text:
# A violation of universality in anomalous Fourier’s law

## Abstract

Since the discovery of long-time tails, it has been clear that Fourier’s law in low dimensions is typically anomalous, with a size-dependent heat conductivity, though the nature of the anomaly remains puzzling. The conventional wisdom, supported by renormalization-group arguments and mode-coupling approximations within fluctuating hydrodynamics, is that the anomaly is universal in 1d momentum-conserving systems and belongs in the Lévy/Kardar-Parisi-Zhang universality class. Here we challenge this picture by using a novel scaling method to show unambiguously that universality breaks down in the paradigmatic 1d diatomic hard-point fluid. Hydrodynamic profiles for a broad set of gradients, densities and sizes all collapse onto an universal master curve, showing that (anomalous) Fourier’s law holds even deep into the nonlinear regime. This allows to solve the macroscopic transport problem for this model, a solution which compares flawlessly with data and, interestingly, implies the existence of a bound on the heat current in terms of pressure. These results question the renormalization-group and mode-coupling universality predictions for anomalous Fourier’s law in 1d, offering a new perspective on transport in low dimensions.

## Introduction

It’s going to be 200 years since Fourier stated his seminal law1, but its microscopic understanding still poses one of the most important and challenging open problems in nonequilibrium statistical physics, with no rigorous mathematical derivation to date2,3,4,5,6,7. Fourier’s law establishes the proportionality between the heat current and the local temperature gradient in a material, with the proportionality factor defining the heat conductivity κ, a key material property. While for bulk, three-dimensional materials κ is well characterized and measured, its status in low-dimensional structures is far from clear. In particular, for low-dimensional systems (d = 1, 2) with momentum conservation, the effective conductivity κ grows with the system size L, diverging in the thermodynamic limit and thus leading to anomalous heat transport3,4,5,6,7. The understanding of this anomaly has attracted a lot of attention in recent years3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, not only because it is expected to shed light on the key ingredients behind Fourier’s law at a fundamental level, but also because of its technological relevance in low-dimensional real-world materials, the most noteworthy being graphene8,9,10,11,12, but with other important examples ranging from molecular chains13 and carbon nanotubes14 to polymer fibers15,16, nanowires17,18 and even spider silk19, to mention just a few; see7 for a recent review. From a theoretical perspective, the low-dimensional anomaly in heat transport can be linked to the presence of strong dynamic correlations in these fluids and lattices20,21,22, though a detailed understanding has remained elusive for decades.

In 1d, clear signatures of this anomaly appear in a number of different phenomena42. For instance, the steady state heat current J of a 1d momentum-conserving system driven by a small boundary temperature gradient (i.e. in the linear regime) typically scales as L−1+γ for large enough system sizes L, with 0 ≤ γ < 1 an anomaly exponent, which can be interpreted in terms of a finite-size heat conductivity κL ~ Lγ. An exponent γ = 0 corresponds to standard diffusive transport, but typically γ > 0 is observed in 1d implying superdiffusive heat transport42. The low-dimensional transport anomaly is also apparent in equilibrium. In particular, the long-time tail of the equilibrium time correlation of the energy current decays in 1d in a nonintegrable, power-law way, 〈J(0)J(t)〉 ~ t−1+δ as t → ∞, with 0 ≤ δ < 1 another exponent. Green-Kubo relations for the transport coefficients hence imply a divergent value for the heat conductivity, in agreement with nonequilibrium results6. Additional signatures of anomalous transport have been also reported in the superdiffusive spreading of energy perturbations in equilibrium40,41,42,43. A range of different values for the exponents γ and δ have been measured in simulations and experiments for different model systems4,5,6,7, the main difficulty being extracting the large L asymptotics due to the strong and poorly understood finite-size effects affecting these measurements (which mix bulk and boundary finite-size corrections). The prevailing picture, however, is that the transport anomaly exponents are universal and within the Lévy/Kardar-Parisi-Zhang (L/KPZ) universality class3,4,5,6,7, a conjecture based on renormalization-group23 and mode-coupling24 calculations, and reinforced by recent related breakthroughs from nonlinear fluctuating hydrodynamics25,26,27,28,29 which predict Lévy (KPZ) scaling for the heat (sound) modes of the equilibrium time correlators of conserved fields. In particular, for the transport anomaly γ = 1/3 = δ is expected in the general case, though a second universality class with γ = 1/2 = δ seems to appear under special circumstances (as e.g. for zero-pressure systems with symmetric potential24,30,31,32,33,34). Special cases with convergent κ in 1d, as the coupled rotors model44,45, can be also accounted for by fluctuating hydrodynamics after noticing that these models have less than three locally-conserved fields46.

In this work we challenge the universality conjecture for anomalous Fourier’s law by using a novel scaling method to offer a high-precision measurement of the conductivity anomaly in a paradigmatic 1d model of transport. Compared to previous attempts at characterizing the transport anomaly, most based on linear response theory and hence critically-dependent on a large system-size limit (which is in fact never attained)39, our method takes full advantage of the nonlinear character of the heat conduction problem in a natural way, allowing to disentangle the crucial bulk size scaling from the artificial boundary finite-size corrections. Our model is the archetypical 1d diatomic hard-point gas in a temperature gradient47,48,49,50,51,52,53,54,55,56,57, which is characterized by the mass ratio μ = M/m > 1 between neighboring particles. We unambiguously show below that, contrary to the standard lore, this model does obey an anomalous version of Fourier’s law, namely

for a broad range of temperature gradients (from the linear response domain to the deeply nonlinear regime), with the heat current J proportional to the local temperature gradient via a conductivity functional

Note that Eqs (1)–(2), are not Fourier’s law in the usual sense, as the latter implies a size-independent κ, while the conductivity in this case grows with the system size as Lα, with α a new exponent characterizing anomalous transport in 1d. The validity of Eqs (1)–(2), is proven below by collapsing onto a striking universal master curve the density and temperature profiles measured for a large set of system sizes, number densities and temperature gradients. Such compelling collapse offers a high-precision measurement of the anomaly exponent α, which remarkably turns out to be non-universal, depending non-monotonously on the mass ratio μ. The observed scaling allows to solve the macroscopic transport problem for this model, and we obtain analytic expressions for the universal master curve (as well as for the hydrodynamic profiles, current, pressure, etc.) which exhibit an excellent agreement with measurements. Interestingly, this solution immediately implies the existence of a nontrivial bound on the current in terms of pressure P.

A natural question concerns the relation of the new anomaly exponent α with the standard ones defined in literature, namely γ in the linear response regime and δ from equilibrium current time correlations (see description above). This relation can be easily established by studying the linear response limit of the anomalous Fourier’s law (1)–(2), a particular regime of the broad range of temperature gradients where these equations hold with high accuracy, as we demonstrate below. Indeed, for small enough boundary temperature difference ΔT the local temperature gradient can be written as dT/dx ≈ −ΔT/L, and this together with Eqs (1)–(2), leads to JL−1+α, an argument which strongly suggests the conjecture α = γ(= δ). In this way, the surprising but clear-cut dependence of α on the mass ratio μ reported below hence signals the breakdown of the universality claimed for 1d anomalous Fourier’s law. We maintain here however the different notation for the various (but related) anomaly exponents to stress out their distinct definitions.

## Results

We hence consider a 1d Hamiltonian model fluid consisting in N hard-point particles with alternating masses, m = 1 and M = μm > 1, moving ballistically in a line of length L in between elastic collisions with neighboring particles. The fluid is coupled to two stochastic thermal walls at the boundaries, x = 0, L, which reflect particles upon collision with a velocity modulus randomly drawn from a Maxwellian distribution defined by the wall temperature T0,L3,4,5,6. For T0 ≠ TL, the temperature gradient drives the system to an inhomogeneous nonequilibrium steady state characterized by nonlinear density and temperature profiles, ρ(x) and T(x) respectively3,4,5,6,7. Interestingly, these profiles can be shown to follow from an universal master curve, independent of the driving gradient and the fluid’s density, if and only if (i) Fourier’s law (1)–(2) and (ii) macroscopic local equilibrium (MLE) hold (see Section I of the Supplementary information for a detailed proof), an equivalence which holds for general d-dimensional systems35. MLE implies that the stationary density and temperature fields are locally coupled via the equilibrium equation of state (EoS)58, which for the 1d diatomic hard-point fluid simply takes the ideal gas form, P = ρT. In this way, iff hypotheses (i)-(ii) hold, we expect all density and temperature profiles to scale as

with ψ = J/P3/2 the reduced current and ζ a constant, see Section I of the Supplementary information. This scaling defines an universal master curve F(u) from which all profiles follow. Alternatively, Eq. (3) implies that all measured density and temperature profiles can be collapsed onto an universal master curve after appropriately scaling space by Lαψ, with ψ measured in each case, and shifting the curve by a constant ζ. The resulting collapse is expected to be very sensitive to the anomaly exponent α, and this suggests a simple scaling procedure to measure both α and the universal master curve in simulations, confirming at the same time our starting hypotheses.

