its two squares on different sides of this strip.We have arrived at a contradiction. By symmetry, it follows that no illicit pair can occur within b^i ∪ b^i + 1, b^i ∪ b^i + 2, and b^i ∪ b^i + 3.Consider, then, b^b^ and b^b^. By the separation property of D and Lemma <ref>, the strip bounded by the lines x = -p and x = p strictly separates the former and the latter. As above, by symmetry it follows that no illicit pair can occur within b^i ∪ b^i + 4.This completes the proof of (f).For (g), by Lemma <ref> the strip bounded by the lines x = p + q and x = 3p + q strictly separates f(D) and f(D) + (4p + 2q, 0). However, since s ≤ p < 2p, no two squares on different sides of this strip can be joined by a move of M.Suppose, then, that a square a in f(D) is joined by a move of M to a square b in f(D) + (3p + q, 3p + q).By Lemma <ref>, f(D) + (3p + q, 3p + q) lies strictly to the right of the line x = 2p and strictly above the line y = 2p. As above, it follows from this that a lies in the portion of f(D) strictly to the right of the line x = 0 and strictly above the line y = 0. By the separation property of f(D) and the translation property of D, this portion of f(D) is a subpath of D + (p, p).Analogously, b lies in D + (2p + q, 2p + q). However, by the protocoherence property of D, no square of D + (p, p) is joined by a move of M to a square of D + (2p + q, 2p + q). We have arrived at a contradiction and the proof of (g) is complete.Case g. This case is analogous to Case f.Case h. Let b^, b^, …, b^ be as in the definition of h.Consider the counterclockwise portion (b^_Mid; b^_Mid) of h(D).By the separation property of D and Lemma <ref>, -[a^_Start; a^_Mid) + (p + q, 0) lies in the interior of the 45^∘ angle between α^ and α^. By the separation property of D and Lemma <ref>, so does [a^_Start; a^_Mid) + (q, q). Since it is the concatenation of those two paths, so does (b^_Mid; b^_Mid) as well. This and the symmetry property of h(D) settle (d).As in Case f, for (e) it suffices to establish that the broken line κ([b^_Mid; b^_Mid]) does not intersect itself.Let o = a^_End + (p + q, 0). By the translation property of D, [o; b^_Mid] is a subpath of a^a^a^a^ + (q, q). By the simplicity property of D, no self-intersection can occur within κ([b^_Mid; b^_End]) and κ([o; b^_Mid]). Therefore, any self-intersection within κ([b^_Mid; b^_Mid]) needs to occur between κ([b^_Mid; o]) and κ([b^_End; b^_Mid]).However, by the separation property of D, the strip bounded by the lines y = -x + p + q and y = -x + 2q separates the former and the latter. This settles (e).For (f), letc^ = -a^ + (p + q, 0)andc^ = a^a^a^a^a^a^a^ + (q, q),and define c^, c^, …, c^ symmetrically.By the translation property of D, h(D) = c^c^… c^, with c^i a subpath of b^i for i =, , , and , and b^i a subpath of c^i for i =, , , and .Let i be any direction out of , , …, .By the coherence property of D, no illicit pair can occur within b^i and c^i.Suppose that an illicit pair has one square a in c^ and another square b in c^c^.Since no illicit pair can occur within b^, b does not belong to b^.By the separation property of D, the strip bounded by the lines y = -x + p + q and y = -x + 2q strictly separates c^ and b^. However, since r + s ≤ q - p, no illicit pair can have its two squares on different sides of this strip. Thus b does not belong to b^.Since c^ is a subpath of D + (p + q, 0) and b^ is a subpath of D + (0, p + q), by the symmetry and protocoherence properties of D no square of c^ is joined by a move of M to a square of b^. Thus b does not belong to b^ either.Since c^c^ is a subpath of b^b^b^, we have arrived at a contradiction. By symmetry, it follows that no illicit pair can occur within c^i ∪ c^i + 1 for any i and c^i ∪ c^i + 2 for i =, , , and .Suppose that an illicit pair has one square a in c^ and another square b in c^.By Lemma <ref>, the strip bounded by the lines y = x and y = x + q - p strictly separates b^ and c^. However, since r + s ≤ q - p, no illicit pair can have its two squares on different sides of this strip.By the separation property of D and Lemma <ref>, the strip bounded by the lines y = -x + p + q and y = -x + 2q strictly separates b^ and c^. As above, no illicit pair can have its two squares on different sides of this strip.Symmetrically, no illicit pair has one square in c^ and another square in either b^ or b^.We conclude that a must belong to c^∖ b^b^ and b must belong to c^∖ b^b^. However, both of those are subpaths of b^, which contains no illicit pair.We have arrived at a contradiction. By symmetry, it follows that no illicit pair can occur within c^i ∪ c^i + 2 for i =, , , and .Lastly, by Lemma <ref>, the strip bounded by the lines x = 