In order to do so, we performed a large number of event-driven simulations of the 1d diatomic gas for a broad set of boundary temperatures T0 = 2, 5, 10, 20 (with fixed TL = 1), global number densities ηN/L = 0.5, 1, 2, 3, different mass ratios μ = 1.3, 1.618, 2.2, 3, 5, 10, 30, 100, and numbers of particles N = 101, 317, 1001, 3163, 10001, reaching up to N = 105 + 1 in some cases. We measured locally a number of relevant observables including the local kinetic energy, number density, virial pressure and energy current density, as well as the energy current flowing through the thermal reservoirs at x = 0, L and the pressure exerted on these walls. We stress that observables measured at the walls agree in all cases with their bulk counterparts, which are constant along the system. For local measurements, we divided the fluid in 30 virtual cells, a constant number independent of other system parameters. The simulation time unit was set to , the mean free time of a heavy particle in a cool environment, and time averages were performed taking into account the relaxation and correlation timescales of the 1d fluid, which grow strongly with N (see Fig. S4 and Section IIB in the Supplementary information). Statistical errors are computed in all cases at 99.7% confidence level, and error bars are shown if larger than the plotted symbols.

Figure 1 shows the temperature and density profiles measured for μ = 3 and varying T0, η and N (similar data are obtained for all other μ’s). These profiles are clearly nonlinear, and exhibit strong finite-size effects. However, the measured local density and temperature in each case are tightly coupled by the equilibrium EoS, P = ρ(x)T(x), with P the finite-size pressure measured in each simulation, see Fig. S2 and Section IIA in the Supplementary information, validating hypothesis (ii) above and confirming the robustness of MLE far from equilibrium58. Note that the thermal walls act as defects (akin to fixed, infinite-mass particles) which disrupt the structure of the surrounding fluid, defining two boundary layers where finite-size corrections mount up. To analyze below the fluid’s scaling behavior, we neglect data from these boundary layers (up to 7 cells adjacent to each wall), focusing the analysis on the remaining bulk profiles ρ(x) and T(x). For a given exponent α, each bulk density profile ρ(x) is then plotted as a function of Lαψx (with measured in each case, see Supplementary Fig. S3), and shifted by a constant ζ to achieve an optimal collapse among all scaled profiles, see Fig. 2a. The vector of optimal shifts ζ0 for fixed α and μ is obtained by minimizing a standard collapse metric D(ζ; α, μ) for the density profiles (defined in detail in Section III of the Supplementary information), which roughly speaking measures the relative average distance among all pairs of overlapping curves59, and the same shifts are used to collapse reduced temperature profiles, T(x)/P. The resulting data collapses are very sensitive to α, see Fig. 2b, so the the true anomaly exponent α can be measured with high precision for each mass ratio μ by minimizing D(α, μ) ≡ D(ζ0; α, μ) as a function of α. In fact, the distance function D(α, μ) has a pronounced minimum in α for each μ, see inset in Fig. 3a, whose width and depth allow to estimate the exponent error, see Supplemementary information, Section III.

Remarkably, the measured anomaly exponent is non-universal, depending non-monotonously on the mass ratio, α = α(μ), see Fig. 3a and Supplementary Table S1, growing first from small values at low μ to a maximum α ≈ 0.3 < 1/3 for μ = 2.2, and decaying afterwards to an asymptotic value α ≈ 1/4 for large μ. Figure 3b shows the master curves obtained from density and reduced temperature bulk profiles for different μ’s by using the measured α’s, and in all cases the resulting collapses are impressive, confirming that anomalous Fourier’s law (1)–(2) rules heat transport in this 1d model. Moreover, this surprising but unambiguous result also calls into question the prevailing conjecture that the anomaly in 1d heat transport is universal6,23,24,25,26,27,28,29,30.

At this point it is worth emphasizing that standard linear response methods to measure the heat conductivity typically yield an effective anomaly exponent in 1d which changes appreciably with the system size, , slowly converging to the asymptotic value γ at very large N56, see Section IIC in the Supplementary information. A natural question is hence whether the new anomaly exponent α(μ) measured with the novel scaling method introduced here exhibits similar finite-size corrections. A first clue that this is not the case is that, for N [102 + 1, 104 + 1], a slight change in the anomaly exponent measured with our scaling method completely destroys the observed collapse, see Fig. 2b, while the effective anomaly exponent measured with standard methods varies widely with N in the same N-range, e.g. , see Fig. 3b in ref. 56. In any case, in order to test quantitatively this idea, we divided our original data into two different subsets, one for small N [102 + 1, 103 + 1] and another one for large N [103 + 1, 104 + 1]. In this way both data subsets have the same amount of points, thus avoiding possible sampling issues. Next, we perform our scaling analysis on both subsets and obtain the collapse distance metric D(α, μ) as a function of α in each case. In both cases, small N vs large N, this function exhibits a pronounced minimum in α for each μ, and these minima identify the anomaly exponent as measured in each subset. Figure 4a shows the results of this analysis for mass ratio μ = 3, and the conclusion is clearcut: the anomaly exponents measured from the small-N and large-N subsets are fully compatible between them and with our previous measurement based on all N [102 + 1, 104 + 1], so no significant, systematic variation of the anomaly exponent with the system size is found beyond the stringent errorbars of our measurements. We found similar results for all other μ’s.

To further test the robustness of the measured anomaly exponents against order-of-magnitude changes in the system size, we also studied the steady-state heat transport in the diatomic hard-point fluid for N = 31623 and N = 105 + 1, i.e. one order of magnitud beyond our previous simulations. The scale of these simulations is so large that we had to restrict the region of parameter space explored. In particular, we perform simulations of the aforementioned values of N for a large temperature gradient given by T0 = 20, global densities η = 0.5, 1, 2, 3, and two intermediate mass ratios μ = 3 and μ = 2.2 for which relaxation (and correlation) timescales are somewhat shorter (note that for both small and large mass ratios the fluid’s relaxation and correlation times increase drastically60,61). Figure 4b shows the collapse of density profiles for μ = 2.2 and μ = 3 obtained by using the measured anomaly exponent α(μ) in each case, namely α(μ = 2.2) = 0.308 and α(μ = 3) = 0.297, see Supplementary Table S1, once the new data for N = 31623 and N = 105 + 1 have been added. In all cases the excellent collapse of all data for N [102 + 1, 105 + 1], i.e. across three orders of magnitude in the system size, strongly confirms the validity of the measured (non-universal) exponents in the large-N limit. Similar excellent collapses are also obtained for temperature profiles. Moreover, if a different anomaly exponent is used in the previous scaling plots (e.g. α = 1/3) no good collapse is obtained, as observed in e.g. Fig. 2b above, even if we restrict the plot to the largest values of N. These observations thus discard the possibility of a running anomaly exponent (at least within our stringent precision limits), demonstrating the robustness of the anomaly exponent α against order-of-magnitude changes in the system size and hence strengthening our conclusions.

We next focus on the density dependence of the heat conductivity . Interestingly, the dynamics of 1d hard-point fluids remains invariant under different scalings (of temperature, velocities, space, mass, etc.) ref. 5. Using such invariance, it is easy to show rigorously that , with f some adimensional function of N and μ. This in turn implies, via dimensional analysis, that necessarily k(ρ) = α, with a some constant. This is fully confirmed in local measurements of the density dependence of the heat conductivity, from which we determine a = a(μ). Indeed, one can easily show from Eq. (2) that , so for each set (N, T0, η) and fixed μ we performed discrete derivatives of the measured bulk temperature profile to evaluate T′(x) and plotted the previous expression, with J measured in each case, as a function of the associated ρ(x). Figure 5 shows the curves k(ρ) so obtained for different μ, which display the best collapse when the measured exponent α(μ) is used. Interestingly the resulting scaling functions, though somewhat noisy due to discretization effects, exhibit a clear power-law behavior, k(ρ) = β, and the fitted exponent is fully compatible in all cases with the measured anomaly exponent, β = α(μ), see Fig. 3a above and Supplementary Table S1. These measurements thus prove in an independent way that the density dependence of the heat conductivity of the 1d diatomic hard-point gas does reflect the transport anomaly.

The above observation that k(ρ) = α opens the door to a full solution of the macroscopic heat transport problem for this model, see Section I of the Supplementary information. In particular, the universal master curve F(u) of Eq. (3) is

with . This master curve depends on μ through the mass ratio dependence of α and a. Figure 3b displays the predicted master curves, with the only input of the measured α(μ) and a(μ), and the agreement with collapsed data is stunning in all cases. Closed forms for temperature profiles follow as

with density profiles given as ρ(x) = P/T(x), and P and J simply written in terms of external parameters T0, TL, η, and L, see Supplementary information, Section I. Note that this novel macroscopic solution is fully compatible with the known scaling symmetries of 1d hard-point fluids5. Interestingly, the master curve F(u) exhibits a vertical asymptote at u = ν*, see Eq. (4), implying the existence of a bound on the scaled current in terms of pressure,

Eq. (5) for temperature profiles can be readily tested against data. For that we plot T(x)3/2−α vs x, with T(x) the measured temperature profiles for each μ, N, η and T0. This is predicted to be a straight line with slope , with J and P the measured current and pressure, respectively. Such linear dependence is confirmed for bulk temperature profiles in all cases (similar results hold also for density profiles), with the correct slope but with effective boundary temperatures (obtained from the y-intercept of the line) slightly different from the thermal wall temperatures in each case, . Figure 6a shows an example of this test for μ = 3, η = 1, varying T0 [2, 20] and two different system sizes, N = 101 (small) and N = 10001 (large), with excellent agreement in all cases. This shows that the measured bulk temperature (and density) profiles for any finite N are in fact those of a macroscopic diatomic hard-point gas sustaining a current J and a pressure P and obeying Eqs (1)–(2), but subject to some effective N-dependent boundary conditions controlled by the boundary layers. Indeed, the striking collapse of data and the agreement with the macroscopic master curve in Fig. 3b strongly support this conclusion. This is a manifestation of the bulk-boundary decoupling phenomenon already reported in hard disks out of equilibrium35, which enforces the macroscopic laws on the bulk of the finite-sized fluid.

The effective boundary temperatures converge toward T0,L as N increases, but at an exceedingly slow rate, (see Supplementary Fig. S5), with Λ some amplitude, and this explains the persistent finite-size corrections found in the effective anomaly exponents measured with traditional linear response methods. Indeed, these methods approximate the heat conductivity as , with ΔT = T0 − TL, and find that the so-defined empirical conductivity diverges as in 1d, with an effective anomaly exponent which exhibits itself persistent finite-size corrections4,5,6. Noting that the real temperature gradient driving the bulk fluid to sustain a current J is and taking into account the strong finite-size corrections affecting the boundary effective temperatures, it is easy to show (see Section IIC of the Supplementary information) that

so the effective anomaly exponent measured from the empirical conductivity converges at an exceedingly slow rate toward the correct, asymptotic anomaly exponent γ, in a way that closely resembles actual measurements, see e.g. ref. 56. This confirms that the slowly-decaying (and artificial) boundary finite-size corrections associated to the boundary layers are responsible of the strong, persistent finite-size deviations affecting the effective anomaly exponent measured with the standard linear response method. Moreover, as our scaling method is independent of the boundary temperatures driving the system out of equilibrium, this explains why our results for the anomaly exponent α (that we conjecture is equal to γ) are free of these persistent finite-size corrections.

Finally, our macroscopic theory also offers a precise prediction for the heat current, see the Supplementary information, Section I. In particular, it predicts that , with hα(z) a well-defined function

This prediction can be tested against data using the effective boundary temperatures measured above, see Fig. 6b, and the agreement is excellent N, T0, η for each μ.

## Discussion

Some comments are now in order. The excellent collapse of our data confirms that anomalous Fourier’s law (1) holds in this model with a well-defined (albeit size-dependent) conductivity functional . This is true even for finite N (as small as !) and under large temperature gradients, extending the range of validity of anomalous Fourier’s law deep into the nonlinear regime and evidencing the absence of higher-order (Burnett-like) corrections in 1d35.

In addition, we provide strong evidences supporting the breakdown of universality in anomalous Fourier’s law for 1d momentum-conserving systems62. In particular, we show with high accuracy that the new anomaly exponent α for the heat conductivity of the 1d diatomic hard-point fluid depends on the mass ratio μ between neighboring particles. This clear-cut observation, together with the conjectured equality between the different anomaly exponents, α = γ(=δ), calls into question the universality picture for heat transport based on renormalization-group and mode-coupling calculations23,24. Note however that our results do not say anything about or contradict the Lévy/KPZ universality of the equilibrium time correlators of the conserved (hydrodynamic) fields, recently predicted within nonlinear fluctuating hydrodynamics and tested in simulations25,26,27,28.

Different tests of the universality conjecture for the heat transport anomaly have been performed in the past for the diatomic hard-point fluid using a number of methods, including both nonequilibrium simulations of heat transport in the linear response regime and equilibrium measurements of current time-correlation functions4,5,6. All tests confirm the existence of the heat transport anomaly for this model. However, the accuracy of the standard methods to determine the anomaly exponents is severely hampered by the strong finite-size corrections affecting these measurements, making very difficult to discern the breakdown of universality here reported. For instance, determining the heat conductivity via the standard nonequilibrium route leads to a running effective anomaly exponent which exhibits itself persistent finite-size deviations and poor convergence with N56. Our scaling results explain the origin of this extremely slow convergence, which in brief can be traced back to the mixing of the artificial but very strong boundary finite-size corrections with the most important bulk scaling behavior. Since our collapse procedure is independent of the boundary driving, this explains why our scaling results for the anomaly exponent α are free of these persistent finite-size effects, offering very precise measurements which remain robust across three decades in N. On the other hand, the standard equilibrium (Green-Kubo) route to study the anomaly can typically test the compatibility of the long-time tail exponent δ with the universality prediction, but cannot discriminate in most cases the small exponent differences associated to the universality violation here reported. This is particularly relevant for mass ratio μ = 3, for which most equilibrium tests have been performed and where our scaling results yield an anomaly exponent close to (but different from) 1/3, the universality prediction for this model. Therefore it would be desirable to perform standard equilibrium tests also for other mass ratios for which the difference between the universality exponent and the one we measure from scaling are more definite, as e.g. μ = 10 for which α = 0.260(14), see Supplementary Table S1. We note however that some recent and very precise simulations of the equilibrium diatomic hard-point fluid for μ = 3 and N = 4096 suggest28 an equilibrium anomaly exponent δ = 0.33 > α(μ = 3) = 0.297(6). This apparent discrepancy, which needs further investigation, could mean that the relation between the different anomaly exponents is not as straightforward as conjectured.

Which is the origin of the universality breakdown here reported? This violation of universality may hint at the possible existence of hidden slowly-evolving fields in the diatomic hard-point gas other than the standard (locally-conserved) hydrodynamic ones. Remarkably, such intriguing behavior has been already reported in the nonequilibrium response of this model to a shock wave excitation36,37, and suggests that a more convoluted fluctuating hydrodynamics description (including the additional slow fields, as in granular fluids63) may be needed to understand anomalous transport in this model. Moreover, as recently put forward29, the existence of further slowly-evolving fields may give rise to an infinite discrete (Fibonacci) family of anomaly exponents that can coexist in different regions of parameter space for a given model29, changing from one value to another as a control parameter is varied, a behavior reminiscent of our results.

The question remains as to how to reconcile the local nature of Fourier’s law with the non-local Lα-term in κL(ρ, T). Our data suggest that this could be achieved in a nonlinear fluctuating hydrodynamics description of the problem derived via an anomalous, non-diffusive hydrodynamic scaling of microscopic spatiotemporal variables, x → x/L1−α and t → t/L2−3α. We also mention that recent results suggest yet another mesoscopic description of anomalous transport in 1d in terms of fractional diffusion equations and/or heat carriers with Lévy-walk statistics43,64,65,66. As far as we know, this description does not seem compatible with the scaling and data collapses observed in this work. Finally, it would be interesting to apply the scaling method here developed to other paradigmatic models of heat transport in low dimensions, as e.g. the Fermi-Pasta-Ulam model of anharmonic oscillators and the hard-square or -shoulder potentials3,4,5,6, where the reported universality breakdown can be further investigated. The role of conservative noise65,66 as a smoothing mechanism to get rid of non-hydrodynamic, hidden slow fields should be also investigated.

How to cite this article: <NAME>. and <NAME>. A violation of universality in anomalous Fourier’s law. Sci. Rep. 6, 38823; doi: 10.1038/srep38823 (2016).

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

1. <NAME>. Théorie analytique de la chaleur (<NAME>, père et fils, 1822).

2. <NAME>. Nonequilibrium statistical physics today: where shall we go from here? American Institute of Physics Conference Proceedings. 1332, 3–22 (2011).

3. <NAME>., <NAME>. & <NAME>. Mathematical Physics 2000, chap. Fourier’s law: A challenge for theorists 128–150 (Imperial College Press, London, 2000).

4. <NAME>., <NAME>. & <NAME>. Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003).

5. <NAME>. Heat transport in low-dimensional systems. Adv. Phys. 57, 457–537 (2008).

6. <NAME>. (ed.) Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer, vol. 921 of Lectures Notes in Physics (Springer, 2016).

7. <NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems. Eur. Phys. J. B 85, 1–20 (2012).

8. <NAME>. et al. Extremely high thermal conductivity of graphene: Prospects for thermal management applications in nanoelectronic circuits. Appl. Phys. Lett. 92, 151911 (2008).

9. <NAME>. et al. Superior thermal conductivity of single-layer graphene. Nano letters 8, 902–907 (2008).

10. <NAME>. et al. Dimensional crossover of thermal transport in few-layer graphene. Nature Mat. 9, 555–558 (2010).

11. <NAME>. Thermal properties of graphene and nanostructured carbon materials. Nature Mat. 10, 569–581 (2011).

12. <NAME>. et al. Length-dependent thermal conductivity in suspended single-layer graphene. Nature Commun. 5 (2014).

13. Meier, T. et al. Length-dependent thermal transport along molecular chains. Phys. Rev. Lett. 113, 060801 (2014).

14. <NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. Breakdown of Fourier’s law in nanotube thermal conductors. Phys. Rev. Lett. 101, 075903 (2008).

15. <NAME>. & <NAME>. High thermal conductivity of single polyethylene chains using molecular dynamics simulations. Phys. Rev. Lett. 101, 235502 (2008).

16. <NAME>. & <NAME>. Length-dependent thermal conductivity of single extended polymer chains. Phys. Rev. B 86, 104307 (2012).

17. <NAME>. et al. Observation of room-temperature ballistic thermal conduction persisting over 8.3 μm in SiGe nanowires. Nature Nano. 8, 534–538 (2013).