0 and x = p strictly separates c^ from c^ and c^, and the strip bounded by the lines y = -x and y = -x + q - p strictly separates c^ from c^ and c^. As above, by symmetry it follows that no illicit pair can occur within c^i ∪ c^i + 3 and c^i ∪ c^i + 4.This completes the proof of (f).For (g), by the separation property of D and Lemma <ref> the strip bounded by the lines x = 2q and x = 2p + 2q strictly separates h(D) and h(D) + (2p + 4q, 0). However, since s ≤ p < 2p, no two squares on different sides of this strip can be joined by a move of M.Suppose, then, that a square a in h(D) is joined by a move of M to a square b in h(D) + (p + 3q, p + 3q).By the separation property of D and Lemma <ref>, h(D) + (p + 3q, p + 3q) lies strictly to the right of the line x = p + q and strictly above the line y = p + q. Since s ≤ p, it follows from this that a lies in the portion of h(D) strictly to the right of the line x = q and strictly above the line y = q. By the separation property of h(D) and Lemma <ref>, this portion of h(D) is a subpath of c^ and thus of D + (q, q).Analogously, b lies in D + (p + 2q, p + 2q). However, by the protocoherence property of D, no square of D + (q, q) is joined by a move of M to a square of D + (p + 2q, p + 2q). We have arrived at a contradiction and the proof of (g) is complete.Case ħ. This case is analogous to Case h. Lemma <ref> supplies the induction step in the proof of Lemma <ref> below. Let M be an (r, s)-leaper, r ≤ s. Define the initial cycles of M as follows.Provided that r ≠ 0, the initial cycle of M of type , D^M_, is the M-instantiation, centered at O, of the direction graph represented by the oriented cycle labeled . It contains the eight squares (r + s, 0), (r, r), (0, r + s), (-r, r), (-r - s, 0), (-r, -r), (0, -r - s), and (r, -r).Provided that r ≠ s, the initial cycle of M of type , D^M_, is the M-instantiation, centered at O, of the direction graph represented by the oriented cycle labeled . It contains the eight squares (s - r, 0), (s, s), (0, s - r), (-s, s), (r - s, 0), (-s, -s), (0, r - s), and (s, -s).The initial cycle of M of type , D^M_, is the M-instantiation, centered at O, of the direction graph represented by the oriented cycle labeled . It contains the eight squares (r + s, 0), (s, s), (0, r + s), (-s, s), (-r - s, 0), (-s, -s), (0, -r - s), and (s, -s).Let M be an (r, s)-leaper, r ≤ s, o one of the characters , , andsuch that there exists an initial cycle of M of type o, and e = e_1e_2… e_l a string composed of the characters , , , and .Let A_ = A_, A_e = A_e_1A_e_2… A_e_l, and([ p; q ]) = A_eA_o([ r; s ]). Then successively applying lifting transformations of types e_l, e_l - 1, …, e_1 to D^M_o yields a (p, q)-perfect cycle D^M_o(e). We proceed by induction on e.When e is the empty string, the lemma holds by the definition of an initial cycle.Suppose, then, that the lemma holds for e. Let e_0 be one of the characters , , , andand([ p'; q' ]) = A_e_0([ p; q ]). We need to show that applying a lifting transformation of type e_0 to D^M_o(e) yields a (p', q')-perfect cycle D^M_o(e_0e).Case 1. s > p or r + s > q - p. By Lemma <ref>, it suffices to show that D^M_o(e_0e) possesses the coherence and protocoherence properties.By induction on e, if s > p then o = and e is a run of the character . Analogously, if r + s > q - p then o = and e is a run of the character .We consider the former case in detail, and the latter case is analogous.Let o = and e be a run of the characterof length n.Define a horizontal band |a; b| of endpoints a and b and length m + 1, with a and b squares and m a nonnegative integer such that b = a + m · (2r, 0), as the set of all squares of the form a + i · (2r, 0), i = 0, 1, …, m. Analogously, define a vertical band |a; b| of endpoints a and b and length m + 1, with a and b squares and m a nonnegative integer such that b = a + m · (0, 2r), as the set of all squares of the form a + i · (0, 2r), i = 0, 1, …, m.(In the case of o = and e being a run of the character , forward and backward bands are defined analogously, with the translations (2r, 0) and (0, 2r) replaced by (s - r, s - r) and (r - s, s - r).)By induction on n, the vertex set of D^M_(e_0e) is the union of a number of bands as follows. In each case, “rotations” stands for “multiple-of-quarter-turn rotations about O” and “reflections” stands for “reflections in the lines x = 0, y = 0, and y = ± x”. Furthermore, in each case we classify all bands in question as either inner or outer ones.The vertex set of D^M_(e) is the union of four inner bands of length n + 3, namely the rotations of|((n + 2)r, -(n + 2)r); ((n + 2)r, (n + 2)r)|,and four outer bands of length n + 2, namely the rotations of|((n + 2)r + s, -(n + 1)r); ((n + 2)r + s, (n + 1)r)|. The vertex set of D^M_(e) is the union of twelve inner bands of lengths n + 1 and n + 2, namely the rotations of|((3n + 2)r + s, -(n + 1)r); ((3n + 2)r + s, (n + 1)r)|and the rotations and reflections of|((n + 2)r + s, (n + 1)r); ((3n + 2)r + s, (n + 1)r)|,and twelve outer bands of length n + 1, namely the rotations of|((3n + 2)r + 2s, -nr); ((3n + 2)r + 2s, nr)|and the rotations and reflections of |((n + 1)r + s, (n + 1)r + s); ((3n + 1)r + s, (n + 1)r + s)|. The vertex set of D^M_(e) is the union of twenty inner bands of lengths n + 1 and n + 2, namely the rotations of|((n + 2)r, -nr); ((n + 2)r, nr)|and the rotations and reflections of|((n + 3)r + s, (n + 1)r + s); ((3n + 3)r + s, (n + 1)r + s)|and|((3n + 3)r + s, (n + 1)r + s); ((3n + 3)r + s, (3n + 3)r + s)|,and twenty outer bands of lengths n + 1 and n + 2, namely the rotations of|((n + 2)r + s, -(n + 1)r); ((n + 2)r + s, (n + 1)r)|and the rotations and reflections of|((n + 2)r + s, (n + 1)r); ((3n + 2)r + s, (n + 1)r)|and|((3n + 3)r + 2s, (n + 2)r + s); ((3n + 3)r + 2s, (3n + 2)r + s)|. Lastly, the vertex set of D^M_( e) is the union of twelve inner bands of length n + 2, namely the rotations of|((3n + 4)r + s, -(n + 1)r); ((3n + 4)r + s, (n + 1)r)|and the rotations and reflections of|((n + 2)r + s, (n + 1)r); ((3n + 4)r + s, (n + 1)r)|,and twelve outer bands of lengths n + 1 and n + 2, namely the rotations of|((3n + 4)r + 2s, -nr); ((3n + 4)r + 2s, nr)|and the rotations and reflections of |((n + 1)r + s, (n + 1)r + s); ((3n + 3)r + s, (n + 1)r + s)|. Figure <ref> shows the bands making up D^(1, 3)_(), corresponding to M being a (1, 3)-leaper, o =, n = 2, e =, and e_0 =, with all inner bands dotted and all outer bands dashed.To establish the coherence of D^M_o(e_0e), it suffices to show that it does not contain an illicit pair.First we show that no two bands of the same type contain an illicit pair for reasons of parity.Namely, by tracing D^M_o(e_0e), M travels from every square in an inner band to every other square in an inner band in an even number of moves, and similarly for outer bands. Since no odd cycle of a leaper exists, two squares in bands of the same type are never joined by a move of M.Then we show that no two bands of opposite types contain an illicit pair because, roughly, they are too far away.There are three kinds of relative positions that bands of opposite types in D^M_o(e_0e) occupy, as follows.Two bands of opposite types are tied if they run in parallel, at a distance of s, and D^M_o(e_0e) zigzags between them, visiting all of their squares in succession.Two bands of opposite types are adjacent if an edge of D^M_o(e_0e) joins an endpoint a of the first band to an endpoint b of the second band, and no other edges of D^M_o(e_0e) join any of the bands' squares. Two adjacent bands always make obtuse angles with the segment joining the centers of a and b.Two bands of opposite types are independent if no edge of D^M_o(e_0e) joins a square of one band to a square of the other.All three kinds of relative positions are exemplified in Figure <ref>.No illicit pair can occur within two tied or adjacent bands.Define the envelope of a band |a; b| as the regular chessboard of lower left corner a + (-s, -s) and upper right corner b + (s, s). All squares joined by a move of M to a square in a band B belong to the envelope of B.(In the case of o = and e being a run of the character , define the envelope of a band |a; b| as the set of all squares inside the convex hull of the centers of a + (±(r + s), 0), a + (0, ±(r + s)), b + (±(r + s), 0), and b + (0, ±(r + s)).)For every band B in D^M_o(e_0e), the envelope of B does not intersect any bands independent from B, except possibly at endpoints that coincide with the endpoints of bands tied or adjacent to B.It follows, then, that no illicit pair can occur within two independent bands either, and that D^M_o(e_0e) does not contain an illicit pair.We are left to establish the protocoherence of D^M_o(e_0e).Consider first D^M_o(e_0e) and D^M_o(e_0e) + (2q', 0). Suppose that a square a in the former is joined by a move of M to a square b in the latter.Let B_ be the union of all eastmost bands of D^M_o(e_0e) (there is one such band when e_0 =, , or , and two when e_0 =), and define B_ symmetrically. Furthermore, let B'_ = B_ + (2q', 0) be the image of B_ in D^M_o(e_0e) + (2q', 0).Since all squares of D^M_o(e_0e) + (2q', 0) apart from the ones in B'_ lie more than s units to the east of D^M_o(e_0e), b belongs to B'_.Symmetrically, a belongs to B_.Since both of B_ and B_ consist of outer bands and, by induction on n, the translation (2q', 0) is a sum of an even number