18. <NAME>., <NAME>. & <NAME>. Violation of Fourier’s law and anomalous heat diffusion in silicon nanowires. Nano Today 5, 85–90 (2010).

19. <NAME>., <NAME>. & <NAME>. New secrets of spider silk: exceptionally high thermal conductivity and its abnormal change under stretching. Adv. Mat. 24, 1482–1486 (2012).

20. <NAME>. & <NAME>. Velocity autocorrelations for hard spheres. Phys. Rev. Lett. 18, 988–991 (1967).

21. <NAME>. & <NAME>. Decay of velocity autocorrelation function. Phys. Rev. A 1, 18–20 (1970).

22. <NAME>. & <NAME>, M. Classical kinetic theory of fluids (John Wiley & Sons, 1977).

23. <NAME>. & <NAME>. Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys. Rev. Lett. 89, 200601 (2002).

24. <NAME>. Exact results for anomalous transport in one-dimensional hamiltonian systems. Phys. Rev. Lett. 108, 180601 (2012).

25. <NAME>. & <NAME>. Dynamic correlators of Fermi-Pasta-Ulam chains and nonlinear fluctuating hydrodynamics. Phys. Rev. Lett. 111, 230601 (2013).

26. <NAME>. Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–1227 (2014).

27. <NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain. Phys. Rev. E 90, 012124 (2014).

28. <NAME>. & <NAME>. Current fluctuations for anharmonic chains in thermal equilibrium. J. Stat. Mech. P03007 (2015).

29. <NAME>., <NAME>., <NAME>. & <NAME>. Fibonacci family of dynamical universality classes. Proc. Natl. Acad. Sci. USA 112, 12645–12650 (2015).

30. <NAME>. Universality classes for thermal transport in one-dimensional oscillator systems. Phys. Rev. E 91, 032102 (2015).

31. <NAME>., <NAME>., <NAME>. & <NAME>. Self-consistent mode-coupling approach to one-dimensional heat transport. Phys. Rev. E 73, 060201 (2006).

32. <NAME>., <NAME>., <NAME>. & <NAME>. Anomalous kinetics and transport from 1d self-consistent mode-coupling theory. J. Stat. Mech. P02007 (2007).

33. <NAME> al. Energy diffusion in hard-point systems. Eur. Phys. J. Spec. Top. 146, 21–35 (2007).

34. <NAME>. Heat conduction of the hard point chain at zero pressure. J. Stat. Mech. P03028 (2011).

35. <NAME>., <NAME>. & <NAME>. Scaling laws and bulk-boundary decoupling in heat flow. Phys. Rev. E 91, 032116 (2015).

36. <NAME>. Breakdown of hydrodynamics in a simple one-dimensional fluid. Phys. Rev. Lett. 96, 010601 (2006).

37. <NAME>. Strong shock waves and nonequilibrium response in a one-dimensional gas: A Boltzmann equation approach. Phys. Rev. E 72, 041101 (2005).

38. <NAME>., <NAME>. & <NAME>. Thermal conductivity and bulk viscosity in quartic oscillator chains. Phys. Rev. E 72, 031202 (2005).

39. <NAME>. Predicting and identifying finite-size effects in current spectra of one-dimensional oscillator chains. Phys. Rev. E 91, 012138 (2015).

40. <NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. Anomalous heat diffusion. Phys. Rev. Lett. 112, 040601 (2014).

41. <NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. 1d momentum-conserving systems: the conundrum of anomalous versus normal heat transport. New J. Phys. 17, 043064 (2015).

42. <NAME>., <NAME>. & <NAME>. Heat transport in low dimensions: introduction and phenomenology. In Lepri, S. (ed.) Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer, vol. 921 of Lecture Notes in Physics. chap. 1, 1–37 (Springer, 2016).

43. <NAME>., <NAME>. & <NAME>. From anomalous energy diffusion to Lévy walks and heat conductivity in one-dimensional systems. Phys. Rev. Lett. 94, 244301 (2005).

44. <NAME>., <NAME>., <NAME>. & <NAME>. Finite thermal conductivity in 1d lattices. Phys. Rev. Lett. 84, 2144 (2000).

45. <NAME>. & <NAME>. Normal heat conductivity of the one-dimensional lattice with periodic potential of nearest-neighbor interaction. Phys. Rev. Lett. 84, 2381 (2000).

46. <NAME>. & <NAME>. Role of conserved quantities in normal heat transport in one dimenison. arXiv:1411.5247 (2014).

47. <NAME>. Energy transport and the Fourier heat law in classical systems. Found. Phys. 16, 51–61 (1986).

48. <NAME>. & <NAME>. Effective hamiltonian description of nonequilibrium spin systems. Phys. Rev. Lett. 62, 1929–1932 (1989).

49. <NAME>., <NAME>. & <NAME>. Simple one-dimensional model of heat conduction which obeys Fourier’s law. Phys. Rev. Lett. 86, 5486–5489 (2001).

50. <NAME>., <NAME>. & <NAME>. Heat conduction in one-dimensional systems with hard-point interparticle interactions. Phys. Rev. Lett. 88, 154301 (2002).

51. <NAME>., <NAME>. & <NAME>. Heat conduction and entropy production in a one-dimensional hard-particle gas. Phys. Rev. Lett. 89, 180601 (2002).

52. <NAME>., <NAME>. & <NAME>. A one-dimensional hard-point gas and thermoelectric efficiency. J. Stat. Mech. L03004 (2009).

53. <NAME>., <NAME>. & <NAME>. Fluctuations of the heat flux of a one-dimensional hard particle gas. Europhys. Lett. 90, 20004 (2010).

54. <NAME>. Boltzmann equations for a binary one-dimensional ideal gas. Phys. Rev. E 84, 031127 (2011).

55. <NAME>. & <NAME>. Equilibrium time-correlation functions for one-dimensional hard-point systems. Phys. Rev. E 90, 012147 (2014).

56. <NAME>., <NAME>., <NAME>. & <NAME>. Nonintegrability and the Fourier heat conduction law. Phys. Rev. E 90, 032134 (2014).

57. <NAME>., <NAME>., <NAME>. & <NAME>. Finite-size effects on current correlation functions. Phys. Rev. E 89, 022111 (2014).

58. <NAME>., <NAME>. & <NAME>. Probing local equilibrium in nonequilibrium fluids. Phys. Rev. E 92, 022117 (2015).

59. <NAME>. & <NAME>. A measure of data collapse for scaling. J. Phys. A 34, 6375 (2001).

60. <NAME>. & <NAME>. Simplest piston problem. I. Elastic collisions. Phys. Rev. E 73, 016136 (2006).

61. <NAME>. & <NAME>. Simplest piston problem. II. Inelastic collisions. Phys. Rev. E 73, 016137 (2006).

62. <NAME>., <NAME>., <NAME>. & <NAME>. Nonuniversal heat conduction of one-dimensional lattices. Phys. Rev. E 85, 020102 (2012).

63. <NAME>. & <NAME>. Choosing hydrodynamic fields. Math. Model. Nat. Phenom. 6, 19–36 (2011).

64. <NAME>., <NAME>. & <NAME>. Exact solution of a Lévy walk model for anomalous heat transport. Phys. Rev. E 87, 010103 (2013).

65. <NAME>., <NAME>. & <NAME>. 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise. Arch. Rational Mech. Anal. 220, 505–542 (2016).

66. <NAME>., <NAME>. & <NAME>. Superdiffusion of energy in a chain of harmonic oscillators with noise. Commun. Math. Phys. 339, 407–453 (2015).

## Acknowledgements

We thank <NAME>, <NAME>, <NAME>, <NAME>, and <NAME> for useful discussions. Financial support from Spanish project FIS2013-43201-P (MINECO), University of Granada, Junta de Andaluca project P09-FQM4682 and GENIL PYR-2014-13 project is acknowledged.

## Author information

Authors

### Contributions

P.I.H. conceived the project, performed the numerical simulations and prepared the figures. P.I.H. and P.L.G. carried out the calculations and analyzed the data. P.I.H. wrote the main manuscript text. All authors reviewed the manuscript.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Rights and permissions

Reprints and Permissions

<NAME>., <NAME>. A violation of universality in anomalous Fourier’s law. Sci Rep 6, 38823 (2016). https://doi.org/10.1038/srep38823

• Accepted:

• Published:

• ### Anomalous heat diffusion from fractional Fokker–Planck equation

• <NAME>
•  & <NAME>

Applied Mathematics Letters (2020)

• ### Dynamical correlations of conserved quantities in the one-dimensional equal mass hard particle gas

• <NAME>
• , <NAME>
•  & <NAME>

Journal of Statistical Mechanics: Theory and Experiment (2020)

• ### Thermal conductance of one-dimensional disordered harmonic chains

• <NAME>
• , <NAME>
• , <NAME>
• , <NAME>
•  & <NAME>

Physical Review B (2020)

• ### Simulations of Transport in Hard Particle Systems

•  & <NAME>

Journal of Statistical Physics (2020)

• ### Too Close to Integrable: Crossover from Normal to Anomalous Heat Diffusion

• <NAME>
• , <NAME>
•  & <NAME>

Physical Review Letters (2020)


text:
# Math Help - Stuck on trying to find integral of a mixture of trigonometric functions.

1. ## Stuck on trying to find integral of a mixture of trigonometric functions.