of M-translations, it takes M an even number of moves to get from a to b. Therefore, as above, a and b cannot be joined by a move of M. We have arrived at a contradiction.Consider, then, D^M_o(e_0e) and D^M_o(e_0e) + (p' + q', p' + q').In both cases e_0 = and e_0 =, an analogous argument applies as follows.Suppose that a square a in D^M_o(e_0e) is adjacent to a square b in D^M_o(e_0e) + (p' + q', p' + q').Define B_ and B_ symmetrically to B_ and B_. Let B”_ = B_ + (p' + q', p' + q') and B”_ = B_ + (p' + q', p' + q') be the images of B_ and B_ in D^M_o(e_0e) + (p' + q', p' + q').Since all squares of (D^M_o(e_0e) + (p' + q', p' + q')) ∖ B”_ lie at least r + s units to the east of all squares of D^M_o(e_0e) ∖ B_, either a belongs to B_ or b belongs to B”_.Symmetrically, either a belongs to B_ or b belongs to B”_.It follows that a lies in the union of B_ and B_ and b lies in the union of B”_ and B”_.Since all of B_, B_, B_, and B_ consist of outer bands and, by induction on n, the translation (p' + q', p' + q') is a sum of an even number of M-translations, it takes M an even number of moves to get from a to b. Therefore, as above, a and b cannot be joined by a move of M. We have arrived at a contradiction.The cases e_0 = and e_0 = are handled as in the proof of part (g) of Lemma <ref> in Cases g and ħ. Namely, when e_0 = or e_0 =, D^M_o(e_0e) and D^M_o(e_0e) + (p' + q', p' + q') are strictly separated by a diagonal strip of width √(2)(q - p) ≥√(2)(r + s) > 1/√(2)(r + s).Case 2. s ≤ p and r + s ≤ q - p. Then D^M_o(e_0e) is a (p', q')-perfect cycle by Lemma <ref>. We have set up all the tools we need and are ready to construct dual boards by means of f, g, h, and ħ.Given M, o, e, p, and q as in the setting of Lemma <ref>, let L^M_o(e) be a (p, q)-leaper and B^M_o(e) the board formed by the squares of D^M_o(e). Let M be an (r, s)-leaper, r ≤ s, o one of the characters , , andsuch that there exists an initial cycle of M of type o, and e = e_1e_2… e_l a string composed of the characters , , , and . Then the board B^M_o(e) is dual with respect to M and L^M_o(e). By Lemma <ref>, D^M_o(e) is a perfect cycle. By the coherence property of D^M_o(e), the leaper graph of M over B^M_o(e) is a cycle. By Lemma <ref> and analogously to the proofs of Lemmas <ref>, <ref>, and <ref>, so is the leaper graph of L^M_o(e) over B^M_o(e). In particular, we obtain the following corollary. Let L and M be two distinct skew basic leapers such that the descent of one of them is a suffix of the descent of the other. Then there exists a board dual with respect to L and M. Let the descent of L be e_1e_2 … e_l and that of M e_m + 1e_m + 2… e_l, with m ≤ l. By Theorem <ref>, B^M_e_m(e_1e_2… e_m - 1) is a board dual with respect to L and M. We go on to extract dual direction graphs from the dual boards given by Theorem <ref>. Let M be a skew (r, s)-leaper, r < s, o one of the characters , , and , and e = e_1e_2… e_l a string composed of the characters , , , and .Then both the cycle C^M_o(e) of L^M_o(e) over B^M_o(e) and the cycle D^M_o(e) are trivial.The direction graph extracted from C^M_o(e) depends only on e. Extending the definition of a fundamental direction cycle, we refer to this direction graph as the fundamental direction cycle Φ(e) of descent e.The direction graph extracted from D^M_o(e) depends only on o and e. Extending the definition of a second fundamental direction cycle, we refer to this direction graph as the second fundamental direction cycle Φ^II_o(e) of origin o and descent e.(The above continues to apply to non-skew leapers M in the sense that D^M_o(e) is an instantiation of Φ^II_o(e).)Extend the definition of a flip by =, and let the strings e' and e” composed of the characters , , , andbe flips of each other. Then the fundamental direction cycle of descent e' and the three second fundamental direction cycles of descent e” are equivalent.Furthermore, both of Φ(e) and Φ^II_o(e) are dual direction graphs. Each of them is a complement of the other, the duality matrix of Φ^II_o(e) being A_eA_o and that of Φ(e) its inverse. As in the proofs of Theorems <ref> and <ref>, the moves of both C^M_o(e) and D^M_o(e) occur in pairs symmetric with respect to O (those of C^M_o(e) by induction on e and those of D^M_o(e) by its symmetry property) and since the sum of the associated direction matrices over each such pair is the zero matrix, both cycles are trivial.Define the signature of C^M_o(e) as in the proof of Theorem <ref>. Furthermore, define an ħ-rewrite by means of the following system of rewriting rules.+_ → -_-_-_+_ → +_+_+_+_+_-_ → +_+_+_-_ → -_-_-_-_-_ The proof that the direction graph extracted from C^M_o(e) depends only on e is then analogous to the proof of the corresponding part of Theorem <ref>.Define the signatures of D^M_(e) and D^M_(e) as in the proof of Theorem <ref>. Define the signature of D^M_(e) analogously, referring to the following table.Label of aDirections of moves to and from a +_ , , ,+_ , , ,-_ , , ,-_ , , , Furthermore, define an ħ-rearrangement by(s^Corner,s^Side) → (s^Side s^Corner s^Side,s^Sides^Corners^Sides^Corners^Side). The proof that the direction graph extracted from D^M_o(e) depends only on o and e is then analogous to the proof of the corresponding part of Theorem <ref>.That the fundamental direction cycle of descent e' and the three second fundamental direction cycles of descent e” are equivalent is established as in the proof of Theorem <ref>, the equivalence permutationπ_ = ([; ]),induced by the matricesP_ = ([ - 1/√(2) 1/√(2); 1/√(2) 1/√(2) ])andQ_ = ([ 1/√(2) 1/√(2); - 1/√(2) 1/√(2) ]),added alongside π_ and π_.Lastly, the derivation of the duality identity for Φ(e) and Φ^II_o(e) is analogous to the derivation of the duality identity in the proof of Theorem <ref>. The concluding part of the theorem then follows from that identity and Theorem <ref> as in the proof of Theorem <ref>. It can be demonstrated that the family of all dual direction graphs given by Theorem <ref> is closed under equivalence. In fact, it coincides with the closure of the family of all dual direction graphs given by Theorem <ref> under equivalence and the extended f, g, and h.Theorems <ref> and <ref> add a great variety of dual boards and dual direction graphs to the ones given by Theorems <ref> and <ref>.Let us look more closely into what dual direction graphs given by Theorem <ref> are essentially different from all dual direction graphs given by Theorem <ref>, in the sense of not being equivalent to any of them.Since the latter family is a subset of the former, it suffices to obtain a necessary and sufficient condition for two dual direction graphs given by Theorem <ref> to be equivalent. Since every second fundamental direction cycle is equivalent to the fundamental direction cycle of flipped descent, the question reduces to obtaining a necessary and sufficient condition for two fundamental direction cycles to be equivalent.First we show that two fundamental direction cycles never coincide except trivially. Two fundamental direction cycles coincide if and only if their descents do. For all strings e composed of the characters , , , and , let n_(e) be the number of occurrences of the character +_ in the signature of Φ(e), and n_(e) the number of occurrences of the character +_.When e is the empty string, n_(e) = n_(e) = 4.By the definition of an f-rewrite,n_(e) = n_(e) + 2n_(e)andn_(e) = n_(e). Analogously, by the definition of a g-rewriten_(e) = n_(e)andn_(e) = 2n_(e) + n_(e),by the definition of an h-rewriten_(e) = 3n_(e) + 2n_(e)andn_(e) = 2n_(e) + n_(e),and by the definition of an ħ-rewriten_( e) = n_(e) + 2n_(e)andn_( e) = 2n_(e) + 3n_(e). By induction on e, then, n_(e) ≥ 4 and n_(e) ≥ 4 for all e.Let e_0 be one of the characters , , , and , a = n_(e_0e), and b = n_(e_0e).It follows that e_0 = if and only ifa > 2b,e_0 = if and only if2a < b,e_0 = if and only if2b > a > b,and e_0 = if and only ifa < b < 2a. Suppose that the fundamental direction cycles Φ(e') and Φ(e”) coincide, with e' = e'_1e'_2 … e'_l and e” = e”_1e”_2… e”_m. Then their signatures coincide as well. Consequently, n_(e') = n_(e”) and n_(e') = n_(e”).If n_(e') = n_(e”) = 4 and n_(e') = n_(e”) = 4, then both of e' and e” are the empty string.Otherwise, by the above analysis we conclude that e'_1 = e”_1, n_(e'_2e'_3… e'_l) = n_(e”_2e”_3… e”_m), and n_(e'_2e'_3… e'_l) = n_(e”_2e”_3… e”_m).Iteration successively yields e'_2 = e”_2, e'_3 = e”_3, …, l = m, e'_l = e”_l, and e' = e”.Let e be a string composed of the characters , , , and . The companion of e is the string obtained from e by replacing everywith aand vice versa, and everywith anand vice versa. A string e' is the companion of e” if and only if e” is the companion of e'. Let e' and e” be two strings composed of the characters , , , and . Then the fundamental direction cycles of descents e' and e” are equivalent if and only if either e' and e” coincide, or e' and e” are companions. If e' and e” are companions, then by induction on e' the equivalence permutation π_Shift defined by π_Shift(i) = i + 1 for all skew directions i and induced by the matricesP_Shift = ([ 1/√(2) - 1/√(2); 1/√(2) 1/√(2) ])andQ_Shift = ([ - 1/√(2) 1/√(2); 1/√(2) 1/√(2) ])maps Φ(e') onto Φ(e”).Let, then, Φ(e') and Φ(e”) be equivalent and π an equivalence permutation that maps Φ(e') onto Φ(e”).Let C' be an oriented cycle representing Φ(e'), and