Thank so much for all of the help thus far on these problems. This forum has been a life-saver.

So, I have another one ...

So, first things first, here is the problem:
Find the general indefinite integral.

Here are the steps that I have taken so far:

My problem is that I still have the second that I haven’t dealt with. At this point, I am not sure what to do.
Any ideas?
Regards,
<NAME>.

2. If you multiply the orginal integral through the brackets with the sec outside, you have sec squared + sec*tan, split the sum, so tan + sec + c would be the solution?

3. Originally Posted by matt.qmar
If you multiply the orginal integral through the brackets with the sec outside, you have sec squared + sec*tan, split the sum, so tan + sec + c would be the solution?

Is that right?

Kind of simple I guess but perhaps this is a compressed form of what everything would look like if I used the chain rule on my solution?

Thanks again.

Regards,
<NAME>.

format: plain


text:
• ### The Discovery of a Gravitationally Lensed Supernova Ia at Redshift 2.22(1707.04606)

May 1, 2018 astro-ph.CO, astro-ph.GA
We present the discovery and measurements of a gravitationally lensed supernova (SN) behind the galaxy cluster MOO J1014+0038. Based on multi-band Hubble Space Telescope and Very Large Telescope (VLT) photometry of the supernova, and VLT spectroscopy of the host galaxy, we find a 97.5% probability that this SN is a SN Ia, and a 2.5% chance of a CC SN. Our typing algorithm combines the shape and color of the light curve with the expected rates of each SN type in the host galaxy. With a redshift of 2.2216, this is the highest redshift SN Ia discovered with a spectroscopic host-galaxy redshift. A further distinguishing feature is that the lensing cluster, at redshift 1.23, is the most distant to date to have an amplified SN. The SN lies in the middle of the color and light-curve shape distributions found at lower redshift, disfavoring strong evolution to z = 2.22. We estimate an amplification due to gravitational lensing of 2.8+0.6-0.5 (1.10 +- 0.23 mag)---compatible with the value estimated from the weak-lensing-derived mass and the mass-concentration relation from LambdaCDM simulations---making it the most amplified SN Ia discovered behind a galaxy cluster.
• ### An Iwasawa-Taniguchi Effect for Compton-thick Active Galactic Nuclei(1804.01100)

April 6, 2018 astro-ph.GA, astro-ph.HE
We present the first study of an Iwasawa-Taniguchi/X-ray Baldwin effect for Compton-thick active galactic nuclei (AGN). We report a statistically significant anti-correlation between the rest-frame equivalent width (EW) of the narrow core of the neutral Fe K$\alpha$ fluorescence emission line, ubiquitously observed in the reflection spectra of obscured AGN, and the mid-infrared 12$\,\mu$m continuum luminosity (taken as a proxy for the bolometric AGN luminosity). Our sample consists of 72 Compton-thick AGN selected from pointed and deep-field observations covering a redshift range of $z\sim0.0014-3.7$. We employ a Monte Carlo-based fitting method, which returns a Spearman's Rank correlation coefficient of $\rho=-0.28\pm0.12$, significant to 98.7% confidence. The best fit found is ${\rm log}({\rm EW}_{{\rm Fe\,K}\alpha})\,\propto\,-0.08\pm0.04\,{\rm log}(L_{12\,\mu{\rm m}})$, which is consistent with multiple studies of the X-ray Baldwin effect for unobscured and mildly obscured AGN. This is an unexpected result, as the Fe K$\alpha$ line is conventionally thought to originate from the same region as the underlying reflection continuum, which together constitute the reflection spectrum. We discuss the implications this could have if confirmed on larger samples, including a systematic underestimation of the line of sight X-ray obscuring column density and hence the intrinsic luminosities and growth rates for the most luminous AGN.
• ### $HST$ Grism Confirmation of $16$ Structures at $1.4 < z < 2.8$ from the Clusters Around Radio-Loud AGN (CARLA) Survey(1804.01500)

April 4, 2018 astro-ph.GA
We report spectroscopic results from our 40-orbit $Hubble~Space~Telescope$ slitless grism spectroscopy program observing the 20 densest CARLA candidate galaxy clusters at $1.4 < z < 2.8$. These candidate rich structures, among the richest and most distant known, were identified on the basis of $[3.6]-[4.5]$ color from a $408~$hr multi-cycle $Spitzer$ program targeting $420$ distant radio-loud AGN. We report the spectroscopic confirmation of $16$ distant structures at $1.4 < z < 2.8$ associated with the targeted powerful high-redshift radio-loud AGN. We also report the serendipitous discovery and spectroscopic confirmation of seven additional structures at $0.87 < z < 2.12$ not associated with the targeted radio-loud AGN. We find that $10^{10} - 10^{11}\,M_{\odot}$ member galaxies of our confirmed CARLA structures form significantly fewer stars than their field counterparts at all redshifts within $1.4\leq z\leq 2$. We also observe higher star-forming activity in the structure cores up to $z = 2$, finding similar trends as cluster surveys at slightly lower redshifts ($1.0 < z < 1.5$). By design, our efficient strategy of obtaining just two grism orbits per field only obtains spectroscopic confirmation of emission-line galaxies. Deeper spectroscopy will be required to study the population of evolved, massive galaxies in these (forming) clusters. Lacking multi-band coverage of the fields, we adopt a very conservative approach of calling all confirmations "structures", although we note that a number of features are consistent with some of them being bona fide galaxy clusters. Together, this survey represents a unique and large homogenous sample of spectroscopically confirmed structures at high redshifts, potentially more than doubling the census of confirmed, massive clusters at $z > 1.4$.
• ### Magnetic field strength of a neutron-star-powered ultraluminous X-ray source(1803.02376)

March 6, 2018 astro-ph.HE
Ultraluminous X-ray sources (ULXs) are bright X-ray sources in nearby galaxies not associated with the central supermassive black hole. Their luminosities imply they are powered by either an extreme accretion rate onto a compact stellar remnant, or an intermediate mass ($\sim100-10^5$ M$_{\odot}$) black hole. The recent detection of coherent pulsations coming from three bright ULXs demonstrates that some of these sources are powered by accretion onto a neutron star, implying accretion rates significantly in excess of the Eddington limit, a high degree of geometric beaming, or both. The physical challenges associated with the high implied accretion rates can be mitigated if the neutron star surface field is high - in the magnetar regime ($10^{14}$ G), since this suppresses the electron scattering cross section, reducing the radiation pressure that chokes off accretion for high luminosities. One of the few ways to determine surface magnetic fields is through the detection of cyclotron resonance scattering features (CRSFs) produced by the transition of charged particles between quantized Landau levels. To date, CRSFs have only been detected in Galactic accreting pulsars. Here we present the detection at 3.8-{\sigma} significance of a strong absorption line at a rest-frame energy of 4.5 keV in the Chandra spectrum of a ULX in M51. We find that this feature is likely to be a CRSF produced by the strong magnetic field of a neutron star. Assuming scattering off electrons, the magnetic field strength is implied to be $\sim10^{11}$ G, however the line is narrower than any electron CRSFs previously observed, and assuming thermal broadening, the implied temperature is significantly cooler than the accretion disk or column. The line shape is, however, consistent with a proton resonance scattering feature, implying that the neutron star has a magnetic field near the surface of B$\sim10^{15}$ G.
• ### $p$-Harmonic Maps to $S^1$ and Stationary Varifolds of Codimension 2(1802.03053)

Feb. 13, 2018 math.DG, math.AP
We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $$\int_M|du_p|^p=O(\frac{1}{2-p}).$$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the support of a stationary, rectifiable $(n-2)$-varifold $V$ of density $\Theta_{n-2}(\|V\|,\cdot)\geq 2\pi$, given by the concentrated part of the measure $$\mu=\lim_{j\to\infty}(2-p_j)|du_{p_j}|^{p_j}dv_g.$$ When $n=2$, we show moreover that the density of $\|V\|$ takes values in $2\pi\mathbb{N}$. Finally, on every compact manifold of dimension $n\geq 2$ we produce examples of nontrivial families $(1,2)\ni p\mapsto u_p\in W^{1,p}(M,S^1)$ of such maps via natural min-max constructions.
• ### High redshift extremely red quasars in X-rays(1802.04272)

Feb. 12, 2018 astro-ph.GA, astro-ph.HE
Quasars may have played a key role in limiting the stellar mass of massive galaxies. Identifying those quasars in the process of removing star formation fuel from their hosts is an exciting ongoing challenge in extragalactic astronomy. In this paper we present X-ray observations of eleven extremely red quasars (ERQs) with $L_{\rm bol}\sim 10^{47}$ erg s$^{-1}$ at $z=1.5-3.2$ with evidence for high-velocity ($v > 1000$ km s$^{-1}$) [OIII]$\lambda$5007\AA\ outflows. X-rays allow us to directly probe circumnuclear obscuration and to measure the instantaneous accretion luminosity. We detect ten out of eleven extremely red quasars available in targeted and archival data. Using a combination of X-ray spectral fitting and hardness ratios, we find that all of the ERQs show signs of absorption in the X-rays with inferred column densities of $N_{\rm H}\approx 10^{23}$ cm$^{-2}$, including four Compton-thick candidates ($N_{\rm H} > 10^{24}$ cm$^{-2}$). We stack the X-ray emission of the seven weakly detected sources, measuring an average column density of $N_{\rm H}\sim 8\times 10^{23}$ cm$^{-2}$. The absorption-corrected (intrinsic) $2-10$ keV X-ray luminosity of the stack is $2.7\times 10^{45}$ erg s$^{-1}$, consistent with X-ray luminosities of type 1 quasars of the same infrared luminosity. Thus, we find that ERQs are a highly obscured, borderline Compton-thick population, and based on optical and infrared data we suggest that these objects are partially hidden by their own equatorial outflows. However, unlike some quasars with known outflows, ERQs do not appear to be intrinsically underluminous in X-rays for their bolometric luminosity. Our observations indicate that low X-rays are not necessary to enable some types of radiatively driven winds.
• ### Eddington-Limited Accretion in z~2 WISE-selected Hot, Dust-Obscured Galaxies(1703.06888)