define C” analogously.The directions of the moves leading to and from every vertex of C' are of the form i and i ± 3, and similarly for C”. Furthermore, by induction on e', for every skew direction i there exists a vertex in C' such that the directions of the moves to and from it coincide with i and i + 3. (Though, possibly, not in that order.)Therefore, π maps every unordered pair of skew directions of the form i and i + 3 to an unordered pair of the same form.There exist sixteen such permutations of the eight skew directions, namely π^k_Shift and π_Reflect∘π^k_Shift for k = 0, 1, …, 7, whereπ_Reflect = ([; ]). All of those are equivalence permutations. Eight of them preserve Φ(e'), and eight map it onto the fundamental direction cycle of descent the companion of e'.By Theorem <ref>, either e' and e” coincide or e' and e” are companions.A dual direction graph given by Theorem <ref> is not equivalent to any dual direction graph given by Theorem <ref> if and only if its descent contains both of the charactersand . The first dual direction graphs given by Theorem <ref> that are not equivalent to any dual direction graph given by Theorem <ref> are thus the ones of descentsand .Figure <ref> shows the board B^(0, 1)_() overlaid with the associated cycle C^(0, 1)_(), the (5, 12)-instantiation of Φ().Similarly, Figure <ref> shows the board B^(0, 1)_() overlaid with the associated cycle C^(0, 1)_(), the (5, 12)-instantiation of Φ().§ CONSTRUCTIONS OF DUAL BOARDSAND DUAL DIRECTION GRAPHS II We conclude by exhibiting one more construction of dual boards and dual direction graphs.Given a (p, q)-leaper L, introduce a Cartesian coordinate system Oxy over the infinite chessboard such that the integer points are the vertices of the squares if p + q is odd, and the centers of the squares if it is even, and write (x, y) for the square centered at (x, y).Given a positive integer d and real numbers e_1 and e_2, write N_d(e_1, e_2) for the net formed by all squares (x, y) such that x ≡ e_1 and y ≡ e_2 modulo d. Let n be a positive integer and L a (p, q)-leaper with q ≠ 0.Let N^I(L) be the netN_2q(1/2(p + q), 1/2(p - q))and W^I_n(L) the board formed by all squares of N^I(L) whose centers lie on the boundary or in the interior of the rectangle R^I_n(L) bounded byp ≤ x + y ≤ p + 2nqand-(2n - 1)q ≤ y - x ≤ (2n - 1)q. Let N^i(L), R^i_n(L), and W^i_n(L), for i = II, III, and IV, be the rotations of N^I(L), R^I_n(L), and W^I_n(L) by 90^∘, 180^∘, and 270^∘ counterclockwise about O.Then the pinwheel board of order n for L, W_n(L), is the disjoint union of its four wings W^i_n(L), i = I, II, III, and IV. Figure <ref> shows the centers of the squares of W_4(1, 2), overlaid with the associated leaper graph and with the four wings labeled (0, 4)8, □, (0, 4)*8, and ▪. Let n be a positive integer,A^Pinwheel_n = ([01;1 2n ]),L a (p, q)-leaper with p < q, and M an (r, s)-leaper with([ r; s ]) = A^Pinwheel_n ([ p; q ]). Then the pinwheel board of order n for L is dual with respect to L and M. Figure <ref> shows W_n(0, 1), overlaid with the associated leaper graph of a (1, 2n)-leaper, for n = 1, 2, and 3. First we show that the leaper graph of L over W_n(L) is connected.Consider the squares a^I, a^II, a^III, and a^IV of W_n(L) of centers (1/2(p + q), 1/2(p - q)), (1/2(q - p), 1/2(p + q)), (-1/2(p + q), 1/2(q - p)), and (1/2(p - q), -1/2(p + q)). They form a cycle C of L.Let a, centered at (x_a, y_a), be any square of W_n(L). It suffices to show that there exists a path of L from a to C within W_n(L).We consider the case 0 ≤ x_a ≤ y_a in detail, and all other cases are analogous.Place L at a.Suppose first that a belongs to W^I_n(L). Let L move successively in directions , , , , …, ,until it cannot advance any further without leaving the board. At that point, L occupies a square b of the form a^I + m · (2q, 2q) for some nonnegative integer m. From b, let L move successively in directions , , , , , , , , …, , , ,until it arrives at a^I.Suppose, then, that a belongs to W^II_n(L). Then a move in directionbrings L to a square a' in W^I_n(L) and the proof continues as in the previous case.Suppose, lastly, that a belongs to W^IV_n(L). This is only possible if p = 0 and a has the form (1/2(4m + 3)q, 1/2(4m + 3)q) for some nonnegative integer m. In that case, let L move successively in directions , , , , , , , , …, , , ,until it arrives at a^IV.We proceed to show that the leaper graphs of L and M over W_n(L) are isomorphic.Given a square a of W_n(L), define φ(a) as follows. Let i, out of I, II, III, and IV, be such that a belongs to the wing W^i_n(L). Then φ(a) is the reflection of a in the center O^i of R^i_n(L). Two squares a and b of W_n(L) are joined by a move of L if and only if φ(a) and φ(b) are joined by a move of M. Let b = a + u, u = (x_u, y_u), φ(b) = φ(a) + v, and v = (x_v, y_v).Suppose first that u is an L-translation.Since each of the four translations (± 2q, 0) and (0, ± 2q) is the sum of two L-translations, it takes L an even number of moves to travel between two squares in the same net N^i(L), for each i out of I, II, III, and IV. Furthermore, since it takes L two moves to travel between opposite squares in C, it also takes L an even number of moves to travel between two squares in opposite nets.It follows that a and b cannot belong to the same wing or to opposite wings. Therefore, a and b belong to adjacent wings.By symmetry, it suffices to consider the case when a belongs to W^I_n(L) and b belongs to W^II_n(L).Thenx_u ≡1/2(q - p) - 1/2(p + q) = -p 2qandy_u ≡1/2(p + q) - 1/2(p - q) = q 2q. Consequently, x_u = -p and y_u = ± q.Let O^I + o = O^II. Since a and φ(a) are symmetric with respect to O^I and b and φ(b) are symmetric with respect to O^II,u + v = 2o. Therefore, as o = (-p - nq, 0),v = 2o - u = (-p - 2nq, ± q) = (-s, ± r)and v is an M-translation.Suppose, then, that v is an M-translation.As above, it suffices to consider the case when a belongs to W^I_n(L) and b belongs to W^II_n(L), when x_v ≡ -p and y_v ≡ q modulo 2q.If p ≠ 0, then it follows that x_v = -s and y_v = ± r and the proof continues as before.When p = 0, however, we also need to rule out the possibility that x_v = s. This is done as follows.Let φ(a) be at (x'_a, y'_a) and φ(b) at (x'_b, y'_b). Since φ(a) lies on the boundary or in the interior of R^I_n(L), x'_a ≥ -1/2(2n - 1)q. Analogously, x'_b ≤1/2(2n - 1)q. Therefore,x'_b - x'_a ≤ (2n - 1)q < s. Thus necessarily x_v = -s and y_v = ± r even if p = 0, and the proof continues as before. By Lemma <ref>, the leaper graph of L over W_n(L) is isomorphic to the leaper graph of M over φ(W_n(L)).When n is odd, φ(W_n(L)) coincides with W_n(L), and when n is even, φ(W_n(L)) is a reflection of W_n(L) (in each of the lines x = 0, y = 0, and y = ± x).Consequently, the leaper graph of M over φ(W_n(L)) is isomorphic to the leaper graph of M over W_n(L) and the proof is complete. Dual pinwheel boards are fundamentally different from all other dual boards we have encountered thus far.However, they do not uncover any novel pairs of leapers L and M such that there exists a board dual with respect to L and M. SinceA^Pinwheel_n = A_^n - 1A_,all pairs of leapers such that a board dual with respect to them exists by Theorem <ref> are already accounted for by Theorem <ref>.We go on to extract dual direction graphs from dual pinwheel boards. Let n be a positive integer and L a skew (p, q)-leaper with p < q. Then the leaper graph of L over W_n(L) is trivial and the direction graph extracted from it depends only on n. We refer to this direction graph as the pinwheel direction graph W_n of order n.(The above continues to apply to non-skew leapers L in the sense that the leaper graph of L over W_n(L) is an instantiation of W_n.)The pinwheel direction graph of order n is a dual direction graph of complement π_Pinwheel(W_n) when n is odd, whereπ_Pinwheel = ([; ])is the equivalence permutation induced by the unit 2 × 2 matrix and ( [01; -10 ]), and π_Reflect∘π_Pinwheel(W_n) when n is even, and duality matrix A^Pinwheel_n. In the setting of the proof of Theorem <ref>, let a be any square of W_n(L). Then there exist unique i out of I, II, III, and IV and integers k and l such that a = a^i + (2kq, 2lq). Assign to a the ordered triplet σ(a) = (i, k, l).LetA^I = ([ 1/2 1/2; 1/2 - 1/2 ]), A^II = ([ - 1/2 1/2; 1/2 1/2 ]),A^III = ([ - 1/2 - 1/2; - 1/2 1/2 ]), A^IV = ([ 1/2 - 1/2; - 1/2 - 1/2 ]),andA(i, k, l) = A^i + ([0 2k;0 2l ]). We proceed to show that a move of L of direction j leads from a' to a” in W_n(L) if and only ifA(σ(a”)) - A(σ(a')) = A_j. The “if” part holds as, for every square a of W_n(L), the transpose of the coordinates of a is given by A(σ(a))( [ p; q ]).For the “only if” part, let a move of L lead from a' to a”.By the proof of Lemma <ref>, a' and a” belong to adjacent wings. By symmetry, it suffices to consider the case when a' belongs to W^I_n(L) and a” belongs to W^II_n(L).By the proof of Lemma <ref>, the move of L from a' to a” is of the form either (-p, q) or (-p, -q). In the former case, it is of directionand A(σ(a”)) - A(σ(a')) = A_, and in the latter case it is of directionand A(σ(a”)) - A(σ(a')) = A_, as needed.It follows that the leaper graph of L over W_n(L) is trivial and the direction of a move of L from a square a' to a square a” in W_n(L) depends only on σ(a') and σ(a”), and not on p and q.