Dec. 6, 2017 astro-ph.GA
Hot, Dust-Obscured Galaxies, or "Hot DOGs", are a rare, dusty, hyperluminous galaxy population discovered by the WISE mission. Predominantly at redshifts 2-3, they include the most luminous known galaxies in the universe. Their high luminosities likely come from accretion onto highly obscured super massive black holes (SMBHs). We have conducted a pilot survey to measure the SMBH masses of five z~2 Hot DOGs via broad H_alpha emission lines, using Keck/MOSFIRE and Gemini/FLAMINGOS-2. We detect broad H_alpha emission in all five Hot DOGs. We find substantial corresponding SMBH masses for these Hot DOGs (~ 10^{9} M_sun), and their derived Eddington ratios are close to unity. These z~2 Hot DOGs are the most luminous AGNs at given BH masses, suggesting they are accreting at the maximum rates for their BHs. A similar property is found for known z~6 quasars. Our results are consistent with scenarios in which Hot DOGs represent a transitional, high-accretion phase between obscured and unobscured quasars. Hot DOGs may mark a special evolutionary stage before the red quasar and optical quasar phases, and they may be present at other cosmic epochs.
• ### NuSTAR Hard X-ray Observation of the Gamma-ray Binary Candidate HESS J1832-093(1710.02118)

Oct. 5, 2017 astro-ph.HE
We present a hard X-ray observation of the TeV gamma-ray binary candidate HESS J1832-093 coincident with supernova remnant (SNR) G22.7-0.2 using the Nuclear Spectroscopic Telescope Array (NuSTAR). Non-thermal X-ray emission from XMMU J183245-0921539, the X-ray source associated with HESS J1832-093, is detected up to ~30 keV and is well-described by an absorbed power-law model with the best-fit photon index $\Gamma = 1.5\pm0.1$. A re-analysis of archival Chandra and XMM-Newton data finds that the long-term X-ray flux increase of XMMU J183245-0921539 is $50^{+40}_{-20}$% (90% C.L.), much less than previously reported. A search for a pulsar spin period or binary orbit modulation yields no significant signal to a pulse fraction limit of fp < 19% in the range 4 ms < P < 40 ks. No red noise is detected in the FFT power spectrum to suggest active accretion from a binary system. While further evidence is required, we argue that the X-ray and gamma-ray properties of XMMU J183245-0921539 are most consistent with a non-accreting binary generating synchrotron X- rays from particle acceleration in the shock formed as a result of the pulsar and stellar wind collision. We also report on three nearby hard X-ray sources, one of which may be associated with diffuse emission from a fast-moving supernova fragment interacting with a dense molecular cloud.
• ### An Active Galactic Nucleus Caught in the Act of Turning Off and On(1710.00825)

Oct. 2, 2017 astro-ph.CO, astro-ph.GA
We present the discovery of an active galactic nucleus (AGN) that is turning off and then on again in the z=0.06 galaxy SDSS J1354+1327. This episodic nuclear activity is the result of discrete accretion events, which could have been triggered by a past interaction with the companion galaxy that is currently located 12.5 kpc away. We originally targeted SDSS J1354+1327 because its Sloan Digital Sky Survey spectrum has narrow AGN emission lines that exhibit a velocity offset of 69 km s$^{-1}$ relative to systemic. To determine the nature of the galaxy and its velocity-offset emission lines, we observed SDSS J1354+1327 with Chandra/ACIS, Hubble Space Telescope/Wide Field Camera 3, Apache Point Observatory optical longslit spectroscopy, and Keck/OSIRIS integral-field spectroscopy. We find a ~10 kpc cone of photoionized gas south of the galaxy center and a ~1 kpc semi-spherical front of shocked gas, which is responsible for the velocity offset in the emission lines, north of the galaxy center. We interpret these two outflows as the result of two separate AGN accretion events; the first AGN outburst created the southern outflow, and then $<10^5$ yrs later the second AGN outburst launched the northern shock front. SDSS J1354+1327 is the galaxy with the strongest evidence for an AGN that has turned off and then on again, and it fits into the broader context of AGN flickering that includes observations of AGN light echoes.
• ### The Hunt for Red Quasars: Luminous Obscured Black Hole Growth Unveiled in the Stripe 82 X-ray Survey(1709.01578)

Sept. 5, 2017 astro-ph.GA, astro-ph.HE
We present results of a ground-based near-infrared campaign with Palomar TripleSpec, Keck NIRSPEC, and Gemini GNIRS to target two samples of reddened active galactic nucleus (AGN) candidates from the 31 deg$^2$ Stripe 82 X-ray survey. One sample, which is $\sim$89\% complete to $K<16$ (Vega), consists of eight confirmed AGNs, four of which were identified with our follow-up program, and is selected to have red $R-K$ colors ($>4$, Vega). The fainter sample ($K>17$, Vega) represents a pilot program to follow-up four sources from a parent sample of 34 that are not detected in the single-epoch SDSS catalog and have {\it WISE} quasar colors. All twelve sources are broad-line AGNs (at least one permitted emission line has a FWHM exceeding 1300 km s$^{-1}$) and span a redshift range $0.59 < z < 2.5$. Half the ($R-K$)-selected AGNs have features in their spectra suggestive of outflows. When comparing these sources to a matched sample of blue Type 1 AGNs, we find the reddened AGNs are more distant ($z > 0.5$) and a greater percentage have high X-ray luminosities ($L_{\rm X,full} > 10^{44}$ erg s$^{-1}$). Such outflows and high luminosities may be consistent with the paradigm that reddened broad-line AGNs represent a transitory phase in AGN evolution as described by the major merger model for black hole growth. Results from our pilot program demonstrate proof-of-concept that our selection technique is successful in discovering reddened quasars at $z > 1$ missed by optical surveys.
• ### NuSTAR Resolves the First Dual AGN above 10 keV in SWIFT J2028.5+2543(1708.06762)

Aug. 22, 2017 astro-ph.HE
We have discovered heavy obscuration in the dual active galactic nucleus (AGN) in the Swift/Burst Alert Telescope (BAT) source SWIFT J2028.5+2543 using Nuclear Spectroscopic Telescope Array (NuSTAR). While an early XMM-Newton study suggested the emission was mainly from NGC 6921, the superior spatial resolution of NuSTAR above 10 keV resolves the Swift/BAT emission into two sources associated with the nearby galaxies MCG +04-48-002 and NGC 6921 (z = 0.014) with a projected separation of 25.3 kpc (91"). NuSTAR's sensitivity above 10 keV finds both are heavily obscured to Compton-thick levels (N H=(1-2)x10^24 cm-2) and contribute equally to the BAT detection ({L}10-50 {keV}{{int}}= 6x10^42 erg s-1). The observed luminosity of both sources is severely diminished in the 2-10 keV band, illustrating the importance of >10 keV surveys like those with NuSTAR and Swift/BAT. Compared to archival X-ray data, MCG +04-48-002 shows significant variability (>3) between observations. Despite being bright X-ray AGNs, they are difficult to detect using optical emission-line diagnostics because MCG +04-48-002 is identified as a starburst/composite because of the high rates of star formation from a luminous infrared galaxy while NGC 6921 is only classified as a LINER using line detection limits. SWIFT J2028.5+2543 is the first dual AGN resolved above 10 keV and is the second most heavily obscured dual AGN discovered to date in the X-rays other than NGC 6240.
• ### The X-ray and Mid-Infrared luminosities in Luminous Type 1 Quasars(1701.05207)

Aug. 21, 2017 astro-ph.GA
Several recent studies have reported different intrinsic correlations between the AGN mid-IR luminosity ($L_{MIR}$) and the rest-frame 2-10 keV luminosity ($L_{X}$) for luminous quasars. To understand the origin of the difference in the observed $L_{X}-L_{MIR}$ relations, we study a sample of 3,247 spectroscopically confirmed type 1 AGNs collected from Bo\"{o}tes, XMM-COSMOS, XMM-XXL-North, and the SDSS quasars in the Swift/XRT footprint spanning over four orders of magnitude in luminosity. We carefully examine how different observational constraints impact the observed $L_{X}-L_{MIR}$ relations, including the inclusion of X-ray non-detected objects, possible X-ray absorption in type 1 AGNs, X-ray flux limits, and star formation contamination. We find that the primary factor driving the different $L_{X}-L_{MIR}$ relations reported in the literature is the X-ray flux limits for different studies. When taking these effects into account, we find that the X-ray luminosity and mid-IR luminosity (measured at rest-frame $6\mu m$, or $L_{6\mu m}$) of our sample of type 1 AGNs follow a bilinear relation in the log-log plane: $\log L_X =(0.84\pm0.03)\times\log L_{6\mu m}/10^{45}{\rm erg\;s^{-1}} + (44.60\pm0.01)$ for $L_{6\mu m} < 10^{44.79}{\rm erg\;s^{-1}}$, and $\log L_X = (0.40\pm0.03)\times\log L_{6\mu m}/10^{45}{\rm erg\;s^{-1}} +(44.51\pm0.01)$ for $L_{6\mu m} \geq 10^{44.79}{\rm erg\;s^{-1}}$. This suggests that the luminous type 1 quasars have a shallower $L_{X}-L_{MIR}$ correlation than the approximately linear relations found in local Seyfert galaxies. This result is consistent with previous studies reporting a luminosity-dependent $L_{X}-L_{MIR}$ relation, and implies that assuming a linear $L_{X}-L_{MIR}$ relation to infer the neutral gas column density for X-ray absorption might overestimate the column densities in luminous quasars.
• ### BAT AGN Spectroscopic Survey I: Spectral Measurements, Derived Quantities, and AGN Demographics(1707.08123)

July 25, 2017 astro-ph.GA, astro-ph.HE
We present the first catalog and data release of the Swift-BAT AGN Spectroscopic Survey (BASS). We analyze optical spectra of the majority of AGN (77%, 641/836) detected based on their 14-195 keV emission in the 70-month Swift BAT all-sky catalog. This includes redshift determination, absorption and emission line measurements, and black hole mass and accretion rate estimates for the majority of obscured and un-obscured AGN (74%, 473/641) with 340 measured for the first time. With ~90% of sources at z<0.2, the survey represents a significant census of hard-X-ray selected AGN in the local universe. In this first catalog paper, we describe the spectroscopic observations and datasets, and our initial spectral analysis. The FWHM of the emission lines show broad agreement with the X-ray obscuration (~94%), such that Sy 1-1.8 have NH<10^21.9 cm^-2, and Seyfert 2, have NH>10^21.9 cm^-2. Seyfert 1.9 show a range of column densities. Compared to narrow line AGN in the SDSS, the X-ray selected AGN have a larger fraction of dusty host galaxies suggesting these types of AGN are missed in optical surveys. Using the most sensitive [OIII]/Hbeta and [NII]/Halpha emission line diagnostic, about half of the sources are classified as Seyferts, ~15% reside in dusty galaxies that lack an Hbeta detection, but for which the line upper limits imply either a Seyfert or LINER, ~15% are in galaxies with weak or no emission lines despite high quality spectra, and a few percent each are LINERS, composite galaxies, HII regions, or in known beamed AGN.
• ### Looking at A 0535+26 at low luminosities with NuSTAR(1707.05648)

July 18, 2017 astro-ph.HE
We report on two NuSTAR observations of the HMXB A 0535+26 taken toward the end of its normal 2015 outburst at very low $3-50$ keV luminosities of ${\sim}1.4\times10^{36}$ erg/s and ${\sim}5\times10^{35}$ erg/s which are complemented by 9 Swift observations. The data clearly confirm indications seen in earlier data that the source's spectral shape softens as it becomes fainter. The smooth, exponential rollover at high energies present in the first observation evolves to a much more abrupt steepening of the spectrum at $20-30$ keV. The continuum evolution can be well described with emission from a magnetized accretion column, modeled using the compmag model modified by an additional Gaussian emission component for the fainter observation. Between the two observations, the optical depth changes from $0.75\pm0.04$ to $0.56^{+0.01}_{-0.04}$, the electron temperature remains constant, and there is an indication that the column decreases in radius. Since the energy resolved pulse profiles remain virtually unchanged in shape between the two observations, the emission properties of the accretion column, however, reflect the same accretion regime. This conclusion is also confirmed by our result that the energy of the cyclotron resonant scattering feature (CRSF) at ${\sim}45$ keV is independent of the luminosity, implying that the magnetic field in the region in which the observed radiation is produced is the same in both observations. Finally, we also constrain the evolution of the continuum parameters with rotational phase of the neutron star. The width of the CRSF could only be constrained for the brighter observation. Based on Monte-Carlo simulations of CRSF formation in single accretion columns, its pulse phase dependence supports a simplified fan beam emission pattern. The evolution of the CRSF width is very similar to that of the CRSF depth, which is in disagreement with expectations.
• ### Understanding extreme quasar optical variability with CRTS: I. Major AGN flares(1706.03079)

June 9, 2017 astro-ph.GA
There is a large degree of variety in the optical variability of quasars and it is unclear whether this is all attributable to a single (set of) physical mechanism(s). We present the results of a systematic search for major flares in AGN in the Catalina Real-time Transient Survey as part of a broader study into extreme quasar variability. Such flares are defined in a quantitative manner as being atop of the normal, stochastic variability of quasars. We have identified 51 events from over 900,000 known quasars and high probability quasar candidates, typically lasting 900 days and with a median peak amplitude of $\Delta m = 1.25$ mag. Characterizing the flare profile with a Weibull distribution, we find that nine of the sources are well described by a single-point single-lens model. This supports the proposal by Lawrence et al. (2016) that microlensing is a plausible physical mechanism for extreme variability. However, we attribute the majority of our events to explosive stellar-related activity in the accretion disk: superluminous supernovae, tidal disruption events, and mergers of stellar mass black holes.
• ### The role of the most luminous, obscured AGN in galaxy assembly at z~2(1705.02649)

June 6, 2017 astro-ph.CO, astro-ph.GA
We present HST WFC3 F160W imaging and infrared spectral energy distributions for twelve extremely luminous, obscured AGN at $1.8<z<2.7$, selected via "Hot, Dust Obscured" mid-infrared colors. Their infrared luminosities span $2-15\times10^{13}$L$_{\odot}$, making them among the most luminous objects in the Universe at $z\sim2$. In all cases the infrared emission is consistent with arising at least in most part from AGN activity. The AGN fractional luminosities are higher than those in either sub-millimeter galaxies, or AGN selected via other mid-infrared criteria. Adopting the $G$, M$_{20}$ and $A$ morphological parameters, together with traditional classification boundaries, infers that three quarters of the sample as mergers. Our sample do not, however, show any correlation between the considered morphological parameters and either infrared luminosity or AGN fractional luminosity. Moreover, their asymmetries and effective radii are distributed identically to those of massive galaxies at $z\sim2$. We conclude that our sample is not preferentially associated with mergers, though a significant merger fraction is still plausible. Instead, we propose that our sample are examples of the massive galaxy population at $z\sim2$ that harbor a briefly luminous, "flickering" AGN, and in which the $G$ and M$_{20}$ values have been perturbed, due to either the AGN, and/or the earliest formation stages of a bulge in an inside-out manner. Furthermore, we find that the mass assembly of the central black holes in our sample leads the mass assembly of any bulge component. Finally, we speculate that our sample represent a small fraction of the immediate antecedents of compact star-forming galaxies at $z\sim2$.
• ### Sagittarius A* High Energy X-ray Flare Properties During NuSTAR Monitoring of the Galactic Center from 2012 to 2015(1705.08002)

May 22, 2017 astro-ph.HE
Understanding the origin of the flaring activity from the Galactic center supermassive black hole, Sagittarius A*, is a major scientific goal of the NuSTAR Galactic plane survey campaign. We report on the data obtained between July 2012 and April 2015, including 27 observations on Sgr A* with a total exposure of ~ 1 Ms. We found a total of ten X-ray flares detected in the NuSTAR observation window, with luminosities in the range of $L_{3-79~keV}$~$(0.2-4.0) \times 10^{35}~erg~s^{-1}$. With this largest hard X-ray Sgr A* flare dataset to date, we studied the flare spectral properties. Seven flares are detected above 5{\sigma} significance, showing a range of photon indices ({\Gamma} ~ 2.0-2.8) with typical uncertainties of +/-0.5 (90% confidence level). We found no significant spectral hardening for brighter flares as indicated by a smaller sample. The accumulation of all the flare spectra in 1-79 keV can be well fit with an absorbed power-law model with {\Gamma}=2.2+/-0.1, and does not require the existence of a spectral break. The lack of variation in X-ray spectral index with luminosity would point to a single mechanism for the flares and is consistent with the synchrotron scenario. Lastly, we present the quiescent state spectrum of Sgr A*, and derived an upper limit on the quiescent luminosity of Sgr A* above 10 keV to be $L_{Xq, 10-79 keV}$ < $(2.9{\pm}0.2) \times 10^{34}~erg~s^{-1}$.
• ### The Swift/BAT AGN Spectroscopic Survey (BASS) -- VI. The Gamma_X - L/L_Edd relation(1705.01550)