(In fact, the leaper graph of L over W_n(L) contains a single simple cycle, namely C. However, the argument above continues to apply in generalizations of pinwheel boards replacing R^i_n(L), i = I, II, III, and IV, with other regions, as in Theorem <ref>.)Consider, then, the direction graph W_n of vertices σ(W_n(L)) such that an arc labeled i points from σ(a') to σ(a”) if and only if an L-move of direction i leads from a' to a”. It is extracted from the leaper graph of L over W_n(L) and does not depend on L, settling the first part of the theorem.For the second part of the theorem, we consider the case when n is odd, and the opposite case is analogous.Define the mapping η over the vertices of W_n byη(i, k, l) = (i, 12(n - 1) - k, 12(n + 1) - l). By the proof of Lemma <ref>, W_n satisfies the definition of a dual direction graph of complement π_Pinwheel(W_n) and duality matrix A^Pinwheel_n with the one-to-one mapping η between the vertex sets of W_n and π_Pinwheel(W_n). Theorem <ref> does not list all values of n, p, and q such that W_n(L) is dual with respect to L and M. For instance, n = 2, p = 2, and q = 1 yield a pinwheel board dual with respect to a (1, 2)-leaper and a (1, 6)-leaper.The construction of a dual pinwheel board admits a variety of modifications.For instance, in the setting of Theorem <ref> and its proof, adjoining a^I + (2nq, 0) and its multiple-of-quarter-turn rotations about O to W_n(L) yields an augmented pinwheel board dual with respect to L and M.The first dual board discovered by the author, in October 2005, was the augmented W_1(0, 1) in Figure <ref>.A different species of modification proceeds by expanding the regions R^i_n(L), i = I, II, III, and IV, in the definition of a pinwheel board.For instance, in the setting of Theorem <ref>, require additionally that p ≠ 0. Then replacing each of R^i_n(L), i = I, II, III, and IV, with the complete plane yields an infinite board, the disjoint union of N^i(L) over i = I, II, III, and IV, dual with respect to L and M.We give a construction of this type of arbitrarily large finite dual pinwheel-like boards. Let n be a positive integer, d a nonnegative integer, and L a (p, q)-leaper with q ≠ 0.Let W^I_n, d(L) be the board formed by all squares of N^I(L) whose centers lie on the boundary or in the interior of the rectangle R^I_n, d(L) bounded byp - 2dq ≤ x + y ≤ p + 2(n + d)qand-(2n + 2d - 1)q ≤ y - x ≤ (2n + 2d - 1)q. Let R^i_n, d(L) and W^i_n, d(L), for i = II, III, and IV, be the rotations of R^I_n, d(L) and W^I_n, d(L) by 90^∘, 180^∘, and 270^∘ counterclockwise about O.Then the expanded pinwheel board of order n and margin d for L, W_n, d(L), is the (disjoint if p < q) union of its four wings W^i_n, d(L), i = I, II, III, and IV. The expanded pinwheel board of order n and margin 0 for L coincides with the pinwheel board of order n for L. Let n and d be positive integers, L a (p, q)-leaper with p ≠ 0 and p < q, and M an (r, s)-leaper with([ r; s ]) = A^Pinwheel_n ([ p; q ]). Then the expanded pinwheel board of order n and margin d for L is dual with respect to L and M.Furthermore, the leaper graph of L over W_n, d(L) is trivial and the direction graph extracted from it depends only on n and d. We refer to this direction graph as the expanded pinwheel direction graph W_n, d of order n and margin d. It is a dual direction graph of complement π_Pinwheel(W_n, d) when n is odd and π_Reflect∘π_Pinwheel(W_n, d) when n is even, and duality matrix A^Pinwheel_n.Figure <ref> shows W_2, 1(1, 2) overlaid with the associated leaper graphs of a (1, 2)-leaper and a (2, 9)-leaper.The proof of Theorem <ref> is analogous to the proofs of Theorems <ref> and <ref>.In particular, we obtain the following corollary. Let n, p, and q be positive integers with p < q. Then there exist arbitrarily large boards dual with respect to a (p, q)-leaper and a (q, p + 2nq)-leaper and arbitrarily large dual direction graphs of duality matrix A^Pinwheel_n. 99[1]JW George Jelliss and Theophilus Willcocks, The Five Free Leapers, Chessics 2, 1976, <http://www.mayhematics.com/p/chessics_02.pdf>; 6, 1978, <http://www.mayhematics.com/p/chessics_06.pdf>.[2]J1 George Jelliss, Theory of Leapers, Chessics 24, 1985, <http://www.mayhematics.com/p/chessics_24.pdf>.[3]K Donald Knuth, Leaper Graphs, Mathematical Gazette 78, 1994, <http://arxiv.org/abs/math/9411240>.[4]J2 George Jelliss, Knight's Tour Notes, 2001, <http://www.mayhematics.com/t/t.htm>. | http://arxiv.org/abs/1706.08845v1 | {
"authors": [
"Nikolai Beluhov"
],
"categories": [
"math.CO",
"05C38, 05C45, 05C60 (Primary), 11A55, 68Q42 (Secondary)"
],
"primary_category": "math.CO",
"published": "20170627135009",
"title": "The Second Leaper Theorem"
} |
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