We study the observed relation between accretion rate (in terms of L/L_Edd) and shape of the hard X-ray spectral energy distribution (namely the photon index Gamma_X) for a large sample of 228 hard X-ray selected, low-redshift active galactic nuclei (AGN), drawn from the Swift/BAT AGN Spectroscopic Survey (BASS). This includes 30 AGN for which black hole mass (and therefore L/L_Edd) is measured directly through masers, spatially resolved gas or stellar dynamics, or reverberation mapping. The high quality and broad energy coverage of the data provided through BASS allow us to examine several alternative determinations of both Gamma_X and L/L_Edd. For the BASS sample as a whole, we find a statistically significant, albeit very weak correlation between Gamma_X and L/L_Edd. The best-fitting relations we find, Gamma_X=0.15 log(L/L_Edd)+const., are considerably shallower than those reported in previous studies. Moreover, we find no corresponding correlations among the subsets of AGN with different M_BH determination methodology. In particular, we find no robust evidence for a correlation when considering only those AGN with direct or single-epoch M_BH estimates. This latter finding is in contrast to several previous studies which focused on z>0.5 broad-line AGN. We discuss this tension and conclude that it can be partially accounted for if one adopts a simplified, power-law X-ray spectral model, combined with L/L_Edd estimates that are based on the continuum emission and on single-epoch broad line spectroscopy in the optical regime. We finally highlight the limitations on using Gamma_X as a probe of supermassive black hole evolution in deep extragalactic X-ray surveys.
• ### Calibrating the Planck Cluster Mass Scale with Cluster Velocity Dispersions(1704.07891)

April 25, 2017 astro-ph.CO
We measure the Planck cluster mass bias using dynamical mass measurements based on velocity dispersions of a subsample of 17 Planck-detected clusters. The velocity dispersions were calculated using redshifts determined from spectra obtained at Gemini observatory with the GMOS multi-object spectrograph. We correct our estimates for effects due to finite aperture, Eddington bias and correlated scatter between velocity dispersion and the Planck mass proxy. The result for the mass bias parameter, $(1-b)$, depends on the value of the galaxy velocity bias $b_v$ adopted from simulations: $(1-b)=(0.51\pm0.09) b_v^3$. Using a velocity bias of $b_v=1.08$ from Munari et al., we obtain $(1-b)=0.64\pm 0.11$, i.e, an error of 17% on the mass bias measurement with 17 clusters. This mass bias value is consistent with most previous weak lensing determinations. It lies within $1\sigma$ of the value needed to reconcile the Planck cluster counts with the Planck primary CMB constraints. We emphasize that uncertainty in the velocity bias severely hampers precision measurements of the mass bias using velocity dispersions. On the other hand, when we fix the Planck mass bias using the constraints from Penna-Lima et al., based on weak lensing measurements, we obtain a positive velocity bias $b_v \gtrsim 0.9$ at $3\sigma$.
• ### The Complete Calibration of the Color-Redshift Relation (C3R2) Survey: Survey Overview and Data Release 1(1704.06665)

April 21, 2017 astro-ph.CO, astro-ph.GA
A key goal of the Stage IV dark energy experiments Euclid, LSST and WFIRST is to measure the growth of structure with cosmic time from weak lensing analysis over large regions of the sky. Weak lensing cosmology will be challenging: in addition to highly accurate galaxy shape measurements, statistically robust and accurate photometric redshift (photo-z) estimates for billions of faint galaxies will be needed in order to reconstruct the three-dimensional matter distribution. Here we present an overview of and initial results from the Complete Calibration of the Color-Redshift Relation (C3R2) survey, designed specifically to calibrate the empirical galaxy color-redshift relation to the Euclid depth. These redshifts will also be important for the calibrations of LSST and WFIRST. The C3R2 survey is obtaining multiplexed observations with Keck (DEIMOS, LRIS, and MOSFIRE), the Gran Telescopio Canarias (GTC; OSIRIS), and the Very Large Telescope (VLT; FORS2 and KMOS) of a targeted sample of galaxies most important for the redshift calibration. We focus spectroscopic efforts on under-sampled regions of galaxy color space identified in previous work in order to minimize the number of spectroscopic redshifts needed to map the color-redshift relation to the required accuracy. Here we present the C3R2 survey strategy and initial results, including the 1283 high confidence redshifts obtained in the 2016A semester and released as Data Release 1.
• ### The X-ray reflection spectrum of the radio-loud quasar 4C 74.26(1704.03673)

April 12, 2017 astro-ph.GA, astro-ph.HE
The relativistic jets created by some active galactic nuclei are important agents of AGN feedback. In spite of this, our understanding of what produces these jets is still incomplete. X-ray observations, which can probe the processes operating in the central regions in immediate vicinity of the supermassive black hole, the presumed jet launching point, are potentially particularly valuable in illuminating the jet formation process. Here, we present the hard X-ray NuSTAR observations of the radio-loud quasar 4C 74.26 in a joint analysis with quasi-simultaneous, soft X-ray Swift observations. Our spectral analysis reveals a high-energy cut-off of 183$_{-35}^{+51}$ keV and confirms the presence of ionized reflection in the source. From the average spectrum we detect that the accretion disk is mildly recessed with an inner radius of $R_\mathrm{in}=4-180\,R_\mathrm{g}$. However, no significant evolution of the inner radius is seen during the three months covered by our NuSTAR campaign. This lack of variation could mean that the jet formation in this radio-loud quasar differs from what is observed in broad-line radio galaxies.
• ### Extreme Variability in a Broad Absorption Line Quasar(1704.03923)

April 12, 2017 astro-ph.GA
CRTS J084133.15+200525.8 is an optically bright quasar at z=2.345 that has shown extreme spectral variability over the past decade. Photometrically, the source had a visual magnitude of V~17.3 between 2002 and 2008. Then, over the following five years, the source slowly brightened by approximately one magnitude, to V~16.2. Only ~1 in 10,000 quasars show such extreme variability, as quantified by the extreme parameters derived for this quasar assuming a damped random walk model. A combination of archival and newly acquired spectra reveal the source to be an iron low-ionization broad absorption line (FeLoBAL) quasar with extreme changes in its absorption spectrum. Some absorption features completely disappear over the 9 years of optical spectra, while other features remain essentially unchanged. We report the first definitive redshift for this source, based on the detection of broad H-alpha in a Keck/MOSFIRE spectrum. Absorption systems separated by several 1000 km/s in velocity show coordinated weakening in the depths of their troughs as the continuum flux increases. We interpret the broad absorption line variability to be due to changes in photoionization, rather than due to motion of material along our line of sight. This source highlights one sort of rare transition object that astronomy will now be finding through dedicated time-domain surveys.
• ### Energy Concentration for Min-Max Solutions of the Ginzburg-Landau Equations on Manifolds with $b_1(M)\neq 0$(1704.00712)

April 3, 2017 math.DG, math.AP
We establish a new estimate for the Ginzburg-Landau energies $E_{\epsilon}(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4\epsilon^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq 0$, obtained by decomposing the harmonic component $h_u$ of the one-form $ju:=u^1du^2-u^2du^1$ into an integral and fractional part. We employ this estimate to show that, for critical points $u_{\epsilon}$ of $E_{\epsilon}$ arising from the two-parameter min-max construction considered by the author in previous work, a nontrivial portion of the energy must concentrate on a stationary, rectifiable $(n-2)$-varifold as $\epsilon\to 0$.
• ### Measurement of the absolute Crab flux with NuSTAR(1703.10685)

March 28, 2017 astro-ph.IM, astro-ph.HE
We present results from a NuSTAR observation of the Crab made at a large off-axis angle of 1.5\degree. At these angles X-rays do not pass through the optics, but rather illuminate the detectors directly due to incomplete baffling. Due to the simplicity of the instrument response in this configuration and the good absolute calibration of the detectors, we are able to measure the absolute intrinsic flux of the Crab to better than 4%. We find the spectral parameters of the powerlaw to be Gamma=2.106 +/- 0.006, N=9.71 +/-0.16, in agreement with the values measured 42 years ago by Toor & Seward (1974). This suggests that the observed variability of the Crab is not part of a long term trend, but instead results from fluctuations around a steady mean. The NuSTAR observation also enabled improved measurement of the detector absorption parameters without the added complications of the mirror response.
• ### The NuSTAR Hard X-ray Survey of the Norma Arm Region(1703.00021)

Feb. 28, 2017 astro-ph.HE
We present a catalog of hard X-ray sources in a square-degree region surveyed by NuSTAR in the direction of the Norma spiral arm. This survey has a total exposure time of 1.7 Ms, and typical and maximum exposure depths of 50 ks and 1 Ms, respectively. In the area of deepest coverage, sensitivity limits of $5\times10^{-14}$ and $4\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$ in the 3-10 and 10-20 keV bands, respectively, are reached. Twenty-eight sources are firmly detected and ten are detected with low significance; eight of the 38 sources are expected to be active galactic nuclei. The three brightest sources were previously identified as a low-mass X-ray binary, high-mass X-ray binary, and pulsar wind nebula. Based on their X-ray properties and multi-wavelength counterparts, we identify the likely nature of the other sources as two colliding wind binaries, three pulsar wind nebulae, a black hole binary, and a plurality of cataclysmic variables (CVs). The CV candidates in the Norma region have plasma temperatures of $\approx$10-20 keV, consistent with the Galactic Ridge X-ray emission spectrum but lower than temperatures of CVs near the Galactic Center. This temperature difference may indicate that the Norma region has a lower fraction of intermediate polars relative to other types of CVs compared to the Galactic Center. The NuSTAR log$N$-log$S$ distribution in the 10-20 keV band is consistent with the distribution measured by Chandra at 2-10 keV if the average source spectrum is assumed to be a thermal model with $kT\approx15$~keV, as observed for the CV candidates.